Unification of Electromagnetism and Gravity
Grace Shellac
actually hasn't been unified. I asked Jerrold Marsden down at CalTech
if this was true.
He responded: "...there is a well defined set of equations called the
Einstein-Maxwell equations that already unifies E and M with
gravity. They are discussed in standard books on general relativity."
"Thus, I would say that most researchers believe that these theories
are already unified. The big question is how to unify quantum theory
with gravity."
So, what is the deal here. Am I misinformed? Or is Prof. Marsden
mistaken? Doing a google.com search didn't seem to help as there
doesn't seem to be a consensus that gravity and electromagnetism are or
have been satisfactorily unified. I see web pages by people like
Sweetster and others but no definitive comment that indicates that
anyone actually thinks gravity and electromagnetism have been unified.
Help! What's the latest opinion?
G.S.
Zeab Nhoj
presumably aka Ace Schallger,
presumably aka Charles Cagle,
wrote:
>I've been under the impression that gravity and electromagnetism
>actually hasn't been unified.
If they haven't been unified, why are you using the
singular form of the verb in this sentence? :-)
>I asked Jerrold Marsden down at CalTech
>if this was true.
>
>He responded: "...there is a well defined set of equations called the
>Einstein-Maxwell equations that already unifies E and M with
>gravity. They are discussed in standard books on general relativity."
>
>"Thus, I would say that most researchers believe that these theories
>are already unified. The big question is how to unify quantum theory
>with gravity."
>
>So, what is the deal here. Am I misinformed? Or is Prof. Marsden
>mistaken?
There is a well-known set of equations called the Einstein-Maxwell
equations which are adequate for describing all known phenomena
involving gravity and electromagnetism as long as one ignore
quantum mechanics. Marsden is an expert on these.
On the other hand, most people in the "unification" business
don't regard the Einstein-Maxwell equations as "unifying"
gravity and electromagnetism, since they are basically formed
by taking Einstein's equations for gravity and Maxwell's equations
for electromagnetism and slapping one on top of the other.
The stress-energy tensor for electromagnetism fits nicely into
the right-hand side of Einstein's equation. Maxwell's equations
generalize easily to curved spacetime. Voila! That's Einstein-Maxwell.
Starting with Einstein's own search for a "unified field theory",
everyone has been looking for something that will combine gravity
and electromagnetism in a more profoundly integrated way. One nice
attempt was Kaluza and Klein's theory: Einstein's equations on a 5d
spacetime where the extra dimension is curled up in a little circle
gives Einstein's equation in ordinary 4d spacetime, *together*
with Maxwell's equations... *together* with a massless spin-0
particle that, sadly, appears not to exist. Oh well - nice try!
More recent attempts usually take into account the fact that
there are other forces besides electromagnetism and gravity.
They also usually try to take quantum mechanics into account
from the very start.
>Doing a google.com search didn't seem to help as there
>doesn't seem to be a consensus that gravity and electromagnetism are or
>have been satisfactorily unified.
One really can't learn the consensus among physicists by doing
a web search. One needs to read textbooks and journal articles.
However, in this case it sounds like you got the right impression:
most physicists would not say electromagnetism and gravity have
been unified.
Chris Hillman
On Tue, 6 Nov 2001, Grace Shellac wrote:
> I've been under the impression that gravity and electromagnetism
> actually hasn't been unified.
Right; they haven't.
> I asked Jerrold Marsden down at CalTech if this was true.
>
> He responded: "...there is a well defined set of equations called the
> Einstein-Maxwell equations that already unifies E and M with
> gravity. They are discussed in standard books on general relativity."
>
> "Thus, I would say that most researchers believe that these theories
> are already unified. The big question is how to unify quantum theory
> with gravity."
Well, Marsden is a mathematician, not a physicist ;-/
Or in other words, I fear that Homer hath nodded.
> So, what is the deal here. Am I misinformed?
No, you are right.
> Or is Prof. Marsden mistaken?
His statement was misleading.
> Doing a google.com search didn't seem to help as there
> doesn't seem to be a consensus that gravity and electromagnetism are or
> have been satisfactorily unified. I see web pages by people like
> Sweetster and others but no definitive comment that indicates that
> anyone actually thinks gravity and electromagnetism have been unified.
Since Sweetser sometimes posts here, you may get somewhat diverging
answers. Let me try to sketch an answer which I believe fairly represents
the -current mainstream consensus-.
Marsden was referring to the notion of an "electrovacuum solution to the
EFE". This is indeed treated in all the textbooks. An "electrovacuum
solution" is simultaneously
1. a sourcefree solution to the curved spacetime Maxwell equations,
2. a solution to the EFE, in which T^(ab) = G^(ab)/8/Pi arises as the
stress-momentum-energy tensor of the EM field in (1).
These are also called "Einstein-Maxwell solutions" (some authors include
in that phrase things like a charged fluid; this introduces various
complications but doesn't affect the gist of what I am about to say).
The best way to explain the concept is by briefly discussing a few simple
but nontrivial examples.
The best known example of an electrovacuum is the "Reissner-Nordstrom
electrovacuum" (1918). This can be defined by giving the static polar
spherical coordinate chart
ds^2 = -(1-2m/r+q^2/r^2) dt^2 + dr^2/(1-2m/r+q^2/r^2)
+ r^2 (du^2 + sin(u)^2 dv^2),
-infty < t < infty, 0 < r < infty, 0 < u < Pi, -Pi < v < Pi
where q is a nonzero constant and m is a positive constant, m > |q|,
together with EM vector potential
A = -qr/(q^2-2mr+r^2) d/dt
It is convenient to work with the following ONB (orthonormal basis) of
vectors for the chart above:
e_1 = 1/sqrt(1-2m/r+q^2/r^2) d/dt
e_2 = sqrt(1-2m/r+q^2/r^2) d/dr
e_3 = 1/r d/du
e_4 = 1/r/sin(u) d/dv
Then, the vector potential A yields a pure electric field (vanishing
magnetic field)
E = q/r^2 e_2
You can check that this solves the source-free curved spacetime Maxwell
equations (that is, with four-current J^a vanishing identically) and gives
rise to the stress-momentum-energy tensor (wrt the above ONB!)
q^2 [ 1 0 0 0 ]
T^(ab) = -------- [ 0 -1 0 0 ]
8 Pi r^4 [ 0 0 1 0 ]
[ 0 0 0 1 ]
Now, if you compute the Einstein tensor wrt the above ONB, you'll find
that Einstein's equation
G^(ab) = 8 Pi T^(ab)
is -also- satisfied. (In doing these computations, it is best to work as
far as possible with exterior forms. See for example Misner, Thorne, and
Wheeler, Gravitation, Freeman, 1973, for examples of how to compute the
curvature tensors using differential forms, and how to compute the EM
field on a curved spacetime, given the vector potential. "MTW" is no
doubt one of the standard textbooks which Marsden had in mind.)
Thus, we have here an "exact Einstein-Maxwell solution" or
"electrovacuum". More specifically, this is a "nonnull electrovacuum"
because the EM field tensor F_(ab) has the principle Lorentz invariants
F_(ab) F^(ab) = -2 q^2/r^4
F_(ab) (*F)^(ab) = 0
do not -both- vanish. (The fact that the second invariant vanishes shows
that while the "generic observer"-- for example, an observer in a stable
circular orbit around the charged static spherically symmetric massive
object modeled by the RN solution-- will measure nonzero electric and
magnetic fields, some observers--such as the static observers defined by
e_1 above, or radially infalling observers-- will measure only an electric
field.)
Another example of an electrovacuum is a "uniform circularly polarized EM
wave". This can be defined by the "NIL chart" (referring the "Thurson NIL
geometry, aka the Riemannian geometry of the three-dimensional Lie group
of three by three real upper triangular matrices with ones on the
diagonal)
ds^2 = [-dt^2/2 + dt dx + z dt dy]/m + dy^2/4/m^2 + dz^2,
-infty < t,x,y,z < infty
where m > 0 is a constant, together with the EM vector potential
A = m d/dy
Again, it is convenient to work with an ONB of vectors:
e_1 = sqrt(2m) d/dt
e_2 = sqrt(2m) [ d/dt + d/dx ]
e_3 = 2 m [ z d/dx + d/dy ]
e_4 = d/dz
(It isn't supposed to obvious that this ONB does indeed yield the above
metric tensor; in effect I found the ONB first, using the "Farnsworth-Kerr
Ansatz", found the coordinate chart from that, and then figured out what
the solution represents physically. The same day I found a whole bunch of
nonsense "solutions", since the Farnsworth-Kerr trick is a rule of thumb,
not a sure-fire method, this was to be expected.) Then, the EM field
turns out to be
E = sqrt(m/2) e_3
B = -sqrt(m/2) e_4
and this solves the source-free Maxwell equations on the above spacetime.
Furthermore, the EM stress-momentum-energy tensor (wrt the above ONB!) is
m [ 1 -1 0 0 ]
T^(ab) = ---- [ -1 1 0 0 ]
8 Pi [ 0 0 0 0 ]
[ 0 0 0 0 ]
and if you compute the Einstein tensor, you'll find that once again
G^(ab) = 8 Pi T^(ab)
so we have an exact electrovacuum. This one is a "null electrovacuum"
because the principle Lorentz invariants of the EM field tensor F^(ab)
both vanish. As you know, this characterizes "radiative solutions" in
Maxwell's theory.
You might well wonder why I said this is a "uniform -circularly- polarized
EM wave" rather than a "uniform -linearly- polarized EM wave". The reason
is rather subtle: while the ONB I gave above has the property that the
timelike vector field e_1 defines a timelike geodesic congruence (i.e. the
world lines of a family of -inertial- observers), the spatial vectors e_3,
e_4 turn out to be -rotating- with a constant angular frequency around the
spatial vector e_2 wrt a -gryostabilized- ONB. When we switch to the
gyrostabilized ONB, we find that now the E and B fields are -rotating-
with a constant angular frequency, so we have a circularly polarized EM
wave.
Coming back to the question of why "the Einstein-Maxwell theory" is -not-
a "unification" of the EM and gravitational interactions: as you know,
neither Maxwell's theory of EM nor Einstein's theory of gravitation are
quantum theories. Therefore, they are not regarded as theories of
fundamental interactions. However, we know that Maxwell's theory of EM is
the "effective field theory" or "classical limit" of a renormalizable
quantum field theory, QED. The latter -is- regarded as a fundamental
theory.
As you may know, after much work in the last part of the previous century,
it turned out that the QFT which would be the "most straightforward" (not
really straightforward at all, as it turned out!) "gravitational analogue"
of QED is -nonrenormalizable- and therefore unworkable. This doesn't mean
that there is no viable quantum theory of gravity, just that whatever such
a theory might look like, it must be quite different from the QFTs
physicists grew familiar with in the last century, in particular from the
three QFTs for the EM, weak, and strong interactions.
As you no doubt know, in the last century, these three QFTs were
incorporated into a single QFT usually called the "Standard Model", which
predicts that at very high energies, the three interactions become
indistinguishable, but at the kind of energies we can readily observe,
they appear to be different interactions with -very- different properties!
Another way of understanding why the Standard Model can be said to "unify"
the EM, weak, strong interactions is by observing that its gauge group or
fundamental symmetry group includes as subgroups the symmetry groups of
the three specialized QFTs, e.g. the abelian circle group U(1) for QED
(the other two are "nonabelian gauge theories"). Furthermore, the way in
which the symmetry of the big gauge group has been "broken" by a kind of
"phase transformation" which occurs at "low energies" corresponds exactly
to the way in which the three subgroups fit into the big gauge group.
It is universally believed that at still higher energies (near the "Planck
energy", if not before), the gravitational interaction must somehow
treated in a "quantum" manner. It is also possible that at these
energies (or maybe at still higher energies), the gravitational
interaction may become unified with the other three. Superstring theory
attempts to provide a -unified- quantum theory of all four fundamental
interactions; that is, it aims to become a TOE. Its most popular
competitor, loop quantum gravity, "merely" attempts to provide a quantum
theory of gravity. In both cases, it is widely believed that gtr must be
the effective field theory of the gravitational interaction at low
energies, but it is possible that some more complicated classical
relativistic field theory will turn out to be the effective field theory
for one or both of these competitors. If this is the case, this more
complicated theory must be indistinguishable from gtr at all
scales/energies thus far investigated by experimenters or astronomers.
Going back to the issue of what it means to say that the EFE can be solved
-simultaneously- with other classical field equations, I think it is
helpful to think of gtr as being analogous to the most general of all
theories of classical physics, namely classical thermodynamics.
Recall that the latter is a very general theory about (roughly)
"dissipation" and "transportation" of "energy". It applies to any
reasonable "theory of matter". In particular, if you regard ordinary
matter as being made up of atoms, you can try to show how thermodynamic
properties emerge from the atomic theory as statistical phenomena---
that's the basic idea "statistical mechanics", a theory which was
initiated by Maxwell and Kelvin but which is still very much under
development.
In much the same way, you can "feed" any reasonable effective field theory
to the RHS of the EFE (Einstein Field Equation), by adding suitable terms
to the stress-momentum-energy tensor T_(ab).
For example: some quantum fields can be modeled in the classical limit as
a "massless scalar field", which is the simplest kind of effective field
theory. Thus, you can search for something which is simultaneously:
1. a scalar field which satisfies the curved spacetime wave equation,
2. a solution to the EFE in which T^(ab) arises as the
stress-momentum-energy tensor of the massless scalar field in (1).
In particular, you can search for a static spherically symmetric solution
of this type, which is analogous to the Reissner-Nordstrom electrovacuum.
The resulting solution is called the "Janis-Newman-Winacour" solution.
Again, the existence of such solutions doesn't mean that QFTs which can be
modeled in the classical limit as a massless scalar field have been
-unified- with gtr. It just reflects that fact that gtr is a very general
theory of gravitation which is automatically compatible with any
reasonable classical relativistic effective field theory.
(By the way, it may sound like finding such simultaneous solutions is
hard--- in fact, it is really quite easy to find interesting solutions by
an elementary "Ansatz method" in which we stipulate a "geometric Ansatz"
wherein we write down an ONB in terms of two undetermined functions of one
or two coordinates, and then systematically use the field equations to
determine first one, then the other function. When you actually carry out
this process, it often does seem as though "miracles" keep occuring--- but
there is, of course, an underlying explanation: symmetry!)
A third example: one of the most popular types of T^(ab) to place on the
RHS of the EFE is the stress-momentum-energy tensor of a "perfect fluid".
In this case, wrt an appropriate ONB, T^(ab) has the simple form
T^(ab) = diag(rho,p,p,p)
where rho is the mass-energy density and p is the pressure. In
particular, the well known FRW cosmological models are perfect fluid
solutions (isotropic pressure; no viscosity or heat flux) to the EFE.
Or we can contemplate more complicated things like charged fluids (in
which T^(ab) would have two terms, one from the fluid and one from the EM
fields). And so forth.
The analogy between gtr and thermodynamics is more than just an analogy:
as you probably know, Hawking, Press, and Bardeen proved around 1972 that
when two black holes merge, the area of the event horizon new hole is no
smaller than the sum of the areas of the event horizon of the original
pair. In fact, they proved three laws of "black hole mechanics" which
were clearly formally analogous to the three laws of thermodynamics.
This immediately led Bekenstein to suggest that analogy was not a mere
formality and that the area of the event horizon of a black hole should be
identified (up to some scale factor) with its physical entropy. Since
this would require a black hole to have a "uniform temperature", and since
classically, this doesn't make any sense, this proposal was generally
dismissed as being based upon a seductive but faulty analogy. Then, in
1974, Hawking showed that if you take quantum effects into account
("semiclassical approximation" for QFTs on a curved spacetime), black
holes emit black-body radiation, with a temperature corresponding to the
"surface gravity" (this doesn't mean what Newtonian intuition might
suggest; the important point is that it is well-defined and [subject to
some generous conditions] -uniform- over the event horizon) of the hole.
This confirmed that Bekenstein was right: the area of a black hole -can-
be identified with its physical entropy.
Much more recently, superstring theory and loop quantum gravity have each
yielded computations showing that according to these theories, black holes
do indeed have the entropy which is predicted by the semiclassical
approximation. In both cases, this is roughly analogous from passing from
classical thermodynamics to statistical mechanics, since both computations
compute the number of (two different notions of) "microscopic
configurations" for the hole which are consistent with "the macroscopic
data". More precisely, superstring theory gets exactly the right entropy,
but only for near-extremal holes! LQG, on the other hand, gets the entropy
right for -any- hole, but only up to a undetermined scale factor! So at
present, both theories fall a bit short, but in slightly different ways.
Right now, they are pretty much racing neck and neck toward the goal of a
viable quantum theory of gravity, although as I said, superstring theory
has a more ambitious ultimate goal.
Meanwhile, Visser and others have turned this around by suggesting that
none of this implies that thermodynamics and -gtr- are fundamentally
related; rather, it may be that essentially any reasonable gravitation
theory (including the zillions of relativistic classical theories which
are self-consistent but happen not to agree with all
observations/experiments) should obey a generalized thermodynamics
including black holes with entropy and all that. IOW, thermodynamics may
be more fundamental than either quantum theory alone or gravitation alone.
Indeed, they have proposed that Hawking radiation should have physical
analogues in other types of "quantum systems". At least two research
groups are very actively working toward creating a quantum system (using a
"Bose-Einstein condensate") which has an optical (photon) or acoustical
(phonon) analogue of an event horizon, in hope of measuring the expected
blackbody radiation in their lab. If they succeed, this would surely lead
to a Nobel Prize for Hawking and for successful experimenters.
(Ted Jacobson has also made a very interesting proposal regarding the
relationship between generalized thermodynamics and the EFE, but since he
often reads this group, I'll let him describe his suggestion.)
As always, I would welcome correction from the experts here if I have
gotten any of the above wrong. Quite a few regulars here know MUCH more
about black hole thermodynamics, the semiclassical approximation, QFTs,
and the Standard Model, than I do--- so I am sure someone will catch any
errors I may have commited. Indeed, the only reason I tried to talk here
about things I don't know much about is that there is no better way to
learn than to try to explain the stuff you don't know much about, and to
hope that an expert uncovers a misconception in what you said! So keep
your eyes peeled for followups :-/
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
Grace Schalle
wrote:
Zeab Nhoj
presumably aka John Baez wrote:
> Grace Shellac,
> presumably aka Ace Schallger,
> presumably aka Charles Cagle,
> wrote:
Ah, yes, aren't anagrams fun?
> >I've been under the impression that gravity and electromagnetism
> >actually hasn't been unified.
>
> If they haven't been unified, why are you using the
> singular form of the verb in this sentence? :-)
English is my second language, Intuition the first.:-)
> >I asked Jerrold Marsden down at CalTech
> >if this was true.
> >
> >He responded: "...there is a well defined set of equations called the
> >Einstein-Maxwell equations that already unifies E and M with
> >gravity. They are discussed in standard books on general relativity."
> >
> >"Thus, I would say that most researchers believe that these theories
> >are already unified. The big question is how to unify quantum theory
> >with gravity."
> >
> >So, what is the deal here. Am I misinformed? Or is Prof. Marsden
> >mistaken?
>
> There is a well-known set of equations called the Einstein-Maxwell
> equations which are adequate for describing all known phenomena
> involving gravity and electromagnetism as long as one ignore
> quantum mechanics. Marsden is an expert on these.
<snip for brevity>
Indeed, Marsden is an expert, it appears, on many things - a veritable
walking textbook factory.
> One really can't learn the consensus among physicists by doing
> a web search. One needs to read textbooks and journal articles.
> However, in this case it sounds like you got the right impression:
> most physicists would not say electromagnetism and gravity have
> been unified.
That, of course, has been my impression also from reading books and
journal articles. So, Marsden's comments took me aback a bit and I
cannot imagine why he would make such a statement which he first made
via a conversation and then via email??
So, then the question came to me - What exactly would people consider a
reasonable unification of electromagnetism and gravity? After all, it
is difficult to divorce the concept of charged particles from
electromagnetism and the concept of particles seems irrevocably tied to
quanta so that if someone claimed they had unified EM and gravity then
they'd also have to be able to deal with gravity at the level of
quanta. Isn't this correct?
I'm just trying to get things pinned down here so that if one were to
claim that they had unified gravity - how would we know for sure if
they had done it? Do you suppose that if they could make a prediction
about some property of a gravitational field that no one else has made
and then provide a logical argument as to why such a property ought to
be found that such a claim would be sufficient cause for believing
them? (Presuming, of course, that such a property ought to be
manifested in physical phenomena).
G.S.
Eric Alan Forgy
This is a question that confuses me from time to time too. Hopefully, I
didn't get your hopes up because instead of trying to answer your question,
this is a just follow up :)
> He responded: "...there is a well defined set of equations called the
> Einstein-Maxwell equations that already unifies E and M with
> gravity. They are discussed in standard books on general relativity."
>
> "Thus, I would say that most researchers believe that these theories
> are already unified. The big question is how to unify quantum theory
> with gravity."
I come across a lot of papers that claim to have unified gravity and
electromagnetism also. In fact, I was just skimming
"Geometrodynamics" -Wheeler the other day and HE claims that
geometrodynamics unifies gravity and EM also. The idea goes roughly like the
imprint that EM leaves on geometry is unique. Hence, by looking at the
geometry, you can deduce what the EM was. In this way, it seems he is saying
that EM is just geomtry and already incorporated into GR. Have I
misinterpretted what he is saying? It doesn't seem like he is adding in
Maxwell's equations by hand to me. It sounds like Einstein-Maxwell is H_GR +
H_EM (or L_GR + L_EM, whatever :)). Is this the same as what Wheeler is
talking about?
Thanks,
Eric
ark
<furlong.phy...@singtech.com> wrote:
>I've been under the impression that gravity and electromagnetism
>actually hasn't been unified. I asked Jerrold Marsden down at CalTech
>if this was true.
>
>He responded: "...there is a well defined set of equations called the
>Einstein-Maxwell equations that already unifies E and M with
>gravity. They are discussed in standard books on general relativity."
>
>"Thus, I would say that most researchers believe that these theories
>are already unified. The big question is how to unify quantum theory
>with gravity."
>
>So, what is the deal here. Am I misinformed?
No, you are not misinformed. You got a direct information about one
particular point of view. To have a full picture you need to read
what other physicists think of this problem. There are at least three
other ways.
> Or is Prof. Marsden
>mistaken?
Prof Marsden represents here a conservative trend, I would say,
and a short description of one vesrsion of "already unified"
theory (originated by Rainich) you can find in "Gravitation" by Misner
et al.
> Doing a google.com search didn't seem to help as there
>doesn't seem to be a consensus that gravity and electromagnetism are or
>have been satisfactorily unified. I see web pages by people like
>Sweetster and others but no definitive comment that indicates that
>anyone actually thinks gravity and electromagnetism have been unified.
See above. Alternatives are:
a) try to describe gravitation as a function of electromagnetic
phenomena
b) try to describe electromagnetism as a function of gravitational
phenomena
c) try to derive both from more fundamental fields
d) try to derive both by making use of alternative geometries
(Finsler, Einstein-Cartan, gauge theories, more than four space-time
dimensions, noncommutative geometry etc
e) try to derive Theory of Everything, and EM+G will come
in a big bag of all kinds of goodies.
>Help! What's the latest opinion?
Different physicists will have different opinions. There is no
consensus. Marsden is certainly right that we have also
problems with adding quantum theory - and here you will find not so
many physicists that will tell you that there are no problems there,
The above is simply understanding of the present situation, based on
my own research experience, not an autoritative answer.
ark
Simon Clark
> I've been under the impression that gravity and electromagnetism
> actually hasn't been unified. I asked Jerrold Marsden down at CalTech
> if this was true.
>
> He responded: "...there is a well defined set of equations called the
> Einstein-Maxwell equations that already unifies E and M with
> gravity. They are discussed in standard books on general relativity."
This is true, but...
> "Thus, I would say that most researchers believe that these theories
> are already unified.
...this is -very- misleading. This is not what physicists mean when talking
about unifying theory X with theory Y.
Take the vacuum (say) Maxwell equations for the field-strength 2-form F in
differential form notation:
dF = 0 (usually considered an identity since F=dA and d^2=0)
d*F = 0 (= j gives the Maxwell equations with a source.)
which are already valid in curved spacetime (if * is then taken to be the
Hodge map associated with the non-flat metric tensor). Now write the
Einstein equation as:
Ein = T[F]
where Ein is the Einstein tensor and T[F] is the energy-momentum tensor of
the EM field associated with F. (Look up the exact expression for T in a GR
book.) These are the Einstein-Maxwell equations you mentioned above. But I
don't think many physicists would describe this as a unification. (Have we
unified EM with electric charge? Newtonian gravity with mass? etc...)
Generally a field equation takes the form:
[Differential operator] Field = Source (or D F = S for short)
where Source is independent of Field.
Now if we have two (say) sets of field equations:
D_1 F_1 = S_1
D_2 F_2 = S_2
where S_1 might depend on F_2 and S_2 might depend on F_1, an (at least
partial) unification of the two theories would require that we write:
D_u F_u = S_u
where F_u is constructed in some way from F_1 and F_2 and S_u is
independent of F_u. A good example of this would be unifying electricity
and magnetism into electro-magnetism. We combine the electric field
E and magnetic field B into the field-strength F.
It -is- possible to arrange this for GR and EM (google for Kaluza-Klein)
though the is much doubt as to the usefulness of doing this. I'll leave
this for someone else to discuss.
--
Simon Clark
Terry Pilling
>
> Starting with Einstein's own search for a "unified field theory",
> everyone has been looking for something that will combine gravity
> and electromagnetism in a more profoundly integrated way. One nice
> attempt was Kaluza and Klein's theory: Einstein's equations on a 5d
> spacetime where the extra dimension is curled up in a little circle
> gives Einstein's equation in ordinary 4d spacetime, *together*
> with Maxwell's equations... *together* with a massless spin-0
> particle that, sadly, appears not to exist. Oh well - nice try!
The way you say `nice try' at the end, seems to mean that the massless
scalar is a embarrassment and that it somehow disproves the theory. In
fact this isn't not so. Theories that predict new particles are widely
accepted nowadays (unlike the 20s when KK theory was born). The fact
that the Brans-Dicke scalar has not been found in accelerators certainly
doesn't mean it can't exist.
In retrospect it is laughable that they thought back then they could
discredit a theory because of a predicted yet unseen particle.
Especially now that we know how many unseen particles they were
clueless about. At the time, Chadwick had not even discovered
the neutron yet! The standard model of particle physics has many new
particles, *including* an unseen scalar called the Higgs boson which is
integral to the model.
There are many carpets in a QFT to sweep things under. The string
theory, that we all love, has a massless scalar too (the dilaton)
but it is hidden away in the string coupling. That is just the
scalar, string theory sweeps a few other predicted particles under the
carpet too -- infinitely many of them! (anything with non-zero mass).
This implies that extra particles and dimensions that aren't seen
in the lab really don't harm us too much anymore.
The conclusion can only be that the jury is still out on whether
Einstein and Maxwell are unified. The reason the jury is still out
is because people are no longer satisfied with that mere baby step.
They want to use the same or similar ideas to unify everything!
-Ter
John Baez
Charles Cagle wrote:
>> One really can't learn the consensus among physicists by doing
>> a web search. One needs to read textbooks and journal articles.
>> However, in this case it sounds like you got the right impression:
>> most physicists would not say electromagnetism and gravity have
>> been unified.
>That, of course, has been my impression also from reading books and
>journal articles. So, Marsden's comments took me aback a bit and I
>cannot imagine why he would make such a statement which he first made
>via a conversation and then via email??
Well, two options leap to mind. Either he is not an expert on
"unification" and simply fails to realize that most people trying
to unify gravity and electromagnetism want more than the Einstein-
Maxwell theory, or else he has opinions about this subject that
don't fit the consensus. In either case, it's no big deal. Nobody
is an expert on everything, and you can never count on learning the
consensus view by talking to just *one* person. Shop around.
>So, then the question came to me - What exactly would people consider a
>reasonable unification of electromagnetism and gravity?
If we knew, we wouldn't be sitting around here wasting time on
the internet, now, would we?
Somewhat more seriously, the best way to tackle this question is
to learn the various theories people have proposed to answer
this question, starting from Einstein's own unified field theories,
Hermann Weyl's theory, and the original Kaluza-Klein theory, and
working on up to modern attempts like superstring theory. The
reason is that a theory will only be considered worth bothering
with if it does at least as well as all of these.
>After all, it
>is difficult to divorce the concept of charged particles from
>electromagnetism and the concept of particles seems irrevocably tied to
>quanta so that if someone claimed they had unified EM and gravity then
>they'd also have to be able to deal with gravity at the level of
>quanta. Isn't this correct?
Well, let me say it this way: by now it would be very hard to interest
people in a theory unifying electromagnetism and gravity that didn't
also take quantum mechanics into account.
>I'm just trying to get things pinned down here so that if one were to
>claim that they had unified gravity - how would we know for sure if
>they had done it?
The best way is the good old-fashioned way: they would make new
experimental predictions which several independent well-respected
teams of researchers could repeatedly verify.
>Do you suppose that if they could make a prediction
>about some property of a gravitational field that no one else has made
>and then provide a logical argument as to why such a property ought to
>be found that such a claim would be sufficient cause for believing
>them?
If I believed the "logical argument" I might believe the theory.
If I didn't, I wouldn't. But anyway, this is a lot less good than the
old-fashioned approach where a physical theory makes a new prediction
and it's actually VERIFIED BY EXPERIMENT.
Chris Hillman
On 7 Nov 2001, I wrote:
> The best known example of an electrovacuum is the "Reissner-Nordstrom
> electrovacuum" (1918). This can be defined by giving the static polar
> spherical coordinate chart
>
> ds^2 = -(1-2m/r+q^2/r^2) dt^2 + dr^2/(1-2m/r+q^2/r^2)
>
> + r^2 (du^2 + sin(u)^2 dv^2),
>
> -infty < t < infty, 0 < r < infty, 0 < u < Pi, -Pi < v < Pi
^^^^^^^^^^^^^
m + sqrt(m^2-q^2) < r < infty
Sorry for the goof.
Charles Francis
writes
>In article <071120010040475449%furlong.phy...@singtech.com>,
>Charles Cagle wrote:
>
>
>>So, then the question came to me - What exactly would people consider a
>>reasonable unification of electromagnetism and gravity?
>
>If we knew, we wouldn't be sitting around here wasting time on
>the internet, now, would we?
Wouldn't we? What could we be doing about it?
>>I'm just trying to get things pinned down here so that if one were to
>>claim that they had unified gravity - how would we know for sure if
>>they had done it?
>
>The best way is the good old-fashioned way: they would make new
>experimental predictions which several independent well-respected
>teams of researchers could repeatedly verify.
And how would they pursuade several independent well-respected teams to
work on the problem if there was not already some other reason the well-
respected teams should believe them. They could not, expect the
experimenters to invest in belief in an unverified theory.
>>Do you suppose that if they could make a prediction
>>about some property of a gravitational field that no one else has made
>>and then provide a logical argument as to why such a property ought to
>>be found that such a claim would be sufficient cause for believing
>>them?
>
>If I believed the "logical argument" I might believe the theory.
That's the thing about logical argument. It does not have to be
believed, only studied. But one has to believe that the study is worth
while, and if a view point is unorthodox, as we must assume unification
will be (since otherwise it would already be known), then it is
difficult to believe it might be worth serious thought.
>If I didn't, I wouldn't. But anyway, this is a lot less good than the
>old-fashioned approach where a physical theory makes a new prediction
>and it's actually VERIFIED BY EXPERIMENT.
>
Oh dear, more crackpot points. I think there might be some doubt from
the perspective of history of science that scientific revolutions have
worked like this, though it is probably correct for normal science.
Regards
--
Charles Francis
Charles Cagle
> In article <071120010040475449%furlong.phy...@singtech.com>,
> Charles Cagle wrote:
>
> >> One really can't learn the consensus among physicists by doing
> >> a web search. One needs to read textbooks and journal articles.
> >> However, in this case it sounds like you got the right impression:
> >> most physicists would not say electromagnetism and gravity have
> >> been unified.
>
> >That, of course, has been my impression also from reading books and
> >journal articles. So, Marsden's comments took me aback a bit and I
> >cannot imagine why he would make such a statement which he first made
> >via a conversation and then via email??
>
> Well, two options leap to mind. Either he is not an expert on
> "unification" and simply fails to realize that most people trying
> to unify gravity and electromagnetism want more than the Einstein-
> Maxwell theory, or else he has opinions about this subject that
> don't fit the consensus. In either case, it's no big deal. Nobody
> is an expert on everything, and you can never count on learning the
> consensus view by talking to just *one* person. Shop around.
Fair enough. And my shopping around has shown me, at least to my
satisfaction that no one really has a 'definitive' and by that I also
mean a 'certain' handle on gravity. Indeed, there are a number of
different theoretical approaches which seem almost to be divided into
'schools' (of thought).
> >So, then the question came to me - What exactly would people consider a
> >reasonable unification of electromagnetism and gravity?
>
> If we knew, we wouldn't be sitting around here wasting time on
> the internet, now, would we?
Hopefully, we're not all wasting time. I, at least, as an amateur
(meaning I do it for the love of it) physicist can at least see a
certain amount of value in everyone hanging their sheets out in the
wind (displaying and discussing their beliefs and pet theories for
others to see and contemplate). As a generalist I'm more interested in
the similarities or commonalities as far as foundational conceptions go
which are woven like threads through all the competing theories.
Perhaps, if the unexamined life is not worth living then maybe those
unexamined foundational 'treads', which together run as a theme through
each of these theories and gives them raison d'etre is not worth
having? With that idea in mind, if you were asked to boil down the
competing theories that you are well versed concerning what might be
left in your pot after the evaporates have departed? This question I
ask in keeping with your next bit of advice below:
> Somewhat more seriously, the best way to tackle this question is
> to learn the various theories people have proposed to answer
> this question, starting from Einstein's own unified field theories,
> Hermann Weyl's theory, and the original Kaluza-Klein theory, and
> working on up to modern attempts like superstring theory. The
> reason is that a theory will only be considered worth bothering
> with if it does at least as well as all of these.
>
> >After all, it is difficult to divorce the concept of charged
> >particles from electromagnetism and the concept of particles seems
> >irrevocably tied to quanta so that if someone claimed they had
> >unified EM and gravity then they'd also have to be able to deal with
> >gravity at the level of quanta. Isn't this correct?
>
> Well, let me say it this way: by now it would be very hard to interest
> people in a theory unifying electromagnetism and gravity that didn't
> also take quantum mechanics into account.
>
> >I'm just trying to get things pinned down here so that if one were to
> >claim that they had unified gravity - how would we know for sure if
> >they had done it?
> The best way is the good old-fashioned way: they would make new
> experimental predictions which several independent well-respected
> teams of researchers could repeatedly verify.
Fair enough.
> >Do you suppose that if they could make a prediction about some
> >property of a gravitational field that no one else has made and then
> >provide a logical argument as to why such a property ought to be
> >found that such a claim would be sufficient cause for believing
> >them?
> If I believed the "logical argument" I might believe the theory.
> If I didn't, I wouldn't. But anyway, this is a lot less good than the
> old-fashioned approach where a physical theory makes a new prediction
> and it's actually VERIFIED BY EXPERIMENT.
Okay, I see you would make the claimant walk a very thin line here.
Now, you know that the longer period of time that we look out into the
universe by the various apparatuses we manage to construct to collect
data that more and more phenomenon are apprehended which have never
been seen before. As a matter of course and habit scientists are prone
to leap upon each new find and immediately clothe it with a new theory
or some variation of an old theory. Sometimes they are forced to admit
that they flat just do not understand the phenonenon that their sensory
apparatus or instrument has discovered.
So, as time goes on it becomed more and more difficult to actually come
up with a new prediction. On that basis wouldn't it also be just as
reasonable to suggest that if the new model could throw its lasso
around a large number of observations and datum at once, some thought
to be well understood and some admitted to not be understood at all,
and provide a common physical understanding for them all that such a
model would be considered at least as equivalent to one which made a
new prediction since in a way the ability to link seemingly disparate
data really is the goal of a general theory or model in the first
place?
C. C.
John Baez
Eric Alan Forgy <fo...@uiuc.edu> wrote:
>I come across a lot of papers that claim to have unified gravity and
>electromagnetism also. In fact, I was just skimming
>"Geometrodynamics" -Wheeler the other day and HE claims that
>geometrodynamics unifies gravity and EM also. The idea goes roughly like the
>imprint that EM leaves on geometry is unique. Hence, by looking at the
>geometry, you can deduce what the EM was.
Hmm. Certainly nobody has succeeded in getting anywhere
with *this* idea. I hadn't even known Wheeler went that
far. I'm more familiar with his ideas on "charge without
charge" - a trick for trying to get charged particles to
show up as an automatic consequence of the Einstein-Maxwell
equations when you allow topologies with wormholes. Nobody
has ever worked out the details of this idea of "charge
without charge"; the idea of getting Maxwell as an automatic
consequence of Einstein seems even more speculative.
Louis Crane's latest paper on the physics preprint server
contains a new trick for trying to get electromagnetism
and other gauge fields to arise naturally from the spin foam
approach to quantum gravity. I like it, but the details
(ahem) remain to be worked out.
Toby Bartels
>As you no doubt know, in the last century, these three QFTs were
>incorporated into a single QFT usually called the "Standard Model", which
>predicts that at very high energies, the three interactions become
>indistinguishable, but at the kind of energies we can readily observe,
>they appear to be different interactions with -very- different properties!
>Another way of understanding why the Standard Model can be said to "unify"
>the EM, weak, strong interactions is by observing that its gauge group or
>fundamental symmetry group includes as subgroups the symmetry groups of
>the three specialized QFTs, e.g. the abelian circle group U(1) for QED
>(the other two are "nonabelian gauge theories").
I'd never heard before that the SM unified the strong force with the others.
It was my understanding that the strong and electroweak forces
had simply been, as someone put it before, "slapped together".
You can see this in the gauge group U(1) x SU(2) x SU(3),
which is simply a direct product of the electroweak U(1) x SU(2)
and the strong SU(3). In contrast, the electroweak U(1) x SU(2)
is not (as people are sometimes fooled into thinking)
the direct product of an EM U(1) and a weak SU(2) --
the EM U(1) is a different copy of U(1) sitting inside SU(2),
and the weak force can't be fully described divorced from EM at all.
Similarly, while everybody seems to believe that
the 3 forces are, as you say, indistinguishable at high energies,
I've never heard anyone say that the SM explains this,
only that it explains the unification at the electroweak scale.
The goal of theories like the SU(5) (and worse) GUTs
is to unify the elctroweak interaction with the strong,
and these go beyond merely the SM. At least that's what I've read --
perhaps I am behind the latest news.
-- Toby
to...@math.ucr.edu
Charles Cagle
<cha...@clef.demon.co.uk> wrote:
> That's the thing about logical argument. It does not have to be
> believed, only studied. But one has to believe that the study is worth
> while, and if a view point is unorthodox, as we must assume unification
> will be (since otherwise it would already be known), then it is
> difficult to believe it might be worth serious thought.
Don't you mean that it is difficult to believe that it might be
considered to be worth serious thought?
Ben
Charles Francis wrote:
> In article <9sc5k3$lkt$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
> writes
> >In article <071120010040475449%furlong.phy...@singtech.com>,
> >Charles Cagle wrote:
> >
> >
> >>So, then the question came to me - What exactly would people consider a
> >>reasonable unification of electromagnetism and gravity?
> >
> >If we knew, we wouldn't be sitting around here wasting time on
> >the internet, now, would we?
>
> Wouldn't we? What could we be doing about it?
Building things. If you can come up with a means of uniting electromagnetism
and gravity, it seems to me that you could construct a device to convert
electrical energy into a gravitational field. Tractor beams, possibly warp
drive, or worm holes, or other kinds of cool toys.
> >>I'm just trying to get things pinned down here so that if one were to
> >>claim that they had unified gravity - how would we know for sure if
> >>they had done it?
> >
> >The best way is the good old-fashioned way: they would make new
> >experimental predictions which several independent well-respected
> >teams of researchers could repeatedly verify.
>
> And how would they pursuade several independent well-respected teams to
> work on the problem if there was not already some other reason the well-
> respected teams should believe them. They could not, expect the
> experimenters to invest in belief in an unverified theory.
Or in other words, build things that make use of the unique properties of the
new theory.
> >>Do you suppose that if they could make a prediction
> >>about some property of a gravitational field that no one else has made
> >>and then provide a logical argument as to why such a property ought to
> >>be found that such a claim would be sufficient cause for believing
> >>them?
> >
> >If I believed the "logical argument" I might believe the theory.
>
> That's the thing about logical argument. It does not have to be
> believed, only studied. But one has to believe that the study is worth
> while, and if a view point is unorthodox, as we must assume unification
> will be (since otherwise it would already be known), then it is
> difficult to believe it might be worth serious thought.
And of course all logical arguments are based on axioms, or postulates, that
have to be true, in order for the logical argument itself to be true or valid
for the real world. Good logic with false postulates still yields false
results, inconsistent with reality.
Pudding is the best evidence to support a theory. The proof of the pudding is
in the eating. The proof of a theory, ultimately is in the technical uses
that it can be applied to and the new technology that results.
Which brings up an interesting question, as to how much serious thought
should be invested in such ideas. The potential reward, but technically and
economically, are great. Very great. The risk is also great too, if past
performance is any judge. As you note, it looks to be very difficult. But
then again, all you have to do is get it right once, and there is a very
large reward for that.
Ben
Charles Cagle
> In article <9sc5k3$lkt$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
> writes
> >In article <071120010040475449%furlong.phy...@singtech.com>,
> >Charles Cagle wrote:
> >
> >
> >>So, then the question came to me - What exactly would people consider a
> >>reasonable unification of electromagnetism and gravity?
> >
> >If we knew, we wouldn't be sitting around here wasting time on
> >the internet, now, would we?
>
> Wouldn't we? What could we be doing about it?
Well, by saying 'What exactly would people consider a reasonable
unification of electromagnetism and gravity?' I meant to illustrate
that perhaps we don't even know what questions to ask on our road to
unification? I think there's a lot of questions that we could ask but
haven't and so I believe that I've never seen the issues addressed
properly.
Quantum gravity seems to be the obvious place to start. I mean really,
the magnitude of the gravitational field of a planet (per our present
understanding) at a point is the vector sum of all the gravitational
fields of all of the bits of particulate mass that comprises the
planet. So, it is obvious that we need to examine questions concerning
the gravitational fields produced by elementary particles.
Now, as far as I know, we *assume* that each bit of mass in the
universe attracts every other bit of mass or that each bit of mass in
the universe is a gravitational field source. But is this truly so?
So, I think that a reasonable question might be: "Are some particles
attracted to gravitational fields but are not themselves the source of
a gravitational field"? We can ask the corollary to that also which
is: "Are some particles repelled by gravitational fields but are not
themselves the source of a gravitational field"?
Do we have any definitive experiment which proves that protons, for
example, are the source of a gravitational field as opposed to just
being an object of attraction towards a gravitational field?
Isn't this somewhat equivalent to asking the question of whether or not
all inertial mass also produces a gravitational field?
Do we have any definitive experiment which proves that electrons are
attracted toward a gravitational field as opposed to being attracted
towards protons which are attracted towards a gravitational source or
even that electrons are the source of a gravitational field?
Aren't these the sort of questions that we ought to be asking, that we
ought to be addressing in our experiments?
Don't you think that it is possible that neutrons produce a
gravitational field but protons and electrons do not?
> >>I'm just trying to get things pinned down here so that if one were to
> >>claim that they had unified gravity - how would we know for sure if
> >>they had done it?
> >
> >The best way is the good old-fashioned way: they would make new
> >experimental predictions which several independent well-respected
> >teams of researchers could repeatedly verify.
>
> And how would they pursuade several independent well-respected teams to
> work on the problem if there was not already some other reason the well-
> respected teams should believe them. They could not, expect the
> experimenters to invest in belief in an unverified theory.
Well, I think the key is not to expect belief in an unverified theory
at all but rather to examine the validity of the claims which would
emerge from it - especially so if they are interesting questions,
hopefully like those I asked above. I personally believe that more
than half of the battle is asking the right questions.
> >>Do you suppose that if they could make a prediction
> >>about some property of a gravitational field that no one else has made
> >>and then provide a logical argument as to why such a property ought to
> >>be found that such a claim would be sufficient cause for believing
> >>them?
> >
> >If I believed the "logical argument" I might believe the theory.
>
> That's the thing about logical argument. It does not have to be
> believed, only studied. But one has to believe that the study is worth
> while, and if a view point is unorthodox, as we must assume unification
> will be (since otherwise it would already be known), then it is
> difficult to believe it might be worth serious thought.
Mr. Frances, this is sort of amusing and pathetic at the same time
because it is the classic 'Catch-22' sort of situation. I'm going to
clip and save this in my well beloved quotes file because I believe you
have concisely captured the problem in just a few words. Thanks.
> >If I didn't, I wouldn't. But anyway, this is a lot less good than the
> >old-fashioned approach where a physical theory makes a new prediction
> >and it's actually VERIFIED BY EXPERIMENT.
> >
> Oh dear, more crackpot points. I think there might be some doubt from
> the perspective of history of science that scientific revolutions have
> worked like this, though it is probably correct for normal science.
I don't get it. Help me out here, Charles. I'm slow. What did you
mean by "Oh dear, more crackpot points." Are you saying that we place
too much significance on finding data that is consistent with a theory
and that John Baez's capitalization of 'VERIFIED BY EXPERIMENT' implies
in your mind that the normal view that experimental data verifies
theories is a form of crackpottery? I see that it is perfectly
plausible to have data that might be predicted by theory to actually
have its source from processes not related to the theory at all so that
the researchers are misled into believing that the theory is 'verified'
(which means established as true) when no such thing has actually
occurred. Is this what you meant?
C.C.
Brian J Flanagan
> > I've been under the impression that gravity and electromagnetism
Simon Clark wrote in message:
> Take the vacuum (say) Maxwell equations for the field-strength 2-form F in
> differential form notation:
>
> dF = 0 (usually considered an identity since F=dA and d^2=0)
> d*F = 0 (= j gives the Maxwell equations with a source.)
[Flanagan]
'Gravitation', by Misner, Thorne, and Wheeler provides a very nice
treatment of the material referenced above. This big book has two
tracks; the first takes you from very easy beginnings up through more
advanced topics. The second features material suitable for grad
students and professionals. Both tracks offer many helpful
illustrations.
> It -is- possible to arrange this for GR and EM (google for Kaluza-Klein)
> though there is much doubt as to the usefulness of doing this. I'll leave
> this for someone else to discuss.
Kaluza-Klein theory has been enjoying a revival thanks to
string/M-theory, where adding more spatial dimensions to the metric
promises to yield all the known interactions. This is in direct
analogy with the original K-K theory, where the addition of a fifth
dimension was shown to yield both gravitation and EM. Also in line
with the original theory is the notion that the extra dimensions must
be "compactified" (very small) because we do not "see" them.
I highly recommend O'Raifeartaigh's beautiful little book on 'The
Dawning of Gauge Theory', which contain the original papers, together
with excerpts from the seminal works by Weyl, London, Schrodinger,
Fock, et al. Also of interest is 'Modern Kaluza-Klein Theories' (ed.,
Applequist) where you will find an easy overview of the subject by
Witten entitled "Search for a Realistic Kaluza-Klein Theory'. Witten
illuminates the relationship between relativity, K-K theories, gauge
theory, and hidden variables theory. Also of great interest in this
regard is Cao's essay "Gauge Theory" in 'Philosophical Foundations of
QFT'. (ed., Brown & Harre)
t...@rosencrantz.stcloudstate.edu
Ben <dra...@nospam.worldnet.att.net> wrote:
>
>
>
>Charles Francis wrote:
>
>> In article <9sc5k3$lkt$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
>> writes
>> >If we knew, we wouldn't be sitting around here wasting time on
>> >the internet, now, would we?
>>
>> Wouldn't we? What could we be doing about it?
>
>Building things. If you can come up with a means of uniting electromagnetism
>and gravity, it seems to me that you could construct a device to convert
>electrical energy into a gravitational field. Tractor beams, possibly warp
>drive, or worm holes, or other kinds of cool toys.
While I suppose this is possible, it doesn't seem at all likely
to me. I predict that the theory of everything, once it arrives,
will have few if any immediate technological applications. Maybe
way down the road it will, but not right away.
>Pudding is the best evidence to support a theory. The proof of the pudding is
>in the eating. The proof of a theory, ultimately is in the technical uses
>that it can be applied to and the new technology that results.
I strongly disagree with this. By this measure, the standard
model of particle physics is worth precisely nothing, since
no technology has come from it, and general relativity is worth
next to nothing. (As many readers of this group know, the Global
Positioning System does involve general-relativistic corrections,
so it's not strictly true to say that general relativity has
no technological uses. But it's pretty close!)
-Ted
Ben
t...@rosencrantz.stcloudstate.edu wrote:
Oh, very bad argument to make, especially if you are trying to obtain
government grants. What you are saying is that high energy physics is
really nothing, not worth the time and other people's money invested
in it. If there is no practical application for the knowledge, then
why should I spend my money, or tax dollars to support it? So you feel
better, have something to do? That is hardly a reason that will
satisfy most taxpayers, or their elected officials.
It is becoming increasingly apparent that the Standard model is in
trouble. Neutron mass alone is problematic for it, and it has never
been a satisfactory model of fundamental forces anyway. It has too
many constants, or factors that have to be determined empirically,
rather than predicted by theory. SLAC's own work on CP violation
threatens the Standard Model.
So what will the eventual big TOE look like, explain or even predict?
Neither of us know. What new technologies will a new theory of
everything enable? Again, we are really only guessing. It is all that
we can do until and unless that theory comes around.
What use is pure research? Well, one answer was to respond what is the
use of a new born baby. It is a far more politic answer than
"nothing". Asking others to pay for the hobbies of the few, thereby
allowing them to generate papers they will not understand, is not
going to sit well with Joe Taxpayer. He will see your statements for
what it is, a demand to pay for your hobbies, your worthless (to him)
pursuits. He will simply reject it, and take his money to pay for his
own hobbies. Which leaves you under funded, if funded at all.
BTW it should be pointed out that while GR may be of limited utility
today, there is no guarantee that it will stay that way. And that GR
is founded on SR, which brought us E=mc^2. Which in turn resulted in
the nuclear reactor and the atomic bomb.
Ben
t...@rosencrantz.stcloudstate.edu
Ben <dra...@nospam.worldnet.att.net> wrote:
>t...@rosencrantz.stcloudstate.edu wrote:
>
>> In article <3BEE4E47...@nospam.worldnet.att.net>,
>> Ben <dra...@nospam.worldnet.att.net> wrote:
>> >in the eating. The proof of a theory, ultimately is in the technical uses
>> >that it can be applied to and the new technology that results.
>>
>> I strongly disagree with this. By this measure, the standard
>> model of particle physics is worth precisely nothing, since
>> no technology has come from it, and general relativity is worth
>> next to nothing. (As many readers of this group know, the Global
>> Positioning System does involve general-relativistic corrections,
>> so it's not strictly true to say that general relativity has
>> no technological uses. But it's pretty close!)
>
>Oh, very bad argument to make, especially if you are trying to obtain
>government grants.
As you may have noticed, I wasn't trying to obtain a government grant
in the above post. :-) (Although if an NSF program officer happens to
wander by and offer me one on the basis of that post, I won't turn it
down!) I was simply making the statement that technology is a poor
yardstick to use in judging physical theories, since there are
physical theories (e.g., the standard model and general relativity)
that are towering achievements of the human intellect and that have
had little or no technological impact.
The question of what funding agencies should look for, and the
question of what they actually do look for, in deciding what research
to fund are a different matter. Even there, though, I claim that
technology is a poor yardstick. It's one criterion that the agencies
do and should look at, but not the only one or even the primary one.
For instance, I have an NSF grant to support my research in cosmological
theory and data analysis. This is work that clearly has precisely
no technological benefit, but the NSF funded it.
You may disagree with the proposition that funding agencies *should*
use criteria other than technological or practical benefits in
deciding what to fund. That's fine with me. I think you'll have a
harder time disagreeing with the statement that funding agencies
actually *do* rely heavily on criteria other than technological or
practical benefits in deciding what to fund. The former is an
opinion; the latter is a statement of fact.
It's true that funding agencies do like to tout technological benefits
when funding big things like particle accelerators. I suspect that
that's largely PR on everyone's part -- neither the scientists nor the
funding agencies really think that that's a primary goal of the
research, but they all think that it's what the public wants to hear.
(Whether they're right about this is another matter -- I'll say
something about that below.) It seems quite obvious to me that if you
really want to fund things that will lead to technological benefits,
you should *not* fund a particle accelerator. The technological
payoff will be much higher if you dump that funding into, say,
materials science research.
Personally, I think that the case for government funding of basic
research is a lot like the case for government funding of the arts and
humanities. Human intellectual achievement is intrinsically valuable.
General relativity, like "Hamlet," will be a supremely important
human achievement regardless of whether it ever yields anything
of technological importance.
>What you are saying is that high energy physics is
>really nothing, not worth the time and other people's money invested
>in it. If there is no practical application for the knowledge, then
>why should I spend my money, or tax dollars to support it? So you feel
>better, have something to do? That is hardly a reason that will
>satisfy most taxpayers, or their elected officials.
I suspect you underestimate taxpayers. The Hubble Space Telescope,
for instance, is enormously popular, and nothing it ever does is
likely to yield technological benefits.
Part of the reason for this is that NASA has gone to quite a bit of
trouble to publicize the results coming out of HST. All large-scale
research projects should do this, so that the results of the research
will be clear to the people who funded it. That's not only good PR;
it's also the right thing to do.
>BTW it should be pointed out that while GR may be of limited utility
>today, there is no guarantee that it will stay that way. And that GR
>is founded on SR, which brought us E=mc^2. Which in turn resulted in
>the nuclear reactor and the atomic bomb.
Incidentally, I wonder how defensible the claim is that special
relativity, through E=mc^2, "resulted in" discoveries about nuclear
energy. Imagine special relativity had not been invented. People
would still have been doing nuclear physics and would still have
discovered that nuclei have different binding energies. I doubt that
special relativity accelerated the development of nuclear bombs
significantly at all.
-Ted
Chris Hillman
On Mon, 12 Nov 2001, Toby Bartels wrote:
> I'd never heard before that the SM unified the strong force with the others.
> It was my understanding that the strong and electroweak forces
> had simply been, as someone put it before, "slapped together".
> You can see this in the gauge group U(1) x SU(2) x SU(3),
[snip]
> Similarly, while everybody seems to believe that
> the 3 forces are, as you say, indistinguishable at high energies,
> I've never heard anyone say that the SM explains this,
> only that it explains the unification at the electroweak scale.
OK, as I said in my post I didn't know what I was talking about, really,
when I talked about the SM, so thanks for the corrections. I am sure you
have it right (I take the point about the symmetry groups of SM versus
electroweak).
Charles Francis
<pro...@singtech.com> writes
>In article <9ska0a$e...@gap.cco.caltech.edu>, Charles Francis
><cha...@clef.demon.co.uk> wrote:
>> In article <9sc5k3$lkt$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
>> writes
>> >In article <071120010040475449%furlong.phy...@singtech.com>,
>> >Charles Cagle wrote:
>> >>So, then the question came to me - What exactly would people consider a
>> >>reasonable unification of electromagnetism and gravity?
>> >If we knew, we wouldn't be sitting around here wasting time on
>> >the internet, now, would we?
>> Wouldn't we? What could we be doing about it?
>Well, by saying 'What exactly would people consider a reasonable
>unification of electromagnetism and gravity?' I meant to illustrate
>that perhaps we don't even know what questions to ask on our road to
>unification?
That is largely true. People will not understand how to unify gravity
and electromagnetism until they understand how to ask the right
questions in the right way. But that is very much the sort of thing we
can discuss on the internet.
> I think there's a lot of questions that we could ask but
>haven't and so I believe that I've never seen the issues addressed
>properly.
Are you sure you would recognise it if you had? Addressing the issues
properly means doing it formally and mathematically, so that, given
enough study, there is no room for error and doubt. That may be done in
papers, not in discussion.
>Now, as far as I know, we *assume* that each bit of mass in the
>universe attracts every other bit of mass or that each bit of mass in
>the universe is a gravitational field source. But is this truly so?
>So, I think that a reasonable question might be: "Are some particles
>attracted to gravitational fields but are not themselves the source of
>a gravitational field"? We can ask the corollary to that also which
>is: "Are some particles repelled by gravitational fields but are not
>themselves the source of a gravitational field"?
Actually this has been addressed fairly well on s.p.r during the last
couple of years. If you run a search for threads featuring both myself
and Steve Carlip, you will see that he has had a lot of useful
information on the subject. For example, here is a post from Steve
Carlip I have kept:
*************
Charles Francis <cha...@clef.demon.co.uk> wrote:
> But is there any way we can say for sure on the basis of
> our current knowledge that photons and/or neutrinoes affect
> curvature, or is it just something we take as read from the field
> equation, but which is actually an open question?
If you mean free photons, then I don't think there's any direct
experimental test. But
(1) There is *extremely* good evidence that electrostatic and
magnetostatic fields contribute an amount E/c^2 to ``passive''
gravitational mass, from tests of the principle of equivalence.
These are accurate at a level of a few parts in 10^10 for the
electrostatic contribution, and better than a part in 10^5 for
the magnetostatic contribution.
(Note that a test of the principle of equivalence is automatically
a test of the contribution of various forms of energy to the
passive gravitational mass. We know that electromagnetic
binding energy contributes to inertial mass, for example. If
it did not contribute equally to gravitational mass, then two
objects with different proportions of binding energy would
have different ratios of inertial and gravitational mass.)
(2) The effect on curvature is actually related to ``active''
gravitational mass rather than ``passive'' gravitational mass.
Tests here are more difficult, though I think you'll find it very
hard to construct a theory in which active and passive masses
are unequal---this leads to a violation of Newton's third law,
and to nonconservation of momentum.
(3) There is at least one direct test of the principle of equivalence
for active gravitational mass (Kreuzer, Phys. Rev. 169 (1968)
1007). It is considerably less accurate than the tests for passive
gravitational mass, but still good enough to show that electrostatic
binding energy contributes E/c^2 to active gravitational mass, and
thus to spacetime curvature, within 5% or so.
For details, start with Will's book _Theory and Experiment in
Gravitational Physics_.
Steve Carlip
******* End of Steve Carlip's post
Likewise there have been discussions of the relative gravitational
masses of quarks. It seems pretty certain that every elementary particle
gravitates in the same way because we can show that any differences in
the gravitational mass of substances with different compositions are
smaller than would be expected from theories in which elementary
particles gravitate differently.
>Do we have any definitive experiment which proves that protons, for
>example, are the source of a gravitational field as opposed to just
>being an object of attraction towards a gravitational field?
Yes.
>Isn't this somewhat equivalent to asking the question of whether or not
>all inertial mass also produces a gravitational field?
Yes.
>Do we have any definitive experiment which proves that electrons are
>attracted toward a gravitational field as opposed to being attracted
>towards protons which are attracted towards a gravitational source or
>even that electrons are the source of a gravitational field?
The way they do this is to fill huge salt caverns with liquids, and
measure the active gravitational mass.
>
>Aren't these the sort of questions that we ought to be asking, that we
>ought to be addressing in our experiments?
Yes, but physicists do ask these questions, and do whatever experiments
they can think of.
>Don't you think that it is possible that neutrons produce a
>gravitational field but protons and electrons do not?
I have considered such possibilities. Any difference as marked as that
has been excluded by experiment.
>> >If I believed the "logical argument" I might believe the theory.
>> >If I didn't, I wouldn't. But anyway, this is a lot less good than the
>> >old-fashioned approach where a physical theory makes a new prediction
>> >and it's actually VERIFIED BY EXPERIMENT.
>> Oh dear, more crackpot points. I think there might be some doubt from
>> the perspective of history of science that scientific revolutions have
>> worked like this, though it is probably correct for normal science.
>I don't get it. Help me out here, Charles. I'm slow. What did you
>mean by "Oh dear, more crackpot points."
It's simply a friendly dig at John Baez, whose crackpot index awards
points for capitals.
Regards
--
Charles Francis
Charles Francis
>Charles Francis wrote:
>> In article <9sc5k3$lkt$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
>> writes
>> >Charles Cagle wrote:
>> >>So, then the question came to me - What exactly would people consider a
>> >>reasonable unification of electromagnetism and gravity?
>> >If we knew, we wouldn't be sitting around here wasting time on
>> >the internet, now, would we?
>> Wouldn't we? What could we be doing about it?
>Building things.
Maxwell didn't build anything. He spent about five years (approx
1860-65) sitting around trying to putting Maxwell's equations and the
surrounding argument into a form that anyone could understand. Building
things, such as radio, came much later. I actually think the internet is
a forum where we can discuss unification of electromagnetism and
gravity, what may or may not be a reasonable about it, and to discuss
the preconceived ideas which are undoubtedly preventing us from
understanding it.
>> >If I believed the "logical argument" I might believe the theory.
>> That's the thing about logical argument. It does not have to be
>> believed, only studied. But one has to believe that the study is worth
>> while, and if a view point is unorthodox, as we must assume unification
>> will be (since otherwise it would already be known), then it is
>> difficult to believe it might be worth serious thought.
>And of course all logical arguments are based on axioms, or postulates, that
>have to be true, in order for the logical argument itself to be true or valid
>for the real world. Good logic with false postulates still yields false
>results, inconsistent with reality.
That is why we need to choose axioms which can be seen to be true of the
physical world. By and large this was so of the axioms of Euclidean
geometry. Once one has allowed for the facts that lines and points are
idealisations, only the parallel postulate could not be seen to be true
for thousands of years and could not be proven either. Finally it was
found that it is not necessarily true, and the line of thought lead to
gtr. The line of thought that points and lines are idealisations might
also have lead to quantum mechanics - the germ of the ideas were already
known to philosophrs, but in fact experiment necessitated quantum
mechanics before anyone clearly came to grips with many valued logic and
relational systems of measurement.
>Pudding is the best evidence to support a theory. The proof of the pudding is
>in the eating. The proof of a theory, ultimately is in the technical uses
>that it can be applied to and the new technology that results.
The proof of a theory is whether it works in a unified and consistent
way to describe everything we observe in nature. We do not have a theory
that does that, and we know it is an extremely difficult challenge to
produce one.
Regards
--
Charles Francis
Zachary Uram
> Hmm. Certainly nobody has succeeded in getting anywhere
> with *this* idea. I hadn't even known Wheeler went that
> far. I'm more familiar with his ideas on "charge without
> charge" - a trick for trying to get charged particles to
> show up as an automatic consequence of the Einstein-Maxwell
> equations when you allow topologies with wormholes. Nobody
> has ever worked out the details of this idea of "charge
> without charge"; the idea of getting Maxwell as an automatic
> consequence of Einstein seems even more speculative.
Didn't Einstein labor the last decades of his life trying to
pursue (unsuccessfully) a unified theory of gravity and EM? His
unified field theory. So geometry is the most fundamental thing?
> Louis Crane's latest paper on the physics preprint server
What is URL for this server?
Zach
zu...@andrew.cmu.edu
"Blessed are those who have not seen and yet have faith." - John 20:29
John Baez
Zachary Uram <zu...@andrew.cmu.edu> wrote:
>On Mon, 12 Nov 2001, John Baez wrote:
>> Hmm. Certainly nobody has succeeded in getting anywhere
>> with *this* idea. I hadn't even known Wheeler went that
>> far. I'm more familiar with his ideas on "charge without
>> charge" - a trick for trying to get charged particles to
>> show up as an automatic consequence of the Einstein-Maxwell
>> equations when you allow topologies with wormholes. Nobody
>> has ever worked out the details of this idea of "charge
>> without charge"; the idea of getting Maxwell as an automatic
>> consequence of Einstein seems even more speculative.
>Didn't Einstein labor the last decades of his life trying to
>pursue (unsuccessfully) a unified theory of gravity and EM?
Yup. He made several attempts at a unified field theory.
So did Schroedinger! In fact, Schroedinger once prematurely
announced to the press that he had found a good unified field
theory, and was later forced to retract it. Einstein then
made a sarcastic crack about how doing this makes physics
look like those little countries where each week you read
in the paper that the government has been overthrown by
another coup. Einstein was far more careful about not
announcing his ideas until they really worked.
Anyway, Wheeler came along considerably later, and came up
with some much sketchier but still influential ideas about
unification, which (unlike Einstein's) took quantum theory
into account.
>So geometry is the most fundamental thing?
Who knows? I hope we'll see someday. That's what Einstein
thought, and in a sense this dream motivates string theory as
well - although there's a very different kind of "geometry"
going on here.
>> Louis Crane's latest paper on the physics preprint server
>What is URL for this server?
Every mathematician and physicist has this bookmarked, and
they eagerly read the new preprints on their specialty each day.
Everyone except a few reactionaries, that is. If you're not
with it, you're out of it.
John Baez
>On Wed, 7 Nov 2001, Zeab Nhoj [secretly John Baez] wrote:
>> One nice
>> attempt was Kaluza and Klein's theory: Einstein's equations on a 5d
>> spacetime where the extra dimension is curled up in a little circle
>> gives Einstein's equation in ordinary 4d spacetime, *together*
>> with Maxwell's equations... *together* with a massless spin-0
>> particle that, sadly, appears not to exist. Oh well - nice try!
>The way you say `nice try' at the end, seems to mean that the massless
>scalar is a embarrassment and that it somehow disproves the theory. In
>fact this isn't not so. Theories that predict new particles are widely
>accepted nowadays (unlike the 20s when KK theory was born). The fact
>that the Brans-Dicke scalar has not been found in accelerators certainly
>doesn't mean it can't exist.
I'm a bit confused for two reasons. First, I was talking about
the scalar particle in the original Kaluza-Klein theory, not the
scalar particle in the Brans-Dicke scalar-tensor theory of gravity.
Are you intimating that these are somehow the same thing, or related?
They're certainly not *the same*: the Kaluza-Klein theory is just
ordinary general relativity on spacetime with an extra curled-up
dimension, while the Brans-Dicke theory is a modification of general
relativity with an extra scalar field, but no extra dimensions.
Kaluza-Klein theory unifies electromagnetism and gravity (which is
what we were talking about), while Brans-Dicke theory only covers
gravity. The scalar fields in the two theories could be related
in some way I've failed to notice, though.... and if so, please
tell me about it! It would be very interesting.
Secondly, unlike massive particles, which devious theorists are free
to postulate as long as the masses are so big no accelerator can
produce them yet, massless particles have an annoying tendency to be
noticeable! If massless or even light particles of some sort exist,
they should be all over the place, created at some point in early
universe, so the only way they could go unnoticed is if they interact
very weakly with other matter... like neutrinos, or worse. Thus,
while the Brans-Dicke theory can never be definitively ruled out
(it gets more and more like general relativity as you turn up the
value of a certain parameter), it has been made to look very unlikely
by astrophysical experiments which keep failing to detect the effects
of the scalar field. Are you suggesting that the Kaluza-Klein theory
with *its* massless scalar has not yet been ruled out? That would
also be very interesting.
Anyway, I agree with your more robust point:
>In retrospect it is laughable that they thought back then they could
>discredit a theory because of a predicted yet unseen particle.
Right; in this sense the Kaluza-Klein theory simply came too
early, back before people started discovering new particles.
According to the pop physics histories I've read, people more
or less threw out the Kaluza-Klein theory as soon as they saw
it predicted a new particle, without wondering too much if that
particle could exist.
>At the time, Chadwick had not even discovered the neutron yet!
Zounds! That's pretty funny.
>The standard model of particle physics has many new
>particles, *including* an unseen scalar called the Higgs boson
>which is integral to the model.
Right. Of course this isn't a *massless* scalar.
>There are many carpets in a QFT to sweep things under.
Right. Hmm! Now that you mention it, scalars have an annoying
tendency to pick up a mass from particles they interact with, so
maybe nowadays someone could try to save the Kaluza-Klein theory
by showing the scalar becomes very massive. Maybe Demian Cho
knows about this - his thesis is all about trying to save the
Kaluza-Klein theory.
>The string
>theory, that we all love, has a massless scalar too (the dilaton)
>but it is hidden away in the string coupling.
Well, I *don't* love string theory, and this is yet another
reason why! In fact when I last checked, the dilaton
had an annoying tendency to make the theory unstable.
Does anyone know the latest word here?
Charles Torre
>>> Hmm. Certainly nobody has succeeded in getting anywhere
>>> with *this* idea. I hadn't even known Wheeler went that
>>> far. I'm more familiar with his ideas on "charge without
>>> charge" - a trick for trying to get charged particles to
>>> show up as an automatic consequence of the Einstein-Maxwell
>>> equations when you allow topologies with wormholes. Nobody
>>> has ever worked out the details of this idea of "charge
>>> without charge"; the idea of getting Maxwell as an automatic
>>> consequence of Einstein seems even more speculative.
I missed the start of this thread, but I think JB is
referring to the old Rainich, Misner, Wheeler "already
unified theory" of GR + EM. I am not too clear on the
history, but here is my take on it. Somewhere in the 1920s
Rainich asked which Einstein tensors could possibly be
equated to a Maxwell energy-momentum tensor. It seems that
the Maxwell energy-momentum tensor satisfies some algebraic
identities. These impose necessary conditions on the
Einstein tensor, which were what Rainich derived. In the
1950s Misner and Wheeler picked up on this work and found a
complicated set of field equations for the metric (I think
they involve the metric and its first 3 or 4 derivatives)
which are equivalent to the Einstein-Maxwell equations. I
don't think there is anything wrong with their results in
this regard. But I am sure someone will correct me if I am
mistaken (heh, heh). Anyway, they (M & W) used this in
their big paper on "physics as geometry" to view EM as
geometry in disguise. In fact, I believe this was the same
paper that tried to model electrically charged matter in
terms of topological structures in spacetime. The whole
physics=geometry program has, of course, largely fizzled
out. But I am a little surprised that the RMW theory does
not seem to be very well-known.
-charlie
Danny Ross Lunsford
which theory you regard as the unifier and which the unificee (pardon my
French :)
1) Gravity as geometry is regarded as the fundamental idea. Needed: an
inclusive geometry with an essentially geometrical explanation of
electromagnetism. This means an irreducible geometry in which both g_{mn} and
A_m live on equal terms. There are two such:
a) A space based on "path invariance" instead of interval invariance,
essentially a form of projective differential geometry. The paths play the
role of geodesics in this space. The details were worked out by
mathematicians T.Y. Thomas, L.P. Eisenhart, and O. Veblen in the 20s and
30s. This geometry turns out to be isomorphic to Kaluza-Klein theory (see
Bergmann's relativity book). Historical trivia: Pauli worked on these ideas
early in his career.
b) A metrical space based on non-transferrable directions (gravity) AND
lengths (EM). Lengths are transferrable iff the EM field vanishes (Am is a
gradient). The geometry then reduces to Riemannian geometry. The full
geometry is known as a Weyl space and still represents the most cogent
unification of gravity and EM - only it doesn't work in 4 dimensions due to
lack of a suitable variational principle. Still the geometry is so
beautiful, it somehow has to be right.
2) The other definition takes EM as fundamental and tries to make gravity a
fundamental quantum field as well, and one has some kind of theory in which
photons and gravitons are equivalent members of some multiplet, in much the
same manner as electroweak theory. Since there is no viable quantum gravity,
not much progress has been made in this direction (in spite of the claims of
stringers and Mers) (donning asbestos undergarments at this time...)
In any case, in a truly unified theory, gravitational phenomena *must* be
accompanied by electromagnetic phenomena. So for example, a black hole would
have a certain intrinsic temperature, and charge would distort space. This
may be a way of thinking of the temperature of horizons (CMB).
-ross
Grace Shellac <furlong.phy...@singtech.com> wrote in message
news:051120011734341087%furlong.phy...@singtech.com...
> I've been under the impression that gravity and electromagnetism
> actually hasn't been unified. I asked Jerrold Marsden down at CalTech
> if this was true.
>
> He responded: "...there is a well defined set of equations called the
> Einstein-Maxwell equations that already unifies E and M with
> gravity. They are discussed in standard books on general relativity."
>
> "Thus, I would say that most researchers believe that these theories
> are already unified. The big question is how to unify quantum theory
> with gravity."
>
> So, what is the deal here. Am I misinformed? Or is Prof. Marsden
> mistaken? Doing a google.com search didn't seem to help as there
> doesn't seem to be a consensus that gravity and electromagnetism are or
> have been satisfactorily unified. I see web pages by people like
> Sweetster and others but no definitive comment that indicates that
> anyone actually thinks gravity and electromagnetism have been unified.
>
Terry Pilling
I can see that this thread has gotten way off the original subject
but I thought I would inject a comment anyway.
Since we were talking about Kaluza-Klein theories originally
I want to mention a very cool variation of the theme, that
people seem to love now and may possibly lead to some unifications,
called `Randall-Sundrum' models (or brane worlds) where the
extra dimensions are *not compact* but instead they are fully extended
just like the rest. The difference is that there are mechanisms in place
so that the matter and gauge fields are confined (on or very near) to a
3-brane which is our universe, whereas Gravity is free to propagate in
the bulk space-time. This may solve some of the heirarchy problems
encountered when trying to unify gravity to the rest.
[See Phys. Rev. Lett. 83, 3370 (1999)]
Also, there is a recent paper [hep-th/0111115] which explores the
possibility that the extra KK dimensions are `fractal dimensions'
which is way weird, but it seems to give particle masses much lower
than the plank mass (unlike conventional KK).
Wild and wonderful stuff...
-Ter
<http://www.phys.ndsu.nodak.edu/people/terry.htm>
Alfred Einstead
> >Building things.
>
> Maxwell didn't build anything. He spent about five years (approx
> 1860-65) sitting around trying to putting Maxwell's equations and the
> surrounding argument into a form that anyone could understand.
Maxwell invented color photography, took the first color photograph and
made the device to do it.
Some people rule it out on a technicality: one of the primary colors in the
prints was actually coming from slightly outside the optical spectrum, so
the pictures looked funny.
Danny Ross Lunsford
when in fact they are vastly different things - one embodying coordinate
invariance and the other gauge invariance. As far as I can tell, gauge
invariance is incidental to KK theory as a founding principle. I'm not
saying that KK's A is not a proper gauge vector, only that if one writes
down the principles on which it is based, gauge invariance isn't one of
them. In contrast, Weyl's theory puts gauge invariance in the foreground on
an equivalent footing with coordinate invariance.
-ross
Terry Pilling <te...@offshell.phys.ndsu.nodak.edu> wrote in message
news:Pine.LNX.4.10.101111...@offshell.phys.ndsu.nodak.edu.
...
Chris Hillman
On Fri, 16 Nov 2001, Charles Torre wrote:
> I missed the start of this thread, but I think JB is referring to the
> old Rainich, Misner, Wheeler "already unified theory" of GR + EM.
Yeah.
> I am not too clear on the history, but here is my take on it.
> Somewhere in the 1920s Rainich asked which Einstein tensors could
> possibly be equated to a Maxwell energy-momentum tensor.
Right, he gave three necessary conditions (two algebraic and one involving
covariant derivatives) for a spacetime to have an interpretation in gtr as
an "electrovacuum" (aka "Einstein-Maxwell solution").
("An interpretation": null electrovacuums are also formally "null dusts"
and can be interpreted as such. Also, many null electrovacuums can also
be interpreted as exact solutions to the EFE with source for the
gravitational field a "minimally coupled massless scalar field".)
> It seems that the Maxwell energy-momentum tensor satisfies some
> algebraic identities. These impose necessary conditions on the
> Einstein tensor, which were what Rainich derived. In the 1950s Misner
> and Wheeler picked up on this work and found a complicated set of
> field equations for the metric (I think they involve the metric and
> its first 3 or 4 derivatives) which are equivalent to the
> Einstein-Maxwell equations.
The three necessary conditions are much easier to state: the two algebraic
conditions are:
R = 0
R_(am) R^(mb) =/= 0
and, putting
V_a = e_(abcd) R^b_m R^m_d
the differential condition is
V_(a;b) = V_(b;a)
(Here, the round brackets denote subscripts in index gymnastics notation,
not symmetrization!)
> I don't think there is anything wrong with their results in this
> regard. But I am sure someone will correct me if I am mistaken (heh,
> heh).
"Rainich theory" at least is fine as far as the math goes. A good
discussion can be found in the monograph
author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},
title = {Exact Solutions of {E}instein's Field Equations},
publisher = {Cambridge University Press},
year = 1980}
Unfortunately I don't have this book in front of me as I write and I
haven't thought about this in quite a while, but hopefully I haven't
misstated anything.
> But I am a little surprised that the RMW theory does not seem to be
> very well-known.
Indeed. But a new generation of students will get a chance to learn it!
This post is pretty much only an excuse for me to "break" the news (or was
I the only one who didn't know this?!) that Hans Stephani, Dietrich
Kramer, Malcolm MacCallum, Cornelius Hoenselaers and Eduard Herlt have
almost finished the much expanded -updated- and corrected second edition
of the above monograph (which since its publication has been -the- single
most fundamental resource in the field) and it should be published by CUP
later this year! :-)
I hear the second edition will be about 700 pages long, reflecting the
tremendous developments in the field since the first edition was
published. I hear that at least at first it will only be available in
hardcover, so it might be pricey, but serious students will certainly want
to consider obtaining a personal copy.
Since this second edition is coming out at the same moment that (we
expect) the age of gravitational wave astronomy is dawning, I expect that
gtr is about to undergo another wave of renewed popularity as a field for
Ph.D. dissertations and the like, since the astronmers will almost surely
:-/ provide many unexpected observations for the theorists to try to
explain, and the natural place to begin trying to explain gravitation
mysteries is with our gold standard theory of gravitation (the simplest
viable theory with gravitational radiation), namely gtr.
Eric Alan Forgy
> My issue with KK theories is that they force g and A into the same box -
> when in fact they are vastly different things - one embodying coordinate
> invariance and the other gauge invariance.
Hmm... I thought coordinate invariance WAS a gauge invariance.
Eric
Brian J Flanagan
> Didn't Einstein labor the last decades of his life trying to
> pursue (unsuccessfully) a unified theory of gravity and EM? His
> unified field theory. So geometry is the most fundamental thing?
The geometric appoach is certainly alive and well. On the other hand,
Peter Bergmann quotes Einstein to the effect that the geometrization
of physics per se is not the important thing, but rather a fusing of
physics with the appropriate mathematics. By way of illustration, I
recently heard an informal lecture by Ashtekar, who is a big gun in
quantum gravity; on his view, it seems to make sense to think of
space-time as arising in a secondary way from what sounds like a
generalized spin connection. But John Baez and others here are far
more conversant in these matters than I am, and so I will refer you to
them, and to the web site for the Center for Gravitational Physics and
Geometry at Penn State:
http://www.phys.psu.edu/news/seminars_cgpg.html
Charles Francis
>t...@rosencrantz.stcloudstate.edu wrote:
>> In article <3BEE4E47...@nospam.worldnet.att.net>,
>> Ben <dra...@nospam.worldnet.att.net> wrote:
>> >The proof of a theory, ultimately is in
>> >the technical uses that it can be applied to and the new technology
>> >that results.
>> I strongly disagree with this. By this measure, the standard
>> model of particle physics is worth precisely nothing, since
>> no technology has come from it, and general relativity is worth
>> next to nothing.
>Oh, very bad argument to make, especially if you are trying to obtain
>government grants. What you are saying is that high energy physics is
>really nothing, not worth the time and other people's money invested
>in it. If there is no practical application for the knowledge, then
>why should I spend my money, or tax dollars to support it?
We are not all concerned solely with the technological and material
benefits which science can bring us, but would like to understand more
about the nature of creation and our place in it, since this question is
highly relevant to the way we choose to live. That is quite a good
enough cause to spend taxes.
>BTW it should be pointed out that while GR may be of limited utility
>today, there is no guarantee that it will stay that way. And that GR
>is founded on SR, which brought us E=mc^2. Which in turn resulted in
>the nuclear reactor and the atomic bomb.
Historically I don't think there was any relationship at all between the
purely theoretical discovery of E=m^2 by Einstein in 1905, and the
splitting of the atom by Rutherford. In fact Rutherford was already
studying radioactivity and proposed that it was due to a decay which
transformed elements with the release of energy in 1902, prior to
Einstein's relativity paper.
Regards
--
Charles Francis
Squark
<AmOhIfL$T7...@cc.usu.edu>):
>
>...It seems that
>identities. These impose necessary conditions on the
>Einstein tensor, which were what Rainich derived. In the
>1950s Misner and Wheeler picked up on this work and found a
>complicated set of field equations for the metric (I think
>they involve the metric and its first 3 or 4 derivatives)
>which are equivalent to the Einstein-Maxwell equations.
Am I right concluding a one can reconstruct the electromagnetical
field from its energy-momentum tensor? Interesting, is the same true
for gauge theory, at least for some gauge groups?
Best regards,
Squark.
-------------------------------------------------------------------------------
Write to me at:
[Note: the fourth letter of the English alphabet is used in the following
exclusively as anti-spam]
dSdqudarkd_...@excite.com
Doug B Sweetser
I've been quiet the last few months in SPR, taking the time to rethink
my proposals. In this post I will lay out the latest sketch for a
classical unification of gravity and electromagnetism. Most work in
this area is done with quantum mechanics, because there are far more
opportunities to be creative, particularly with regards to new
mathematical insights and structures. I will stick with a purely
classical approach. Tight constraints are good :-)
Here is the outline:
1. Define the unified Lagrangian [L = -J^u A_u - 0.5 d^u A^v d_u A_v]
2. Find the equations of motion [(d^2/dt^2 - Del^2) A^u = Junified]
3. Find a solution. [A^u = (1/tau^2, 0, 0, 0)]
4. Connect the solution to the classical Newtonian gravitational
potential.
5. Write the 4-force equation.
[F^u = dp^u/dtau = m dU^u/dtau + U^u dm/dtau = km d^u A^v U_v/|A|]
6. Solve the force equation for the solution constrained to be like
the classical Newtonian gravitational potential in two cases:
A. if dm/dtau = 0. This solution has 8 constants, which if
eliminated leads to a metric
[dtau^2 = e^(-2GM/c^2 tau) dt^2 - e^(2GM/c^2 tau) dR^2]
B. if dU^u/dtau = 0. This case can be used to explain the flat
velocity profile seen in disk galaxies where the mass falls
off exponentially.
The metric in 6A is different from the Schwarzschild metric of general
relativity. If one does a Taylor series expansion for a weak
gravitational field on the Schwarzschild metric in isotropic
coordinates (MTW, Eq. 31.22), the second-order spatial term has a
coefficient of 2.5, contrasted with 2.0 for the metric in 6A. This
will be very hard to test, but makes the proposal subject to
experimental confirmation or rejection. The solution for 6B asserts
that no dark matter is needed to explain the velocity distribution and
stability of disk galaxies.
My central hypothesis is that a 4-potential should have enough degrees
of freedom to describe two transverse waves for electromagnetism and
two waves for gravity. Classically, the source of electromagnetism
(electric charge) and gravity (mass) are independent of each other.
The classical unification is in the structure of the equations, not
for sources or test charges.
Start with the following Lagrangian which will hopefully unify
equations for gravity with the Maxwell equations:
L = -J^u A_u - 0.5 d^u A^v d_u A_v
There are only two terms, a source term, -J^u A_u, and a field
strength term, -0.5 d^u A^v d_u A_v. Contrast this Lagrangian with
the one for the Maxwell equations used by Gupta and Bleuler.
L = -J^u A_u - 0.5 (d^u A_u)^2
*******
-0.25 (d^u A^v - d^v A^u)(d_u A_v - d_v A_u)
******* *******
This Lagrangian has 3 parts: a source, a gauge-fixing term, and an
antisymmetric field strength tensor. The underlined parts make up the
unified Lagrangian. If fixing the gauge is viewed as a constraint,
then the unified Lagrangian may be able to explain more than just the
Maxwell equations because the choice of gauge is eliminated. That
hunch will be tested.
Find the equations of motion. Expand the Lagrangian:
L = -rho phi + J_x A_x + J_y A_y + J_z A_z
+ .5(-(dphi/dt)^2 + (dphi/dx)^2 + (dphi/dy)^2 + (dphi/dz)^2
+ (dA_x/dt)^2 - (dA_x/dx)^2 - (dA_x/dy)^2 - (dA_x/dz)^2
+ (dA_y/dt)^2 - (dA_y/dx)^2 - (dA_y/dy)^2 - (dA_y/dz)^2
+ (dA_z/dt)^2 - (dA_z/dx)^2 - (dA_z/dy)^2 - (dA_z/dz)^2)
The next step is to take derivatives of this Lagrangian with respect
to the field variables. Fortunately, the only thing that one needs to
pay attention to is the sign. All the derivatives follow this
pattern:
dL/d(dphi/dt) = -dphi/dt
or more generally:
dL/d(...) = +/- (...)
Use the Euler-Lagrange equation to get the equations of motion:
dL/dphi - d/ dL/ - d/ dL/ - d/ dL/ - d/ dL/
dt d(dphi/dt) dx d(dphi/dx) dy d(dphi/dy) dz d(dphi/dz)
= -rho + d^2phi/dt^2 - d^2phi/dx^2 - d^2phi/dy^2 - d^2phi/dz^2 = 0
dL/dA_x - d/ dL/ - d/ dL/ - d/ dL/ - d/ dL/
dt d(dA_x/dt) dx d(dA_x/dx) dy d(dA_x/dy) dz d(dA_x/dz)
= J_x - d^2A_x/dt^2 + d^2A_x/dx^2 + d^2A_x/dy^2 + d^2A_x/dz^2 = 0
dL/dA_y - d/ dL/ - d/ dL/ - d/ dL/ - d/ dL/
dt d(dA_y/dt) dx d(dA_y/dx) dy d(dA_y/dy) dz d(dA_y/dz)
= J_y - d^2A_y/dt^2 + d^2A_y/dx^2 + d^2A_y/dy^2 + d^2A_y/dz^2 = 0
dL/dA_z - d/ dL/ - d/ dL/ - d/ dL/ - d/ dL/
dt d(dA_z/dt) dx d(dA_z/dx) dy d(dA_z/dy) dz d(dA_z/dz)
= J_z - d^2A_z/dt^2 + d^2A_z/dx^2 + d^2A_z/dy^2 + d^2A_z/dz^2 = 0
or in a more compact notation:
(d^2/dt^2 - Del^2) A^u = Junified^u
These are the equations of motion reached without presuming the Lorenz
gauge. If I want to see if this equation contains the Maxwell
equations, to be logically consistent with the Lagrangian, the Lorenz
gauge must not be presumed. Start with the scalar field equation:
d^2phi/dt^2 - Del^2 phi
= d/dt(dphi/dt + Del.A) + Del.(-dA/dt - Delphi)
= dg/dt + Del.E
= rhounified
where g === dphi/dt + Del.A
There are proofs that a scalar field such as g cannot account for
curvature of spacetime. It is important to note that g is the trace
of the diagonal of the second-rank field strength tensor, d^u A^v,
which also contains E. The scalar field g is not fundamental, but the
second-rank field strength tensor d^u A^v is. That tensor must be
used in the force equation, not the scalar field g, to arrive at the
metric equation. This work is consistent with the proofs that a
second rank tensor is required for a metric theory of gravity.
To understand the source term, two cases will be considered. First,
let g=0. Experimental test have confirmed Gauss' law,
Del.E = rhoelectric, to a high degree of precision. To be consistent
with this experimental observation, for the case were g=0:
Del.E = Del.(-dA/dt - Delphi) = rhoelectric
The second case is where E=0. Based on the same experimental test of
Gauss' law, the remaining field cannot make a contribution to
rhoelectric or else Gauss' law would be violated. We will give this
different source a different label:
dg/dt = d/dt(dphi/dt + Del.A) = rhomass
so that
rhounified = rhomass + rhoelectric
In the classical region, charge is independent of mass. This
classical separation of sources for charge and mass leads to separate
equations of Gauss' law and gravity. The unity only happens at the
level of the potential, where the cross terms containing d/dt Del.A
from dg/dt and Del.E field equations cancel each other.
A similar exercise works for the 3-vector equations:
d^2A/dt^2 - Del^2 A
= d/dt(dA/dt + Delphi) - Del dphi/dt - Del^2 A
= -dE/dt + Delx(DelxA) - Del(dphi/dt + Del.A)
= -dE/dt + DelxB - Delg
= Junified
= Jcharge + Jmass
This has Ampere's law and another equation that may turn out to be
connected to gravity. The no monopoles equation and Faraday's law are
vector identities, so will still be true assuming a simple topology.
Gravity involves the distance between a source and a test mass,
R=(x^2 + y^2 + z^2)^0.5. The problem with Newton's approach is that
a change in the mass density can propagate instantaneously. To make
make the distance R more "relativistic", work instead with a timelike
distance tau such that
tau = (c^2 t^2 - x^2 - y^2 - z^2)^0.5 = Rsourcetotestmass
This interval is the amount of time required to cover the distance
between the source and the test mass. Information about changes in
the mass density cannot propagate infinitely fast. Still, the
numerical value of tau is almost the same as Rsourcetotestmass. For a
special case, ct = Rsourcetotestmass if x = y = z = 0. [Having the
distance Rsourcetotestmass in the slot for t can be disorienting, so I
will continue to use this very long name for R which is so strongly
linked to x, y, and z, to indicate the distance between the source and
test mass.]
Find a solution to the equations of motion using this more
relativistic measure of distance, tau. Here is one possibility:
A^u = (1/tau^2, 0, 0, 0)
Take the derivatives of A^u to prove this:
d(1/tau^2)/dt = -2t/tau^4
d(-2t/tau^4)/dt = -2/tau^4 + 8t^2/tau^6
d(1/tau^2)/dx = 2x/tau^4
d( 2t/tau^4)/dx = 2/tau^4 + 8x^2/tau^6
d(1/tau^2)/dy = 2y/tau^4
d( 2t/tau^4)/dy = 2/tau^4 + 8y^2/tau^6
d(1/tau^2)/dz = 2z/tau^4
d( 2z/tau^4)/dz = 2/tau^4 + 8z^2/tau^6
Subtract the last three second-order derivatives from the first:
(d^2/dt^2 - Del^2)(1/tau^2)
= -8/tau^4 + 8(t^2 - x^2 - y^2 -z^2)/tau^6 = 0 QED
The inverse interval squared is a solution to the equations of motion.
How the solution behaves numerically must mimic the classical
gravitational potential force field:
Del phi = -GM/R^2 Rhat
where Rhat is a unit vector in the R direction
To make the connection between the purely mathematical potential
solution A^u = (1/tau^2, 0, 0, 0) and the gravitational potential
force field requires two assumptions:
1. The timelike interval tau is the sum of both a distance,
Rsourcetotestmass, and the geometric length of the source mass,
GM/c^2, or tau = Rsourcetotestmass + GM/c^2.
2. The force field must be normalized to the potential, d^u A^v/|A|
The classical gravitational force field depends on only two numbers:
the distance Rsourcetotestmass and the source mass M. Yet for the
potential under study, there is only one place to plug a value in,
tau. A simple, direct hypothesis is that the interval tau is the sum
of Rsourcetotestmass and the source mass M expressed in units of distance.
At this point, only the units of mass are being manipulated by
universal constants. No connection to event horizons is asserted.
Event horizons only arise if there is a metric to work with, and that
stage will happen later.
For classical systems, Rsourcetotestmass >>> GM/c^2. Any value that
depends on Rsourcetotestmass + GM/c^2 will be approximately
Rsourcetotestmass. For the gravitational field of the Sun one Earth
orbit away, Rsourcetotestmass = 1.5 x 10^11 m, while GMsun/c^2 = 1.5 x
10^3 m. The force field involves the derivative of the potential.
Any small local change will have a much larger effect on the smaller
number, GMsun/c^2, than the far larger R. To make that clear when
taking the derivative, use the following change of variables:
t -> t' = A + t GM/(2 c^2 A)
R -> R' = B + R GM/(2 c^2 |B|)
where A^2 - B^2 = Rsourcetotestmass^2
This change in variables is valid only locally, not globally, since it
breaks down for arbitrarily long times t or distances R away. This is
consistent with general relativity which has only local, not global
solutions. Take the derivative of the normalized interval squared:
1/|1/tau^2| d(1/tau^2)/dt ~= -GM/(c^2 tau^2)
1/|1/tau^2| d(1/tau^2)/dR ~= GM/(c^2 tau^2)
This should look familiar. Since tau is the sum of Rsourcetotestmass
and GM/c^2, for the Sun's gravitational field, tau will equal
Rsourcetotestmass to one part in 10^8. It has the linear dependence
on the source mass. This proposal is more relativistic that Newton's
because there are four terms. This also indicates the force is not
conservative, which is consistent with general relativity.
The next step is to write out the force equation.
F^u = dp^u/dtau = m dU^u/dtau + U^u dm/dtau = kq d^u A^v U_v/|A|
I wish I had a nice logically fail-proof way of discussing the proper
way of handling the test charge "q" in the equation above, but at this
point I don't. From the earlier discussion, in the classical region,
the sources for electricity and magnetism are not unified, but at the
level of the potential there are terms that cancel out between the
gravitational and electrical field equations. I will continue this
work under the assumption that the classical test charges for
electricity and magnetism also are separate. Since I am focusing on a
gravitational potential, the charge must be a gravitational charge, a
mass. I must assume the gravitational and inertial mass are equal.
This assumption has been tested and confirmed experimentally, and is a
logical basis for general relativity.
Calculate the right-hand side of the force equation:
m d^u A^v U_v/|A| =
|d/tau^-2/dt -dtau^-2/dx -dtau^-2/dx -dtau^-2/dx| | dt/dtau|
m | 0 0 0 0 | |-dx/dtau| tau^2
| 0 0 0 0 | |-dy/dtau|
| 0 0 0 0 | |-dz/dtau|
= m (-GM/(c^2 tau^2) dt/dtau, GM/(c^2 tau^2) dx/dtau,
GM/(c^2 tau^2) dy/dtau, GM/(c^2 tau^2) dz/dtau)
= m (-GM/(c^2 tau^2) dt/dtau, GM/(c^2 tau^2) dR/dtau)
There are two simple cases to solve for the force equation:
A. if dm/dtau = 0
B. if dU^u/dtau = 0
The first will lead to a metric equation for gravity consistent to
post-Newtonian accuracy with experimental tests of general relativity.
The second explains the flat velocity profiles seen in disk galaxies.
From here on out, it is nothing but calculus.
Case A. If dm/dtau = 0, the left hand side of the 4-force equation
looks like this:
m dU^u/dtau = m (d^2 t/dtau^2, d^2 R/dtau^2)
Gather all the terms on one side and assume the equivalence principle
to generate two equations of motion:
d^2 t/dtau^2 + GM/(c^2 tau^2) dt/dtau = 0
d^2 R/dtau^2 - GM/(c^2 tau^2) dR/dtau = 0
Solve this second order differential equation:
t = k1 (tau e^( GM/(c^2 tau)) - GM/c^2 Ei( GM/(c^2 tau)) + k5
R = K234 (tau e^(-GM/(c^2 tau)) + GM/c^2 Ei(-GM/(c^2 tau)) + K678
where Ei(t) = Integral from -inf to t of e^t/t
Check the solution:
d(tau e^(GM/(c^2 tau))/dtau
= e^(GM/(c^2 tau)) - GM/(c^2 tau) e^(GM/(c^2 tau))
d(- GM/c^2 Ei( GM/(c^2 tau)))/dtau
= GM/(c^2 tau) e^(GM/(c^2 tau))
dt/dtau = e^(GM/(c^2 tau))
d^2 t/dtau^2 = -GM/(c^2 tau^2) e^(GM/(c^2 tau))
d^2 t/dtau^2 + GM/(c^2 tau^2) dt/dtau = 0 QED
Similar math applies to 3-vector equation.
Eliminate the 8 constants, k1, K234, k5, K678. First, take the
derivative of t and R with respect to the interval to generate the
4-velocity and eliminate the constants k5 and K678.
dt/dtau = k1 e^( GM/(c^2 tau))
dR/dtau = K234 e^(-GM/(c^2 tau))
In flat spacetime, U^u U_u = 1. Space time is flat if M = 0 or the
distance tau gets infinitely large:
(dt/dtau)^2 - (dR/dtau)^2 = k1^2 - K234^2 = 1
Solve for k1^2 and K234^2
k1^2 = e^(-2GM/(c^2 tau)) (dt/dtau)^2
K234^2 = e^( 2GM/(c^2 tau)) (dR/dtau)^2
Plug this back into the flat spacetime constraint and rearrange:
dtau^2 = e^(-2GM/(c^2 tau)) dt^2 - e^( 2GM/(c^2 tau)) dR^2
Do a Taylor series expansion to second order in GM/(c^2 tau):
dtau^2 = (1 - 2 GM/(c^2 tau) + 2 (GM/(c^2 tau))^2) dt^2
- (1 + 2 GM/(c^2 tau) + 2 (GM/(c^2 tau))^2) dR^2
Remember that one of the assumptions used in this derivation was that
the numerical value of the timelike interval tau is nearly the same
as the distance R between the source and the test mass. Contrast this
with the Schwarzschild metric in isotropic coordinates to second order
in GM/(c^2 R):
dtau^2 = (1 - 2 GM/(c^2 R) + 2 (GM/(c^2 R))^2) dt^2
- (1 + 2 GM/(c^2 R) + 2.5 (GM/(c^2 R))^2) dR^2
When the Schwarzschild metric is tested to "post-Newtonian accuracy",
this means the first five of the six coefficients are tested. It is
the sixth coefficient which is 20% lower in the unification
proposal. A technically challenging experiment could be set up to
confirm or refute the proposal.
At this point, I do not understand singularities in the metric, other
than it is clearly different from the Schwarzschild metric. When tau
equals zero, the interval is lightlike. The behavior of particles
that travel over lightlike intervals is handled by the Maxwell
equations, which are part of the mathematical structure. This may
represent a fundamental shift in the physics of gravitational
singularities.
Case B: if dU^u/dtau = 0, then the velocity is constant, and not
necessarily zero. The force equation describes the distribution of
the test mass over spacetime. Use the same potential as before, with
exactly the same assumptions. Collect the force terms on one side of
the equation:
dt/dtau dm/dtau + dt/dtau GM/(c^2 tau^2) = 0
dR/dtau dm/dtau - dR/dtau GM/(c^2 tau^2) = 0
Solve the first-order differential equation for m:
m = k1 e^( GM/(c^2 tau))
m = K234 e^(-GM/(c^2 tau))
For a static system, k1 = 0. The value of tau by the assumptions used
in this derivation are almost equal to R, so this solution represents
how the mass m changes over space. The distribution of mass decays
exponentially with distance.
Astronomers have made examinations of the mass and velocity
distributions of disk galaxies. Near the core, a maximum speed is
achieved which is consistent with Newton's law of gravity and a
reasonable ratio between the amount of light seen and the mass
present. The mass of the disk decays exponentially, ignoring the core
region which often has a spherical component. The velocity profile
however stays flat at the maximum speed. By Newton's theory, the
velocity should drop off with a Keplarian decline. Another
independent problem is that the solution for disk galaxies is not
stable to axisymmetric disturbances.
Near the core, the effect of gravity is generated by the m dU^u/dtau
term. That term generates the close approximation to the
Schwarzschild metric, which reduces to Newton's law of gravity under
classical condition of low speed and mass densities. After the
maximum speed is reached, the U^u dm/dtau term takes over, which
dictates a flat velocity profile and a mass distribution that falls
off exponentially with distance. That model is consistent with the
data today. I don't know how to demonstrate the stability of the
solution, but exponential decay is ubiquitous in Nature. Computer
modeling will be required to see if this proposal works with all the
data at a fine level of detail. The big picture is clear: no dark
matter is required to explain the velocity profile and mass
distribution of disk galaxies.
As is has been pointed out in this thread, there will be little
interest in a proposal unless something is said about quantum
mechanics. What I will have to learn is exactly how the Gupta/Bleuler
quantization of the Maxwell equations works. Once I understand that,
the method for quantizing the proposal will hopefully be straight
forward, since the proposed Lagrangian involves a subset of
Gupta/Bleuler Lagrangian. With fewer terms, it should be simpler.
Something concrete to work on in the new year.
doug <swee...@TheWorld.com>
quaternions.com
Terry Pilling
> Terry Pilling [secretly Gnillip Yrret] wrote:
>
> >On Wed, 7 Nov 2001, Zeab Nhoj [secretly John Baez] wrote:
Holy anagrams batman! :) I feel like a would be high school bully who
foolishly decided to pick on Clark Kent, only to find out... :(
> >The way you say `nice try' at the end, seems to mean that the massless
> >scalar is a embarrassment and that it somehow disproves the theory. In
> >fact this isn't not so. Theories that predict new particles are widely
> >accepted nowadays (unlike the 20s when KK theory was born). The fact
> >that the Brans-Dicke scalar has not been found in accelerators certainly
> >doesn't mean it can't exist.
>
> I'm a bit confused for two reasons. First, I was talking about
> the scalar particle in the original Kaluza-Klein theory, not the
> scalar particle in the Brans-Dicke scalar-tensor theory of gravity.
> Are you intimating that these are somehow the same thing, or related?
> They're certainly not *the same*: the Kaluza-Klein theory is just
> ordinary general relativity on spacetime with an extra curled-up
> dimension, while the Brans-Dicke theory is a modification of general
> relativity with an extra scalar field, but no extra dimensions.
> Kaluza-Klein theory unifies electromagnetism and gravity (which is
> what we were talking about), while Brans-Dicke theory only covers
> gravity. The scalar fields in the two theories could be related
> in some way I've failed to notice, though.... and if so, please
> tell me about it! It would be very interesting.
Argh. No, I just made a mistake. I am in the habit of calling the
scalar field either the Brans-Dicke scalar or the dilaton, and
in this case I should have simply said `scalar field'.
There is probably some way that a person could relate them though.
Actually what I should say is that there is _always_ a way to
relate things like this. Give me the two lagrangians and I will
form a single lagrangian that reduces to each of the originals in
some limit, and where the scalar field is the same field in both.
The real question is whether it makes some physical sense. I wonder.
But anyway. One thing that Kaluza-Klein shows us is that a theory
containing a bunch of interacting fields in 4 dimensions may look
*exactly the same* to us, as a theory with less fields with
compact extra dimensions. So I am convinced that one could form a
KK type theory that gives you Brans-Dicke gravity apon reduction.
>
> Secondly, unlike massive particles, which devious theorists are free
> to postulate as long as the masses are so big no accelerator can
> produce them yet, massless particles have an annoying tendency to be
> noticeable! If massless or even light particles of some sort exist,
> they should be all over the place, created at some point in early
> universe, so the only way they could go unnoticed is if they interact
> very weakly with other matter... like neutrinos, or worse. Thus,
> while the Brans-Dicke theory can never be definitively ruled out
> (it gets more and more like general relativity as you turn up the
> value of a certain parameter), it has been made to look very unlikely
> by astrophysical experiments which keep failing to detect the effects
> of the scalar field. Are you suggesting that the Kaluza-Klein theory
> with *its* massless scalar has not yet been ruled out? That would
> also be very interesting.
I would say that it hasn't been ruled out. I really believe this too.
The reason is because of things like you mention in the above
paragraph and what I said before about sweeping things under the carpet.
I can remember perusing a book for the layman about modern particle
physics and I read that this scalar is predicted by KK but can't exist
because we haven't seen it in the lab. Then, a few pages later, he is
talking about renormalization theory, tachyons and anomaly cancellation
in string theory needing 26 dimensions! :) This just makes me smile.
>
> Anyway, I agree with your more robust point:
>
> >In retrospect it is laughable that they thought back then they could
> >discredit a theory because of a predicted yet unseen particle.
>
> Right; in this sense the Kaluza-Klein theory simply came too
> early, back before people started discovering new particles.
> According to the pop physics histories I've read, people more
> or less threw out the Kaluza-Klein theory as soon as they saw
> it predicted a new particle, without wondering too much if that
> particle could exist.
Yep! I agree, it was one of those ideas that were ahead of their
time. (I wonder what other gems are waiting to be resurected from the
dusty old journals.)
If we are going to make the statement that KK theory is not
correct because of the scalar then we have to throw away extensions
of the KK idea. For example: string theory. Where is the antisymmetric
tensor field? I don't see it in the accelerators.. where is the dilaton?
Where are the 6 compact dimensions? Where are the supersymmetric
partners? ....etc. etc. etc.
> >There are many carpets in a QFT to sweep things under.
>
> Right. Hmm! Now that you mention it, scalars have an annoying
> tendency to pick up a mass from particles they interact with, so
> maybe nowadays someone could try to save the Kaluza-Klein theory
> by showing the scalar becomes very massive. Maybe Demian Cho
> knows about this - his thesis is all about trying to save the
> Kaluza-Klein theory.
That is true! Or the scalar may be confined in some way. That can
also happen.
>
> >The string
> >theory, that we all love, has a massless scalar too (the dilaton)
> >but it is hidden away in the string coupling.
>
> Well, I *don't* love string theory, and this is yet another
> reason why! In fact when I last checked, the dilaton
> had an annoying tendency to make the theory unstable.
> Does anyone know the latest word here?
I was actually just being dramatic. I don't love string theory as
a unification theory either (I love the math involved though!).
I don't really think that the idea is all that beautiful and I think
that they are always inventing patches to cover up the holes that
pop up. I *do* love the idea of higher dimensions though and I love all
of the work that the string theorists have been doing to advance
this notion. But the idea that the fundamental entity of the
universe is a little string just doesn't sit well with me...
unfortunately, my only reason is aesthetics.
The problem is that if you want to disagree with an aspect
of string theory, you have to know the theory as well as it's
advocates do, or you won't be heard... and so, I learn.
PS: I am starting to think that the best way to learn string theory is to
pick a branch of mathematics that is so far removed from the physical
world that there could not possibly be applications for it. Then,
by the time you have learned it, Witten will have shown that it is
actually an integral part of string theory, and you will have an
advantage for a few months. :)
-Ter
<http://www.phys.ndsu.nodak.edu/people/terry.htm>
John Baez
Ben <dra...@nospam.worldnet.att.net> wrote:
>It is becoming increasingly apparent that the Standard Model is in
>trouble. Neutron mass alone is problematic for it [...]
Eh?? Maybe you mean "neutrino mass". There were some interesting
recent puzzles about how much of the neutron's spin comes from
virtual quarks - the so-called "spin crisis" - but I don't know of
any big problems associated with its mass.
For info on the spin crisis see:
http://www.rarf.riken.go.jp/rarf/rhic/phys/SF/SF.html
For some reason most people talk about the "proton spin crisis",
even though I *believe* these problems arise for neutrons as well.
Does someone know why? Also, what's the latest story on this issue?
Is it still a crisis, or has it been solved?
By the way, the Standard Model with nonzero neutrino masses
and mixing angles is a lot nicer, in my opinion, than the old
version back when people thought all those things were zero.
Quarks come in both handednesses and have an interesting mass
matrix, so why not the neutrinos? We now see they do, too!
This strengthens the impression that quarks and leptons are
really two aspects of the same thing... a nice clue for would-be
unifiers.
Jeff
> http://www.arXiv.org/
>
> Every mathematician and physicist has this bookmarked, and
> they eagerly read the new preprints on their specialty each day.
If it's of use to anyone, I've designed a cover page to the arXiv which
makes it easier, at least for me, to keep up with different areas. I
thought others might be interested in taking a look:
http://physics.hyperjeff.net/lanl/
-Jeff
Steve McGrew
<cha...@clef.demon.co.uk> wrote:
>In article <3BEE4E47...@nospam.worldnet.att.net>, Ben
><dra...@nospam.worldnet.att.net> writes
>>...
>>And of course all logical arguments are based on axioms, or postulates, that
>>have to be true, in order for the logical argument itself to be true or valid
>>for the real world. Good logic with false postulates still yields false
>>results, inconsistent with reality.
>
>That is why we need to choose axioms which can be seen to be true of the
>physical world. By and large this was so of the axioms of Euclidean
>geometry. Once one has allowed for the facts that lines and points are
>idealisations, only the parallel postulate could not be seen to be true
>for thousands of years and could not be proven either. Finally it was
>found that it is not necessarily true, and the line of thought lead to
>gtr. The line of thought that points and lines are idealisations might
>also have lead to quantum mechanics - the germ of the ideas were already
>known to philosophrs, but in fact experiment necessitated quantum
>mechanics before anyone clearly came to grips with many valued logic and
>relational systems of measurement.
What are some sets of axioms that have been tested as a basis
for a theory unifying GR, QM and EM, and how have they failed?
Steve
Terry Pilling
> Terry Pilling <te...@offshell.phys.ndsu.nodak.edu> wrote:
>
> >accepted nowadays (unlike the 20s when KK theory was born). The fact
> >that the Brans-Dicke scalar has not been found in accelerators certainly
> >doesn't mean it can't exist.
>
> I'm a bit confused for two reasons. First, I was talking about
> the scalar particle in the original Kaluza-Klein theory, not the
> scalar particle in the Brans-Dicke scalar-tensor theory of gravity.
> Are you intimating that these are somehow the same thing, or related?
> They're certainly not *the same*: the Kaluza-Klein theory is just
> ordinary general relativity on spacetime with an extra curled-up
> dimension, while the Brans-Dicke theory is a modification of general
> relativity with an extra scalar field, but no extra dimensions.
I have been thinking more about this and I am starting to
believe that the massless scalar in KK and the Brans-Dicke scalar
not as far removed as you indicate. Browsing Physics Reports 238 (1997)
there is a review by J.M. Overduin and P.S. Wesson entitled
`Kaluza-Klein Gravity' and on page 316 there is a section entitled
3.5. The case A_z = 0: Brans-Dicke theory
which begins, "If one does not set \phi = constant, then Kaluza's
five-dimensional theory contains besides electromagnetic effects
a Brans-Dicke-type scalar field theory, as becomes clear when one
considers the case in which the electromagnetic potentials vanish,
A_z = 0." So it seems from this section that they also consider the
scalar to be the same thing in both contexts.
If I continue calling the scalar a Brans-Dicke scalar would
I be totally wrong? Half wrong? or mostly right? I am not sure
anymore. What are your thoughts?
-Ter
Daniel Doro Ferrante
> On Fri, 16 Nov 2001 22:00:16 +0000 (UTC), Charles Torre wrote (in
> <AmOhIfL$T7...@cc.usu.edu>):
> >
> >...It seems that
> >the Maxwell energy-momentum tensor satisfies some algebraic
> >identities. These impose necessary conditions on the
> >Einstein tensor, which were what Rainich derived. In the
> >1950s Misner and Wheeler picked up on this work and found a
> >complicated set of field equations for the metric (I think
> >they involve the metric and its first 3 or 4 derivatives)
> >which are equivalent to the Einstein-Maxwell equations.
>
> Am I right concluding a one can reconstruct the electromagnetical
> field from its energy-momentum tensor? Interesting, is the same true
> for gauge theory, at least for some gauge groups?
>
From the top of my head, i belive that the answer for your first
question would be "Yes!" and, for the second, would also be "Yes.", at
least, for the U(1) case. =;)
You see, once the energy-momentum tensor can be viewed as the
derivative of the Lagrangian with repect to the metric (either g^{mu nu}
in GR or eta_{mu nu} in SR - which implies QFT's), you can see that, in
the EM case, it'll be something like F^{mu nu}. And, by knowing F you
can find E and B out. Now, i just never saw it done before for the weak
and strong forces but, if the same argument is true (that the
energy-momentum tensor will be given by the derivative of the
Lagrangian with respect to eta - the SR metric), i believe that, just as
in EM, the answer would be something like F (the curvature in the
appropriate space), in which case you'd be able to recover the fields
you want to. You see, the whole trick hides itself in the fact that
"energy/momentum" means something related to spacetime and, in all of
those cases, spacetime is just Minkowski (SR). Thus, the gauge group has
no particular role. (Other than changing "stuff" into
"covariant-stuff", e.g., the derivative and so on... However, all this
happens at a "fibre level" and not with respect to spacetime.)
--
Daniel
,-----------------------------------------------------------------------------.
> | www.physics.brown.edu www.fma.if.usp.br <
> Daniel Doro Ferrante | <
> dani...@het.brown.edu | I want you to organize my PASTRY trays ... <
> Linux Counter #34445 | my TEA-TINS are gleaming in <
> | formation like a ROW of DRUM MAJORETTES -- <
> | please don't be FURIOUS with me -- <
`-----------------------------------------------------------------------------'
Matt McIrvin
whop...@csd.uwm.edu (Alfred Einstead) wrote:
> Maxwell invented color photography, took the first color photograph and
> made the device to do it.
>
> Some people rule it out on a technicality: one of the primary colors in the
> prints was actually coming from slightly outside the optical spectrum, so
> the pictures looked funny.
He was also a major figure in the study of human color vision, and
invented a color classification system that was a pretty direct
ancestor of the CIE system still widely used today in printing
and computer graphics.
--
Matt McIrvin
Mark William Hopkins
>I've been under the impression that gravity and electromagnetism
>actually hasn't been unified. I asked Jerrold Marsden down at CalTech
>if this was true.
More accurately: that such unification is largely irrelevant since
electromagnetism is not a fundamental theory but only an approximation
of QED, which in turn is an approximation of the more general gauge quantum
field theory that underlies matter and energy.
>So, what is the deal here.
That (1) unifying gravity with an obsolete theory is an exercise in spinning
your wheels. You might as well unify gravity with phogiston theory for all
that matters.
And that (2), the whole issue is passe' since both theories are Classical
Relativistic Physics and the REAL problem is reconciling the conflicting
views of Relativistic Physics with Quantum Physics.
Matthew Nobes
wrote:
> For some reason most people talk about the "proton spin crisis", even
> though I *believe* these problems arise for neutrons as well. Does
> someone know why? Also, what's the latest story on this issue? Is it
> still a crisis, or has it been solved?
One of the profs here at SFU is involved in the HERMES experiment at
DESY. They are working on this stuff. Last talk he gave he downgraded
``crisis'' to ``problem''.
I'm not entirely sure why this was ever regarded as a crisis in the first
place. We know that the valence quarks don't carry all of the proton's
momentum, why should they carry all the spin?
Check out the HERMES webpage (http://www-hermes.desy.de/) for some
details. Last I heard, they were attempting to measure the gluonic
contribution to the proton's spin. This is very hard.
--
Matthew Nobes,c/o Physics Dept. Simon Fraser University,
8888 University Drive Burnaby, B.C., Canada.
Get my PGP public key, and more,at http://www.sfu.ca/~manobes
Danny Ross Lunsford
itself. So the question is, given
FmnFnp + 1/4 gmp FabFab
can one reconstruct Fmn?
The answer is no - locally you can make a duality transformation
E' = E cos W(x) + B sin W(x)
B' = -E sin W(x) + B cos W(x)
and the resulting energy tensor is the same. For example the energy density
is
1/2( E'^2 + B'^2) = 1/2(E^2 + B^2)
because the crossterms in EdotB cancel out.
If the duality angle is the same everywhere, then F is defined up to a
phase - physically, every charge has a fixed ratio of electric to magnetic
charge. If the duality angle is *not* constant, then one has an interesting
idea to play with - a U(1) gauge theory of duality rotations. One could
locally assign all charge as electric charge, but not globally. I smell
torsion.
-ross
"Daniel Doro Ferrante" <dani...@olympus.het.brown.edu> wrote in message
news:Pine.LNX.4.32.011119...@olympus.het.brown.edu...
> On Mon, 19 Nov 2001, Squark wrote:
> > Am I right concluding a one can reconstruct the electromagnetical
> > field from its energy-momentum tensor? Interesting, is the same true
> > for gauge theory, at least for some gauge groups?
> From the top of my head, i belive that the answer for your first
> question would be "Yes!" and, for the second, would also be "Yes.", at
> least, for the U(1) case. =;)
[unnecessary further quoted text deleted by angry gods]
Danny Ross Lunsford
condition one has an extra Lorentz scalar field to deal with. Considerations
of invariance imply that this field must enter into the gravitational
equations in more or less the same fashion as the Brans-Dicke scalar field.
In fact Pascual Jordan looked at this possibility and used it to build up a
theory with a varying gravitational "constant", following Dirac's idea. See
his book "Schwerkraft und Weltall", 1955 (referened in Pauli, supplementary
note 23, p. 231). This of course is the main upshot of the Brans-Dicke
theory, where G is the reciprocal of the BD scalar field.
Already in Pauli's relativity book there is a crippling argument against KK
theory, which again goes back to the fact that gauge invariance is a side
effect and not a founding principle in this theory.
-ross
Terry Pilling <te...@offshell.phys.ndsu.nodak.edu> wrote in message
news:Pine.LNX.4.10.101111...@offshell.phys.ndsu.nodak.edu.
...
> I have been thinking more about this and I am starting to
Steve Carlip
> I have been thinking more about this and I am starting to
> believe that the massless scalar in KK and the Brans-Dicke
> scalar not as far removed as you indicate.
If you have only gravity and a scalar field, there's a lot of
flexibility. That's because you can do a field redefinition,
g_{ab} -> f(\phi) g_{ab}
that moves the scalar coupling around in the action.
In Brans-Dicke theory, for instance, there's a term in the
Lagrangian of the form
\phi R
but by redefining the metric to \phi^{-1}g_{ab} you can
absorb this into R, at the expense of changing the ``kinetic
energy'' term for \phi. In string theory, such a choice of
scaling is sometimes called a (conformal) frame; the scaling
in which the purely gravitational part of the Lagrangian is
just R, with no scalar coupling, is the ``Einstein frame.''
If your theory couples to other forms of matter, though, this
kind of field redefinition is less innocuous. That's because
the metric appears in the matter couplings as well, and a field
redefinition involving g_{ab} changes the couplings of matter
to gravity. In general, this can lead to a breakdown of the
principle of equivalence---different types of matter will
couple to different effective metrics, because of the different
dependence on \phi in each term. This leads to some important
experimental constraints, which essentially require that any
Brans-Dicke-like scalar field be very nearly constant.
There's one exception, though: if your matter fields happen
to have a ``conformal'' coupling, that is, a coupling that's
invariant under local rescalings of the metric, then you can
do your field redefinitions without harming anything. In
particular, electromagnetism in four dimensions is invariant
under such rescalings---the metric appears only in the form
\sqrt{g} g^{ab}g^{cd}
and it's easy to check its invariance.
So the upshot is that in pure Kaluza-Klein theory, the scalar
field really is equivalent to a Brans-Dicke scalar field, at
least after appropriate field redefinitions. The equivalence
can break down if you couple additional matter---spinors,
for eample---but the difference can always be pushed off
into the matter couplings to the scalar field and to gravity.
Steve Carlip
Charles Francis
<ste...@iea.com> writes:
> What are some sets of axioms that have been tested as a basis
>for a theory unifying GR, QM and EM, and how have they failed?
There are several pages of MTW devoted to explaining why axiomatising
gtr is no longer considered a good idea, and even discussing whether
physics should be axiomatised at all. In part this is because we have no
consistent unified model, and therefore we cannot have an axiom set for
one. But mainly it is because of the constraint which I imposed at the
beginning of the paragraph, "we need to choose axioms which can be seen
to be true of the physical world". This has proved to be an extremely
difficult thing to do. For example these days physics is mostly
formulated in terms of a Lagrangian or Hamiltonian principle, but the
justification for a the assumption of a given Lagrangian is that it
produces right answers, not that it is intuitively possible to see that
it is physically true.
Axioms for quantum mechanics were produced by Birkhoff & Von Neumann in
1936, and these are generally thought correct. But only some of these
axioms can be seen to be true. For example it is a truism to say that at
any time we can define states in accordance with a probability
interpretation. But it is not obvious why a wave equation should relate
the state at one time with that at another. Nor is it obvious that
states should be a continuum. We need them to be a continuum because we
want to do mathematical operations like differentiation and describe
wave equations, but we have no a priore physical reason to define a
continuum.
As for other axioms sets, I don't know that anyone else has really tried
to tackle the foundations of quantum mechanics from this viewpoint since
Birkhoff & Von Neumann. The philosophy espoused by MTW that it is no
longer appropriate to physics has become too greatly ingrained. I have
tried producing a discrete form of quantum electrodynamics in a fairly
axiomatic approach, getting rid of the unjustified continuum assumptions
in the Birkhoff - Von Neumann axioms, and I would say it only fails
because no one can understand it. The big sticking point is that a
discrete model cannot be covariant in the normal sense, and I claim that
there is a relaxed form of covariance with satisfies the fundamental
requirement, that the laws of physics are everywhere the same. Of course
everyone thinks I don't understand the requirement for covariance, and I
think everyone does not understand what I am saying.
Regards
- --
Charles Francis
Charles Cagle
the "To," "Cc," and "Newsgroups" headers for details. ]]
In article <9tddkt$f76$1...@uwm.edu>, Mark William Hopkins
<whop...@alpha2.csd.uwm.edu> wrote:
Whoa. I guess that tells me. Here all along I really thought that
knowledge had survival value and that because of this then knowledge
could be subdivided into false knowledge which didn't have survival
value and true knowledge which did. What you're telling me is that
what is important is not coming to an understanding of the universe
(which is what I thought was the point of unification) but really to
find ways to reconcile conflicting views. You know, here all along I
stupidly supposed that the reason that they were conflicting is because
they were, in fact, irreconcilable. Thanks for that clear vision
you've given me of the true purpose of modern physics.
CC
John Baez
>I have been thinking more about this and I am starting to
>believe that the massless scalar in KK and the Brans-Dicke scalar
>not as far removed as you indicate. Browsing Physics Reports 238 (1997)
>there is a review by J.M. Overduin and P.S. Wesson entitled
>`Kaluza-Klein Gravity' and on page 316 there is a section entitled
>
>3.5. The case A_z = 0: Brans-Dicke theory
>
>which begins, "If one does not set \phi = constant, then Kaluza's
>five-dimensional theory contains besides electromagnetic effects
>a Brans-Dicke-type scalar field theory, as becomes clear when one
>considers the case in which the electromagnetic potentials vanish,
>A_z = 0." So it seems from this section that they also consider the
>scalar to be the same thing in both contexts.
Zounds! As I said in my last post,
:The scalar fields in the two theories could be related
:in some way I've failed to notice,
and I guess it's true!
This is great, because it reduces by one the number of completely
unrelated theories of gravity that I need to learn more about someday! :-)
>If I continue calling the scalar a Brans-Dicke scalar would
>I be totally wrong? Half wrong? or mostly right? I am not sure
>anymore. What are your thoughts?
Don't ask me; I'm no expert on this stuff - Steve Carlip understands
it infinitely better, and his post makes it sound like you're right!
Anyway, I'm really glad you made the original "mistake" and
I made my "correction", because now we're seeing a cool connection
between things I thought were unrelated. This is one thing newsgroups
are good for - unleashing the power of serendipity.
Happy Thanksgiving to all,
jb
John Baez
Charles Cagle <pro...@singtech.com> wrote:
>In article <9tddkt$f76$1...@uwm.edu>, Mark William Hopkins
><whop...@alpha2.csd.uwm.edu> wrote:
Charles Cagle wondered:
>> >So, what is the deal here[?]
>> That (1) unifying gravity with an obsolete theory is an exercise in spinning
>> your wheels. You might as well unify gravity with phogiston theory for all
>> that matters.
>>
>> And that (2), the whole issue is passe' since both theories are Classical
>> Relativistic Physics and the REAL problem is reconciling the conflicting
>> views of Relativistic Physics with Quantum Physics.
>Whoa. I guess that tells me. Here all along I really thought that
>knowledge had survival value and that because of this then knowledge
>could be subdivided into false knowledge which didn't have survival
>value and true knowledge which did. What you're telling me is that
>what is important is not coming to an understanding of the universe
>(which is what I thought was the point of unification) but really to
>find ways to reconcile conflicting views.
No, he's telling you, in his own sweet and gentle way, :-)
that most physicists regard unifying the classical Maxwell
equations with classical Einstein equations as a somewhat
misguided goal, because these two theories are known to be WRONG
in two important ways:
1) electromagnetism is really just part of a bigger package
involving the weak and strong nuclear forces,
2) classical physics is wrong; we need quantum physics.
In short, we are trying to get an understanding of the actual
universe we find ourselves in, not just unify two somewhat obsolete
theories. This is why most physicists are not content with the
Einstein-Maxwell theory mentioned by Marsden.
>Thanks for that clear vision
>you've given me of the true purpose of modern physics.
You're welcome. No charge this time.
Arkadiusz Jadczyk
Baez) wrote:
>1) electromagnetism is really just part of a bigger package
>involving the weak and strong nuclear forces,
>
>2) classical physics is wrong; we need quantum physics.
Ad 1) It may happen that physicist will find unification of gravitation
and electromagnetism (say: UFT1) which will have many advantages
and yet it will not be to get unified with unification of weak,
electromagnetic and strong interactions. We will have then UFT1
and UFT2 and we will be still having a problem of unifying both
unified field theories. Although the above is not within the
"mainstream thinking" - nevertheless it is still probable (and
assigning probability here is subjective)
Ad 2) There is no clear distinction between classical and
quantum physics. Fine structure constant involves c,e,h
- it connects classical and quantum world. It is here that
unifying classical and quantum view is needed, rather than
abandoning classical for quantum.
I could quote theories attempting to derive quantum theory
from "classical theories" - for instance from fractal geometry,
from classical stochastic theories in many dimensions and with
non-trivial topologies, etc. The very definition of "classical
physics" is fuzzy. Some physicists will even say that quantum theory
is a particular case of a classical theory - the symplectic form there
is one on the unit ball of the infinite-dimensional Hilbert space.
Or that quantum theory is a particular case of a classical theory,
where only certain class of variables is "observable" (namely those
functions on the state space that are bilinear), etc. etc.
Again, this is not a mainstream view. But these are facts.
One can easily imagine theories that are neither classical
nor quantum - yet they can produce both in some limits.
I am not acting here as advocatus diavoli. I am simply
presenting a broader view. What we think we know about the
universe may well be only tip of the iceberg .
ark
Danny Ross Lunsford
range forces live together, because a good theory would say a great deal
about how to deal with quantum gravity from what we also know about quantum
light. Futhermore, GR is far sounder than QED mathematically. There is no
disputing this. As of now, there is no mathmatically consistent theory of
light. There is a mathematically consistent theory of gravity.
> ...which in turn is an approximation of the more general gauge quantum
> field theory that underlies matter and energy.
And what might that be? SU(5)?
-ross
"Mark William Hopkins" <whop...@alpha2.csd.uwm.edu> wrote in message
news:9tddkt$f76$1...@uwm.edu...
> In article <051120011734341087%furlong.phy...@singtech.com> Grace
Shellac <furlong.phy...@singtech.com> writes:
> >I've been under the impression that gravity and electromagnetism
> >actually hasn't been unified. I asked Jerrold Marsden down at CalTech
> >if this was true.
>
> More accurately: that such unification is largely irrelevant since
> electromagnetism is not a fundamental theory but only an approximation
> ...which in turn is an approximation of the more general gauge quantum
Demian H.J. Cho
> gravity. The scalar fields in the two theories could be related
> in some way I've failed to notice, though.... and if so, please
> tell me about it! It would be very interesting.
>
Well, this whole subject is more of ark's territory, but...
Yes, they are related. The clearest presentation is either
Appelquist&Chodos, PRD28, 772-784 (1983)
Cho&Freund, PRD12, 1711-1720 (1975)
BTW, I am NOT an author of the second paper.
> Right. Hmm! Now that you mention it, scalars have an annoying
> tendency to pick up a mass from particles they interact with, so
> maybe nowadays someone could try to save the Kaluza-Klein theory
> by showing the scalar becomes very massive. Maybe Demian Cho
> knows about this - his thesis is all about trying to save the
> Kaluza-Klein theory.
Thanks for raising me into a crusader of KK :-)
I saw a couple claims that massless scalar will acquire mass in
full quantum theory. The scale of the mass depends on the scale
of stable minimum of quantum effective potential due to Casimir
effect in internal dimension, which typically assumed to be close to
the Planck scale. I said claim because I never saw
the actual calculation, but it seems it's just a standard result of
the symmetry breaking. Reference is in the first article I quote
above section V.
Of course ,as you know , we don't know how
to stabilize internal dimensions for pure KK theory.
--
Demian H.J. Cho
Center for Gravitation and Cosmology
University of Wisconsin-Milwaukee
Aaron J. Bergman
>
>Thanks for raising me into a crusader of KK :-)
>I saw a couple claims that massless scalar will acquire mass in
>full quantum theory.
That would be nice. As far as I know, though, no one knows how to do it.
IIRC, the most common thing that can happen is that instantons give the
scalar a runaway potential. Stabilizing these sorts of vacua is a major
unsolved problem.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
Kevin A. Scaldeferri
Danny Ross Lunsford <antima...@worldnet.att.net> wrote:
>Futhermore, GR is far sounder than QED mathematically. There is no
>disputing this. As of now, there is no mathmatically consistent theory of
>light. There is a mathematically consistent theory of gravity.
This is comparing apples and oranges. There is a mathematically
consistent theory of light, but it is a classical theory and is known
(experimentally) to be wrong (i.e. an approximation to the correct
theory).
QED is not, we suspect, consistent, but no one knows a consistent
theory of quantum gravity in 4 dimensions either.
The only issue is that there is no experimental evidence that GR is
wrong, so one could hold out some belief that one doesn't actually
need to quantize gravity in the course of reconciling GR and QFT, but
this seems naive.
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Danny Ross Lunsford
... or something like that :) - "The electron is a stranger in
electrodymamics". He meant, E&M with Lorentz electrons is a bad theory,
that is, inconsistent. I think Fritz Rohrlich worked on bettering the
classical theory, but I'm not sure how that all worked out.
Over and over again, Dirac in particular stressed that it isn't
surprising that QED is plagued, because the classical theory before it
is even worse! That is, the level of divergence in QED is somewhat
ameliorated, but not removed.
Dirac made some interesting stabs at "getting a better classical theory"
but these are, like Einstein's later work, all but forgotten. One very
interesting one was the "superconducting vacuum", where there was a sort
of equation of state like J - -kA between the current and the potential,
which made the vacuum a dynamic place. He did this in the '50s if I
recall correctly.
-ross
Charles Francis
<ke...@blinky.its.caltech.edu> writes:
>There is a mathematically
>consistent theory of light, but it is a classical theory and is known
>(experimentally) to be wrong (i.e. an approximation to the correct
>theory).
>
>The only issue is that there is no experimental evidence that GR is
>wrong
I am not sure about that. Firstly because if GR and QFT are not
reconciled then there is an incompatibility in the experimental evidence
for each which at the least suggests that both are only approximate (I
refuse to say 'wrong'). But also I know of two anomalous measurements of
GR, first the prediction of gravitational waves, which as I understand
should have been detected if they occur at the predicted magnitude of
gr, and have not been (though it is a close call) and second the
anomalous measurement of super nova red shift, which I now believe has
to do with the assumption of the geodesic motion of light.
>, so one could hold out some belief that one doesn't actually
>need to quantize gravity in the course of reconciling GR and QFT, but
>this seems naive.
One has to do something with gravity, but whether that something is
correctly described as quantisation is another matter. As I say I think
the reconciliation of GR & QFT has to do with the assumption of geodesic
motion of light, because we cannot second quantise a wave equation
obeying general covariance. But if we use parallel displacement of
momentum in a flat co-ordinate space over large distances, as distinct
from parallel transport, then the incompatibility is removed. The
difference between parallel displacement and parallel transport is that
in parallel transport the magnitude of the vector is restored after each
infinitesimal parallel displacement, whereas I am proposing that the
magnitude of the momentum vector should only be restored in the initial
and final measurements, not at points in the hypothetical "path" between
the two.
This sounds pretty weird, but it is in line with interpretations of QM
which say that we can only talk about observational quantities such as
momentum in the measurements, as properties of an extant wave between
measurements. It is straightforward to show that parallel displacement
is independent of choice of co-ordinate space, and that this still gives
a model which is the same for all observers.
Regards
--
Charles Francis
Demian H.J. Cho
>
> That would be nice. As far as I know, though, no one knows how to do it.
> IIRC, the most common thing that can happen is that instantons give the
> scalar a runaway potential. Stabilizing these sorts of vacua is a major
> unsolved problem.
>
Aaron or other stringy brains out thetre,
Can you lead me into most recent review paper on moduli stability
problem in stringy theories?
Paul D. Shocklee
> Aaron or other stringy brains out thetre,
> Can you lead me into most recent review paper on moduli stability
> problem in stringy theories?
Here are a bunch of papers on possible moduli stabilization mechanisms:
hep-th/0002047
Towards a Solution of the Moduli Problems of String Cosmology
Author: Michael Dine
http://arXiv.org/abs/hep-th/0002047
hep-th/0001112
A Cosmological Mechanism for Stabilizing Moduli
Authors: Greg Huey, Paul J. Steinhardt (Princeton), Burt A. Ovrut (Penn),
Daniel Waldram (CERN)
http://arXiv.org/abs/hep-th/0001112
hep-th/9906246
Remarks on the Racetrack Scheme
Authors: Michael Dine, Yuri Shirman
http://arXiv.org/abs/hep-th/9906246
The following is also a nice review, though it's a bit older:
hep-th/9503114
Modular Cosmology
Authors: T.Banks, M.Berkooz, G.Moore, S.H.Shenker, P.J.Steinhardt
http://xxx.lanl.gov/abs/hep-th/9503114
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Phone: (703) 704-9109
Kevin A. Scaldeferri
Charles Francis <cha...@clef.demon.co.uk> wrote:
>In article <9tucap$dqa$1...@blinky.its.caltech.edu>, Kevin A. Scaldeferri
><ke...@blinky.its.caltech.edu> writes:
>
>>The only issue is that there is no experimental evidence that GR is
>>wrong
>
>I am not sure about that. Firstly because if GR and QFT are not
>reconciled then there is an incompatibility in the experimental evidence
>for each which at the least suggests that both are only approximate (I
>refuse to say 'wrong').
Indeed, it suggests this. But, that is not the same a experimental
evidence which is in conflict with GR. One might imagine some
reconciliation in which GR remains unmodified and only QFT has to be
modified or extended. I don't suspect this is likely, but I don't
think you can rule it out completely.
> But also I know of two anomalous measurements of
>GR, first the prediction of gravitational waves, which as I understand
>should have been detected if they occur at the predicted magnitude of
>gr, and have not been (though it is a close call)
I'm not aware of any conflict. The matter of calculating the
frequency and magnitude of gravity waves incident on the earth is
tricky and requires some amount of estimation. If you really pin down
the LIGO team, I've never heard them claim that they seriously expect
to detect anything with the first version of the detector.
Note, though that the Hulse-Taylor binary pulsar shows energy loss
which is dead on the predictions of GR.
> and second the
>anomalous measurement of super nova red shift, which I now believe has
>to do with the assumption of the geodesic motion of light.
I'm not sure what measurements you're refering to. The ones I know of
are consistent with standard, well-understood cosmological models, but
maybe I'm forgetting about something at the moment.
Jim Carr
"Matthew Nobes" <man...@sfu.ca> writes:
>
... regarding the "proton spin crisis", now downgraded to "problem" ...
>
>I'm not entirely sure why this was ever regarded as a crisis in the first
>place. We know that the valence quarks don't carry all of the proton's
>momentum, why should they carry all the spin?
This question was asked in the early days, when it was already
evident that this "crisis" was like many others: a spectacular
headline over a comparison of good data to a naive model.
>Check out the HERMES webpage (http://www-hermes.desy.de/) for some
>details. Last I heard, they were attempting to measure the gluonic
>contribution to the proton's spin. This is very hard.
Hence the headline. The best way to get funding for a large project
that will have to make a very hard measurement is to declare it a
crisis. Color me (and my elders who pointed it out at the time)
cynical. ;-) It also probably helps keep up the morale of the
troops who have to do the hard work.
--
James Carr <j...@scri.fsu.edu> http://www.scri.fsu.edu/~jac/
SirCam Warning: read http://www.cert.org/advisories/CA-2001-22.html
e-mail info: new...@fbi.gov pyr...@ftc.gov enfor...@sec.gov
Doug B Sweetser
In this post I would like to clarify the relationship between the
source of the unified field, the electric charge, and the mass charge
densities discussed in my previous post.
The proposed unified field strength tensor, d^u A^v, is a second-rank
asymmetric tensor. Any asymmetric tensor can be expressed as a
combination of a symmetric and an antisymmetric tensor. This will be
done for the unified Lagrangian:
L = -J^u A_u - 0.5 d^u A^v d_u A_v
= -J^u A_u - 0.25 (d^u A^v d_u A_v + d^v A^u d_v A_u)
= -J^u A_u - 0.125 ((d^u A^v d_u A_v + d^v A^u d_v A_u) +
(d^u A^v d_u A_v + d^v A^u d_v A_u))
= -J^u A_u - 0.125 ((d^u A^v d_u A_v + d^v A^u d_v A_u +
- d^u A^v d_v A_u - d^v A^u d_u A_v)
(d^u A^v d_u A_v + d^v A^u d_v A_u
+ d^u A^v d_v A_u + d^v A^u d_u A_v))
= -J^u A_u - 0.125 (F^uv F_uv + G^uv G_uv)
where F^uv is the electromagnetic field strength tensor
G^uv === d^u A^v + d^v A^u
Now the asymmetric tensor can be viewed as the sum of a very familiar
tensor, the antisymmetric field strength tensor F^uv, and a symmetric
tensor I have called G^uv. The unified current density J^u is the
source for both of these tensors with different symmetries. It sounds
reasonable to me that at least classically (which is all I am doing),
the source for the symmetric tensor is separate from the source of the
antisymmetric tensor. There should not be an issue connecting the
electric current density source to F^uv. There is a longer path to
connect the symmetric tensor source to a metric, and thus to gravity.
Now the equality
Junified = Jcharge + Jmass
can be viewed as arising the division of an asymmetric unified tensor
into two second rank field strength tensors with different symmetries.
doug <swee...@TheWorld.com>
http://quaternions.com
Kevin A. Scaldeferri
Danny Ross Lunsford <antima...@worldnet.att.net> wrote:
>"Kevin A. Scaldeferri" wrote:
>>
>> In article <ttxK7.119165$WW.75...@bgtnsc05-news.ops.worldnet.att.net>,
>> Danny Ross Lunsford <antima...@worldnet.att.net> wrote:
>>
>> >Futhermore, GR is far sounder than QED mathematically. There is no
>> >disputing this. As of now, there is no mathmatically consistent theory of
>> >light. There is a mathematically consistent theory of gravity.
>>
>> This is comparing apples and oranges. There is a mathematically
>> consistent theory of light, but it is a classical theory and is known
>> (experimentally) to be wrong (i.e. an approximation to the correct
>> theory).
>>
>> QED is not, we suspect, consistent, but no one knows a consistent
>> theory of quantum gravity in 4 dimensions either.
>
>
>... or something like that :) - "The electron is a stranger in
>electrodymamics". He meant, E&M with Lorentz electrons is a bad theory,
>that is, inconsistent. I think Fritz Rohrlich worked on bettering the
>classical theory, but I'm not sure how that all worked out.
Hmm... well, Maxwell's equations with point sources... I guess they
are somewhat ill. I didn't realize that was what you meant.
Electrons aren't a part of classical electromagnetism, in my mind. I
was talking about continuous source distributions, in which case I
claim that Maxwell's equations are a completely consistent and
rigorously describable system of differential equations.
>Over and over again, Dirac in particular stressed that it isn't
>surprising that QED is plagued, because the classical theory before it
>is even worse! That is, the level of divergence in QED is somewhat
>ameliorated, but not removed.
Whoa, whoa, whoa... the problem with QED is not the (obvious)
divergences. That doesn't have anything to do with the consistency of
the theory. It's the Landau pole or the renormalization flow, the
behavior _after_ we deal with the divergences, that's the problem with
QED.
To bang on this from another direction, classical, non-abelian gauge
theories have the same problems with point sources that Maxwell theory
does. And quantized non-abelian gauge theories have the same sort of
divergences. But, the renormalization group behavior is different --
they are asymptotically free, and this makes us think that they are
most likely mathematically consistent.
Ralph E. Frost
problem is or how it fits in, and what the various approaches are to
stabilizing whatever is unstable? Some of us do not have access to all
resources that you list.
Thanks in advance.
Paul D. Shocklee <shoc...@phoenix.Princeton.EDU> wrote in message
news:9ubu4j$dta$1...@cnn.Princeton.EDU...
> Demian H.J. Cho (q...@uwm.edu) wrote:
>
> > Aaron or other stringy brains out thetre,
> > Can you lead me into most recent review paper on moduli stability
> > problem in stringy theories?
>
> Here are a bunch of papers on possible moduli stabilization mechanisms:
[...]
t...@rosencrantz.stcloudstate.edu
Kevin A. Scaldeferri <ke...@inky.its.caltech.edu> wrote:
>In article <a7Of$gKFSL...@clef.demon.co.uk>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>> But also I know of two anomalous measurements of
>>GR, first the prediction of gravitational waves, which as I understand
>>should have been detected if they occur at the predicted magnitude of
>>gr, and have not been (though it is a close call)
>
>I'm not aware of any conflict.
Nor am I. Gravitational radiation has not yet been directly detected,
but as far as I know all realistic calculations show that existing
detectors aren't nearly sensitive enough for a detection to be
expected.
>> and second the
>>anomalous measurement of super nova red shift, which I now believe has
>>to do with the assumption of the geodesic motion of light.
>
>I'm not sure what measurements you're refering to. The ones I know of
>are consistent with standard, well-understood cosmological models, but
>maybe I'm forgetting about something at the moment.
I assume that this is about the evidence from high-redshift supernovae
that the expansion of the Universe is accelerating. That evidence is
inconsistent with the following combination of hypotheses:
1. General relativity is correct.
2. The Universe is gravitationally dominated by low-pressure matter
(p << rho in units where c=1).
So one of those statements must be wrong. The conventional wisdom is
that it's number 2 that's got to go, but if you regard 2 as sacrosanct
then I guess you could take the supernova data as evidence against GR.
-Ted
Charles Francis
>In article <a7Of$gKFSL...@clef.demon.co.uk>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>>In article <9tucap$dqa$1...@blinky.its.caltech.edu>, Kevin A. Scaldeferri
>><ke...@blinky.its.caltech.edu> writes:
>>>The only issue is that there is no experimental evidence that GR is
>>>wrong
>>I am not sure about that. Firstly because if GR and QFT are not
>>reconciled then there is an incompatibility in the experimental evidence
>>for each which at the least suggests that both are only approximate (I
>>refuse to say 'wrong').
>Indeed, it suggests this. But, that is not the same a experimental
>evidence which is in conflict with GR. One might imagine some
>reconciliation in which GR remains unmodified and only QFT has to be
>modified or extended. I don't suspect this is likely, but I don't
>think you can rule it out completely.
I think you have to make the modifications at a fundamental level, in
the underlying philosophy for GR and QM. I find it highly significant
that both relativity and quantum mechanics are theories of measurement,
and on that, philosophical, level they are not incompatible theories,
but different sides of the same coin, relativity based on measurement of
events with ruler and clock, QM based on measurement of particles with
apparatus, and in both cases the theories describe the general
relationships found in such measurement, rather than the specific data
coming from particular measurements.
The natural modification to GR is to say that since in QM observable
quantities do not exist between measurements, the space-time manifold
also does not exist between measurements, and our measurements only tell
us of the existence of a finite number of points on it. This represents
a modification to GR in itself, but I think there is a significant
resulting effect, with vital implications to unification. If the wave
function does not have independent physical existence, but is dependent
on an observer's reference frame, then it is not necessary to insist
that it satisfies a generally covariant wave equation, and once that is
acknowledged one of the big barriers to unification, the impossibility
of second quantising a generally covariant wave equation, goes away.
However, since light can be viewed as wave functions for photons this
means that the law of geodesic motion of light is modified, and red
shift measurements recalibrated.
>>But also I know of two anomalous measurements of
>>GR, first the prediction of gravitational waves, which as I understand
>>should have been detected if they occur at the predicted magnitude of
>>gr, and have not been (though it is a close call)
>I'm not aware of any conflict. The matter of calculating the
>frequency and magnitude of gravity waves incident on the earth is
>tricky and requires some amount of estimation. If you really pin down
>the LIGO team, I've never heard them claim that they seriously expect
>to detect anything with the first version of the detector.
I'm sure you're right. I'm afraid I listened to extravagant claims by a
poster who should have known better.
>Note, though that the Hulse-Taylor binary pulsar shows energy loss
>which is dead on the predictions of GR.
Yes. In any case following the modification I was suggesting above I get
the same equations in the flat space approximation which is usually used
for gravitational waves. I had forgotten about that.
>>and second the
>>anomalous measurement of super nova red shift, which I now believe has
>>to do with the assumption of the geodesic motion of light.
>I'm not sure what measurements you're refering to. The ones I know of
>are consistent with standard, well-understood cosmological models, but
>maybe I'm forgetting about something at the moment.
As I understand the most distant supernovas have been observed with red
shifts which are greater than expected as compared to luminosity. It is
possible to fix this by choosing a value of the cosmological constant,
but this means that the expansion of the universe is accelerating. Not
impossible, but certainly uncomfortable. As I say, it is an anomalous
measurement, not a contradictory one. Nonetheless I think it would make
one feel better about a unification theory if it were able to provide us
with a more "reasonable" value of the cosmological constant.
Regards
--
Charles Francis
Charles Francis
t...@rosencrantz.stcloudstate.edu writes
>I assume that this is about the evidence from high-redshift supernovae
>that the expansion of the Universe is accelerating. That evidence is
>inconsistent with the following combination of hypotheses:
>
>1. General relativity is correct.
>2. The Universe is gravitationally dominated by low-pressure matter
> (p << rho in units where c=1).
>
>So one of those statements must be wrong. The conventional wisdom is
>that it's number 2 that's got to go, but if you regard 2 as sacrosanct
>then I guess you could take the supernova data as evidence against GR.
I am only suggesting a fairly minor revision to GR to incorporate a QM
effect, not what I would describe as evidence against!. The law of
geodesic motion of light is separate from the other assumptions of GR,
and can be tweaked slightly without affecting anything much else. But it
does affect this prediction, which I can attempt to motivate as follows:
If we take a synchronous space-like slice such that we have a defined
set of probability amplitudes for a particle to be found at any point on
that slice then we have fully defined the state. (the particle may be
photon or other, so long as it is described by a wave function). Then we
can take a transform to find a momentum space wave function, which is an
integral of exp(i p.x), where p and x are vectors in 3-space and dot is
the dot product using the three space metric.
In curved space p is a vector field, rather than a vector. But we may
define the field from the value of p at the origin by parallel
displacement in any co-ordinate space. Unlike parallel transport, in
which the length of the vector is adjusted to the manifold at each point
along the path, in parallel displacement we do not readjust the vector
except at the end point of the path. It is straightforward to show that
this is both path independent, and independent of co-ordinate space
(done in http://arXiv.org/abs/physics/0110007).
Now, the wave function originally chosen depended on the synchronous
slice, but this was arbitrary. In another frame we would have had to
parallel displace p to other times. Hence p must be parallel displaced
from initial measurement to final measurement, not parallel transported
as geodesic motion would suggest.
For classical motions the motion is the same as if there were continuous
measurement, as p is adjusted at each measurement the classical motion
is found as a series of infinitesimal parallel displacements, i.e. by
parallel transport. And for small regions of space time there will be no
detectable difference between parallel transport and parallel
displacement. However if we regard light from distant supernovas as a
wave function in which there is no measurement between emission by the
supernova and absorption by our detector then we will have to
recalibrate the red shift to luminosity correlation accordingly.
There is also theoretical support for this idea, in that it is not
possible to second quantise a field on curved space time, whereas this
law of time evolution enables everything to work just as easily as it
does in flat space.
Regards
--
Charles Francis MA PhD
Jim Carr
... upside-down posting corrected ...
"Mark William Hopkins" <whop...@alpha2.csd.uwm.edu> wrote
in message news:9tddkt$f76$1...@uwm.edu...
}
} More accurately: that such unification is largely irrelevant since
} electromagnetism is not a fundamental theory but only an approximation
} ...which in turn is an approximation of the more general gauge quantum
} field theory that underlies matter and energy.
}
} >So, what is the deal here.
}
} That (1) unifying gravity with an obsolete theory is an exercise in
} spinning your wheels. You might as well unify gravity with phogiston
} theory for all that matters.
}
} And that (2), the whole issue is passe' since both theories are Classical
} Relativistic Physics and the REAL problem is reconciling the conflicting
} views of Relativistic Physics with Quantum Physics.
In article <ttxK7.119165$WW.75...@bgtnsc05-news.ops.worldnet.att.net>
"Danny Ross Lunsford" <antima...@worldnet.att.net> writes:
>
>This is absolutely false.
I thought Kevin's response to your article was far too mild; perhaps
he did not pay attention to what was below the upside-down reply
and ask which part of the above is false. Certainly there can
be no question that Maxwell's electrodynamics is obsolete
because it gives incorrect answers to basic questions (such as
what one should observe for the Compton effect) that are of
relevance to physics on the scale of the cosmos. I also cannot
see why you would not think that the real problem is how you
fit quantum phenomena into a picture with gravity.
Now, to be fair, one might *guess* (based on the quoted fragment
that appears later in your reply) that your comment and what
follows was directed at
"More accurately: that such unification is largely irrelevant
since electromagnetism is not a fundamental theory but only
an approximation ..."
but even then I fail to see why you think that Maxwell's
electrodynamics is not an approximation given that it gives
wrong answers to well-posed problems.
>There is compelling reason to try to make the long
>range forces live together, because a good theory would say a great deal
>about how to deal with quantum gravity from what we also know about quantum
>light.
By "good" you mean something better than what is already in hand
for electrodynamics and GR? But how could it tell us a "great
deal" if it cannot tell us how to do Compton scattering?
>Futhermore, GR is far sounder than QED mathematically.
To carry forward the analogy above, so is phlogiston theory.
Somehow the physics ends up being more important than the
mathematics. What I fail to see is how, if the problem
is that QED is not mathematically sound enough to try to
merge it with GR, you will make progress on this problem
by unifying Maxwell with GR.
Kevin A. Scaldeferri
What would you consider a more reasonable value? I know only of
extremely naive estimates, which call for an enormously larger value,
and aesthetic prejudices, which call for a value of exactly zero.
Alfred Einstead
> Over and over again, Dirac in particular stressed that it isn't
> surprising that QED is plagued, because the classical theory before it
> is even worse! That is, the level of divergence in QED is somewhat
> ameliorated, but not removed.
There's nothing wrong with classical E/M, per se, other than the
conversion of the force law into one involving densities. That's
the unwarranted extrapolation.
In fact, without it, you can write down a perfectly sensible
classical theory and even a perfectly consistent semi-classical
quantum theory (which has quantized free E/M field + quantized
particles). That's described below.
Generally, the classical theory (a' la Lorentz) posits a bunch of
point sources; i.e., worldlines r1(t), r2(t), ..., rn(t), with
masses m1, m2, ..., mn; and charges q1, q2, ..., qn.
Each source is the source of a field Ei, Bi; for i = 1, 2, ..., n;
solutions to the Maxwell equations with charge and current sources:
Ji(x, t) = qi ri'(t) delta(x - ri(t))
rhoi(x, t) = qi delta(x - ri(t))
and there's also the free field E0, B0, the solution to the
homogeneous Maxwell equations.
The force law for mass i would be given by:
dpi/dt = sum Fij; j = 0, 1, ..., i-1, i+1, ..., n
Fij = qi (Ej(ri(t), t) + ri'(t) x Bj(ri(t), t))
dei/dt = sum Pij; j = 0, 1, ..., i-1, i+1, ..., n
Pij = qi Ej(ri(t), t) . ri'(t)
with pi = mi gammai ri'(t); ei = mi c^2 gammai
1/gammai^2 = 1 - (ri'(t)/c)^2
treating factors of c = mu_0 = epsilon_0 = 1.
The unwarranted extrapolation is passing over to the densit-ized
version of the force law and adding the self-force term to the
right hand sides, which is infinite. No such thing ever appeared
in any of the experiements that led up to Maxwell's Theory.
You can even write down a stress tensor, which consists of
taking the usual expression for it, but subtracting out all the
quadratic combinations of the forms (Ei Ei, Bi Ei, Bi Bi; for
i = 1, 2, ..., n).
What Lorentz was trying to do was add in the self-force term and
use it to actually derive the law of inertia, itself, and even
the masses of the particles by an appropriately framed charge
distribution model.
(Of course, using a charge distribution model to explain the
ultimate constituency of charge is a SERIOUS begging of the
question. Nobody ever said that charge had to be made out
of charge or that it had to be anything more than a collective
macroscopic manifestation of a microscopic phenomenon that,
itself, has nothing at all to do with charge!)
If you wanted to derive a semi-classical quantum theory for
the fields + particles you could, in fact, start from the
above model, which is classical E/M without the infinite
self-force, and work from there.
The inhomogeneous Maxwell equations can be solved uniquely
in terms of the particle degrees of freedom, up to an overall
free field ... which can be thrown in with the rest of E0 and B0.
So, we're free to take either the retarded or advanced solution
or any combo of them that adds up to 1, since their difference
is a free field. With that, both Ei and Bi are defined for
i = 1, 2, 3, ..., n.
So, the only degrees of freedom to quantize are the free
field degrees of freedom (2 polarization degrees per mode),
and the particle degrees of freedom (6 per particle; 3 position
and 3 momentum components). These are all independent and
mutually unconstrained. So you get a nice clean-cut quantization
of the whole system.
What you get is particles acting directly on each other
across one anothers' light cones and acting with a background
quantum electromagnetic medium.
The particles comprise a finite number of degrees of freedom,
plus the infinite number of degrees of freedom of the free
field.
The free field, interestingly, is not constrained by anything.
As a medium, it's totally independent of the particles resting
on top of it, but yet acts on these particles.
Also, since it contains the differences between advanced and
retarded solutions, you can also say that part of what it's
encoding is the future causality effects whereby one particle
acts on another through its past lightcone; encoding them as
random field modes in the free field.
So, you get most of the standard QED 2nd quantized-corrected
results (e.g., I think the Lamb shift comes out correctly), but
not phenomena related to pair processes, because there's no
feedback to the free field from the particle degrees of
freedom.
The divergence that's in QED is ultimately inherited by naively
taking the (incorrect) Lorentz force law in density form (complete
with its self-force terms F11, F22, F33, ...) and and quantizing
that.
Of course, when you quantize infinity, you usually get infinity.
So QED has problems which come almost entirely from here.
When you pass over to a fully 2nd quantized formulation, you might
think that the self-force problem may arize anew, since particles
will not be being replaced by fields which are smeared out in
space. But Fermions satisfy the exclusion principle, so they
can't occupy the same space. So that, itself, is going to
limit or even eliminate the self-force problem even here too.
Charles Francis
<ke...@its.caltech.edu> writes
>In article <s4liP6LhyID8EwX$@clef.demon.co.uk>,
>Charles Francis <cha...@clef.demon.co.uk> wrote:
>>
>>As I understand the most distant supernovas have been observed with red
>>shifts which are greater than expected as compared to luminosity. It is
>>possible to fix this by choosing a value of the cosmological constant,
>>but this means that the expansion of the universe is accelerating. Not
>>impossible, but certainly uncomfortable. As I say, it is an anomalous
>>measurement, not a contradictory one. Nonetheless I think it would make
>>one feel better about a unification theory if it were able to provide us
>>with a more "reasonable" value of the cosmological constant.
>
>What would you consider a more reasonable value? I know only of
>extremely naive estimates, which call for an enormously larger value,
>and aesthetic prejudices, which call for a value of exactly zero.
>
other that it is determined on a cosmological scale and takes a value
sufficient to compensate for missing mass at least sufficiently to give
a finite closed universe.
Regards
--
Charles Francis
Patrick
about solving the EFE with E&M fields. That was an extremely
informative post. I just wanted to correct a statement:
>However, we know that Maxwell's theory of EM is
>the "effective field theory" or "classical limit" of a renormalizable
>quantum field theory, QED. The latter -is- regarded as a fundamental
>theory.
What we refer to as "effective field theories", are *not* classical
limits of quantum field theories. They are full fledged quantum field
theories themselves, except that it is understood that they are to be
used at energies much below some physical scale which plays the role
of a cutoff. One computes quantum effects (loop diagrams) as in any
quantum field theory.
I hope this does not come out as nitpicking. I though it was an
important detail to point out, in order to prevent confusion.
Patrick
Alfred Einstead
> What you're telling me is that what is important is not coming to
> an understanding of the universe (which is what I thought was the
> point of unification) but really to find ways to reconcile
> conflicting views.
No. That what is important is coming to an understanding of the
universe by finding ways to reconcile conflicting views of it,
not trying to unify obsolete theories that are wrong; which
has nothing more to do with the goal of understanding
than the task of trying to unify Descartes' vortex theory with
Newton's laws of Optics (and the analogy is both pertinent and apt).
> You know, here all along I stupidly supposed that the reason that
> they were conflicting is because they were, in fact, irreconcilable.
GR and QFT are neither irreconcilable, nor mutually contradictory
as far as anyone can tell.
> Thanks for that clear vision you've given me of the true purpose
> of modern physics.
It's worth pointing out that part and parcel of what was being said
by what you quoted is that the entire issue of GR/EM unification is
moot. Any 4-D Riemannian manifold can be represented as a
4-D section of a 5-D Riemannian vacuum; i.e., every classical theory
of matter is representable as a theory of pure gravity in 5-D.
That's Campbell's Theorem. There is, also, already a well-known
standard geometric interpretation of Yang-Mills theory in terms
of fibre bundles.
In terms of the old problem of finding a unified field theory
they already clinched the deal long ago and are Old News. But
at the same time, it means absolutely nothing because this has
nothing to do with the ACTUAL problem of unification -- which
is to provide a cogent synthesis of Quantum Theory and
General Relativity (and even Quantum Theory and Speical Relativity
(!); considering that general locally flat spaces can be achronal
and QT is chronal as presented in standard textbook treatments
and the problem of formulating a bona fide achronal QFT -- which
also happens to be a large part of the QFT/GR reconciliation
issue -- is still unresolved even in the QFT/SR context).
Danny Ross Lunsford
> use it to actually derive the law of inertia, itself, and even
> the masses of the particles by an appropriately framed charge
> distribution model.
Not true at all. This was a separate problem completely from the dynamical
force laws - that is, the union of mechanics and the EM field.
> (Of course, using a charge distribution model to explain the
> ultimate constituency of charge is a SERIOUS begging of the
> question.
That WAS the question. (It still is.)
> If you wanted to derive a semi-classical quantum theory for
> the fields + particles you could, in fact, start from the
> above model, which is classical E/M without the infinite
> self-force, and work from there.
Oh? That was tried many times, and it always failed. See the Born-Infeld
theory.
> The inhomogeneous Maxwell equations can be solved uniquely
> in terms of the particle degrees of freedom, up to an overall
> free field ... which can be thrown in with the rest of E0 and B0.
> So, we're free to take either the retarded or advanced solution
> or any combo of them that adds up to 1, since their difference
> is a free field. With that, both Ei and Bi are defined for
> i = 1, 2, 3, ..., n.
>
> So, the only degrees of freedom to quantize are the free
> field degrees of freedom (2 polarization degrees per mode),
> and the particle degrees of freedom (6 per particle; 3 position
> and 3 momentum components). These are all independent and
> mutually unconstrained. So you get a nice clean-cut quantization
> of the whole system.
>
> What you get is particles acting directly on each other
> across one anothers' light cones and acting with a background
> quantum electromagnetic medium.
This is Feynman's relativistic absorber theory. It's as flawed as Dirac's
sea, which it re-interprets in a quainter fashion, just as the Feynman
positron theory did.
> The particles comprise a finite number of degrees of freedom,
> plus the infinite number of degrees of freedom of the free
> field.
>
> The free field, interestingly, is not constrained by anything.
> As a medium, it's totally independent of the particles resting
> on top of it, but yet acts on these particles.
This is simply not true. "totally independent" "acts on these particles".
These statements are ipso facto contradictory.
You are simply describing gauge invariance, i.e. the currents are the
constraints implied by the Lorentz condition on the potentials. JmAm term is
basically a Lagrange multiplier term.
> Also, since it contains the differences between advanced and
> retarded solutions, you can also say that part of what it's
> encoding is the future causality effects whereby one particle
> acts on another through its past lightcone;
etc. etc. Feynman and Wheeler...
> encoding them as
> random field modes in the free field.
Where do you get this? How can a free degree be encoded? If so it was not
really free.
> So, you get most of the standard QED 2nd quantized-corrected
> results (e.g., I think the Lamb shift comes out correctly), but
> not phenomena related to pair processes, because there's no
> feedback to the free field from the particle degrees of
> freedom.
> The divergence that's in QED is ultimately inherited by naively
> taking the (incorrect) Lorentz force law in density form (complete
> with its self-force terms F11, F22, F33, ...) and and quantizing
> that.
This is wrong, no matter what it means. In fact, the form of the Lorentz
force law is mandated by invariance principles and does not require point
particles, because it arises naturally as a result of a contraction
operation of a larger, unconstrained group than the Poincare group. In any
case if what you said were true, you would have to be able to quantize
gravity, because it is really a statement about inertia you are making.
> Of course, when you quantize infinity, you usually get infinity.
> So QED has problems which come almost entirely from here.
You made the very point, the same one Einstein made - then missed it. It is
however interesting that you point out, corrrectly if inadvertently, the
close connection between Feynman's wacko absorber theory and the virtual
particles of QED. The Feynman rules are then nothing but a local restatement
of the assumptions about the absorber.
-drl
Danny Ross Lunsford
> universe by finding ways to reconcile conflicting views of it,
> not trying to unify obsolete theories that are wrong; which
> has nothing more to do with the goal of understanding
> than the task of trying to unify Descartes' vortex theory with
> Newton's laws of Optics (and the analogy is both pertinent and apt).
I vehemently disagree with this statement.
> GR and QFT are neither irreconcilable, nor mutually contradictory
> as far as anyone can tell.
The reason there is not even a decent candidate for quantum gravity is that
the principles of GR, and indeed even of special relativity is a sense,
*are* directly in contradiction to those of QFT. This is almost too obvious
to state. It is of course politically correct to claim otherwise. This gives
the claimant a certain "panache" and marks him with the "in-crowd".
-drl
Chris Hillman
On Tue, 11 Dec 2001, Patrick wrote:
> What we refer to as "effective field theories", are *not* classical
> limits of quantum field theories. They are full fledged quantum field
> theories themselves, except that it is understood that they are to be
> used at energies much below some physical scale which plays the role
> of a cutoff. One computes quantum effects (loop diagrams) as in any
> quantum field theory.
>
> I hope this does not come out as nitpicking.
Not at all.
> I though it was an
> important detail to point out, in order to prevent confusion.
Yes, I agree, and thanks for the correction! I obviously misunnderstood
the terminology.
So, I should have said "Maxwell's theory of EM is the classical limit of
QED", right? How about "a flux of pions can be modeled classically as a
[a modifier which I have forgotten goes here] scalar field"?
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
Demian H.J. Cho
For some reason I always thought there are two
different ussage of "effective".
What Patrick tells sounds like a "effective action"
of full quantum theory. It is an action for a vacuum
expectation value of the fields, and turns out to be
a generator of 1PI green's functions. So, in that sense
Maxwell's equation is a classical limit (take h bar -> 0)
of , say, one loop quantum effective action, of QED.
Now, about the Chris's comment on pion.
I think that the word "effective field theory" means any theory
that includes all possible terms allowed by symmetry of the
theory. In quantum gravity, you can certainly construct an
action which includes all possible diffeomorphism inv. terms.
(which I think is different from including one loop correction
from gravitons to classical action.)
I am no expert here, but isn't pion modelled as a sigma model
with SU(2) in "effective field theory"?
Am I missing something?
Cheers everyone,
Douglas B Sweetser
Nearly all work on gravity done today, even issues discussed in SPR,
is concerned with reconciling the dividing line between general
relativity and quantum mechanics. In this post, I will focus on a
different division: between Newton's law of gravity and general
relativity for the very specific case of a non-rotating, uncharged,
spherical mass. The gap is small but subtle because the classical
limit of general relativity is Newton's law. In my first post in this
thread, step 4 which "connect[s] the solution to the classical
Newtonian gravitational potential" is one that needs careful
reconsideration. I hope to show that a local four-dimensional simple
harmonic oscillator is the engine behind gravity.
First acknowledge just how excellent Newton's law of gravity happens
to be:
F = m_inertial d^2 R/dt^2 = - G M_source m_gravitational/R^2
This formula can accurately predict the precession of the perihelion
of Mercury to within 43 arc seconds per century. The other planets
in the solar system add over 500 arc seconds in the same time.
The first step in moving from Newton's law toward relativity is to
cancel m_inertial with m_gravitational by assuming the equivalence
principle. The equivalence principle is backed by experimental data.
There is also the wonderful thought experiment of the person in the
windowless elevator, unable to tell the difference between the effect
of gravity and a uniformly accelerating rocket-elevator (ignoring the
tidal effect). The major mathematical impact of the equivalence
principle is that a second rank tensor is required to curve spacetime.
The second step is trivial, but has wonderful implications - divide
both side by the constant c:
1/c d^2 R/dt^2 = - G M_source/c R^2
This equation has units of 1/time, a frequency. Instead of viewing
gravity as a force, it is connected to measurements of a wave. This
is a perfectly classical idea. The planets travel around the sun at a
certain frequency, in the Earth's case once a year. Consider a ball
that falls through an evacuated tunnel trough the center of a
non-rotating Earth. That is an example of a simple harmonic oscillator
driven by gravity. When an apple falls to the Earth, it begins the
~84 minute round-trip journey. As we are familiar, the planned trip
is cut short after less than a second (5 m or 16 ft) due roughly to
the transverse waves of electromagnetism. One can view the stuff of
the Earth as the ultimate case of gridlock, with all the particles
trying to oscillate around the center, but going no where.
The importance of changing perspective, and viewing classical gravity
as a wave phenomena, is that it requires a far smaller step to unify
with the wave equations describing electromagnetism. The hunt is for
a wave equation which leaves the Maxwell equations unaltered for an
electrically charged source and includes another wave equation
consistent with the success of Newton's law, and the profound
adjustments required to be consistent (but not necessarily identical)
with general relativity.
Both Newton's law of gravity and the Maxwell equations can be
expressed as a potential. There have been threads in SPR on what that
may mean, particularly with respect to quantum mechanics, but I think
it is a deep observation that both can be expressed as potential
theories. Since the classical gravity and electromagnetism field
theories have significant differences, it may only be at the potential
level where the two theories unite. That is my operational
assumption.
The Maxwell equations can be formulated as a 4-vector potential.
Newton's law can be expressed as a scalar potential. From general
relativity, it is clear that a scalar potential is not enough to
curve spacetime. One can embed a scalar potential into a 4-vector
potential by setting three of the four components to zero:
A^u = (phi, 0, 0, 0)
Again this is a trivial step with fun implications. By stapling on a
few zeros, one can view classical gravity and EM as arising from
4-vector potentials. Certainly there will be cases for gravity when
the potential is more complicated, but I have chosen to focus on the
simplest, non-trivial case first. For the non-rotating sphere, the
potential is this:
A^u = (GM/cR, 0, 0, 0)
Del A^u = (-GM/cR^2, 0, 0, 0)
The potential is similar to something that appears in the
Schwarzschild metric of relativity, but not quite. Since there never
is any harm in dividing by a universal constant, divide the above
equations by the speed of light c:
Del (GM/c^2R, 0, 0, 0) = (-GM/(c^2 R^2), 0, 0, 0)
Now the potential, GM/c^2R, is the measure of the gravitational field
used in the Schwarzschild metric (a technical aside: R can either be
based on Schwarzschild coordinates which makes the math easier or
isometric coordinates which makes a connection to a grid simpler, but
whichever coordinate system is chosen, the ratio GM/c^2R is in the
metric). The units of the above equation are 1/distance, which are
the units of curvature. The Schwarzschild metric defines how
spacetime is curved by the factor GM/c^2R.
The Newtonian potential written for a force equation is the ratio of
two dissimilar things: the source mass to the distance R between the
test and the source mass. Einstein's great leap was to say that
everything is geometry (albeit complicated geometry, being local and
nonlinear). Energy density happens to curve the geometry of
spacetime. The Newtonian potential over c^2 is the ratio of a
geometric source mass, GM/c^2, and the distance between the source and
the test mass, R. Both have the same units. Time is not space, but
ct has the same units as a distance and both are included in
spacetime. Energy is not momentum, but the energy-momentum 4-vector
is a composite of the two. The geometric mass of a source and a
displacement are not the same thing, but they will be used together.
Classically, although potentials are useful in calculations, it is the
fields that make up the measurable fabric of our world. The
electromagnetic field is an antisymmetric 2-rank tensor:
F^uv = d^u A^v - d^v A^u
There are only two forces with an infinite range because they are
mediated by massless particles: gravity and electromagnetism. There
should be a role for the symmetric counterpart to the electromagnetic
field strength tensor, call it G^uv:
G^uv = d^u A^v + d^v A^u
The combined asymmetric unified field strength tensor would be the
sum:
U^uv = d^u A^v
With a specific classical unified question, we must immediately worry
if the Maxwell equations of motion have been lost in the process. In
a previous post, I worked with this Lagrangian:
L = - J^u A_u - .5 d^u A^v d_u A_v
By direct calculation of the Euler-Lagrange equations, I was able to
show that the equations of motion for a vacuum were:
(d^2/dt^2 - Del^2) A^u = 0
What is most fun about this equation is that if you wrote it on a
blackboard in front of physicists, they would all nod along bored,
mumbling that this is just the Maxwell equations written in the Lorenz
gauge (most people actually make the error of crediting Lorentz). Yet
it should be clear from the Lagrangian that no gauge was fixed. The
gauge-fixing term has the form (d^u A_u)^2.
The potential has four degrees of freedom, but the Maxwell equations
describe two transverse waves. By removing a constraint on the
Gupta/Bleuler Lagrangian (gauge-fixing), it is possible that the
prosaic four dimensional wave equation can describe both the Maxwell
equations and gravity.
Rewrite the four dimensional wave equations in terms of classical
fields for a vacuum:
dg/dt + Del.A = 0
-Delg - dE/dt + DelxA = 0
where
g = dphi/dt + Del.A
E = -dA/dt - Delg
B = DelxA
The homogeneous Maxwell equations are vector identities. If the
topology is simple, they will still be valid.
What happens if there are sources? Look at the Lagrangian. The
asymmetric unified tensor d^u A^v can be expressed as the combination
of a symmetric 2-rank tensor G^uv and an antisymmetric 2-rank
electromagnetic field strength tensor F^uv. The classical source may
also be the sum of a symmetric source (mass) and an antisymmetric
source (electric charge).
The goal of this post is to look for a connection between Newton's law
and general relativity within the context of this classical
unification proposal. There are almost no choices that can be made!
We must work with a potential of the form A = (phi, 0, 0, 0). The
potential must solve the equations of motion in a vacuum, Box^2 A^u=0.
The potential must involve some type of inverse distance, 1/(x^2 + y^2
+ z^2)^0.5.
The potential must also confront the criticism Einstein leveled at
Newton's very useful classical law: that changes in the mass density
must not instantaneously propagate. The only way to do this
simply is to bring in a factor of time on equal footing with the space
contributions. Newton's law predicts the bending of space for photons
traveling around the Sun correctly. General relativity predicts the
same bending of space, along with an equal bending of time, and
together the bending matches the experimental evidence.
The above collection of constraints dictates investigation of the
following potential:
A^u = (1/(c^2 t^2 - x^2 - y^2 - z^2), 0, 0, 0) = (1/tau^2, 0, 0, 0)
As shown in a previous post, this potential solves the wave equation.
The potential is invariant under a Lorentz transformation. Time is
explicitly included, so nothing will happen instantaneously.
There is only one distance that might be interpreted as an interval
involved in a gravitational system: the distance between the source
and the test mass. Therefore, to first approximation, 1/tau^2 =~
1/R^2. Intervals come in three different types: timelike, lightlike,
and spacelike. No doubt Nature understands the different uses of all
three, but the goal in this post is to stay rooted to classical
physics. That goal dictates a pair of timelike events that happen to
be separated a distance R away.
This is easy to visualize. Imagine a laser beam is fired from the source
to the test mass (event F). Choose a reference frame such that there is
simultaneously an event on the test mass (event 0). When the laser
photons arrive, that is event 1:
.1
/| tau = R
/ | time
F.--.0
R
The interval between events 0 and 1 is timelike, and its magnitude is
about R. It is not quite that for two reasons already discussed: the
system needs to behave like a simple harmonic oscillator, and it needs
to do this in four dimensions. How does one construct a simple
harmonic oscillator? One inspiration, not to be taken too literally,
is Hooke's law, F = k r. This has a spring constant and a linear
displacement of distance. Hooke's law is global in the sense that the
spring constant is not sensitive to where it is in spacetime. It is
also linear in r.
Two other aspect of general relativity must be folded into this
construction: that the field equations are local and non-linear. We
are more acquainted with the oscillators that have constant spring
constants and a linear dependence on space.
Construct a four-dimensional local, non-linear simple harmonic
oscillator out of the variables in Newton's law of gravity. Start by
separating out parts of the potential into those that are constant,
and those that contribute to the simple harmonic oscillator:
tau^-2 = 1/(t^2 - r^2) = 1/((A + SHO(t))^2 - (B + SHO(r))^2)
where A^2 - B^2 ~= R^2
A, B != f(t, r)
This model will only work locally. By that I mean for small amounts
of change in either t or r. Given enough time or distance, A and B
will need to change. But very close in spacetime to the test mass,
all the change will be due to the contribution of the simple harmonic
oscillators in time and 3-space. Form a dimensionless spring constant
using the constant geometric source mass:
t' = A + (GM/(2 c^2 A)) t
r' = B + (GM/(2 c^2 B)) r
Substitute this back into the potential:
tau^-2 = 1/(A^2 - B^2 + GMt/c^2 - GMr/c^2 + O((GM/c^2)^2)
For the classical approximation, the second order displacement terms
are dropped. Take the derivative with respect to t and r:
dtau^-2/dt = -GM/(c^2 tau^4)
dtau^-2/dr = GM/(c^2 tau^4)
Since tau^2 ~= R^2, there is too much dependence on distance for this
to represent a classical gravitational field in a "more relativistic"
way. In my previous post, I noted that normalizing the field to the
potential created the correct dependence on distance. That
manipulation may be connected to another mystery. One can build a
device to measure mass or electric field density. One forms a ratio
between the measuring device and some sort of external standard.
The potential cannot be directly measured. Why can't the potential be
compared directly to an outside standard? If the change in the
potential gets normalized to the potential itself, perhaps the
self-referential comparison eliminates the ability to compare to an
outside standard, which itself will also be normalized to itself.
Take the normalized force field, and pug it into a relativistic force
equation:
dp^u/dtau = k q d^u A^v U_v/|A|
If dm/dtau ~= 0, then the 1/tau^2 4-D oscillator potential leads to
this set of differential equations:
d^2 t/dtau^2 + GM/(c^2 tau^2) dt/dtau = 0
d^2 r/dtau^2 - GM/(c^2 tau^2) dr/dtau = 0
These can be solved using standard calculus. When the eight constants
are eliminated, a metric equation is the result (done in a previous
post). To be consistent with the equivalence principle almost requires
a metric equation for gravity. That metric is testably different that
the Schwarzschild metric of general relativity.
Based on this post, there is now an ontological (why is it that way)
difference between general relativity and this classical unified field
proposal beyond the inclusion of EM. According to general relativity,
an energy density happens to curve spacetime. No reason is given for
this, just equations that work. At the core of this proposal is a
very old friend - a four-dimensional wave equation - seen in the new
light of a Lagrangian freed of the gauge choice constraint. A truism
in physics is that "everything is a simple harmonic oscillator." Now
gravity joins that list. The periodic motion of the planets around
the Sun, of Suns around galaxies, of apples trying to oscillate around
the center of the Earth - all may arise from the Maxwell equations
written as a potential wave equation without assuming the Lorenz
gauge.
doug
quaternions.com
Kevin A. Scaldeferri
>> limits of quantum field theories. They are full fledged quantum field
>> theories themselves, except that it is understood that they are to be
>> used at energies much below some physical scale which plays the role
>> of a cutoff. One computes quantum effects (loop diagrams) as in any
>> quantum field theory.
>>
>> I hope this does not come out as nitpicking.
>
>
>> I though it was an
>> important detail to point out, in order to prevent confusion.
>
>the terminology.
The confusion is probably not entirely your fault. While the term
"effective field theory" does have a fairly narrow definition, people
do frequently use the term "effective theory" much more casually.
So, I might use "effective theory" to mean any theory which is a limit
of another theory (or a "power series expansion" of another theory).
However, I would restrict "effective field theory" for the cases where
we obtain the limit by integrating out some degrees of freedom.
Kevin A. Scaldeferri
there, so let me try to be a little more honest here to clarify
something in this post.
In article <3C18F05D...@uwm.edu>, Demian H.J. Cho <q...@uwm.edu> wrote:
>
>I think that the word "effective field theory" means any theory
>that includes all possible terms allowed by symmetry of the
>theory.
This is a touch backwards.
The Lagrangians of effective field theories _do_ in fact typically
contain all interaction terms consistent with the symmetries of the
theory. (And, if they are missing some, that usually means you failed
to notice a symmetry.)
So, what we would like to do is to explicitly integrate out degrees of
freedom, and if you could do this, you would obtain an effective
Lagrangian with all these additional, non-renormalizable terms.
Sadly, this is almost never possibly to do.
The next best thing is to write down the full Lagrangian and the
effective Lagrangian (with all terms allowed by symmetry, but unknown
coefficients) and start calculating diagrams in both theories. Now
you can determine the coefficients in the effective Lagrangian by
matching the full theory onto the effective theory. There are an
infinite number of coefficients, but there is also a power counting so
the process is systematic and you are only limited by how strong you
are. An example of this is the theory of only photons obtained by
integrating the electrons out of QED. You can calculate the
coefficient of the F^4 term by matching to the box diagram in QED.
Sadly, there are theories where you cannot do even this. For example,
the effective field theory which contains only nucleons and pions (or
only nucleons). The problem here is that you can't calculate in the
full theory (QCD) at the relevant (low) energy scale. So, in this
case, you can only fix the coefficients by comparing to experiment.
>I am no expert here, but isn't pion modelled as a sigma model
>with SU(2) in "effective field theory"?
Yes, this is one of the canonical examples of an effective field theory.
Danny Ross Lunsford
> cancel m_inertial with m_gravitational by assuming the equivalence
> principle.
This is false. Relativity theory is independent of the nature of particular
forces.
The first step to relativity theory is recognition of the non-existence of
global simultaneity.
>The major mathematical impact of the equivalence
> principle is that a second rank tensor is required to curve spacetime.
This also is false, as far as getting a relativistic generalization of
Newtonian gravity is concerned (there is more to GR than generalizing
Newton). Nordstrom's scalar theory of gravitation, which is a relativistic
generalization of Newtonian potential theory, can be brought into generally
covariant form, that is, brought into line with the equivalence principle.
The key fact that leads to a 2nd rank tensor theory is the *assumption* that
the *entire* energy tensor be the source of gravitation, and *that* source,
in its nature, is a second-rank tensor. There is the further assumption that
only second derivatives of the metric should occur on the left side - thus
a Rmn + b R gmn + c gmn ~ Tmn
As Hilbert showed, considerations of causality now determine the constants
a, b, and c. The argument goes like this. Given any solution, a general
coordinate transformation with 4 arbitrary functions can be made and general
covariance guarantees that this too is a solution. Hence, for a proper
physical theory, there *must* exist 4 (differential) identities among the
gmn to reduce this indeterminateness. These of course turn out to be the
Bianchi identities.
Physics requires Tmn,n = 0, which are 4 equations, so it is "extremely
plausible" (Pauli) to further *assume* that the divergence of the LHS is
*identically* zero and that the differential conservation law of energy and
momentum is identically satisfied as a *consequence* of the field equations
of gravitation. With the help of the Bianchi identities, this puts the
equations in the form
Rmn - 1/2 gmn R + L gmn = Tmn
where the constant L remains undetermined.
I emphasize that all these things are based on *assumptions* backed up by
physical insight and supported *afterward* by the mathematical structure.
>By stapling on a
> few zeros, one can view classical gravity and EM as arising from
> 4-vector potentials.
Gravity
1) cannot be a simple scalar theory (Nordstrom), since that theory gives
predictions that are wrong.
2) cannot be a vector field theory, since gravity is not a polar theory,
i.e. everything attracts everything else.
Therefore this assumption is eo ipso wrong and must be discarded.
> There are only two forces with an infinite range because they are
> mediated by massless particles: gravity and electromagnetism. There
> should be a role for the symmetric counterpart to the electromagnetic
> field strength tensor, call it G^uv:
>
> G^uv = d^u A^v + d^v A^u
For the thousandth time, this is in and of itself, quite apart from
connection with real physics, WRONG because it is REDUCIBLE. It is like
saying that cooking and piano playing are a unified art because
C = 0, P = 0 implies C+P = 0
that is, excellent cooking (C=0) and excellent piano playing (P=0) lead to a
theory of unified cooking and playing (C+P=0).
You cannot slap theories together and make physics like this. A unified POV
requires an irreducible base *physical principle*, and not just some
arbitrary symbol chopping.
-drl
Jim Carr
}
} Over and over again, Dirac in particular stressed that it isn't
} surprising that QED is plagued, because the classical theory before it
} is even worse! That is, the level of divergence in QED is somewhat
} ameliorated, but not removed.
In article <e58d56ae.01120...@posting.google.com>
whop...@csd.uwm.edu (Alfred Einstead) writes:
>
>There's nothing wrong with classical E/M, per se, other than the
>conversion of the force law into one involving densities. That's
>the unwarranted extrapolation.
Why is it unwarranted to do the same thing Newton did in his
classical model for gravity when calculating the force between
extended objects? However, I will also add that Lunsford let
you off the hook by not asking you how to calculate the
potential energy of the electron and what you get.
>In fact, without it, you can write down a perfectly sensible
>classical theory and even a perfectly consistent semi-classical
>quantum theory (which has quantized free E/M field + quantized
>particles). That's described below.
I did not see you describe how to get a result for Compton
scattering that is perfectly consistent with observation.
>What Lorentz was trying to do was add in the self-force term and
>use it to actually derive the law of inertia, itself, and even
>the masses of the particles by an appropriately framed charge
>distribution model.
Lorentz was not alone in doing so, probably because Abraham and
others also realized a collection of charge has a self energy
and that a point charge would have an infinite one.
> ... without the infinite self-force ...
It is not a self-force, it is the potential energy of the charge
calculated consistently in the theory.
>The inhomogeneous Maxwell equations can be solved uniquely
>in terms of the particle degrees of freedom, up to an overall
>free field ... which can be thrown in with the rest of E0 and B0.
It seems that here is where you sweep your infinity under the rug.
Brian J Flanagan
>Relativity theory is independent of the nature of particular forces.
I'm not sure what you mean by "independent", here. Surely the special
theory has much to do with EM (in re: the constancy of c), just as the
general theory has much to do with gravitation?
Danny Ross Lunsford
Not at all. In fact the framework of relativity is derives from a
group-theoretic analysis based on very basic and simple assumptions about
space, time, and motion, taken from experience. The conclusion is that the
allowed transformations depend on parameter with the dimensions of a
velocity that is either finite or not. In the real world, it turns out to
be finite. Of course, it is c. That light goes at this speed is incidental
to the analysis, which would still be correct if light went at some other
speed. Light happens to go at c because the photon is massless.
To give this derivation, we assume
1) The allowable transformations are linear between frames in uniform
relative motion, so that they depend on the relative (vectorial) velocity V
2) That time can enter into the transformations in an essential way, that
is we do not assume t' = t.
3) That space and time are isotropic, that is, time flows evenly and there
are no preferred directions.
Spatial isotropy eliminates the directional character of V and so we can
concentrate on the simple case of one spatial dimension.
Temporal isotropy implies that what is true of frame A in relation to frame
B for V, is true of frame B in relation to frame A for -V.
Now, writing
x' = a(V) x + b(V) t
t' = c(V) x + d(V) t
x = a(-V) x' + b(-V) t'
t = c(-V) x' + d(-V) t'
so
a(V) a(-V) + b(-V) c(V) = 1
d(V) d(-V) + b(V) c(-V) = 1
a(-V) b(V) + b(-V) d(V) = 0
a(V) c(-V) + c(V) d(-V) = 0
and of course we can replace V -> -V and these still hold. So
a(V) b(-V) + b(V) d(-V) = 0
a(V) c(-V) + c(V) d(-V) = 0
so b(V) = c(V).
Now
a(V) a(-V) + b(V) b(-V) = 1
a(V) b(-V) + b(V) d(-V) = 0
thus
b(V) [ b(-V)^2 - a(-V)d(-V) ] = b(-V)
and this also holds for V -> -V, which implies either
a(V) d(V) - b(V)^2 = 1, b(V) = -b(-V)
a(V) d(V) - b(V)^2 = -1, b(V) = b(-V)
The latter is ruled out by letting V=0 in which case
x = x'
t = t'
We can solve the equations now with
a(V) = d(V) = a(-V) = d(-V)
and so
a(V)^2 - b(V)^2 = 1
and
a(0) = 1, b(0) = 0
The origin x=0, which moves at speed V in the other frame, transforms as
x' = b(V) t
t' = a(V) t
so
dx'/dt' = V = b(V)/a(V)
so
a(V) = 1 / sqrt(1 - V^2)
b(V) = V / sqrt(1 - V^2)
Finally we dimensionalize time vs. space and replace
V -> V/C
t -> Ct
and write
x' = 1/sqrt(1 - (V/C)^2) ( x + (V/C) Ct )
Ct' = 1/sqrt(1 - (V/C)^2) ( Ct + (V/C) x )
If we let C go to infinity,
x' = x + Vt
t' = t
So either C is finite, or not. Experience shows that it is finite.
Of course, historically relativity emerged from the contradictions in
electron theory implied by the tacit, wrong assumptions about the nature of
simultaneity.
-drl
Danny Ross Lunsford
equivalent.
-drl
"Danny Ross Lunsford" <antima...@yahoo.com> wrote in message
news:9CNU7.175553$WW.11...@bgtnsc05-news.ops.worldnet.att.net...
[Moderator's note: Quoted text deleted. Please quote judiciously. -TB]
mike james
> If we let C go to infinity,
>
> x' = x + Vt
> t' = t
>
> So either C is finite, or not. Experience shows that it is finite.
>
Just to make sure I'm understanding - what "experience" shows that C is
finite?
Clearly you aren't assuming that C is anything to do with the speed of light
so when does the connection arise?
(I've seen this derivation before but I've never thought to ask the
question.)
mikej
[Moderator's note: There are huge piles of experimental evidence that
special relativity works and that the maximum allowed speed is c =
2.9979x10^8 m/s, which is finite. -TB]
Brian J Flanagan
> Brian J Flanagan wrote:
> > I'm not sure what you mean by "independent", here. Surely the special
> > theory has much to do with EM (in re: the constancy of c), just as the
> > general theory has much to do with gravitation?
>
> Not at all. In fact the framework of relativity is derives from a
> group-theoretic analysis based on very basic and simple assumptions about
> space, time, and motion, taken from experience.
The choice of group and the "simple assumptions about space, time, and
motion" must conform to the observed facts of EM and G. Our
"experience" is conditioned by both these interactions.
Danny Ross Lunsford
> Clearly you aren't assuming that C is anything to do with the speed of
> light so when does the connection arise?
> (I've seen this derivation before but I've never thought to ask the
> question.)
> mikej
Let's assume Maxwell's vacuum theory is modified like this:
As before
div B = 0
curl E + dB/dt = 0
which implies we can write
E = - grad phi - dA/dt
B = curl A
But now suppose
div E = - m^2 phi
curl B - dE/dt = - m^2 A
where m is very, very small, and a scalar in regard to the new group. Then
A and phi satisfy modified equations
((d/dt)^2 - div^2 + m^2) (A,phi) = 0
with div A + dphi/dt = 0 (not optional!)
All this is still invariant under the new group, but now light is massive.
A simple radially symmetric, time-independent solution is
A = 0
phi = k/r exp -(mr) (k=arbitrary constant)
This goes over to the Coulomb potential for m->0. So here is a theory
(Proca) that is Lorentz invariant and goes over to the Maxwell theory in
the limit. In other words, the actual behavior of light can only be decided
by experiments to determine m, and special relativity exists independently
of it. If m happens to be 0, then light will go at C. If it goes at C in
one frame, it goes at C in all frames, by the simple composition of
reference frames (addition theorem for velocities). This seems to be what
actually happens in the world.
Note that m<>0 forces on us the equation
div A + dphi/dt = 0
which is not required in the Maxwell theory. We lost general gauge
invariance by assuming m<>0. So, the masslessness of the light is
equivalent to exact gauge invariance.
-drl
Doug B Sweetser
Let me address the last issue you raised with a degree of passion.
Write out the symmetric second-rank field strength tensor G^uv
explicitly in terms of its components:
G^uv = G^vu = d^u A^v + d^v A^u
= (d/dt, -d/dx, -d/dy, -d/dz)(phi, Ax, Ay, Az) + ...
| dphi/dt dAx/dt dAy/dt dAz/dt|
= |-dphi/dx -dAx/dx -dAy/dx -dAz/dx|
|-dphi/dy -dAx/dy -dAy/dy -dAz/dy|
|-dphi/dz -dAx/dz -dAy/dz -dAz/dz|
| dphi/dt -dphi/dt -dphi/dy -dphi/dz|
+ | dAx/dt -dAx/dx -dAx/dy -dAx/dz|
| dAy/dt -dAy/dx -dAy/dy -dAy/dz|
| dAz/dt -dAz/dx -dAz/dy -dAz/dz|
| 2 dphi/dt dAx/dt-dphi/dx dAy/dt-dphi/dy dAz/dt-dphi/dz|
= |dAx/dt-dphi/dx -2 dAx/dx -dAy/dx-dAx/dy -dAz/dx-dAx/dz|
|dAy/dt-dphi/dy -dAx/dy-dAy/dx -2 dAy/dy -dAz/dy-dAy/dz|
|dAz/dt-dphi/dz -dAx/dz-dAz/dx -dAy/dz-dAz/dy -2 dAz/dz |
You claimed this matrix could not play a role in physics because it
was "WRONG because it is REDUCIBLE" [the capitalization was in the
original post]. Pardon my ignorance, but I don't see how to reduce
this tensor. There are ten unique derivatives in the matrix, as one
would expect of a symmetric second-rank field strength tensor
constructed from two 4-vectors (n^2 - n(n-1)/2).
If, after a quick read, one concluded that G^uv was reducible, then
the conclusion that the proposal was not interesting may be
justified. The details of writing out G^uv explicitly show that G^uv
is not reducible.
To continue in this explicit mode, write out the electromagnetic field
strength tensor, F^uv, which is easy to do since it is just like G^uv
except that it is traceless and a few signs are different:
F^uv = -F^uv = d^u A^v - d^v A^u
| 0 dAx/dt+dphi/dx dAy/dt+dphi/dy dAz/dt+dphi/dz|
= |-dAx/dt-dphi/dx 0 -dAy/dx+dAx/dy -dAz/dx+dAx/dz|
|-dAy/dt-dphi/dy -dAx/dy+dAy/dz 0 -dAz/dy+dAy/dz|
|-dAx/dt-dphi/dz -dAx/dz+dAz/dx -dAy/dz+dAz/dy 0 |
A few things to notice. First, one does not say this anti-symmetric
field strength tensor is reducible. F^uv is not reducible, and
neither is G^uv. There is no linear combination of F^uv that can
create the information contained in G^uv, or visa versa. There are
six unique derivatives in the matrix, as one would expect of a
symmetric second rank tensor constructed from two 4-vectors
(n(n-1)/2).
There are interesting points regarding unification. First, this
second-rank tensor is usually written in terms of the classical fields
E and B:
| 0 -Ex -Ey -Ez|
= |Ex 0 -Bz By|
|Ey Bz 0 -Bx|
|Ez -By Bx 0|
There is no prima facia reason to suppose a 3-vector E and a
3-pseudo-vector B should be unified, but when expressed in as a second
rank antisymmetric tensor, it should be obvious: they arise from
derivatives of the same 4-potential, (phi, Ax, Ay, Az). If there is a
force in Nature that uses G^uv, it will be unified with E and B at the
same level, that of the 4-potential. This does not make the G^uv a
"4-vector" theory - it is a second-rank tensor. Yet it depends on the
same 4-potential that the electromagnetic field strength tensor does.
If the first element of the 4-vectors are flipped - d/dt and phi - then
the sign of E changes. No element in G^uv flips signs. If d/dx,
d/dy, d/dz, Ax, Ay, and Az flip signs, then the signs of E flip again.
Again, no element in G^uv flips signs. It may turn out that because
the field strength tensor F^uv has elements that can flip signs, a
force based on such a tensor can attract or repel, but the field
strength tensor G^uv does not change signs under time or spatial
inversion, and a force based on this tensor is unidirectional.
The classical field, (Ex, Ey, Ez, Bx, By, Bz) are really just a
renaming exercise for the six unique derivatives in F^uv. They can be
divided into two categories depending on where they appear in the
matrix. A similar exercise could be repeated for G^uv, but there
would be a third category for the elements on the diagonal. No one
likes to deal with ten new names, so we won't, but please remember
they are there in the symmetric second-rank field strength tensor.
For a second rank tensor, there are at most 16 (n^2) degrees of
freedom. The asymmetric classical unified field U^uv is the linear
combination of G^uv and F^uv:
| dphi/dt dAx/dt dAy/dt dAz/dt|
U^uv = 2 |-dphi/dx -dAx/dx -dAy/dx -dAz/dx|
|-dphi/dy -dAx/dy -dAy/dy -dAz/dy|
|-dphi/dz -dAx/dz -dAy/dz -dAz/dz|
This again is irreducible. It is a linear combination of G^uv and
F^uv, so U^uv has the same amount of information in it as G^uv and
F^uv. It is this tensor that was part of the Lagrangian, along with a
source term. Using the Euler-Lagrange equation, I was able to
demonstrate this had the equations of motion of a four dimensional
wave equation. This is not "arbitrary symbol chopping". [I was
actually taken aback on that Saturday morning after churning through
the Euler-Lagrange equations and finding the 4D wave equation].
In the road to a matrix, it was the second rank tensor that was used
in the force equation:
dp^u/dtau = k q d^u A^v U_v/|A|
For a particular choice of a potential, a solution to this force
equation was found. When the eight constants were eliminated, the
result was a metric equal to post-Newtonian accuracy of the
Schwarzschild metric of general relativity. This path to a metric is
clearly distinct from the methods used for general relativity. Since
this path is different, one must use more caution in mapping work done
on general relativity directly to this work.
Fortunately, there is a good explanation for Danny's error. When the
second-rank tensor is contracted using a differential operator to
generate a 4-vector of field equations, here is the result:
d_u U^uv = d/dt(dphi/dt + Del.A) + Del.(-dA/dt - Delphi)
+ d/dt(dA/dt + Delphi) - Del dphi/dt - Del^2 A
= dg/dt + Del.E
-d E/dt + DelxB - Delg
= rhomass + rhoelectric + Jcharge + Jmass
This has the nonhomogenous Maxwell equations: Gauss' and Ampere's
laws. There are also two equations involving a scalar field g, where
g = dphi/dt + Del.A. The field equations of general relativity
involve second rank tensors. Since the unified field equations and
Einstein's field equations are tensors of different ranks, they cannot
be compared directly. Instead, the properties of the solutions to
equations containing the fields can be compared. In my previous post
working through an explicit solution to the force equation for the
unified field equation, the solution had the following properties: it
was local, nonlinear, depended on the equivalence principle, and could
generate a testable metric equation for gravity. Those are all
properties shared by solutions to Einstein field equations.
The first point of contention was a miscommunication, not an issue of
substance. The relativity of simultaneity was realized by Einstein
while working on special relativity. For general relativity, one
critical insight was Einstein's careful consideration of the
cancellation of the inertial and the gravitational mass.
We both agree that physical theories must reflect the relativity of
simultaneity and the equivalence principle. The precise role each
idea played in developing special and general relativity is not
important. My comment was about general relativity, not special
relativity, but I omitted the key adjective.
I don't disagree with this assessment:
>I emphasize that all these things are based on *assumptions* backed up
>by physical insight and supported *afterward* by the mathematical
>structure.
There are inspired guesses involved in general relativity about second
order differential equations. I must confess I don't care about the
details, because I am not trying to "regenerate" the field equations
of general relativity as is done in string theory as a validation of
that work. As a specific example, the Bianchi identities were not
invoked in the road to the classical unified field metric work out in
detail in my posts. My work makes a different prediction about the
metric for a spherically-symmetric, non-rotating mass source than
general relativity for a coefficient beyond the standard
post-Newtonian accuracy. No matter how much you disagree with the
logical presentation, at least it can be experimentally confirmed or
denied (a big experimental undertaking, but not as impossible as
having to work on the Planck scale).
Bill Hobba
Brian J Flanagan wrote
> The choice of group and the "simple assumptions about space, time, and
> motion" must conform to the observed facts of EM and G. Our
> "experience" is conditioned by both these interactions.
>
Not true. The argument presented is very similar to that found in
Introduction to Special Relativity by Rindler. From the POR the maximum
speed of physical iteration (it may be infinite) must be the same in all
inertial frames. Let this speed be c. Using that and the simple augments
presented previously derives the Lorentz transformation containing this c.
The structure of this transformation shows there can be only one c that is
the same in all reference frames. Determination of its numerical value
rests on other physical arguments. Probably the simplest is that we know
light behaves like waves an a fundamental property of waves is they travel
at the same speed regardless of the speed of the source. From the POR this
leads to the speed of light being the same in all inertial frames so must be
the c above. This is not the only reason it must be the speed of light.
Analysis of magnetic fields, the drag effect, etc all lead to the same
result, the c above must be the speed of light ie the speed of light is the
maximum speed physical effects can propagate. But note this is the
experimental fixing of a value that naturally follows from the POR.
Bill
Danny Ross Lunsford
> You claimed this matrix could not play a role in physics because it
> was "WRONG because it is REDUCIBLE" [the capitalization was in the
> original post]. Pardon my ignorance, but I don't see how to reduce
> this tensor.
For an arbitrary tensor T, one can always write
Tmn = 1/2 (Tmn - Tnm) + 1/2 (Tmn + Tnm) = Tsym + Tantisym
Given an arbitrary transformation of coordinates, the symmetric and
antisymmetric parts retain their symmetry. The split is invariant so the
tensor has been reduced. This is why the fundamental tensors that show up
in realistic field theories are always either symmetric or antisymmetric.
If you have some asymmetric tensor Tmn satisfying, say,
Tmn,n = 0
then what you are really saying is
Tsym mn,n = 0
Tantisym mn,n = 0
and you have done nothing but add zero to zero. Thus, regardless of the
details, there is no information content in your idea. You could add the
Maxwell stress tensor to the field tensor if you wanted, set the derivative
to zero, and claim to have "unified" matter and field. Of course, it would
mean nothing at all, because the stress tensor and the field tensor
represent irreducible physical objects.
Note that asymmetric tensors may arise in some other context - for example,
in Weyl's gauge invariant geometry, the Ricci tensor is asymmetric - but
that is an object derived from the irreducible field variables gmn and Am.
> There is no prima facia reason to suppose a 3-vector E and a
> 3-pseudo-vector B should be unified
This is false. In fact E and B transform together as the bivector part of
the Dirac algebra on spacetime, an irreducible object, and can be
represented
(Ei alphai - Bi sigmai)
where alpha1 = gamma4 gamma1, sigma1 = gamma2 gamma3 etc.
Since alpha has 1 space leg and sigma has 2, E and B have opposite spatial
parity.
> This does not make the G^uv a
> "4-vector" theory - it is a second-rank tensor. Yet it depends on the
> same 4-potential that the electromagnetic field strength tensor does.
The very fact that E and B can be derived from a potential at all goes back
to the Maxwell set
Fab,c + Fbc,a + Fca,b = 0
and is essentially dependent on the antisymmetry of F.
> If the first element of the 4-vectors are flipped - d/dt and phi - then
> the sign of E changes. No element in G^uv flips signs. If d/dx,
> d/dy, d/dz, Ax, Ay, and Az flip signs, then the signs of E flip again.
> Again, no element in G^uv flips signs.
You have discovered here the reducibility of your tensor.
The problem you are having is lack of sufficient mathematical tools to deal
with invariant structures on spacetime. All you have at hand is the
quaternions - that is, the even subalegbra of the Dirac algebra (if you
prefer, the Pauli matrices). You don't have tensors (at least, you aren't
using them), you don't have spinors, and you only have half the algebra of
spacetime. Any "theory" that you concoct from the quaternions alone is
going to be worthless.
-drl
Etherman
"Doug B Sweetser" <swee...@world.std.com> wrote in message
news:Gp2Fv...@world.std.com...
> Hello Danny:
>
> Let me address the last issue you raised with a degree of passion.
> Write out the symmetric second-rank field strength tensor G^uv
> explicitly in terms of its components:
>
> G^uv = G^vu = d^u A^v + d^v A^u
Perhaps I've missed something along the way, but why do you consider
this to be a tensor? It appears you are using ordinary differentiation
here, which means that G^uv is not a tensor. This is different from
the case of F^uv because (assuming the connection is symmetric) F_uv =
A_v;u - A_u;v = A_v,u - A_u,v. The terms involving the connection
vanish identically whereas they add together in G_uv.
Of course this can easily be rectified by redefining G_uv so that:
G_uv = A_u;v + A_v;u
--
Etherman
AA # pi
EAC Director of Ritual Satanic Abuse Operations
AMTCode(v2): [Poster][TÆ][A5][Lx][Sx][Bx][FD][P-][CC]
Brian J Flanagan
> Brian J Flanagan wrote
>
> > The choice of group and the "simple assumptions about space, time, and
> > motion" must conform to the observed facts of EM and G. Our
> > "experience" is conditioned by both these interactions.
> >
>
> speed of physical iteration (it may be infinite) must be the same in all
> presented previously derives the Lorentz transformation containing this c.
Do you suppose you would obtain a Lorentzian which conformed to
observed fact if you took c to be infinite?
> The structure of this transformation shows there can be only one c that is
> the same in all reference frames. Determination of its numerical value
> rests on other physical arguments.
Would such physical arguments as you have in mind have anything to do
with the observed facts of EM and G?
Probably the simplest is that we know
> light behaves like waves an a fundamental property of waves is they travel
> at the same speed regardless of the speed of the source.
Would this qualify as an observed fact?
Oz
>The problem you are having is lack of sufficient mathematical tools to deal
>with invariant structures on spacetime. All you have at hand is the
>quaternions - that is, the even subalegbra of the Dirac algebra (if you
>prefer, the Pauli matrices). You don't have tensors (at least, you aren't
>using them), you don't have spinors, and you only have half the algebra of
>spacetime. Any "theory" that you concoct from the quaternions alone is
>going to be worthless.
I wonder if you would mind just slightly amplifying the above?
What are the structures on spacetime modelled by:
Dirac algebra
Spinors
Tensors (outside GR)
and what is the 'missing half' of the algebra of spacetime mentioned.
A brief simple discussion on each topic would be most gratefully
received.
NB I have been trying to find these out for some years ..
PS I have found your posts wonderfully clear and helpful, this may or
may not be a compliment here.
--
Oz
This post is worth precisely what you paid for it.
Today Oz is a happy bunny ...
Danny Ross Lunsford
> Do you suppose you would obtain a Lorentzian which conformed to
> observed fact if you took c to be infinite?
If C were infinite and light just happened to go at a very high speed, then
the Michelson-Morley experiment would not have a null result. There is only
one speed that has to be the same in all frames.
-drl
Danny Ross Lunsford
> What are the structures on spacetime modelled by:
>
> Dirac algebra
> Spinors
> Tensors (outside GR)
> and what is the 'missing half' of the algebra of spacetime mentioned.
The most important equation in physics is the Dirac equation, which is
based on taking the square root of the spacetime metric with the algebraic
ansatz
{ ym, yn } = 2 gmn
The elements ym, the 4x4 Dirac matrices, make up what is known as a
Clifford algebra. The specific Clifford algebra derived from this metric is
known as the Dirac algebra, or sometimes (David Hestenes) spacetime
algebra.
Any 4x4 matrix can be written as a linear combination of the 16 matrices
s) I
p) y5 = i y1 y2 y3 y4
v) ym
a) y5 ym
b) a1,2,3 = y4 y1, y4 y2, y4 y3
s1,2,3 = y2 y3, y3 y1, y1 y2
The relavisitic invariance of the Dirac equation shows that the ym
transform like vectors - one can think of the 4 ym as an orthonormal basis
for spacetime. The square of each space leg is -1, that of the time leg is
+1. Given this interpretation, the five classes of products of y's
represent s) scalars (1), p) pseudoscalars (1) (which change sign under
parity), v) vectors (4), a) pseudovectors (4), and b) bivectors (6).
The even subalgebra consists of the 8 elements in s), p), and b) . Products
of linear combinations of these are again linear combinations of the same
set. The "other half" of spacetime algebra is the set of 8 elements v) and
a).
To show that the utility and generality of this algebra is not restricted
to the Dirac equation, note that Maxwell's equations can be written
(ym Dm) (a.E - s.B) = ym Jm
Historically, the first example of a Clifford algebra was Hamilton's
quaternions. These first surfaced in a physically essential way in Pauli's
non-relativistic theory of spin, where the quaternions emerge as the Pauli
matrices.
Rather than extend a long post on the key features of vector and spinor
representations and the role of Clifford algebras, please see the following
excellent lecture notes:
http://tph16.tuwien.ac.at/~kreuzer/inc/gmtp.pdf
The most amazing aspects of Clifford algebras are
1) their 8-periodicity in the signature. Any Clifford algebra depends only
on the dimension D and (p-q) mod 8, where the signature of the underlying
quadratic form is (p,q)
2) the central importance of the unit pseudoscalar in their recursive
construction by tensor products. To answer Gauss, the true metaphysics of i
is now understood, and it is parity.
It is fashionable to play down Dirac's role, but taking the square root of
gmn was as great a stride as was ever made in physics.
-drl