Is general relativity incompatible with the Newtonian limit?
Juan R.
Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
Most physicists are very satisfied with this situation [refer to
divergences of QFT]. They argue that if one has rules for doing
calculations and the results agree with observation, that is all that
one requires. But it is not all that one requires. One requires a
single comprehensive theory applying to all physical phenomena. Not one
theory for dealing with non-relativistic effects and a separate
disjoint theory for dealing with certain relativistic effects.
Furthermore, the theory has to be based on sound mathematics, in which
one neglects only quantities that are small. One is not allowed to
neglect infinitely large quantities [...]
The agreement [QED] with observation is presumably a coincidence, just
like the original calculation of the hydrogen spectrum with Bohr
orbits. Such coincidences are no reason for turning a blind eye to the
faults of a theory. One must seek a new relativistic quantum mechanics"
I, like others are working in a new relativistic quantum mechanics
which was more succesfull than RQFT... However let me focus now in
gravitation. I always thought that general relativity had been
*completely* verified in experiments until one day i studied GR...
There are several obvious flaws in GR, including discrepancies with
data especially in the extragalactic regime. Still my main motivation
is if GR is incompatible with NG in the same form that RQFT is
incompatible with NRQM like claimed by Dirac.
When one ask by the Newtonian limit of GR one heard in Baez page
http://math.ucr.edu/home/baez/RelWWW/wrong.html
that "The theorem stating that gtr does indeed go over to Newtonian
gravitostatics in the very weak field, very slow motion limit is proven
in detail in almost every gtr textbook."
Then one read one of those textbooks and discover like NG is 'obtained'
from GR, but one finds that derivation is NOT rigorous.
Then one takes a more rigorous textbook (Wald) and one discovers that,
effectively, NG is NOT derived from GR in the linear regime. Those
textbooks are wrong, in the linear regime a = 0. Then Wald explains
that one may go beyond the linear regime, he does and obtains eq. 4.21
of Wald
a = - GRAD (Phy)
whereas the Newtonian law is
a = - GRAD (Phy)
A priori one has obtained Newtonian law, but is not. a in GR is not a
in NG, time in GR is not the time in NG. Phy in GR is a retarded field.
Phy in NG is an instantaneous potential, The GR above formula contains
c, Newtonian formula does not contain c, etc.
Of course, if one takes limit c --> infinite then tGT --> tNG and Phy
transforms into an instantaneous field (still is not a potential) but
the Schwartzild metric
g00 = (1 - 2Phy/c^2) --> 1
and
gRR = -(1 - 2Phy/c^2)^-1 --> -1
and Rab = 0. In fact, the curvature of spacetime is zero like in the
flat metric (1, -1, -1, -1).
Then one discovers that the so called 'derivation' given in textbook is
incorrect.
One searchs relevant literature and all one finds is the Cartan-like
approach (Anyone who has studied this know that specialized literature
on Newtonian limit clearly states that textbook derivation is not
rigorous by several motives), where one first geometrizes NG, after
reformulates GR in a 3+1 manner and then one obtains a flat derivative
more a 'potential'.
The spliting is not defined by GR and then one needs introduce ad hoc
'things'. There are two approaches:
- Introduce ad hoc equations for fixing the 'gauge' of the 'potential'.
Those equation are not derived from GR field equations. Therefore NG is
not derived from GR, at the best derived from (GR + adittional
equations).
- Introduce a global boundary. This is Ehlers approach, which is based
in flatness, but that boundary is unphysical. (arXiv:gr-qc/9810078 v3)
"However, physical evidence clearly suggests that we are not living in
an 'island universe' (cf. Penrose 1996, 593-594) - i.e., universe is
not 'an island of matter surrounded by emptiness' (Misner et al. 1973,
295)."
Even if a correct boundary was found, or even if finally it is admited
by relativists that one needs ad hoc equations does not contained in
GR, one is not deriving NG. Problems:
i) The Phy derived from GR has the functional form
Phy = Phy (x, t)
whereas the Phy of NG has the functional form
Phy = Phy (R(t))
and there is no posibility for transforming one in other simply via a
limit. By arguments similar to one showed in PRE 1996 53(5), 5373. we
can see that the field approach is inconsistent and cannot explain all
the phenomena.
ii) In the non relativistic limit c --> infinite, there is no posible
explanation of gravity in the basis of spacetime curvature. It can be
seen from the Schwartzild metric that when c is more and more large,
the curvature of spacetime is more and more small, until vanishes. In
the c --> infinite limit the metric is (1, -1, -1, -1)
Therefore, one recovers Dirac's criticism also in gravitation.
At one hand NG, at the other hand GR which does not reduces to NG and
therefore cannot explain data all already explained by NG.
If this is correct, GR is not correct and then we may develop a new
theory of gravity where gravitation WAS a force. Then one can explain
phenomena already explained by GR and explained by NG.
Please, do not reply that "it is impossible to obtain a force theory of
gravity, or similar" because i already obtained one from the new
relativistic mechanics (it explains usual solar system test:
perihelion, radar delay, etc. and reduces exactly to NG in the
nonrelativistic limit).
The emphasis of this thread is on if really GR alone is able to obtain
NG like a particular case or if, copying Dirac, today we are using two
theories NG and GR ("a separate disjoint theory for dealing with
certain relativistic effects") and we need a new theory of gravity
explaining both relativistic and nonrelativistic phenomena.
If this new theory of gravity was correct, then the main problem in the
quantization of GR was that one was quantizing the incorrect theory.
Juan R.
Center for CANONICAL |SCIENCE)
Igor Khavkine
> In one of his last works Mathematical Foundations of
> Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
>
> Most physicists are very satisfied with this situation [refer to
> divergences of QFT]. They argue that if one has rules for doing
> calculations and the results agree with observation, that is all that
> one requires. But it is not all that one requires. One requires a
> single comprehensive theory applying to all physical phenomena. Not one
> theory for dealing with non-relativistic effects and a separate
> disjoint theory for dealing with certain relativistic effects.
> Furthermore, the theory has to be based on sound mathematics, in which
> one neglects only quantities that are small. One is not allowed to
> neglect infinitely large quantities [...]
However great a theorist was Dirac, he was wrong about this assesment
of renormalization. Perturbative renormalization is based on sound
mathematics *and* is capable of produce correct veriable (and verified)
predictions.
> The agreement [QED] with observation is presumably a coincidence, just
> like the original calculation of the hydrogen spectrum with Bohr
> orbits. Such coincidences are no reason for turning a blind eye to the
> faults of a theory. One must seek a new relativistic quantum mechanics"
The derivation of the Hydrogen spectrum by Bohr was no coincidence. It
has been put on solid footing on by topological aspects of the
semiclassical WKB approximation. If one is allowed to say that the
agreement of QED with observation is merely an coincidence, then the
same can be said about each and every successful theory in physics.
> When one ask by the Newtonian limit of GR one heard in Baez page
>
> http://math.ucr.edu/home/baez/RelWWW/wrong.html
>
> that "The theorem stating that gtr does indeed go over to Newtonian
> gravitostatics in the very weak field, very slow motion limit is proven
> in detail in almost every gtr textbook."
That is indeed true.
[...]
> Then one discovers that the so called 'derivation' given in textbook is
> incorrect.
It is completely unclear to me what your objection to these derivations
is. Having looked through such derivations in a few books, I can't
admit to having found any objections myself.
Without support for the this claim of yours, the rest of the
speculations in your post fall through and perhaps disapear in a black
hole.
Igor
tes...@um.bot
> In one of his last works Mathematical Foundations of
> Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
[delete Dirac's critique of QFT circa 1978]
> I always thought that general relativity had been *completely* verified
> in experiments
Did you perhaps mean that you always thought that "to date, no
widely-accepted experiment or observation has been generally agreed to
clearly -violate- gtr"?
(Which is probably a fair statement of the current situatoin, but see also
the Pioneer Effect thread. This mystery is gaining ever more attention,
but of course "something went wrong with gtr" is just one of many
explanations which have been proposed, and still seems to most of us, I
think, still fairly unlikely to ultimately become the accepted
explanation, although only time will tell.)
> There are several obvious flaws in GR, including discrepancies with data
> especially in the extragalactic regime.
GR does have flaws which are widely recognized, but discrepancies with
data is emphatically -not- one of them! And to be fair, GR also has many
widely recognized -virtues-, including this: it's so darned good that it's
been so intensively studied that scads of -theoretical- flaws have been
identified. None of its competitors have come in for that kind of
scrutinity (not a good thing, but understandable).
What about experimental flaws? As in disagreement with Nature?
Well, there are all kinds of mysteries in cosmology, and the possibility
that our Gold Standard Theory of Gravitation is finally beginning to fail
at large scales, after decades of sturdy use, must always be considered.
Indeed, this has been repeatedly suggested over the decades for various
things which were eventually put down to other causes. Not many would say
that a possible failure of gtr is the leading contender among seriously
discussed possible explanations of any given contemporary cosmological
mystery, however.
> Still my main motivation is if GR is incompatible with NG in the same
> form that RQFT is incompatible with NRQM like claimed by Dirac.
Apples and oranges, surely? Dirac was talking (I guess) about divergent
integrals; nothing like that is happening here; you just misunderstood
what you read in Wald.
> When one ask by the Newtonian limit of GR one heard in Baez page
>
> http://math.ucr.edu/home/baez/RelWWW/wrong.html
>
> that "The theorem stating that gtr does indeed go over to Newtonian
> gravitostatics in the very weak field, very slow motion limit is proven
> in detail in almost every gtr textbook."
Just curious: did you visit that page after reading the Physics Forums
"sticky note:" pointing to it? BTW, see the byline before complaining to
Baez.
> Then one read one of those textbooks and discover like NG is 'obtained'
> from GR, but one finds that derivation is NOT rigorous.
It is well known that "limits" can indeed be tricky. But if you read the
textbooks and understand what statement is being proven, you can usually
see that the alleged proof is good enough for physics :-/
> Then one takes a more rigorous textbook (Wald) and one discovers that,
> effectively, NG is NOT derived from GR in the linear regime. Those
> textbooks are wrong, in the linear regime a = 0.
What do you mean, "derived from"?
It sounds like you might have confused linearized gtr (weak fields; very
roughly speaking first order in the mass parameter, which does indeed kill
off a lot of interesting and important behavior) with the slow motion weak
field limit, which is even more stringent.
This might not be your fault if you are only reading Wald, since in my
experience his discussion in chapter 4 confuses many readers exactly on
this point, because the distinction between weak field and slow motion
weak field is not stated clearly enough. Reading one of the more
elementary textbooks such as Carroll, Schutz, or Stephani might help here.
> Then Wald explains that one may go beyond the linear regime,
Where exactly does he say "beyond the linear regime"? I can't seem to
find this in section 4.4a.
> he does and obtains eq. 4.21 of Wald
(4.4.21) (roughly, equation of motion of test particles, aka force law),
right? And don't forget (4.4.17) (Laplace equation).
> a = - GRAD (Phy)
^^^^^
Phi
> whereas the Newtonian law is
>
> a = - GRAD (Phy)
> A priori one has obtained Newtonian law, but is not. a in GR is not a
> in NG,
I think you might be thinking of relativistic momentum versus Newtonian
momentum. If so, right, these -are- different, but they do agree -up to
first order- in the velocity, as Einstein pointed out. That is, in the
slow motion limit, they do agree:
mv/sqrt(1-v^2) = mv + mv^3/3 + O(v^5)
= mv + O(v^2)
> time in GR is not the time in NG.
No, but for slowly moving test particles, up to first order the proper
time counted off by the test particle does agree with the Newtonian time
approximately:
(Delta t)/sqrt(1-v^2) = (Delta t) + O(v^2)
> Phy in GR is a retarded field.
I have always been careful to point out (as does Wald, on p. 76) that this
approximation also assumes -slowly changing- fields. Or even static
fields, if you are just analyzing the motion of test particle and checking
that in weak static fields, slowly moving test particles to approximately
obey Newtonian laws, as they should. That is what Wald is verifying.
> Phy in NG is an instantaneous potential, The GR above formula contains
> c, Newtonian formula does not contain c, etc.
Wald uses geometric or relativistic units in which c=G=1 by fiat.
> Of course, if one takes limit c --> infinite
Ah. This might be the problem. Wald is thinking of the magnitude of the
velocity of the test particles as being "small" wrt c=1; you want to let c
tend to infinity. Either do it Wald's way or reinsert the c,G factors
(see the appendix) to do it your way.
> the Schwartzild metric
^^^^^^^^^^^
Schwarzschild
> g00 = (1 - 2Phy/c^2) --> 1
>
> and
>
> gRR = -(1 - 2Phy/c^2)^-1 --> -1
Try it again with c=G=1. Try expanding the metric components to first
order in m. Can you pick off the Newtonian potential? Now try the
general case following Wald.
> Then one discovers that the so called 'derivation' given in textbook is
> incorrect.
It looks fine to me; I think you just misunderstood something.
"T. Essel" (hiding somewhere in cyberspace)
Ilja Schmelzer
<tes...@um.bot> schrieb
> > There are several obvious flaws in GR, including discrepancies with data
> > especially in the extragalactic regime.
>
> GR does have flaws which are widely recognized, but discrepancies with
> data is emphatically -not- one of them!
Looking at the Einstein equations G_mn = T_mn it is clear that a
disagreement of the Einstein equations with nature may be described as "dark
matter"defined by
T_mn^dark = G_mn - T_mn^obs
In the extragalactic regime we need dark matter, and a lot of it. We need it
for galaxies, on the large scale today, on the large scale in the early
universe. Moreover, part of the dark matter does not have the properties of
usual matter (violates the strong energy condition).
In this sense, what we observe is exactly what we have to expect if there
are discrepancies of GR with Nature in the extragalactic regime.
Ilja
Eugene Stefanovich
news:1127529917.8...@g44g2000cwa.googlegroups.com...
> Juan R. wrote:
> > In one of his last works Mathematical Foundations of
> > Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
> >
> > Most physicists are very satisfied with this situation [refer to
> > divergences of QFT]. They argue that if one has rules for doing
> > calculations and the results agree with observation, that is all that
> > one requires. But it is not all that one requires. One requires a
> > single comprehensive theory applying to all physical phenomena. Not one
> > theory for dealing with non-relativistic effects and a separate
> > disjoint theory for dealing with certain relativistic effects.
> > Furthermore, the theory has to be based on sound mathematics, in which
> > one neglects only quantities that are small. One is not allowed to
> > neglect infinitely large quantities [...]
>
> However great a theorist was Dirac, he was wrong about this assesment
> of renormalization. Perturbative renormalization is based on sound
> mathematics *and* is capable of produce correct veriable (and verified)
> predictions.
.. for the S-matrix and related properties, but not for the time dependence
of observables.
Eugene.
carlip...@physics.ucdavis.edu
[...]
> Of course, if one takes limit c --> infinite then tGT --> tNG and Phy
> transforms into an instantaneous field (still is not a potential) but
> the Schwartzild metric
> g00 = (1 - 2Phy/c^2) --> 1
> and
> gRR = -(1 - 2Phy/c^2)^-1 --> -1
> and Rab = 0. In fact, the curvature of spacetime is zero like in the
> flat metric (1, -1, -1, -1).
You're making a couple of mistakes here. First, remember that the
value of the components of the metric has no physical significance --
it's highly coordinate-dependent. For example, the flat metric
diag(1,-1,-1,-1) and the flat metric diag(1000,-1,-1,-1) are not
different. What really matters is second derivatives of the metric
(in the form of the curvature tensor), or, if you have a preferred
coordinate system, first derivatives (in the form of the connection
appearing in the geodesic equation).
Second, your g_00 is wrong. There's an extra c^2:
g_00 = c^2(1 - 2GM/rc^2)
In the limit c->infinity, g_00 goes to infinity, which is fine;
that's the same behavior as the flat Minkowski metric. But the
*derivative* of g_00 is well behaved, and the connection goes over
to a nice limiting value, in which the geodesic equation becomes
the standard Newtonian force law.
[...]
> One searchs relevant literature and all one finds is the Cartan-like
> approach (Anyone who has studied this know that specialized literature
> on Newtonian limit clearly states that textbook derivation is not
> rigorous by several motives), where one first geometrizes NG, after
> reformulates GR in a 3+1 manner and then one obtains a flat derivative
> more a 'potential'.
> The spliting is not defined by GR and then one needs introduce ad hoc
> 'things'. There are two approaches:
> - Introduce ad hoc equations for fixing the 'gauge' of the 'potential'.
> Those equation are not derived from GR field equations. Therefore NG is
> not derived from GR, at the best derived from (GR + adittional
> equations).
These additional equations are just a coordinate choice. GR is defined
in arbitrary coordinates. You can't expect an *easily recognizable*
Newtonian limit unless you work in coordinates that go over to the
coordinates that you use in Newtonian gravity.
> - Introduce a global boundary. This is Ehlers approach, which is based
> in flatness, but that boundary is unphysical. (arXiv:gr-qc/9810078 v3)
> "However, physical evidence clearly suggests that we are not living in
> an 'island universe' (cf. Penrose 1996, 593-594) - i.e., universe is
> not 'an island of matter surrounded by emptiness' (Misner et al. 1973,
> 295)."
This is a nitpick. The issue is discussed in MTW on pages 292-299. GR
gives the Newtonian limit *in a particular set of coordinates*. If you
are foolish enough to choose accelerating coordinates, you will get
Newtonian gravity with an additional "inertial" acceleration. The
simplest way to avoid this is to require that the acceleration goes to
zero far away from any source of gravity -- this is your "unphysical
boundary," which is the same as the "boundary" that is always used in
Newtonian gravity to determine the potential. If you don't do that, then
even in standard Newtonian gravity you don't have a unique potential.
> Even if a correct boundary was found, or even if finally it is admited
> by relativists that one needs ad hoc equations does not contained in
> GR, one is not deriving NG. Problems:
> i) The Phy derived from GR has the functional form
> Phy = Phy (x, t)
> whereas the Phy of NG has the functional form
> Phy = Phy (R(t))
> and there is no posibility for transforming one in other simply via a
> limit.
Of course there is. This is your c->infinity limit.
> ii) In the non relativistic limit c --> infinite, there is no posible
> explanation of gravity in the basis of spacetime curvature. It can be
> seen from the Schwartzild metric that when c is more and more large,
> the curvature of spacetime is more and more small, until vanishes. In
> the c --> infinite limit the metric is (1, -1, -1, -1)
That's not correct. As I pointed out above, you have the c's in the
wrong place in g_00. Fix that, and you'll find that the curvature
tensor does not go to zero -- and, more than that, that it gives the
right Newtonian limit for the equation of geodesic deviation.
Steve Carlip
tes...@um.bot
> In the extragalactic regime we need dark matter, and a lot of it. We
> need it for galaxies, on the large scale today, on the large scale in
> the early universe. Moreover, part of the dark matter does not have the
> properties of usual matter (violates the strong energy condition).
Are you objecting that it is unwise to try to explain galactic rotation
curves by keeping the EFE (indeed, by assuming that Newtonian gravitation
is not -totally- out of whack at galactic scales) but introducing a
concept of dark matter, in the absence of direct evidence for such stuff?
If so, I'd agree that at this point the "dark matter" concept is
-speculative-. I'd probably assess the chances that galactic rotation
curves will ultimately put down to a gross failure of gtr differently from
you, however.
> Looking at the Einstein equations G_mn = T_mn it is clear that a
> disagreement of the Einstein equations with nature may be described as
> "dark matter"defined by
>
> T_mn^dark = G_mn - T_mn^obs
It might be clear to you, but not so clear to many others :-/
You'd have to be much more specific about what alleged "disagreement with
Nature" you have in mind (are we still talking about galactic rotation
curves? were we -ever- talking about galactic rotation curves), and what
constraints if any you intend place on T_mn^dark (Segre type, for
example?) before I could comment except in generalities. And if this
refers to some prior thread or a discussion elsewhere, you should assume
that I missed this discussion if you want to pursue this, because I am
lost.
Actually, -I- don't really want to pursue this, Ilja, so I hope you will
be willing to leave it at this:
Dark matter is currently a speculative concept, some would even say a
dubious concept, but nonetheless most contemporary cosmologists seem to be
disinclined to abandon gtr as our Gold Standard Theory of Gravitation.
These are judgement calls which could be profoundly affected by new
observations, or possibly even by new theoretical developments, although
at the moment many would probably agree that some independent confirmation
of the existence of dark matter (if it does exist), new tests of the
alleged "Pioneer effect", etc., would be more helpful than yet another
gravitation theory, classical or otherwise. Fair enough?
"T. Essel"
Homo Lykos
news:dh89ep$5ba$1...@beech.fernuni-hagen.de...
Not only in this general sense: There exist a great lot more very specific
arguments against GR, because its until now not possible to describe
satisfingly the galactic observations with exotic (or ghost) dark
matter-models: For this it would be necessary to explain simultaneously the
constant curves of rotation in galaxies, the Tully/Fisher-rule and the
universality of the gravitational accelaration, at which the influence of
exotic dark matter begins to be important. More about the astrophysical
constraints for a good theory of gravitation you find in (GR is by far not
fulfilling these constraints):
Astrophysical Constraints on Modifying Gravity at Large Distances by
Aguirre, Burgess, Friedland und Nolte, 25. Mai 2001,
http://arxiv.org/abs/hep-ph/0105083
Homo Lykos
Ilja Schmelzer
> On Tue, 27 Sep 2005, Ilja Schmelzer wrote:
> > In the extragalactic regime we need dark matter, and a lot of it. We
> > need it for galaxies, on the large scale today, on the large scale in
> > the early universe. Moreover, part of the dark matter does not have the
> > properties of usual matter (violates the strong energy condition).
>
> Are you objecting that it is unwise to try to explain galactic rotation
> curves by keeping the EFE (indeed, by assuming that Newtonian gravitation
> is not -totally- out of whack at galactic scales) but introducing a
> concept of dark matter, in the absence of direct evidence for such stuff?
No. We have a disagreement between theory (GR + theory of visible matter)
with observation. It is wise to consider different possible explanations.
This includes, of course, dark matter. But, as long as we have not found
this dark matter, the dark matter hypothesis is only an ad hoc explanation
for an observed discrepancy with data. And it is unwise to make claims like
this:
>>> GR does have flaws which are widely recognized, but discrepancies with
>>> data is emphatically -not- one of them!
> If so, I'd agree that at this point the "dark matter" concept is
> -speculative-. I'd probably assess the chances that galactic rotation
> curves will ultimately put down to a gross failure of gtr differently from
> you, however.
I'm not evaluating probabilities for the different solutions for the
observed discrepancy. One thing is the belief that some form of dark matter
allows to explain the observed discrepancy, another one that there is no
discrepancy.
> > Looking at the Einstein equations G_mn = T_mn it is clear that a
> > disagreement of the Einstein equations with nature may be described as
> > "dark matter"defined by
> >
> > T_mn^dark = G_mn - T_mn^obs
>
> It might be clear to you, but not so clear to many others :-/
>
> You'd have to be much more specific about what alleged "disagreement with
> Nature" you have in mind (are we still talking about galactic rotation
> curves? were we -ever- talking about galactic rotation curves), and what
> constraints if any you intend place on T_mn^dark (Segre type, for
> example?) before I could comment except in generalities.
I'm talking about a general principle. We can observe (via length and time
measurements) the metric and, therefore, G_mn. We can observe T_mn of
observable matter, but not of dark matter - by definition of "dark" matter.
That means, whatever we can observe, in principle, we can always define
T_mn^dark = G_mn - T_mn^obs
and, as a consequence, the EFE holds. Thus, the EFE in itself cannot be
falsified by observation. There cannot be a discrepancy between EFE and
data, as long as we do not restrict the type of dark matter.
Judging from your answer (where you refer to _constraints_ on T_mn^dark) you
seem to be aware of this. I only want to emphasize the point: A dark matter
explanation without nontrivial constraints on T_mn^dark can explain
everything.
And, as an additional point, let's note that the GR equation of motion for
the dark matter nabla T^dark = 0 is not a nontrivial constraint. Instead, it
is a consequence of nabla T^obs = 0 (which means that visible matter behaves
like predicted in the gravitational field) and the tautology nabla
(T^dark+T^obs) = nabla G = 0.
> Dark matter is currently a speculative concept, some would even say a
> dubious concept, but nonetheless most contemporary cosmologists seem to be
> disinclined to abandon gtr as our Gold Standard Theory of Gravitation.
> These are judgement calls which could be profoundly affected by new
> observations, or possibly even by new theoretical developments, although
> at the moment many would probably agree that some independent confirmation
> of the existence of dark matter (if it does exist), new tests of the
> alleged "Pioneer effect", etc., would be more helpful than yet another
> gravitation theory, classical or otherwise. Fair enough?
As formulated (as a judgement of most contemporary cosmologists) I see no
reason for disagreement.
But, as observed by Kuhn, theories will be abandoned only if there is a
replacement which is superior. If people refuse to look at "yet another
gravitation theory", GR will never be abandoned, independend of any
data.
Ilja
mark...@yahoo.com
Juan R. wrote:
> When one ask by the Newtonian limit of GR one heard in Baez page
>
> http://math.ucr.edu/home/baez/RelWWW/wrong.html
>
> that "The theorem stating that gtr does indeed go over to Newtonian
> gravitostatics in the very weak field, very slow motion limit is proven
> in detail in almost every gtr textbook."
>
> Then one read one of those textbooks and discover like NG is 'obtained'
> from GR, but one finds that derivation is NOT rigorous.
>
> Then one takes a more rigorous textbook (Wald) and one discovers that,
> effectively, NG is NOT derived from GR in the linear regime.
In effect, you're looking for the solution to
lim c->infinity (The Theory of GR)
which would be a curved spacetime theory of gravity for Newtonian
spacetime. That's Galilean General Relativity. Wheeler developed this
early on, I believe, in the 1960's and in more recent times Jadczyk (an
occasional poster here) has touched on it, as well.
> Of course, if one takes limit c --> infinite then tGT --> tNG and Phy
> transforms into an instantaneous field (still is not a potential) but
>
> the Schwartzild metric
>
> g00 = (1 - 2Phy/c^2) --> 1
>
> and
>
> gRR = -(1 - 2Phy/c^2)^-1 --> -1
Things get a little subtle in the limit since the spacetime underlying
Minkowski is of signature (+++-), whereas the spacetime underlying
Newtonian physics is of signature (+++0), which might be considered as
the "hyperbolic" and "parabolic" cases of a general family that
includes as its "elliptical" case the Euclidean 4-space (++++).
For the signature (+++0), the metric splits into 2 independent
constructs. In Lorentzian spacetime, you can equivalently write the
metric as (+++-) as a "space-like" metric; or as (---+) as a measure of
proper time. This is the equivalence that splits up as you go from
(+++-) to (+++0).
The Riemannian spacetime of General Relativity goes over into a
spacetime that is NOT Riemannian. The main task of Wheeler's
development was to show what kind of spacetime that underlying Galilean
General Relativity is.
Juan R.
> Juan R. wrote:
> > When one ask by the Newtonian limit of GR one heard in Baez page
> >
> > http://math.ucr.edu/home/baez/RelWWW/wrong.html
> >
> > that "The theorem stating that gtr does indeed go over to Newtonian
> > gravitostatics in the very weak field, very slow motion limit is proven
> > in detail in almost every gtr textbook."
> >
> > Then one read one of those textbooks and discover like NG is 'obtained'
> > from GR, but one finds that derivation is NOT rigorous.
> >
> > Then one takes a more rigorous textbook (Wald) and one discovers that,
> > effectively, NG is NOT derived from GR in the linear regime.
>
> In effect, you're looking for the solution to
> lim c->infinity (The Theory of GR)
> which would be a curved spacetime theory of gravity for Newtonian
> spacetime. That's Galilean General Relativity. Wheeler developed this
> early on, I believe, in the 1960's and in more recent times Jadczyk (an
> occasional poster here) has touched on it, as well.
Main points i noted continue to be unsolved in any recent research i
know:
1) Can the curved spacetime view be maintained in the Newtonian regime?
(+++-) directly derived from my above c--> infinite limit applied to
standard GR (Carlip claims that 'my' metric is wrong but that is
clearly incorrect since 'my' metric can be found in C. Moller
well-known manual on relativity) does not permit the interpretation of
gravity like curvature of spacetime because the spacetime is (+++-) as
in SR.
(+++0) in the c--> infinite limit re-geometrization of GR (NC theory)
does not permit the interpretation of gravity like curvature of
spacetime. In fact, one works with a flat derivative MORE a potential
and the 'gauge' is fixed /a posteriori/ via direct comparison with
Newtonian gravity because GR cannot unambiguously obtains the correct
answer (one obtains correct and wrong answers and the *correct* is
recently obtained via ad hoc complementary equations as that used by
specialist Chrystian).
2) How field theoretic quantities are transformed into Newtonian ones.
for example how any Phi(x,t) of GR is transformed into Phi(R(t)) in the
limit c--> infinite. All i know is that 'potential' U introduced ad hoc
in papers is /a posteriori/ indentified with Newtonian potential.
Really U(x, t) is replaced *by hand* by U(R(t)) in final steps of
papers 'proofs'.
3) how is the 'derivation' done without unphysical boundaries or ad hoc
equations do NOT derived from field equations of GR?
> > Of course, if one takes limit c --> infinite then tGT --> tNG and Phy
> > transforms into an instantaneous field (still is not a potential) but
> >
> > the Schwartzild metric
> >
> > g00 = (1 - 2Phy/c^2) --> 1
> >
> > and
> >
> > gRR = -(1 - 2Phy/c^2)^-1 --> -1
>
> Things get a little subtle in the limit since the spacetime underlying
> Minkowski is of signature (+++-), whereas the spacetime underlying
> Newtonian physics is of signature (+++0), which might be considered as
> the "hyperbolic" and "parabolic" cases of a general family that
> includes as its "elliptical" case the Euclidean 4-space (++++).
>
> For the signature (+++0), the metric splits into 2 independent
> constructs. In Lorentzian spacetime, you can equivalently write the
> metric as (+++-) as a "space-like" metric; or as (---+) as a measure of
> proper time. This is the equivalence that splits up as you go from
> (+++-) to (+++0).
>
> The Riemannian spacetime of General Relativity goes over into a
> spacetime that is NOT Riemannian. The main task of Wheeler's
> development was to show what kind of spacetime that underlying Galilean
> General Relativity is.
Yes, one has post-Newtonian expansions and post-Minkoski expansions of
GR, many differerent spacetimes, simetries, etc. But where are my basic
1), 2), and 3) solved?
Juan R.
> > Still my main motivation is if GR is incompatible with NG in the same
> > form that RQFT is incompatible with NRQM like claimed by Dirac.
>
> integrals; nothing like that is happening here; you just misunderstood
> what you read in Wald.
Dirac emphasized that there is two *disjoint* theories: nonrelativistic
QM and relativistic QFT. I am asking if that is also correct for
gravitation. At one hand, nonrelativistic Newtonian gravity at the
other Einstein(Hilbert) GR.
> > Then Wald explains that one may go beyond the linear regime,
>
> Where exactly does he say "beyond the linear regime"? I can't seem to
> find this in section 4.4a.
It clearly states that in 4.4a. In the same form that a first reading
you are unable to check that derivation in the linear regime is wrong
(as explicitely stated by Wald, in the linear regime a=0, and one does
not derive NG), you may also be unaware of others flaws of the
derivation that are discussed in specialized literature in Newtonian
limit.
> I think you might be thinking of relativistic momentum versus Newtonian
> momentum. If so, right, these -are- different, but they do agree -up to
> first order- in the velocity, as Einstein pointed out. That is, in the
> slow motion limit, they do agree:
>
> mv/sqrt(1-v^2) = mv + mv^3/3 + O(v^5)
>
> = mv + O(v^2)
This equation is incorrect you are using SR.
> > time in GR is not the time in NG.
>
> No, but for slowly moving test particles, up to first order the proper
> time counted off by the test particle does agree with the Newtonian time
> approximately:
>
> (Delta t)/sqrt(1-v^2) = (Delta t) + O(v^2)
This equation is incorrect you are using SR.
> > Phy in GR is a retarded field.
>
> I have always been careful to point out (as does Wald, on p. 76) that this
> approximation also assumes -slowly changing- fields. Or even static
> fields, if you are just analyzing the motion of test particle and checking
> that in weak static fields, slowly moving test particles to approximately
> obey Newtonian laws, as they should. That is what Wald is verifying.
That is incorrect. Even in the static limit a field U(x, t) is not
transformed into a potential U(R(t)).
> > Phy in NG is an instantaneous potential, The GR above formula contains
> > c, Newtonian formula does not contain c, etc.
>
> Wald uses geometric or relativistic units in which c=G=1 by fiat.
>
> > Of course, if one takes limit c --> infinite
>
> Ah. This might be the problem. Wald is thinking of the magnitude of the
> velocity of the test particles as being "small" wrt c=1; you want to let c
> tend to infinity. Either do it Wald's way or reinsert the c,G factors
> (see the appendix) to do it your way.
No one cannot do that. This is the resaon that research in Newtonian
limit does not follow textbooks way.
tes...@um.bot
> It is wise to consider different possible explanations. This includes,
> of course, dark matter. But, as long as we have not found this dark
> matter, the dark matter hypothesis is only an ad hoc explanation for an
> observed discrepancy with data.
So far I think I'd agree.
> And it is unwise to make claims like this:
>
>>>> GR does have flaws which are widely recognized, but discrepancies with
>>>> data is emphatically -not- one of them!
As you know, gtr appears to be consistent with many, many observations at
the solar system scale, but rotation curves would seem to suggest neither
gtr nor Newtonian gravitation can be even approximately accurate on
galactic scales, -if- you assume that what we can see is all there is.
While this is clearly a judgement call, right now most physicists seem to
think the lesser of two evils is to try to work with an "ad hoc"
hypothesis, of the form that what we see is not all there is. Don't
forget, completely "ad hoc" suggestions have sometimes been right on the
money, for example Planck's quantum hypothesis.
>> If so, I'd agree that at this point the "dark matter" concept is
>> -speculative-. I'd probably assess the chances that galactic rotation
>> curves will ultimately put down to a gross failure of gtr differently from
>> you, however.
>
> I'm not evaluating probabilities for the different solutions for the
> observed discrepancy.
You and I are making judgement calls. We happen to be calling it
different ways, but I think you are in fact trying to express here the
same idea which I am: neither of us really knows who made the right call
yet.
> One thing is the belief that some form of dark matter allows to explain
> the observed discrepancy, another one that there is no discrepancy.
Well, if you go that route, you have to explain why gtr works so well for
all those other predictions, in fact you even have to explain why
Newtonian gravity is not too bad on solar system scales and below, but
fails grossly at larger scales. I know you think you have an explanation,
but my judgement is that dark matter is less implausible.
> I'm talking about a general principle. We can observe (via length and
> time measurements) the metric and, therefore, G_mn.
Indeed, we can estimate the Riemann tensor directly from sufficiently
detailed observations of test particle motion.
> We can observe T_mn of observable matter, but not of dark matter - by
> definition of "dark" matter. That means, whatever we can observe, in
> principle, we can always define
>
> T_mn^dark = G_mn - T_mn^obs
>
> and, as a consequence, the EFE holds. Thus, the EFE in itself cannot be
> falsified by observation. There cannot be a discrepancy between EFE and
> data, as long as we do not restrict the type of dark matter.
I think you are saying that if we declare it OK to toss in "completely
arbitrary" ad hoc new "stress-energy" terms whenever we like, -any-
Lorentzian manifold could become a "solution of the EFE". I agree, and in
fact I have often stressed this very point. But I think you are
overlooking that fact that, while the proposed dark matter term may be ad
hoc, it is -not-, as I understand it, "completely arbitrary".
> But, as observed by Kuhn, theories will be abandoned only if there is a
> replacement which is superior. If people refuse to look at "yet another
> gravitation theory", GR will never be abandoned, independend of any
> data.
Not sure what you are saying, since I see preprints almost every day in
which physicists are looking at "yet another gravitation theory".
Neither gtr nor Newtonian gravity are likely to ever be -abandoned-
since they are clearly useful where they are sufficiently accurate. But
Newtonian gravity has already been -dethroned- as our gold standard
theory of gravitation (since not relativistic) and everyone expects gtr
will eventually be dethroned in turn (since not quantum).
I was saying that I expect that taking this step will be avoided until
there is clear evidence that -gtr- specifically is failing. As in a
clean -test- of a specific prediction which appears to be have
essentially -no other explanation- than a failure of gtr. Taking the
plunge will be much easier, of course, if we have a workable alternative
theory already at hand which explains everything which gtr does, but
doesn't fail this hypothetical future test!
I would add that I happen to doubt that gtr will ever be dethroned by
another -classical- field theory, but agree that this -might- happen.
OK, I don't really want to continue this conversation because apart from
a few judgement calls I think our positions are almost identical. Since
I think we agree that either of us could be wrong about these judgement
calls, I don't see that we have anything to discuss, in the absence of
startling new data.
Ilja Schmelzer
> On Sat, 1 Oct 2005, Ilja Schmelzer wrote:
> > One thing is the belief that some form of dark matter allows to explain
> > the observed discrepancy, another one that there is no discrepancy.
>
> Well, if you go that route, you have to explain why gtr works so well for
> all those other predictions, in fact you even have to explain why
> Newtonian gravity is not too bad on solar system scales and below, but
> fails grossly at larger scales.
I don't understand. I argue against claims of type
>>>>> GR does have flaws which are widely recognized, but discrepancies with
>>>>> data is emphatically -not- one of them!
IMHO it would be correct to say that we have discrepancies between GR and
data on the galactic scale, with dark matter/energy as a possible ad-hoc
explanations. That's all.
> I know you think you have an explanation,
> but my judgement is that dark matter is less implausible.
Hm, the purpose of my theory of gravity (gr-qc/0205035) was not to solve
the dark matter/energy problems. That it gives some terms which allow to
solve some part of the problem I have observed only later. The purpose
of the terms was to obtain the harmonic condition as an Euler-Lagrange
equation. And the original motivation for introducing the harmonic
condition as an additional equation was a quantum gravity thought
experiment (considered in gr-qc/0001101).
Given the derivation of the Lagrangian starting with simple "ether
axioms", there is certainly nothing ad-hoc in my additional terms.
> But I think you are
> overlooking that fact that, while the proposed dark matter term may be ad
> hoc, it is -not-, as I understand it, "completely arbitrary".
I like to emphasize the point that even a completely arbitrary dark
matter term looks much less arbitrary because nabla_m T_mn^dark holds
tautologically. That the dark energy has to violate the strong energy
condition IMHO makes dark energy highly arbitrary. OTOH, I agree, some
other energy conditions may be, possibly, preserved.
> > But, as observed by Kuhn, theories will be abandoned only if there is a
> > replacement which is superior. If people refuse to look at "yet another
> > gravitation theory", GR will never be abandoned, independend of any
> > data.
> Not sure what you are saying, since I see preprints almost every day in
> which physicists are looking at "yet another gravitation theory".
Maybe there is some interest in alternative theories of gravity
somewhere. I have not seen any interest in my theory of gravity. Once
you see such preprints every day, maybe you want to take a look at
gr-qc/0205035?
> Neither gtr nor Newtonian gravity are likely to ever be -abandoned-
> since they are clearly useful where they are sufficiently accurate. But
> Newtonian gravity has already been -dethroned- as our gold standard
> theory of gravitation (since not relativistic) and everyone expects gtr
> will eventually be dethroned in turn (since not quantum).
Agreement. My theory has a GR limit, thus, I don't suggest a complete
abandonement of GR.
> I was saying that I expect that taking this step will be avoided until
> there is clear evidence that -gtr- specifically is failing. As in a
> clean -test- of a specific prediction which appears to be have
> essentially -no other explanation- than a failure of gtr.
Which "clean test" do you have in mind here? Given that
> ... if we declare it OK to toss in "completely
> arbitrary" ad hoc new "stress-energy" terms whenever we like, -any-
> Lorentzian manifold could become a "solution of the EFE"...
no test will be completely clean. It always includes a judgement call
that the ad hoc stress-energy term is in some sense "too arbitrary".
> Taking the
> plunge will be much easier, of course, if we have a workable alternative
> theory already at hand which explains everything which gtr does, but
> doesn't fail this hypothetical future test!
>
> I would add that I happen to doubt that gtr will ever be dethroned by
> another -classical- field theory, but agree that this -might- happen.
This was my starting point too. But the classical limit of quantum
gravity may have some minor differences. Like, for example, a hidden
preferred background which becomes observable only in the quantum
domain.
Ilja
Juan R.
> Second, your g_00 is wrong. There's an extra c^2:
> g_00 = c^2(1 - 2GM/rc^2)
This is not true, 'my' metric g_00 = (1 - 2GM/rc^2) is correct. Not
only it is a *standard* metric choosen by relativist Moller in his
well-known textbook on relativity. It is also a standard metric
*RECOMMENDED* by the /XXIVth International Astronomical Union General
Assembly/ see
-Resolution B1.3 Definition of Barycentric Celestial Reference System
and Geocentric Celestial Reference System.
- Resolution B1.5 Extended relativistic framework for time
transformations and realisation of coordinate times in the solar
system.
http://danof.obspm.fr/IAU_resolutions/Resol-UAI.htm
In both cases, the STANDARD g_00 component of the metric does not
contain the c^2 term claimed by Carlip, and at second order in c, the
g_00 coincide exactly with metric i chosed. Moreover 'my' STANDARD
metric g_00 = (1 - 2GM/rc^2) is natural because:
i) metric is a geometrical quantity with no units. Carlip g_00 has
units of c^2, whereas his g_RR has no units. 'My' full metric g has no
units.
ii) metric i choose is the natural choice for a spacetime (ct, x) that
any textbook on SR takes. Carlip uses the strange 'spacetime' (t, x),
where the 4D manifold is formed by a time dimension more a 3 spatial
dimension. The reason for choosing x^0 = ct and working with (ct, x) is
that one has four-space with spatial dimensions x^0, x^1, x^2, and x^3
and one can easily appply 4D-geometrical formulas.
iii) metric i choose is compatible with all experimental data and is a
standard.
Carlip *now* agrees that taking metric without the c^2 term one obtains
Newtonian law of motion
*************************************************************
From: carlip-nos...@physics.ucdavis.edu
Newsgroups: sci.physics.relativity
Subject: Re: Does the 'Curvature of Spacetime' cause gravity?
Date: Fri, 7 Oct 2005 18:51:33 +0000 (UTC)
Organization: University of California, Davis
Lines: 149
Sender: Steve Carlip <car...@dirac.physics.ucdavis.edu>
Message-ID: <di6g3l$dnm$2...@skeeter.ucdavis.edu>
> carlip-nos...@physics.ucdavis.edu wrote:
>> If you want to keep c as an adjustable parameter, the standard form
>> of the Minkowski metric is
>> ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2
>> You can, of course, define a coordinate x^0=ct, and hide the c dependence.
>> If you do so, though, derivatives d/dx^0 have to be rescaled as you vary c,
>> and go to zero as c goes to to infinity. If you then look at the equations
>> of motion of GR in that limit without paying attention to what you're
>> doing,you get 0=0.
> I do not understand. Can I do x^0 = ct and next take care with limits
> or cannot?
You can, if you're careful.
>> But this, of course, is a foolish thing to do. At any finite value of c,
>> the equations are of the form
>> (1/c^2)(something independent of c)=(1/c^2)(something else independent
>> of c)
>> and to get a limit, you should first multiply both sides by c^2. If you do
>> so, you will recover the standard Newtonian equations of motion.
> What is the difference in the geodesic equation taking x^0 = ct or x^0 = t?
> Is not the c component cancelled in left and right terms?
Let's leave x^0 free for now -- it could be t or ct. In the
approximation
we need -- velocities all much less than c -- the geodesic equation is
d^2x^i/ds^2 = -\Gamma^i_{00}(dx^0/ds)^2
Furthermore, in this approximation, dx^0/ds is constant (either 1 or
c), so
d^2x^i/(dx^0)^2 = -\Gamma^i_{00}
Furthermore, in this approximation
\Gamma^i_{00} = (1/2)\partial_i g_{00}
so
d^2x^i/(dx^0)^2 = -(1/2)\partial_i g_{00}
Now, with the conventional choice x^0=t, g_{00}=c^2(1 - 2Gm/rc^2)=c^2 -
2Gm/r.
The right-hand side is the standard acceleration d^2x^i/dt^2; the
left-hand
side has a derivative of c^2, which is zero independent of the value of
c,
and in the next term, which is finite as c->infinity, a derivative of
the
Newtonian potential.
If, as you prefer, you set x^0=ct, then g_{00}=1 - 2Gm/rc^2, and the
derivative on the right-hand side goes as 1/c^2. But the derivative on
the left-hand side is no longer a derivative with respect to t, but
rather
a derivative with respect to ct. So one obtains
(1/c^2)d^2x^i/dt^2 = -(1/c^2)\partial_i U
It's true that both sides go to zero as c->infinity, but for a trivial
reason -- they have a common factor of 1/c^2. If you do the sensible
thing and multiply both sides by c^2 before taking the limit, you get
the usual Newtonian limit.
*************************************************************
iv) metric i choose reduces to standard Minkoski metric (1 -1 -1 -1)
when gravity vanishes. Carlip nonstandard metric reduces to (c^2 -1 -1
-1), which is *not* a standard in special relativity, electromagnetism
or even in relativistic quantum field theory. See for example Weinberg
manual on QFT volume 1. Weinberg chooses trace 2 and the metric that he
uses in particle physics is (-1 1 1 1). Other authors prefer (1 -1 -1
-1).
Moreover, Carlip chossing (c^2 -1 -1 -1) or spacetime (t, x) instead of
(ct, x) is wrong in the nonrelativistic limit.
In the limit c--> infinite, one CANNOT obtain a spacetime (t, x) as
Carlip claims, because t is NOT a dimension in Newtonian limit, t is a
parameter. Taking 'my' STANDARD convention (1 -1 -1 -1) or spacetime
(ct, x), in the limit c--> infinite x^0 collapses like a dimension, and
the only physical dimensions are x, y, and z. One can prove that in the
limit c --> infinite dtau = ds/c transforms into a parameter of the
trajectory and this parameter is Newtonian theory one.
One recovers, a 3D-space (x ,y , z) and an evolution parameter:
Galilean time. One *cannot* obtain a spacetime (t, x) in the Newtonian
limit. This is the reason that Carlip cannot obtain Newtonian
potentials U(R(t)) and only can obtain 'fields' U(x, t).
See PRE 1996 53(5), 5373 for some detailed discussion of why
field-theoretic potentials A(x t) cannot explain data explained by
Newtonian-like potentials A(R(t)).
In PRA 2002 65, 0341041 one can see that equation (1) is Newtonian
equation of motion. The potential explicitely written is U(x(t)), what
is
just i wrote, with *implicit* time dependence, because time is not a
dimension, there is nothing like U = U(x, t) in the Newtonian limit.
In a recent Solvay conference, i find exactly the same potential for
quantum mechanics. (2.1) of Adv Chem Phys 1997, 99, 1. The potential
has dependence V(|q_i -q_j|) that is EXACTLY i wrote U(R(t)) because R
= |q_i -q_j|.
In Phys Lett A 1988 128(3,4) 123, authors (one of them one of most
respected particle physicists) study a relativistic generalization of
Newtonian-like potentials. Note the explicit dependence V = V(rho) in
equation (3). doing c--> infinite (they explicitely do in other
published paper) one EXACTLY obtains V = V(R), with 'MY' implicit time
dependence R = R(t). They do not obtain the incorrect (x, t) dependence
that Carlip argue. In posterior works authors generalized this to
gravity and
'proved' that classical gravity (GR) is an eikonal approximation to
their theory.
In Prigogine, I. Non-equilibrium statistical mechanics 1962, John Wiley
and sons, we can see that for classical systems the dependence is again
V(R(t)) equation 1 of chapter 4.
That spacetime quantities are wrong in the Newtonian limit is clearly
seen from the Poisson equation for stationary states -See for example
Handbook of molecular physics and quantum chemistry; John Wiley & Sons
Ltd: West Sussex, 2003; Volume 2, Chapter 21, equation 25-
Grad^2 U = 4 pi delta(x-y)
solving it the non-relativistic potential is U = U(R(t)) NOT U = U(x
t) which is also the criticism of authors of PRE 1996 53(5), 5373.
Even Goldstein, at a very elementary level, discusses the two-body
problem in his celebrated textbook on mechanics. In the chapter 3 of
Spanish version, Goldstein discusses the Lagrangian and explicitely
does U (energy) function of R, dR/dt, and higgher order derivatives of
**** R **** /in the general case/ WHITOUT TIME DEPENDENCE. Of course,
in the Newtonian limit dR/dt and higgher order derivatives vanish and
Newtonian potential is just V = V(R)
As discussed in this thread, *derivation* of Newtonian limit from GR is
still unproved.
Arnold Neumaier
> Juan R. wrote:
>
>>In one of his last works Mathematical Foundations of
>>Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
>>
>>Most physicists are very satisfied with this situation [refer to
>>divergences of QFT]. They argue that if one has rules for doing
>>calculations and the results agree with observation, that is all that
>>one requires. But it is not all that one requires. One requires a
>>single comprehensive theory applying to all physical phenomena. Not one
>>theory for dealing with non-relativistic effects and a separate
>>disjoint theory for dealing with certain relativistic effects.
>>Furthermore, the theory has to be based on sound mathematics, in which
>>one neglects only quantities that are small. One is not allowed to
>>neglect infinitely large quantities [...]
>
> However great a theorist was Dirac, he was wrong about this assessment
> of renormalization. Perturbative renormalization is based on sound
> mathematics *and* is capable of produce correct veriable (and verified)
> predictions.
True and not true. It produces verifiable predictions if you restrict
attention to the first few terms of a (most probably divergent)
asymptotic series, but it has no way to make sense of the whole
series. This is what Dirac found deficient in the foundations.
An asymptotic series is a series such as
f(x) = sum_{k=0:inf} k! x^k
with radius of convergence zero. For small enough x, the first few
terms give seemingly good approximations, but if one includes for
any fixed nonzero x enough terms, the series diverges. Thus, as Dirac
asserts, one neglects arbitrarily large terms to get the approximations
which work so well in QED.
There are infinitely many different ways to assign to an
asymptotic series a function with this series as Taylor expansion.
The problem is to have a way to choose the right one. Borel summation
is often taken as default, but seems to be no cure for QFT in view
of the so-called renormalon problem.
For more details, see the entry ''Summing divergent series'' in my
theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt
At present, there is no sound mathematical foundation of relativistic
quantum field theory. Who finds one will be awarded one of the
1 Million Dollar Clay Millenium prizes...
Arnold Neumaier
Eugene Stefanovich
news:43785F95...@univie.ac.at...
> > However great a theorist was Dirac, he was wrong about this assessment
> > of renormalization. Perturbative renormalization is based on sound
> > mathematics *and* is capable of produce correct veriable (and verified)
> > predictions.
>
> True and not true. It produces verifiable predictions if you restrict
> attention to the first few terms of a (most probably divergent)
> asymptotic series, but it has no way to make sense of the whole
> series. This is what Dirac found deficient in the foundations.
>
> An asymptotic series is a series such as
> f(x) = sum_{k=0:inf} k! x^k
> with radius of convergence zero. For small enough x, the first few
> terms give seemingly good approximations, but if one includes for
> any fixed nonzero x enough terms, the series diverges. Thus, as Dirac
> asserts, one neglects arbitrarily large terms to get the approximations
> which work so well in QED.
I think that Dirac was frustrated not by the zero radius of convergence
of the perturbation series for the renormalized S-matrix. I think his
frustration was directed at the basic renormalization algorithm that is
at work in each given perturbation order. This algorithm has two
flavors. One is to simply throw away some infinite terms on the pretext
that they are "physically unacceptable", e.g., they violate the gauge
invariance or they give infinite contributions to the electron mass or
charge. Another, slightly more agreeable approach, is to add certain
infinite counterterms to the interaction Hamiltonian, so that
contributions from these counterterms exactly cancel other infinite
contributions to the S-matrix. I think Dirac disliked both these
approaches. I agree with him completely.
There is another quote from a leading theoretician that seems to confirm
my interpretation:
"Thus, present quantum electrodynamics is one of the strangest
achievements of the human mind. No theory has been confirmed by
experiment to higher precision; and no theory has been plagued by
greater mathematical difficulties which have withstood repeated attempts
at their elimination. There can be no doubt that the present agreement
with experiments is not fortuitous. Nevertheless, the renormalization
procedure can only be regarded as a temporary crutch which holds up the
present framework. It should be noted that, even if the renormalization
constants were not infinite, the theory would still be unsatisfactory,
as long as the unphysical concept of "bare particle" plays a dominant
role. If one considers quantum electrodynamics as a phenomenological
theory with respect to the mass and charge of the interacting particles,
and if one consequently condones the necessity of infinite mass and
charge renormalizations, one is tempted to consider quantum
electrodynamics as a pretty satisfactory theory..." F. Rohrlich in
http://www.philsoc.org/1962Spring/1526transcript.html
It is clear that Rohrlich was worrying about the difference between bare
and physical particles, rather than about the convergence of the
perturbation series. Eugene.
Norm Dresner
>
>> Juan R. wrote:
>>
>>>In one of his last works Mathematical Foundations of
>>>Quantum Theory. (Academic Press, Inc., 1978) Dirac claimed:
>>>
>>>Most physicists are very satisfied with this situation [refer to
>>>divergences of QFT]. They argue that if one has rules for doing
>>>calculations and the results agree with observation, that is all that
>>>one requires. But it is not all that one requires. One requires a
>>>single comprehensive theory applying to all physical phenomena. Not one
>>>theory for dealing with non-relativistic effects and a separate
>>>disjoint theory for dealing with certain relativistic effects.
>>>Furthermore, the theory has to be based on sound mathematics, in which
>>>one neglects only quantities that are small. One is not allowed to
>>>neglect infinitely large quantities [...]
>
> attention to the first few terms of a (most probably divergent) asymptotic
> series, but it has no way to make sense of the whole
> series. This is what Dirac found deficient in the foundations.
>
There's a [IMHO] fascinating paper in the (most) recent Sept 2005 SIAM
Reviews on Divergent Asymptotic Series including an elementary discussion on
(some of) the reasons for divergence and a discussion of how useful they can
still be as a valid approximation technique.
While I have no mathematical justification for ascribing this type of
behavior to QFT perturbation results, it still is a good read on a very
useful mathematical technique -- and if it is applicable, it's very
revealing.
Norm
Arnold Neumaier
> It is clear that Rohrlich was worrying about the difference between bare
> and physical particles, rather than about the convergence of the
> perturbation series. Eugene.
The concept of bare particles (and infinite masses and coupling
constants) does not figure at all in modern settings of renormalization
based on the renormalization group. (See, e.g., Salmhofer's book).
Thus these worries are no longer relevant.
The only persisting worries are those about the meaning of the
scattering matrix which so far exists only as an asymptotic series
rather than as a mathematically well-defined operator.
Arnold Neumaier
Eugene Stefanovich
Arnold Neumaier wrote:
> The concept of bare particles (and infinite masses and coupling
> constants) does not figure at all in modern settings of renormalization
> based on the renormalization group. (See, e.g., Salmhofer's book).
>
> Thus these worries are no longer relevant.
We discussed this a lot, but I am still puzzled by your words.
Let me take for definiteness the expression (11.1.1) from Weinberg's
vol. 1. This is the Lagrangian (the Hamiltonian H is
qualitatively similar) in the renormalized QED.
Could you please tell me which of these three statements are not
true?
1) The Hamiltonian (11.1.1) is expressed in terms of bare particles
that have very distant relationship to the electrons and photons
we observe in real life.
2) The Hamiltonian (11.1.1) contains infinite counterterms
(OK, to be more precise I'll say that the matrix elements of
the counterterm operators on bare states tend to infinity as
the momentum cutoff tends to infinity).
3) Quantum theory with a deficient Hamiltonian (as in 1) and 2))
cannot be considered complete. For example, in this theory,
the spectrum of stationary states cannot be found by direct
diagonalization of the Hamiltonian. The time evolution of
non-stationary states cannot be described by the time evolution
operator exp(iHt). The bright side is that Feynman-Dyson
perturbation series for the S-matrix is finite
(all infinities cancel out in each perturbation order)
and accurate.
If the above statements are true, then does Salmhofer's book address
these problems? If yes, then I should definitely get a copy.
Thank you.
Eugene.
Igor Khavkine
> Could you please tell me which of these three statements are not
> true?
> 3) Quantum theory with a deficient Hamiltonian (as in 1) and 2))
> cannot be considered complete. For example, in this theory,
> the spectrum of stationary states cannot be found by direct
> diagonalization of the Hamiltonian. The time evolution of
> non-stationary states cannot be described by the time evolution
> operator exp(iHt).
This one.
> The bright side is that Feynman-Dyson
> perturbation series for the S-matrix is finite
> (all infinities cancel out in each perturbation order)
> and accurate.
The same perturbation theory gives a series expansion, which is also
finite at each order, for all other quantities of interest. These
include the energy levels, states, dynamical quantities, etc. I know
that you will now ask for references. However, references were given
given many times along with explanations for how to use the information
provided in them. You'll just have to look for them in this group's
archives.
Igor
Arnold Neumaier
>
> Arnold Neumaier wrote:
>
>> The concept of bare particles (and infinite masses and coupling
>> constants) does not figure at all in modern settings of renormalization
>> based on the renormalization group. (See, e.g., Salmhofer's book).
>>
>> Thus these worries are no longer relevant.
>
> We discussed this a lot, but I am still puzzled by your words.
> Let me take for definiteness the expression (11.1.1) from Weinberg's
> vol. 1.
If you want to have precise statements, you must look into the
literature on mathematical physics, not into Weinberg who proceeds
with lots of handwaving and uses the traditional fuzzy motivations to
which physicists are accustomed. I am tired of repeating old things
to deaf ears.
> If the above statements are true, then does Salmhofer's book address
> these problems? If yes, then I should definitely get a copy.
You should definitely get a copy. He does renormalization theory without
ever mentioning bare particles or infinities in the formal exposition.
Everything he does is mathematically sound. He constructs the expansion
of the S-matrix with the same rigor as one constructs the expansion of
the exponential function in undergraduate math.
Arnold Neumaier