Maxwell equation model of gravitation
William D. Walker
set of Maxwell type equations, predicting that a moving mass produces
electric and magnetic field components of gravity. In electromagnetic
theory, Maxwell Equations reduces to Coulomb's law for quasistatic
charges, whereas in the gravitational case it reduces to Newton's equation.
For more details refer to following paper
http://folk.ntnu.no/williaw/Forward.pdf
R. Forward, `General relativity for the experimentalist',
Proceedings of the IRE, 49, May (1961)
How accurate is the model compared to General Relativity, and have the
differences been detected experimentally yet?
robert bristow-johnson
> It is not commonly known that gravity can be modeled using an equivalent
> set of Maxwell type equations, predicting that a moving mass produces
> electric and magnetic field components of gravity. In electromagnetic
> theory, Maxwell Equations reduces to Coulomb's law for quasistatic
> charges, whereas in the gravitational case it reduces to Newton's equation.
i've always thunked that they called this "GravitoElectroMagnetism" or
"GEM". there's a Wikipedia article about it.
to me, it's a sorta natural expectation if both static gravity and
static electric fields are inverse square and share identical form of
equations (Newton vs. Coulomb) and if the effect of both travel at some
finite speed, c, the 4 Maxwell's equations can plausibly be used,
subtituting mass (or mass density) for charge (or charge density) and
-G for 1/(4*pi*epsilon_0) (after eliminating mu_0 with
1/(c^2*espilon_0)), then you have equations that will boil down to the
known inverse square static effect and speed of c. real physicists (of
which i am not) like Mashoon, et al. will derive it from the Einstein
equation with an assumption of reasonably flat spacetime.
> For more details refer to following paper
>
> http://folk.ntnu.no/williaw/Forward.pdf
>
> R. Forward, `General relativity for the experimentalist',
> Proceedings of the IRE, 49, May (1961)
wow, i wonder if Mashoon and company know of this old paper and cite
it?
> How accurate is the model compared to General Relativity, and have the
> differences been detected experimentally yet?
if GR is true, i don't think there would be measurable differences
detectable unless you in some place with nastily curved space. if GR is
true (which i presume it is), my understanding from the physicists is
that GR and GEM will get you very similar results in flat spacetime.
r b-j
(not an expert, but an interested amateur.)
Timo Nieminen
Well, Forward's paper basically takes a weak-field, low-velocity limit of
GR. As it retains the idea of curvature, I wouldn't call it a real
Maxwell-type model like Heaviside's 1894 version (Appendix B in vol 1 of
his Electromagnetic Theory).
Heaviside's version would predict gravitomagnetic effects and
gravitational waves (actually, he assumed the existence of gravitational
waves in the ether in his derivation), but not deflection of
electromagnetic waves, gravitational redshift, or any effect on time.
Forward's version appears to give all of this, reasonably correct for weak
fields.
There's a paper by Kirk McDonald discussing this, with a number of
references that might be of interest:
http://www.physics.princeton.edu/~mcdonald/examples/vectorgravity.pdf
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
dougsw...@gmail.com
Gravitomagnetism looks like field equations, and field equations
only. To be considered a field _theory_, one would need a Lagrange
density, and how to vary that so as to generate the field equations,
and solutions to the field equations that are consistent with
experiments to date. Has any of these sources proposed a Lagrange
density to generate the field equations? My bet would be no, almost
based on the dates alone (McDonald referenced Heaviside the most). I
know Feynman was clear that a symmetric rank 2 tensor was needed for a
universally attractive force using the graviton. A symmetric tenor
does not have a clone of the EM B inside it, and will not have the no
monopoles vector identity. Anyone know when that particle physics
based observation about the symmetric rank 2 tensor was first made?
It might have been early on, physicists are smart.
Thanks,
doug
quaternions.com
Rich L.
beginning
to appreciate some of the important differences between gravity and
electromagnetism.
For one thing, dispite the ability to model the inverse square force
using an analog of
Maxwell's equations, these equations do not model the regenerative
nature of GR. That
is, the gravitational "force" is only aproximately inverse square in
the weak field limit.
E&M is inverse square at even very high fields. Another important
difference is time
dilation. The argument for time dilation that Einstien used does not
work for electric
fields, and the quality of the predictions of quantum mechanics argues
that there really isn't
any time dilation with EM fields. Finally, recently I've come to
realize that the gravitational
field of GR does not have a potential in the same sense as EM. It is
a strictly geometric
theory and if gravity is properly treated in QM then it has to be done
via the non-Euclidean
geometry rather than a potential. (The time/energy part of the wave
equation would likewise
incorporate time dilation as part of the 4 dimensional geometry.)
It was interesting to think about a Maxwell version of gravity, but it
is clear to me now that
it would only work in the weak field limit, and even then would miss
some easy to observe
GR effects like the gravitational red shift.
Rich L.
Tim Nerub
>Anyone know when that particle physics based observation about the
>symmetric rank 2 tensor was first made? It might have been early on,
>physicists are smart.
Yes, it was basically Maxwell who first pointed out the fundamental
problem with trying to explain gravity with a theory that looks like
electromagnetism. He didn't phrase it in terms of tensor ranks,
obviously, but his comments amount to the same thing, as people like
Pais have noted. Try googling on "Why Maxwell Couldn't Explain
Gravity".
Uncle Al
EM is virtual spin-1 vector bosons and gravitation might be virtual
spin-2 tensor bosons. There is no reason to believe gravitation can
be quantized at all - classical theories appear errorless to the
limits of observation, quantum gravitations are hopeless disasters.
Not a good start.
Teleparallel gravitation has spacetime torsion that looks like Lorentz
force in EM rather than spacetime curvature as in GR. There is only
one observed reality. The answers from different correct theories
must be identical in all cases.
They aren't.
Spacetime curvature and torsion diverge on matters of angular
momentum. GR gives wrong answers but the largest divergence is very
small, no larger than 10^(-12) relative and probably 10X smaller than
that for starters. This and its detection have been discussed in
other posts here and in the last URL below.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
Peter
wrote:
Dear William Walker,
Thank you very much for posting this question and providing the URL to
this paper!
I'm interested in gravito-electromagnetism since I have heard of and
red Sciama's first paper about using it for explaining Mach's
principle.
Is it correct that the gravito-electromagnetic equations are, (i),
isomorphic with the microscopic Maxwell equations and, (ii), a better
approximation to GR than Newton's theory of gravity?
My point is that the microscopic Maxwell equations have got its form
not by chance, but by the necessity that classical charged bodies move
according to the laws of classical mechanics. The same applies to
gravito-electromagnetism.
Thus, while Newton's law for the force of gravity needs this force to
be conservative, Lorentz's magnetic force needs Lipschitz's answer to
Helmholtz question about a force not to change the kinetic energy of a
body.
In other words, the microscopic Maxwell equations can be derived by
purely mechanical arguments (as GR would require); can the gravito-
electromagnetic equations, too?
What are, in your view, the implications for the unity of physics?
Thank you very much in advance,
Peter Enders
dougsw...@gmail.com
> Try googling on "WhyMaxwellCouldn't Explain Gravity".
That turned up this URL:
http://www.mathpages.com/home/kmath613/kmath613.htm
I have been impressed with Maxwell when I happened to flip through his
Treatise, and felt like much of Jackson's "Classical Electrodynamics"
was covered there, even though the units and nomenclature was
different. One of the problems assigned in the BU graduate level
class on EM was worked out in the treatise.
Maxwell also thought carefully about gravity. Newton's static gravity
law can be written in a potential form as:
rho = Del^2 phi
The general anwser to this equation is f(r)= k/R + C. To make things
attract, toss in a -k. That creates a problem though: the energy
density of the field will be negative unless C is HUGE. I know I
didn't worry about this, but Maxwell did, and I respect him more than
myself! The Newtonian limit of GR involves the g_tt term of the
Schwarzschild solution, which I'll write in a suggestive way:
g_tt = 1/mc^2 (m c^2 - 2 GMm/R)
G is a small number, c^2 is HUGE, so the constant does its job. The
URL will do the explanation much better than I will, but it has
changed my way of thinking. I used to think of a potential in the
form M/R (C=0) as being a solution for gravity. Unfortunately it is
not consistent with a field theory approach to gravity. So now I will
always look for a big fat constant C so the answer is logically
consistent.
doug
quaternions.com
carlip...@physics.ucdavis.edu
[...]
> Maxwell also thought carefully about gravity. Newton's static gravity
> law can be written in a potential form as:
> rho = Del^2 phi
> The general anwser to this equation is f(r)= k/R + C. To make things
> attract, toss in a -k. That creates a problem though: the energy
> density of the field will be negative unless C is HUGE.
That's part of the problem. The other part is that if you flip signs
from E&M to make "like charges" attract -- as you must for gravity --
you find that gravitational *radiation* would carry negative energy
as well. This can't be fixed by adding a constant, and it is in bad
conflict with binary pulsar observations, in which we see gravitational
radiation carrying off positive energy.
You might want to look at Deser, gr-qc/0411026.
Steve Carlip
dougsw...@gmail.com
> You might want to look at Deser, gr-qc/0411026.
Thanks for the lead. For those who didn't, I'll quote a fun line from
the introduction:
[QUOTE=Deser] One of the less heralded triumphs of special relativity
(SR) is that it determines the signs of the interactions between
sources according to the spins of their mediating fields. In
contrast, these signs are arbitrary in non-relativistic physics: the
observed Coulomb/Newtonian repulsion/attraction must be put in by
hand.
[/QUOTE]
I am confused by the inference you draw. We agree on a solution to
part of the problem:
>> Newton's static gravity
>> law can be written in a potential form as:
>> rho = Del^2 phi
>> The general answer to this equation is f(r)= k/R + C. To make things
>> attract, toss in a -k. That creates a problem though: the energy
>> density of the field will be negative unless C is HUGE.
>
>That's part of the problem.
I don't think this implies anything about the spin of the particles
mediating the forces. The equation is classical, the spin of mediate
force particles is quantum and relativistic. The spin of the
particles comes out of looking at the field strength tensor, which is
not in the above expression.
We agree that if the field strength tensor for gravity was rank 2 and
antisymmetric just like EM, that would not work for a variety of
reasons, binary pulsars being one example. In Misner, Thorne, and
Wheeler, exercise 7.2, they look at exactly that case, and it does not
work. One of the cool/worrisome things about the relationship between
a Lagrange density and field equations is that one cannot assume they
are unique: a different Lagrangian can generate the same field
equations.
Deser's paper makes clear that the particles that mediate the force
must be spin 2. That forces the field strength tensor to be symmetric
and rank 2. There is a path from a symmetric field strength tensor to
rho = Del^2 phi, and such a path has a chance of being consistent with
binary pulsar data.
doug
Autymn D. C.
> EM is virtual spin-1 vector bosons and gravitation might be virtual
> spin-2 tensor bosons. There is no reason to believe gravitation can
> be quantized at all - classical theories appear errorless to the
> limits of observation, quantum gravitations are hopeless disasters.
> Not a good start.
fool!: http://groups.google.com/groups?q=Autymn+%22neutron+traps%22
Autymn D. C.
> That's part of the problem. The other part is that if you flip signs
> from E&M to make "like charges" attract -- as you must for gravity --
> you find that gravitational *radiation* would carry negative energy
> as well. This can't be fixed by adding a constant, and it is in bad
> conflict with binary pulsar observations, in which we see
gravitational
> radiation carrying off positive energy.
Scientists are the problem:
http://groups.google.com/groups?q=Autymn+vectors+%22dot+product%22&scoring=d
http://egroups.com/group/free_energy/msearch?query=Autymn+dot+product
So are the cretins-censors at sci.physics.research.
-Aut
Bill Miller
If you have not done so, you might want to look at Oleg Jefimenko's
"Causality, Electromagnetic Induction and Gravitation." He considers a
number of the issues you raise. Interesting reading.
Bill
"swee...@alum.mit.edu" <dougsw...@gmail.com> wrote in message
news:1171237315.3...@a34g2000cwb.googlegroups.com...
dougsw...@gmail.com
I went ahead and ordered Oleg Jefimenko's "Causality, Electromagnetic
Induction and Gravitation." My reaction is less positive. I don't
believe the book is worth the time for readers to these newsgroups.
Folks who write well typeset books with lots of partial differential
equations can be intimidating! Still, there is a major omission, one
I learned from a well-know MIT professor. I chatted with him one day
about my own unified field equation, a one liner that fits on t-
shirt. He said that although I had a field equation, I did not have a
field theory. A modern field theory must have the following:
1. A Lagrange density (or Hamiltonian, or action, there are roads
connecting the three).
2. Field equations derived from the Lagrangian by varying the
action with respect to the potential
3. Force equations derived from the Lagrangian by varying the
action with respect to the velocity.
4. Solutions to the field equations that were physically relevant
5. Solutions to the field equations that were measurably different
from current tests
6. The energy/stress tensor derived from the Lagrangian
7. The field equations must be quantizeable.
It was quite the list! It was also a scary list because I had never
worked with a Lagrangian before. Now that I understand it, the tool
has a simple idea at the core. You want to have a scalar expression
that contains all the ways that a system can exchange energy per unit
volume. Integrate that over volume and an arbitrary amount of time.
That would give the energy times an arbitrary amount of time, not so
interesting. Using the calculus of variations, one looks for how one
could varying the Lagrangian, yet not change the outcome of the
integral, no matter how long one waited. That is a constant for the
equation of motion, and is related to a conserved quantity.
Jefimenko does not have a Lagrangian, a Hamiltonian, or an action, so
his work is not worth studying. I know this is a harsh standard, but
least it can be applied evenly. Einstein's 1915 paper on general
relativity would not have passed this test, but fortunately Hilbert
published the action within weeks. Earlier articles cited in this
thread also don't have Lagrangians: Robert Forward's "General
Relativity for the Experimentalist" or Kirk McDonald's "Vector
Gravity".
Bahram Mashhoon's "Gravitoelectromagnetism: A Brief Review" provides a
partial answer. He write the Lagrangian needed to derive the force
equation (a inertia term and a charge coupling term, so by varying the
equation with respect to the velocity, a force equation results).
What is missing is whatever would make the field strength tensor.
This in not trivial! It is the field strength tensor that will help
you know what particles mediate the force. For gravity, it must be a
symmetric rank 2 tensor: it must be symmetric so like charges attract,
and at least rank 2 to bend light. From my games played with
symmetric rank 2 tensors, I don't get how one could make a gravity
analog to the magnetic field that would have the vector identities of
the magnetic field.
Thanks for the reference. At least I can be clear on my area of
disagreement.
doug
Bill Miller
Thanks for the reply and the critique on Jefimenko cum gravitation.
I have buried a few comments among your notes.
"swee...@alum.mit.edu" <dougsw...@gmail.com> wrote in message
news:1177379424.9...@l77g2000hsb.googlegroups.com...
> Hello Bill:
>
> I went ahead and ordered Oleg Jefimenko's "Causality, Electromagnetic
> Induction and Gravitation." My reaction is less positive. I don't
> believe the book is worth the time for readers to these newsgroups.
For Gravitation, I suspect you are right. Kirk has his reservations also.
BUT I suspect there may be more than a few list readers that still think
that Maxwell's Displacement Current *causes* a magnetic field (even though
there is not a shred of direct measurement evidence to support the idea.) To
my mind, Jefimenko's book (should) finally put a set of silver nails in that
coffin!
>
> Folks who write well typeset books with lots of partial differential
> equations can be intimidating!
If you enjoy being intimidated with page after page of differential
equations, Jefimenko has a companion work, "Electromagnetic Retardation and
Theory of Relativity" that makes "Causality" read like a Grishom novel!
>Still, there is a major omission, one
> I learned from a well-know MIT professor. I chatted with him one day
> about my own unified field equation, a one liner that fits on t-
> shirt. He said that although I had a field equation, I did not have a
> field theory. A modern field theory must have the following:
>
> 1. A Lagrange density (or Hamiltonian, or action, there are roads
> connecting the three).
> 2. Field equations derived from the Lagrangian by varying the
> action with respect to the potential
> 3. Force equations derived from the Lagrangian by varying the
> action with respect to the velocity.
> 4. Solutions to the field equations that were physically relevant
> 5. Solutions to the field equations that were measurably different
> from current tests
> 6. The energy/stress tensor derived from the Lagrangian
> 7. The field equations must be quantizeable.
>
> It was quite the list!
An understatement! Are they necessary *and* sufficient?
I'm puzzled by number 5. If you are talking about physical tests (i.e.
measurements) then this seems wrong. Perhaps you talking about test
conditions, such as a boundary, wherein an existing theory faisl to pass
muster? Or (as often happens) I am mis-understanding completely?
BTW, may I have your unified field equation, please? I have a couple of
no-slogan T Shirts, and feel intimidated when wear them in Home Depot.
>It was also a scary list because I had never
> worked with a Lagrangian before.
#SIGH# Nor have I. Looks like its back to the books again!
>Now that I understand it, the tool
> has a simple idea at the core. You want to have a scalar expression
> that contains all the ways that a system can exchange energy per unit
> volume. Integrate that over volume and an arbitrary amount of time.
> That would give the energy times an arbitrary amount of time, not so
> interesting. Using the calculus of variations, one looks for how one
> could varying the Lagrangian, yet not change the outcome of the
> integral, no matter how long one waited. That is a constant for the
> equation of motion, and is related to a conserved quantity.
>
> Jefimenko does not have a Lagrangian, a Hamiltonian, or an action, so
> his work is not worth studying.
I'm not sure this statement is valid, especially considering your paragraph
that follows. IF Hilbert had not serendipitously shown up, then applying
your standard would mean that Einstein's 1915 paper would not have been
studied. Among other possible outcomes of that lack of study is that we
might still be fighting in Japan. Also, since much of Europe is nuclear,
gasoline would now be about $50 a gallon.
I can think of a few more unpleasant consequences, but I think the point is
made. You, perhaps unwittingly, also made it.
It is not necessary that the individual that conceives a theory MUST also be
the one that meets all the agreed-upon criteria. But I concur that the
appropriate criteria must (ultimately) be met.
>I know this is a harsh standard, but
> least it can be applied evenly. Einstein's 1915 paper on general
> relativity would not have passed this test, but fortunately Hilbert
> published the action within weeks. Earlier articles cited in this
> thread also don't have Lagrangians: Robert Forward's "General
> Relativity for the Experimentalist" or Kirk McDonald's "Vector
> Gravity".
>
> Bahram Mashhoon's "Gravitoelectromagnetism: A Brief Review" provides a
> partial answer. He write the Lagrangian needed to derive the force
> equation (a inertia term and a charge coupling term, so by varying the
> equation with respect to the velocity, a force equation results).
> What is missing is whatever would make the field strength tensor.
> This in not trivial! It is the field strength tensor that will help
> you know what particles mediate the force. For gravity, it must be a
> symmetric rank 2 tensor: it must be symmetric so like charges attract,
> and at least rank 2 to bend light. From my games played with
> symmetric rank 2 tensors, I don't get how one could make a gravity
> analog to the magnetic field that would have the vector identities of
> the magnetic field.
>
> Thanks for the reference. At least I can be clear on my area of
> disagreement.
> doug
And thanks again for your comments. Lagrange, here I come!
Bill
Eric Gisse
wrote:
[...]
In classical mechanics all Lagrangians are is a slightly fancy form of
T - V. It isn't that complicated if you get a halfway decent book to
explain it to you.
Hendrik van Hees
> In classical mechanics all Lagrangians are is a slightly fancy form of
> T - V. It isn't that complicated if you get a halfway decent book to
> explain it to you.
It's already wrong for a simple system like a relativistic point
particle, if T denotes kinetic energy as usual.
--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:he...@comp.tamu.edu
Bill Miller
Can you help me wth a couple of recommendation to keep me from reading a
whole bunch of "not-halfway-decent" books until I find one that I have a
shot at understanding? :)
Thanks!
Bill Miller
"Eric Gisse" <jow...@gmail.com> wrote in message
news:1178084107....@c35g2000hsg.googlegroups.com...
Eric Gisse
wrote:
> OK...
>
> Can you help me wth a couple of recommendation to keep me from reading a
> whole bunch of "not-halfway-decent" books until I find one that I have a
> shot at understanding? :)
>
> Thanks!
>
> Bill Miller
First - no top posting. It fucks up thread structure.
As far as book suggestions go...hmm
The book I learned it from was Symon's _Mechanics_, but I _do not_
suggest that book for you unless you are hugely patient and/or have
someone to work with. My next classical mechanics book will be [I have
spent some time looking at it] Goldstein's _Classical Mechanics_,
which goes _straight to_ Lagrangian/Hamiltonian mechanics but it seems
more like a refresher than anything else.
You need something lighter than Symon, while being understandable,
which still covers Lagrangian mechanics.
My suggestion is to go to the library and just spend some time looking
at books that cover classical mechanics. The Lagrangian/Hamiltonian
formalism dates to the mid to late 19th century, so there is not a
lack of material.
Before you do that, ask those who have experience teaching. Post a
request in sci.physics / sci.physics.relativity for information about
a decent text that covers Lagrangian mechanics at the undergraduate
level.
[...]
dougsw...@gmail.com
> I'm puzzled by number 5
which reads:
>> 5. Solutions to the field equations that were measurably different
>> from current tests
There are many ways to rewrite an accepted area of physics. There are
certainly people who think I am a great example of this, since I am a
fan of quaternions. Quaternions can be viewed as a 4-vector, which
can be added and subtracted from each other, and multiplied by a
scalar. Rewriting physics in terms of quaternions may accomplish
nothing, but it is great way to learn physics. You will find people
who really like writing physics in terms of geometric algebras, or
Clifford algebra, or groupoids. Since my hope was to do new physics,
not
a rewrite in different notation, the way to establish the proposal is
distinct is to have a physical test.
> BTW, may I have your unified field equation, please?
"Always give 2 Brownies to Jim" was how my mailman remembered it. It
looks much like the Maxwell equations in the Lorentz gauge, J = Box^2
A.
The Box^2 is not the D'Alembertian operator by the way, it is two
covariant derivatives applied to the 4-potential. That detail is
important. A D'Alembertian is a scalar operator, so the field
equations could only be about a potential. We know such an approach
is wrong via tests of the equivalence principle. A proposal must have
a way to create a dynamic metric solution, meaning there must be at
least a second order differential equation whose solution is a metric
determined by the distribution of mass sources. A covariant
derivative of a contravariant derivative of a 4-potential will have a
derivative of a connection which is a derivative of a metric (assuming
the connection is metric compatible and torsion free). Second
order derivatives of a metric are a basic requirement if one hopes to
find an equation whose solution is a dynamic metric. Sorry for the
techno-babel tangent, but it is important to me. The T shirt features
an oil painting of mine, "Turquoise Einstein", from cafepress.com, and
the J = Box^2 A. Global sales are zero. Get 'em fast.
>IF Hilbert had not serendipitously shown up, then applying your
>standard would mean that Einstein's 1915 paper would not have been
>studied.
Hilbert and Einstein were corresponding with each other. The
importance of the action/Lagrangian/Hamiltonian has only increased in
time. Particularly if you hope to get a sense of what is going on in
quantum field theory, one takes a good look at the Lagrangian to see
what's there.
> #SIGH# Nor have I. Looks like its back to the books again!
I looked through "Classical Fields" by Landau and Lifshitz dozens of
times before I started to pick up the message. I really was scared to
try and learn this stuff. I found out something amazing about it.
The road from the classical EM Lagrange density to the Maxwell
equations takes up 6 blackboards and an hour and fifteen minutes when
I did it for a friend. I'll presume you know that d xy/dx = y and
d x^2/dx = 2x. It turns out that is the only calculus you need! The
thing is, there is LOTS of details. The Euler-Lagrange equation has
some 20 partial differential equations in it if you were to write out
all the terms. The Lagrangian that gets plugged in has 24 terms. No
one write it all out, that requires too much typesetting. Well,
almost no one. One crazy guy I know pretty well did it (me). If you
want to see all the details, no partial derivatives left out, click
through about a half dozen slides. I only understand things when that
much detail is provided explicitly. I get lost as soon as the phrase
"the details are left as an exercise" appear.
doug
http://theworld.com/~sweetser/quaternions/talks/IAP_2/1004.html
Marc Millis
wrote:
> On May 3, 1:40 pm, Bill Miller <BillMillerKT...@worldnet.att.net>
> wrote:
>> OK...
>>
>> Can you help me wth a couple of recommendation to keep me from
>> reading a whole bunch of "not-halfway-decent" books until I find one
>> that I have a shot at understanding? :)
>>
>> Thanks!
>>
>> Bill Miller
Bill;
When it comes to working through a Lagrangian and onto the Hamiltonian (
which I did less than 2 yrs ago in ernest... finally), I did not find a
best textbook. I had to examine about three books and work through it
myself slowly until it sunk in.
One of the things that threw me off was in the integral for the least
action... I had it so drilled into me that the path of an integration
did not matter (from calculus), that it was hard for me to shift mental
gears and realize that the whole "least action" concept depends on the
path (even when the starting and ending points are the same). Even
though the textbooks say this, it just didn't sink in until I thought
about the physical meaning of each step, rather than just working the
math. The actual math is not hard. Understanding what each step of the
math really means, was harder for me. The conceptual tool that helped
me next was realizing that the whole point of the Lagrangian is to get
the energy into a *minimal* form, while the Hamiltonian is for employing
conservation of energy after that. Again, although the texts say this,
the power of such simple distinctions is not emphasized. The textbooks
basically showed the steps without really explaining which steps where
pivotal and which were merely math tools.
Sorry I cannot recommend a good book that focuses on just this question.
I hope my short experiences help somewhat.
Marc
Bill Miller
Thanks Eric and Doug...
I appreciate the time and recommendations and advice. Bottom posting do I
will.
Lagrange seems to be something that is really simple and useful -- once you
understand it. But the book, "Lagrange For Dummies" has not yet been
written.
I'll check out the authors/sources you suggested.
Thanks again!
Bill
>
Calvin D. Ritchie
H. C. Corben and P. Stehle, "Classical Mechanics", 2nd Ed., Dover
edition published in 1994.
A BIG advantage of Dover books is that they are inexpensive enough
that you can afford to gamble that they will actually be what you're
looking for. I've recommended Corben and Stehle to several beginners
and had no comebacks. It's understandable, I think, to almost anyone
who's had introductory calculus and is willing to work a little.
Don Ritchie
FrediFizzx
news:kLm_h.406952$5j1.2...@bgtnsc04-news.ops.worldnet.att.net...
> OK...
>
> Can you help me wth a couple of recommendation to keep me from reading
> a
> whole bunch of "not-halfway-decent" books until I find one that I have
> a
> shot at understanding? :)
Perhaps a more exciting way to learn about Lagrangians is via particle
physics which uses them alot. Griffiths' "Intro. to Elementary
Particles" will get you started.
Best,
Fred Diether
Moderator: sci.physics.foundations
http://groups.google.com/group/sci.physics.foundations
Michael Moroney
In other words, consider this: Two infinitely long parallel massless
pipes containing a massive fluid. There will be a gravitational
attraction between the pipes because of the fluid in them. Now open some
virtual valve and the fluid in the pipes flows. They'd have to flow at
different rates since one could always choose as a reference frame where
the flow in one of the pipes is zero, if both were zero we're back at the
initial condition.
Does the force between the pipes change, even by an absurdly small amount?
Or has it been proven that it doesn't change?
carlip...@physics.ucdavis.edu
> Is there a gravitational equivalent to magnetism?
Yes. It's called "gravitomagnetism," and for weak fields it has
roughly the same relationship to ordinary Newtonian gravity as
magnetism has to the Coulomb interaction.
The frame-dragging effect that Gravity Probe B hopes to measure
is a direct form of gravitomagnetism. But it's also been measured,
slightly less directly, in the Lunar orbit: see, for example, Murphy,
Nordtvedt, and Turyshev, http://arxiv.org/abs/gr-qc/0702028, Phys.
Rev. Lett. 98 (2007) 071102.
Steve Carlip
dougsw...@gmail.com
> Yes. It's called "gravitomagnetism," and for weak fields it has
> roughly the same relationship to ordinary Newtonian gravity as
> magnetism has to the Coulomb interaction.
I have a question about this. The EM B field arises from an
antisymmetric second rank tensor, A^u;v - A^v;u. If we use Cartesian
coordinates, then for the x direction: Bq_x = d Aq_z/dy - d Aq_y/dz.
If the indexes are switched, then Bq_x flips signs. That is the sign
of an antisymmetric tensor.
Now consider gravity. The field strength tensor has to be symmetric
because like charges attract. The field strength tensor has to be
rank 2 so that gravity can interact with photons and get light to bend
around the Sun. Given these constraints, I would think that the
gravitomagnetic term along the x axis would be: Bg_x = d Ag_z/dy + d
Ag_y/dz, the important difference being the '+' sign. If the two
indexes are swapped, Bg_x would not change signs, as expected for an
element of a symmetric tensor.
I looked up a paper on the archives by Mashhoon , Gronwald , and
Lichtenegger (gr-qc/9912027v1), and they define Bg as being DelxA,
with the standard minus sign coming from the cross product. That
would be consistent with the magnetic monopole equation and Faraday-
like law that is part of gravitomagnetism.
If I am reading these papers correctly, then in the standard approach
to gravitomagnetism, the gravity-magnetic field can be represented by
a rank 2 antisymmetric tensor. Even as a weak field approximation, it
looks to me like such a field would have the spin of the mediating
particles wrong, and like charges would repel.
Would gravity probe B be able to tell the difference between a B field
defined by a + sign instead of a -?
Thanks,
doug
carlip...@physics.ucdavis.edu
>> Yes. It's called "gravitomagnetism," and for weak fields it has
>> roughly the same relationship to ordinary Newtonian gravity as
>> magnetism has to the Coulomb interaction.
> I have a question about this. The EM B field arises from an
> antisymmetric second rank tensor, A^u;v - A^v;u.
Right. That's an antisymmetric field strength, obtained from
derivatives of a potential.
> Now consider gravity. The field strength tensor has to be symmetric
> because like charges attract. The field strength tensor has to be
> rank 2 so that gravity can interact with photons and get light to bend
> around the Sun.
You're misusing the term "field strength." The argument that the metric
has to be a rank two symmetric tensor ("spin 2") is analogous to the
argument that the gauge potential A for electromagnetism has to be a
rank one tensor ("spin 1").
If you are looking at linearized gravity around a flat background, the
analog of A_u is h_{uv}, the deviation of the metric from the Minkowski
metric. The analog of F_{uv} is the Christoffel connection, or more
precisely the set of components Gamma^i_{0j}. These are antisymmetric
(in this approximation); this is completely different from the symmetry
of h.
In a broader context, if you're not linearizing around a fixed background,
the closest analog to A is the connection Gamma, and the closest analog
to F is the curvature tensor. (This doesn't contradict the picture in
linearized gravity. If you have a fixed background, you can talk about
the deviation from straightness of the path of a single particle, while
if you don't, you have to talk about relative paths of two nearby
particles, which gives you an extra derivative.)
Steve Carlip
dougsw...@gmail.com
Thanks for reminding me how the game of Maxwell <-> GR is played: what
is a tensor of rank n in EM is a tensor of rank n+1 in GR.
> If you are looking at linearized gravity around a flat background, the
> analog of A_u is h_{uv}, the deviation of the metric from the Minkowski
> metric. The analog of F_{uv} is the Christoffel connection, or more
> precisely the set of components Gamma^i_{0j}. These are antisymmetric
> (in this approximation); this is completely different from the symmetry
> of h.
This sounds like a bad approximation to me. The point of any
approximation is that it is the first step in the correct direction.
A basic property of gravity is that like charges attract. If two like
particles exchange a graviton, the field for that graviton must be
symmetric if the two particles attract each other, no matter how good
or bad the approximation happens to be.
I understand you are following a well worn and documented line of
reasoning. Breaking symmetry is part of the approximation process -
so long as you can see where the missing part of the true symmetry
live. In this case, it might be the components of Gamma^i_{0j}. To
get the _opposite_ kind of symmetry, that makes me skeptical. Think
about the Lorentz versus the Galilean symmetry. At low speed, time
cannot rotate into space. The transition from one to the other would
be smooth, with initially a wee bit of mixing of time and space. For
the case at hand, with the antisymmetric DelxA for the gravitomagnetic
field, like charges would repel, exactly has happens for EM. For the
full theory based on h, like charges attract. I see no smooth
transition between the approximation and the full theory. I think it
is wrong.
It is disturbing - but standard - to see the tensor F_{uv} compared to
that which is not a tensor, Gamma^i_{0j}. I don't know what it means
to find links between something that transforms like a tensor so it
has a chance of being measured in physics, with a math widget that
arose to make a derivative transform like a tensor. In my obscure
corner of the world of physics, I never write Gamma^i_{0j} outside the
context of a tensor, which would involve either the Riemann curvature
tensor or my favorite, a covariant derivative.
doug
carlip...@physics.ucdavis.edu
> Thanks for reminding me how the game of Maxwell <-> GR is played: what
> is a tensor of rank n in EM is a tensor of rank n+1 in GR.
>> If you are looking at linearized gravity around a flat background, the
>> analog of A_u is h_{uv}, the deviation of the metric from the Minkowski
>> metric. The analog of F_{uv} is the Christoffel connection, or more
>> precisely the set of components Gamma^i_{0j}. These are antisymmetric
>> (in this approximation); this is completely different from the symmetry
>> of h.
> This sounds like a bad approximation to me. The point of any
> approximation is that it is the first step in the correct direction.
If you have a particular theory, the point of an approximation is
to get a good enough solution to a problem you can't solve exactly.
In GR, the expansion around a flat background is, observably, a good
approximation for weak fields. The symetries of the connection are
then a conclusion.
> A basic property of gravity is that like charges attract. If two like
> particles exchange a graviton, the field for that graviton must be
> symmetric if the two particles attract each other, no matter how good
> or bad the approximation happens to be.
Yes. And if you learn *why* this is true, you will understand that the
symmetry in question is the symmetry of the metric (and not, say, the
connection). This symmetry is present in the approximation I'm describing;
the (approximate) antisymmetry is in the *connection*, not the metric.
There is no contradiction here. In E&M, the vector potential has no
symmetries on its indices (since it has only one index); do you object
to the fact that the field strength tensor is antisymmetric, and
therefore has different symmetry properties?
> I understand you are following a well worn and documented line of
> reasoning. Breaking symmetry is part of the approximation process -
> so long as you can see where the missing part of the true symmetry
> live. In this case, it might be the components of Gamma^i_{0j}. To
> get the _opposite_ kind of symmetry, that makes me skeptical.
That's because you don't understand the role of the symmetry. Look
at chapter 3 of the _Feynman Lectures on Gravitation_, and at the
description of representations of the Poincare group in any decent
QFT textbook. The issue is not "symmetry vs. antisymmetry," it's
the right way to represent a field of a given spin.
Steve Carlip
dougsw...@gmail.com
Thanks for the great Feynman reference! I had fun over Memorial Day
Weekend reading and playing with chapter 3. It made clear that the
main barrier to communication was my lack of understanding about
current-current interactions, which I hope to demonstrate has been
corrected. Most of this post will be a rehash of section 3.2, so if I
am unclear, please read Feynman. I promise to play by the rules, yet
do something new.
Feynman's analysis deals directly with 2 currents that interact, a
viewpoint I have not used often. The charge coupling term, j'^u A_u
has one current. Where is the other? One can take the Fourier
transform of the 4-potential A_u and in the momentum space
representation rewrite the potential like so:
A_u = -j_u/k^2
This is how the 2 currents interact:
interaction = -j'^u j_u/K^2
Make things simpler by having the current move along z:
K_u = (w, 0, 0, k) using the (t, x, y, z) convention
K^2 = w^2 - k^2
Write out the interaction by its components:
- j'^u j_u/K^2 = - (rho' rho - jx' jx - jy' jy - jz' jz) / (w^2 -
k^2)
Charge is conserved, so:
K^u j_u = 0 = w rho - k jz
Use this to eliminate jx:
- j'^u j_u/K^2 = rho' rho/k^2 + (jy' jy + jz' jz) / (w^2 - k^2)
If we are in the rest frame of j' or j, then only the charge density
matters. Move relative to that reference frame, and the other terms
come into play.
Feynman now focuses on the jx and jy terms. This is where physics
becomes math magic. These two currents always involve virtual
photons. Further, Feynman works with the poles, where w->k. These
virtual photons are the sum of two independent terms, jx' jx and jy'
jy. A different way to say this is that there are 2 independent
polarities for photons, a topic being discussed in another thread on
SPR.
That was fun, but I wanted to think more precisely about the product
of two currents. I'm going to use quaternion algebra, but if you are
more comfortable with the Dirac algebra - only a twist of i away - go
ahead. Form the j' j product like so:
(0, jx', jy', 0) (0, jx, jy, 0)*
= (jx' jx + jy' jy, 0, 0, jx' jy - jy' jx)
The phase term is in the z slot. It will require a 2 pi rotation to
get back to go. The current-current interaction is a spin 1 photon,
so like charges repel. Good.
It occurred to me that there might be another distinct product of
these two currents. Consider the conjugate operator, which flips all
the signs except the first one. It is known by mathematicians that
there is more than one anti-involutive automorphism. Let's break down
that jargon. The automorphism means that the function maps back to
the same space. Taking two operations brings the function back home.
The final bit is (a b)* = b* a*. Big words, but here is a simple
idea: fix a term other than the first one, and flip the signs of all
others. This little algebra trick is missing from many professional
physicists tool drawer. Let me define the second conjugate like so:
(t, x, y, z)*2 === ( (0, 0, 1, 0) (t, x, y, z) (0, 0, 1, 0) )* = (-
t, -x, y, -z)
Put this tool to work for two interacting currents:
(0, jx', jy', 0) (0, jx, jy, 0)*2
= (jx' jx - jy' jy, 0, 0, jx' jy + jy' jx)
Take a peek at page 39, and you'll realize this product has the
character of spin 2! I think the idea is that the two parts of the
phase term can add together, able to race back to their initial spot
in pi radians. This product describes with two degrees of freedom a
current interaction where like charges attract. Cool. A 4-current
has 4 degrees of freedom, 2 for a spin 1 current where like charges
repel, 2 for a spin 2 current where like charges attract.
This calculation made my Memorial Day weekend memorable.
doug
hanson
> A 4-current has 4 degrees of freedom, 2 for a spin 1 current where
> like charges repel, 2 for a spin 2 current where like charges attract.
>
I am glad to hear that you felt great about your insight, Doug.
Now, since we do live in a real world of self-similarities,
"it's like" "it's analogous to", etc. could you make a pi-turn
back from your math magic world, back into physics and
give some examples of your calc that do occur in, or do
reflect our real world applications. (mech/dynamics/electr)
I ask this because the EM force carrier is supposed to be of
a spin 1 nature, whereas the gravitational force mediator is
supposed to be a spin 2 entity.
Thanks, Doug
hanson
"swee...@alum.mit.edu" <dougsw...@gmail.com> wrote in message
news:1180666748....@h2g2000hsg.googlegroups.com...
maxwell
wrote:
approach to gravitation. Your predilection for field theories is
based on the wide-spread assumption of physical continuity (especially
with respect to variations in time); this is the metaphysical analogue
of the mathematician's obsession with calculus (variational or
differential). L.V. Lorenz demonstrated in 1867 the complete
equivalence (mathematically) with Maxwell's EM field theory but used
only two retarded functions (scalar & vector potentials) at the local
& remote source locations. Hence there is no absolute need for fields
or Lagrangians, especially Lagrangian densities: Maxwell's classical
theory can be (& today is) recast using continuum electric charge
densities, an idea that does NOT correspond to the reality of
particulate electricity (i.e. electrons).
Accordingly, I go along with Bill Miller's recommendation & encourage
others who are interested in investigating alternatives to a 150 year-
old 'blind-alley' to buy Jefimenko's new book on gravity (his EM &
Retardation book is even better).
PS Jefimenko's style of using a lot of differential equations reflects
the great teacher that he is: he takes you through every step in the
math exposition - unlike so many, who simply write: "it can be shown
that ...".
Bill Miller
"maxwell" <sp...@shaw.ca> wrote in message
news:1180974032.6...@x35g2000prf.googlegroups.com...
I just ordered and read Jefimenko's latest book, Gravitation and
Cogravitation. In a way, it was a bit of a disappointment, since it contains
substantial amounts of material from his "Causality" and "Retardation"
books. I have already read those a couple of times. (I will NOT claim that I
understood them completely!)
HOWEVER, I think I undersatnd why Jefimenko structured the book as he did.
He clearly wanted a book that would stand on its own metaphorical legs,
without supposing that someone already knew something (new) that was not in
evidence.
And because his conclusions are so innovative, anything but a firm
foundation would lead to a total collapse of the basic premises.
Here are SOME of the issues that were new and/or different.
He provides a clear explanation of the mechanism by which the gravitational
potental energy on Newton's apple transforms into kinetic energy as the
apple falls.
He provides an explanation of why the sun's equatorial rotation period (25
days) is different from its near-axis period of 36 days. This concept of
"gravitational torque" may have important consequences in understanding
other phenomenon such as (my idea) that this may be why the Earth's Magnetic
pole is moving.
He provides a non-relatavistic explanation of gravitational bending of light
rays.
Finally (sort of) , he provides an interesting basis for speculating that
Einstein may have "cooked the books" by a factor of 4 in order for
relativity to "explain" the 43 seconds per century residual error in
Mercury's precession.
There is more. But very careful reading is required. (At least by a dullard
like me!)
Bill Miller
Bilge
> On Apr 23, 5:44 pm, "sweet...@alum.mit.edu" <dougsweet...@gmail.com>
> wrote:
>> Hello Bill:
>> Bahram Mashhoon's "Gravitoelectromagnetism: A Brief Review" provides a
>> partial answer. He write the Lagrangian needed to derive the force
>> equation (a inertia term and a charge coupling term, so by varying the
>> equation with respect to the velocity, a force equation results).
>> What is missing is whatever would make the field strength tensor.
>> This in not trivial! It is the field strength tensor that will help
>> you know what particles mediate the force. For gravity, it must be a
>> symmetric rank 2 tensor: it must be symmetric so like charges attract,
>> and at least rank 2 to bend light. From my games played with
>> symmetric rank 2 tensors, I don't get how one could make a gravity
>> analog to the magnetic field that would have the vector identities of
>> the magnetic field.
>>
> I must disagree strongly with your sweeping dismissal of Jefimenko's
> approach to gravitation. Your predilection for field theories is
> based on the wide-spread assumption of physical continuity (especially
> with respect to variations in time);
Actually, I think it's fair to say that most physicists expect
spacetime to be quantized in some fashion, but the quantization to
be irrelevant until gravity becomes important. Your use of the
word ``physical'' presumes your personal metaphysics is part of
some wide-spread assumption, which I seriuosly doubt.
> this is the metaphysical analogue
> of the mathematician's obsession with calculus (variational or
> differential). L.V. Lorenz demonstrated in 1867 the complete
> equivalence (mathematically) with Maxwell's EM field theory but used
> only two retarded functions (scalar & vector potentials) at the local
> & remote source locations. Hence there is no absolute need for fields
> or Lagrangians, especially Lagrangian densities: Maxwell's classical
Incorrect. Maxwell's equations _do_ _not_ contain _all_ of the
observable electromagnetic effects. The aharonov-bohm effect does
not follow from classical E&M. In classical E&M, the potentials
are mathematical artifices with no observable effects. That is not
true for quantum theory. Compare the phase difference around the closed
loop in the aharonov-bohm effect with the gauge field derived from
the invariance of the lagrangian under gauge transformations.
Without a lagrangian formulation, you also have no way to define
conserved quantities in any rigorous way. For example, try to define
a conserved energy without saying anything about invariance under
time translations and some assumptions about spacetime.
> theory can be (& today is) recast using continuum electric charge
> densities, an idea that does NOT correspond to the reality of
> particulate electricity (i.e. electrons).
What does a continuous electric charge density mean if you don't
presume a continum over which it is distributed?
> Accordingly, I go along with Bill Miller's recommendation & encourage
> others who are interested in investigating alternatives to a 150 year-
> old 'blind-alley'
Blind alley? QED has been tested to better precision than any
theory in the history of planet earth without finding any disagreement
with the experimental data.
> to buy Jefimenko's new book on gravity (his EM &
> Retardation book is even better).
> PS Jefimenko's style of using a lot of differential equations reflects
> the great teacher that he is: he takes you through every step in the
> math exposition - unlike so many, who simply write: "it can be shown
> that ...".
If you assume the equivalemce between inertial mass and gravitational
mass, you get a metric theory of gravity - i.e., general relativity.
So far, this equivalence holds to the limits of experimental precision.
maxwell
> Actually, I think it's fair to say that most physicists expect
> spacetime to be quantized in some fashion, but the quantization to
> be irrelevant until gravity becomes important. Your use of the
> word ``physical'' presumes your personal metaphysics is part of
> some wide-spread assumption, which I seriuosly doubt.
>
> observable electromagnetic effects. The aharonov-bohm effect does
> not follow from classical E&M. In classical E&M, the potentials
> are mathematical artifices with no observable effects. That is not
> true for quantum theory. Compare the phase difference around the closed
> loop in the aharonov-bohm effect with the gauge field derived from
> the invariance of the lagrangian under gauge transformations.
> Without a lagrangian formulation, you also have no way to define
> conserved quantities in any rigorous way. For example, try to define
> a conserved energy without saying anything about invariance under
> time translations and some assumptions about spacetime.
> presume a continum over which it is distributed?
> theory in the history of planet earth without finding any disagreement
> with the experimental data.
> mass, you get a metric theory of gravity - i.e., general relativity.
> So far, this equivalence holds to the limits of experimental precision.
same courtesy & respond now to yours.
You seem well acquainted with the current orthodox views of modern
physics; from this I infer that you have passed your exams, as I did
(a good memory is a necessary requirement for any professional). If
all you want from life is to be 'another brick in the wall' then you
will not have to challenge any of the thoughts that your teachers have
impressed upon you - I'm sure there are still jobs out there that only
require that you repeat what you have been told. However, if anyone
desires to make any new contributions to science then I would strongly
encourage them to start to think about the assumptions that underlie
all that they have been taught. For example, it is a historical fact
that the notion of one-to-one mapping between physical reality (the
subject of physics) & the representations (mathematical physics) that
have been used to try to map our ideas of reality have followed the
continuum hypothesis for many centuries. The recent attempts at
'quantum foam' do not deviate from this pattern unless all of the
mathematics used therein is discrete. Any discussion of the 'reality'
of the world implies metaphysics - a subject that is NOT discussed in
modern physics but is intrinsic to the distinction between math &
physics. I prefer to make my metaphysical assumptions explicit, in
the natural philosophical tradition of Newton. Then, like Leibniz,
others can challenge these ideas even if they find the associated
mathematics impeccable.
Your commonly repeated claim that classical electromagnetism (CEM)
does not cover the Aharonov-Bohm effect (actually QM, not CEM) does
not include the work done by that venerable UK professor of Electrical
Engineering, C. John Carpenter, who (like myself) is dedicated to the
Lorenz Current-Potential (or retarded potential) model of EM. You can
track down his papers on Professor McDonald's superb history of
electromagnetism site at: http://puhep1.princeton.edu/~mcdonald/examples/EM/
In this Newtonian-style, action-at-a-distance approach (that traces
its roots back to Gauss, then Riemann & Helmholtz) the potentials are
introduced as the primary concepts & force-densities only as optional
derivatives. The resulting mathematical equations appear identical to
Maxwell's but the underlying model is very different. Your assumption
of the necessity of Lagrangian densities for conservation is also
flawed. Total mass or electric charge in a closed system are also
conserved without invoking continuous mass or charge density
concepts. Pauli (in his lectures on Classical Electrodynamics)
attempted to mathematically justify the continuum limit in CEM, since
he wanted to use the ideas of charge & current densities, rather than
Maxwell's original aether model. Pauli needed to begin with a
quadruple limit: where the spatial volume goes to zero, the number of
volume cells goes to infinity, the unit of charge goes to zero & the
number of charges per unit volume goes to infinity: nice math but no
contact with reality (but as a Pythagorean that shouldn't bother you
at all!). Since I only work with a finite number of finitely charged,
point models of electrons I don't need to introduce a physical
continuum - I continue to work with Newton's model of space & time
quite successfully.
Your hoary old defense of QED based on the calculation of the
anomalous magnetic moment of the electron in the hydrogen atom would
get you a commendation from Pope Urban VIII, if he were still alive
today. When Galileo tried to convince his fellow astronomers of the
reality of the Copernican model of the solar system, it was correctly
pointed out that the calculations of the Ptolemaic system were
significantly better. My lack of respect for QED & field theories is
based on the observations that they have been limited to more & more
refinements about less & less. Yes, we can do incredible calculations
for the hydrogen atom but what about all the other elements of the
periodic table? Be very cautious defending the current views of
science as the final word. Thomas Kuhn has illustrated how scientific
revolutions often occur when most scientists stop questioning their
assumptions & self-limit their activities to adding an extra digit of
accuracy.
Finally, I should add that Einstein's equivalence principle for
inertial & gravitational mass ignores both the reality of tidal
effects (except at the mathematical limit) & all gravitational
theories ignore the overwhelming effects of the EM interaction in the
universe, so that 'experimental' evidence for general relativity
should be suspect from the start - but then this type of science has
always really been about religion, hasn't it?
Eric Gisse
Bilge's example of the Aharonov-Bohm effect was first discovered,
experimentally, in the late 1980s. Hardly something present from day
one.
Your representation of physicists as folks who only repeat what they
are told is moderately insulting. Only moderately because you don't
know what you are talking about.
> However, if anyone
> desires to make any new contributions to science then I would strongly
> encourage them to start to think about the assumptions that underlie
> all that they have been taught. For example, it is a historical fact
> that the notion of one-to-one mapping between physical reality (the
> subject of physics) & the representations (mathematical physics) that
> have been used to try to map our ideas of reality have followed the
> continuum hypothesis for many centuries.
...until the 20th century. The discovery that reality is discrete sort-
of upsets the notion that reality is a continuum.
What you do not realize is that all [good] scientists are taught to
question what they are told. Do you think folks take QM on faith? Are
you totally unaware at how hard QM was, ans is being, fought by folks
much more capable than yourself?
Einstein disagreed strongly with QM and the result were the EPR class
of experiments. Folks thought there were things "behind the scenes"
that resulted in the probabilistic behaviour reality exhibits, which
resulted in the Bell inequality that utterly destroyed an entire swath
of possible theories.
> The recent attempts at
> 'quantum foam' do not deviate from this pattern unless all of the
> mathematics used therein is discrete. Any discussion of the 'reality'
> of the world implies metaphysics - a subject that is NOT discussed in
> modern physics but is intrinsic to the distinction between math &
> physics. I prefer to make my metaphysical assumptions explicit, in
> the natural philosophical tradition of Newton. Then, like Leibniz,
> others can challenge these ideas even if they find the associated
> mathematics impeccable.
> Your commonly repeated claim that classical electromagnetism (CEM)
> does not cover the Aharonov-Bohm effect (actually QM, not CEM) does
> not include the work done by that venerable UK professor of Electrical
> Engineering, C. John Carpenter, who (like myself) is dedicated to the
> Lorenz Current-Potential (or retarded potential) model of EM. You can
> track down his papers on Professor McDonald's superb history of
> electromagnetism site at:http://puhep1.princeton.edu/~mcdonald/examples/EM/
Might want to look at that web page. You will notice a problem.
The claim that the Aharonov-Bohm effect is unexplainable by classical
electrodynamics is often repeated because it has the virtue of being
true. You don't have phase differences in classical mechanics, and you
most certainly do not have a way of measuring the vector potential.
The A-B effect is one of many that classical theory cannot handle.
Good luck explaining how classical E&M quantizes magnetic flux - a
verified effect used in commercial devices called SQUIDs, you may or
may not have heard of them.
> In this Newtonian-style, action-at-a-distance approach (that traces
> its roots back to Gauss, then Riemann & Helmholtz) the potentials are
> introduced as the primary concepts & force-densities only as optional
> derivatives. The resulting mathematical equations appear identical to
> Maxwell's but the underlying model is very different. Your assumption
> of the necessity of Lagrangian densities for conservation is also
> flawed.
What makes the Lagrangian formalism flawed? It is simply a
generalization of Newtonian mechanics. There is no new physics
contained, it is just incredibly easier to get to the physics.
If you have a conserved quantity, the derivative of the Lagrangian
with respect to that quantity will be zero. How much more simple can
you get? Noether's theorem makes it even more rigorous. Every symmetry
buys you a conserved quantity.
Good luck doing that without reference to a Lagrangian.
> Total mass or electric charge in a closed system are also
> conserved without invoking continuous mass or charge density
> concepts. Pauli (in his lectures on Classical Electrodynamics)
> attempted to mathematically justify the continuum limit in CEM, since
> he wanted to use the ideas of charge & current densities, rather than
> Maxwell's original aether model. Pauli needed to begin with a
> quadruple limit: where the spatial volume goes to zero, the number of
> volume cells goes to infinity, the unit of charge goes to zero & the
> number of charges per unit volume goes to infinity: nice math but no
> contact with reality (but as a Pythagorean that shouldn't bother you
> at all!). Since I only work with a finite number of finitely charged,
> point models of electrons I don't need to introduce a physical
> continuum - I continue to work with Newton's model of space & time
> quite successfully.
No...you don't.
Your wrongness just expanded in scope because classical mechanics
excludes quantum mechanics *and* relativity.
Both Bilge and myself can easily give you 50 things each that your
model cannot handle.
> Your hoary old defense of QED based on the calculation of the
> anomalous magnetic moment of the electron in the hydrogen atom would
> get you a commendation from Pope Urban VIII, if he were still alive
> today.
How deliciously stupid. QED gets it right, and all you can do is say
something dumb.
> When Galileo tried to convince his fellow astronomers of the
> reality of the Copernican model of the solar system, it was correctly
> pointed out that the calculations of the Ptolemaic system were
> significantly better. My lack of respect for QED & field theories is
> based on the observations that they have been limited to more & more
> refinements about less & less.
Your lack of respect for QED & field theories is based upon your
complete and utter ignorance. Tell us about your education in math and
physics, if you have any.
> Yes, we can do incredible calculations
> for the hydrogen atom but what about all the other elements of the
> periodic table?
What about them?
It is significantly more complicated due to electron-electron
interactions, but it is routine work to handle multi-electron and
multi-ATOM systems with quantum mechanics. Have you ever heard of
quantum chemistry?
I am willing to bet that you couldn't even solve the Schroedinger
equation for the Hydrogen atom.
> Be very cautious defending the current views of
> science as the final word. Thomas Kuhn has illustrated how scientific
> revolutions often occur when most scientists stop questioning their
> assumptions & self-limit their activities to adding an extra digit of
> accuracy.
Asshat. Nobody is claiming the current views of science are the final
word. You once again reveal your massive ignorance of science.
> Finally, I should add that Einstein's equivalence principle for
> inertial & gravitational mass ignores both the reality of tidal
> effects (except at the mathematical limit) & all gravitational
> theories ignore the overwhelming effects of the EM interaction in the
> universe, so that 'experimental' evidence for general relativity
> should be suspect from the start - but then this type of science has
> always really been about religion, hasn't it?
Oh dear you fucked up the equivalence principle too. Tidal effects
have fuck-all to do with the equivalence principle.
Your whining that all gravitational theories ignore E&M is obvious and
uninteresting. That GR cannot unify with the rest of physics is a well-
known problem and not something that needs to be repeated. There has,
however, no experimental conflict with GR. You can whine about how the
evidence is suspect but that doesn't mean much when you don't
understand any physics.
Comparing religion and science is quite frankly, fucking /stupid/. If
science were religion, we would still be using classical mechanics.
Phineas T Puddleduck
Eric Gisse <jow...@gmail.com> wrote:
> Oh dear you fucked up the equivalence principle too. Tidal effects
> have fuck-all to do with the equivalence principle.
>
> Your whining that all gravitational theories ignore E&M is obvious and
> uninteresting. That GR cannot unify with the rest of physics is a well-
> known problem and not something that needs to be repeated. There has,
> however, no experimental conflict with GR. You can whine about how the
> evidence is suspect but that doesn't mean much when you don't
> understand any physics.
>
> Comparing religion and science is quite frankly, fucking /stupid/. If
> science were religion, we would still be using classical mechanics.
It looks like it is the month for kooks with new theories.
--
COOSN-174-07-82116: Official Science Team mascot and alt.astronomy's favourite
poster (from a survey taken of the saucerhead high command).
Official maintainer of the supra-cosmic space fluid pump (Mon and Tues only).
Phineas T Puddleduck
> > In this Newtonian-style, action-at-a-distance approach (that traces
> > its roots back to Gauss, then Riemann & Helmholtz) the potentials are
> > introduced as the primary concepts & force-densities only as optional
> > derivatives. The resulting mathematical equations appear identical to
> > Maxwell's but the underlying model is very different. Your assumption
> > of the necessity of Lagrangian densities for conservation is also
> > flawed.
>
> What makes the Lagrangian formalism flawed? It is simply a
> generalization of Newtonian mechanics. There is no new physics
> contained, it is just incredibly easier to get to the physics.
>
> If you have a conserved quantity, the derivative of the Lagrangian
> with respect to that quantity will be zero. How much more simple can
> you get? Noether's theorem makes it even more rigorous. Every symmetry
> buys you a conserved quantity.
>
> Good luck doing that without reference to a Lagrangian.
Did he really just question Lagrangian mechanics???
Wow. Good luck working with anything more complex then a slide rule, Maxwell.
Bilge
On 2007-06-09, maxwell <sp...@shaw.ca> wrote:
> On Jun 5, 10:28 pm, Bilge <dubi...@radioactivex.sz> wrote:
> You seem well acquainted with the current orthodox views of modern
> physics; from this I infer that you have passed your exams, as I did
> (a good memory is a necessary requirement for any professional). If
> all you want from life is to be 'another brick in the wall' then you
> will not have to challenge any of the thoughts that your teachers have
Sorry, but your patronizing reply is based on a logical fallacy in which
you assume that because I think you are wrong, I must be satisfied with
the stagnation of science for lack of originality.
> impressed upon you - I'm sure there are still jobs out there that only
> require that you repeat what you have been told. However, if anyone
> desires to make any new contributions to science then I would strongly
> encourage them to start to think about the assumptions that underlie
> all that they have been taught.
And you think that makes you unique? I hate to inform you of this,
but the majority of physicists question assumptions. The way in which
they differ from what you are trying argue is that the rest of us
recognize that whatever assumptions one makes, what follows must match
the data obtained from experiment.
>For example, it is a historical fact
> that the notion of one-to-one mapping between physical reality (the
> subject of physics) & the representations (mathematical physics) that
> have been used to try to map our ideas of reality have followed the
> continuum hypothesis for many centuries. The recent attempts at
> 'quantum foam' do not deviate from this pattern unless all of the
> mathematics used therein is discrete. Any discussion of the 'reality'
> of the world implies metaphysics
No. Metphysics only comes into play when someone invents entities
to conform to preconceptions, treats those entities as real objects
and then explains those objects away by inventing new laws of
physics which apply only to those entities to hide them from observation.
I call that, ``stupid,'' since I think nature ought to have to conform to
its own rules when deciding what it means form something to be real.
>- a subject that is NOT discussed in
> modern physics but is intrinsic to the distinction between math &
> physics.
Oh. You are one of THOSE. Look, you might think there is no physical
meaning to the mathematics, but that is your difficulty, not mine, and
certainly not that of every physicist.
>I prefer to make my metaphysical assumptions explicit, in
> the natural philosophical tradition of Newton.
Had Newton been able to do that, he could have derived all of classical
mechanics from galileos principle. Instead, he posed three laws which
quantified his observations, but contained hidden assumptions and left
questions like the definition of kinetic energy to be determined later by
others and the proof of conservation laws until the beginning of the
twentieth century. It wasn't his fault. What made newton great is that he
did what he did despite lacking lacked the mathematics required to do all of
those things from one simple assumption (or even recognize it as THE
assumption). The reason that no one bothers with the metaphysics that
newton actually wrote in the principia is that, taken literally (rather
than as a naive attempt to justify his physics), it contradicts the
physics he wrote down in the principia.
If there is one thing that makes me wretch and conclude that someone
is a complete crackpot, it's an argument which starts with the assumption
that somehow physics has abandoned the idea of describing nature as nature
is in order to argue that nature is something that fits some personal
philosophy which has no physical import while unexplaining well known
physics.
Physics is what it always has been - an attempt to explain nature
on NATURE'S terms. Nature isn't obligated to make itself conceptually
simple, especially if you disregard the concepts which would make it
simple because you want everything to be made of metaphysical LEGO
blocks that are just little versions of macroscopic ones. Either deal
with the idea that some people (such as myself) consider modern theories
to make physical sense, or be resigned to the fact that those same people
are going to dismiss what you have to say because you've based it on
the blatantly false premise that modern theories are unphysical.
[...]
> Your commonly repeated claim that classical electromagnetism (CEM)
> does not cover the Aharonov-Bohm effect (actually QM, not CEM) does
> not include the work done by that venerable UK professor of Electrical
> Engineering, C. John Carpenter, who (like myself) is dedicated to the
> Lorenz Current-Potential (or retarded potential) model of EM.
Sorry, but the potentials have no physical significance in a
classical theory.
>You can
> track down his papers on Professor McDonald's superb history of
> electromagnetism site at: http://puhep1.princeton.edu/~mcdonald/examples/EM/
I can track down lots of things, but I'm not about to search through
several hundred papers, listed by a vague filename to try and figure out
what you think your point is. If you had a point, you could have spent your
effort making it rather than posting a verbose narrative on what you think
science is and a reference to directory with a lot of meaningless file
names. Make your own point, since your failure (or inability) to do so only
makes me think you don't have one. I'm not going to try and second guess
what it might be.
Bilge
> In article <1181423457.3...@d30g2000prg.googlegroups.com>,
> Eric Gisse <jow...@gmail.com> wrote:
>
>> > In this Newtonian-style, action-at-a-distance approach (that traces
>> > its roots back to Gauss, then Riemann & Helmholtz) the potentials are
>> > introduced as the primary concepts & force-densities only as optional
>> > derivatives. The resulting mathematical equations appear identical to
>> > Maxwell's but the underlying model is very different. Your assumption
>> > of the necessity of Lagrangian densities for conservation is also
>> > flawed.
>>
>> What makes the Lagrangian formalism flawed? It is simply a
>> generalization of Newtonian mechanics. There is no new physics
>> contained, it is just incredibly easier to get to the physics.
>>
>> If you have a conserved quantity, the derivative of the Lagrangian
>> with respect to that quantity will be zero. How much more simple can
>> you get? Noether's theorem makes it even more rigorous. Every symmetry
>> buys you a conserved quantity.
>>
>> Good luck doing that without reference to a Lagrangian.
>
>
> Did he really just question Lagrangian mechanics???
>
> Wow. Good luck working with anything more complex then a slide rule, Maxwell.
Actually, the slide rule is out, too. Without having the means to define a
stright line by its correspondence to a physical measurement, one cannot say
whether linear scale x is a logarithm of some scale x' or whether the linear
scale x' is the exponential of some scale x. One has to assign numbers
to positions based on some principle which assures self-consistency.
maxwell
Dear Bilge (appropriate self-description), sorry I touched on one of
your raw nerves. Freud would have had a field-day (pun intended)
analysing (another pun) why you react explosively to opinions that
differ from your own.
Bilge
On 2007-06-11, maxwell <sp...@shaw.ca> wrote:
>
> Dear Bilge (appropriate self-description), sorry I touched on one of
> your raw nerves. Freud would have had a field-day (pun intended)
> analysing (another pun) why you react explosively to opinions that
> differ from your own.
He would have more of a field day analyzing the ego of someone who
confuses my intolerance of pseudoscientific bullshit with a mere
difference of opinion.
Rock Brentwood
> In sci.physics Michael Moroney <moro...@world.std.spaamtrap.com> wrote:
> > Is there a gravitational equivalent to magnetism?
> Yes. It's called "gravitomagnetism," and for weak fields it has
> roughly the same relationship to ordinary Newtonian gravity as
> magnetism has to the Coulomb interaction.
It's present in Newtonian spacetime, as well. The best way to see this
(directly) is to adopt a 3+1 Newton-Cartan frame (e^0, e^1, e^2, e^3)
= (T, X, Y, Z) = (T, X^1, X^2, X^3) and write out the various items in
it.
Defining (S_1, S_2, S_3) = (Y^Z, Z^X, X^Y), the structure coefficients
arise from
dT = sum (B^a S_a + D_a X^a ^ T)
dX^a = sum (A^{ab} S_b + C^a_b X^b ^ T)
for (a,b = 1,2,3)
Reference Link:
http://federation.g3z.com/Physics/index.htm#Solutions
Section 3.1 of the attached PDF has all the details
For a metrical connection (i.e. one that respects the rank 1 metric
g_{mn} and (independent) rank 3 dual metric g^{{mn}), one has for the
connection 1-forms w^m_n the following conditions
w^0_b = 0
w^a_0 = alpha^a = sum (G^a_b X^b + E^a T)
w^a_b = -w^b_a
(w^2_3, w^3_1, w^1_2) = (sigma^1, sigma^2, sigma^3)
sigma^a = sum (F^a_b X^b + H^a T)
In the absence of torsion, one finds from the Cartan structure
equations
de^m + sum w^m_n ^ e^n = 0
that
B^a = 0, D_a = 0
while
A^{ab} = -F^b_a + delta^{ab} sum F^c_c.
as well as an additional relation linking the C, G and H coefficients.
The fields E and H are unrelated to the structure coefficients and are
independent parameters of the gravity field. This contrasts to GR,
where one has a relation (D = E/c^2, and B related to the anti-
symmetric part of G with a factor of 1/c^2; B also vanishes for ADM
decompositions).
Substituting these into the geodesic equation, one finds that E is the
"electric" part of the gravity field (in the Raychaudhuri equation, it
will satisfy an equation of the general form div E + (...) = 4 pi G
rho, where rho is the mass density); and H the "magnetic" part of the
field. The G coefficients relate to the shear (the symmetric traceless
part), the rotation (the anti-symmetric part) and volume expansion/
contraction (the trace). The C coefficients share the same symmetric
part, with signs reversed, as G. The anti-symmetric part has a
correction to G's rotation from the H field. In effect, the H field
behaves as a kind of intrinsic component to the rotation. The A and F
fields relate to the purely spatial part of the curvature.
The only difference with respect to GR will be in the appearance of
non-zero B and D coefficients (except that non-zero B's and D's can
arise in Newtonian gravity if the torsion is non-zero); and additional
terms in the field law (e.g., the Raychaudhuri equation acquires a
correction rho -> rho + 3 p/c^2.)
Rock Brentwood
wrote:
> > If you are looking at linearized gravity around a flat background, the
> > analog of A_u is h_{uv}, the deviation of the metric from the Minkowski
> > metric. The analog of F_{uv} is the Christoffel connection, or more
> > precisely the set of components Gamma^i_{0j}. These are antisymmetric
> > (in this approximation); this is completely different from the symmetry
> > of h.
The connection one-form Gamma^i_j is anti-symmetric in any frame whose
associated metric is constant. The usual approach is to adopt a
"moving frame" that is orthonormal. There's no need to approximate
anything.
The appropriate analogies (which are not mere analogies) is A <->
(Gamma & e), F <-> (R, tau) where e and tau are, respectively, the
frame and torsion.
But note the proviso way down below (about what's analogous to the
dual Maxwell fields D and H).
> It is disturbing - but standard - to see the tensor F_{uv} compared to
> that which is not a tensor, Gamma^i_{0j}.
That's why. (But, again, note the proviso way down below, which will
put back toward the idea of Gamma as field).
The analogy to Maxwell is tied directly to the structure equations
dA = F
becomes
de + w^e = tau & dw + w^w = R
(skipping index-writing and using w to denote the connection one-form)
The analogy to dF = 0 becomes
d(tau) + w^tau = R^e & dR + w^R = R^w
The fields are F = (E, B) and potentials A -> (-phi, A) in ordinary
3+1 notation. The current J -> (rho, J) and dual fields G = (H, D)
arise from varying the Lagrangian
DL = DA ^ J - DF ^ G.
Since it is assumed in the variational problem that Dd = dD, then
DF ^ G = D(dA) ^ G = d(DA) ^ G = d(DA ^ G) + DA ^ dG
(noting that the product rule for 1-forms A is d(A^(...)) = dA^(...) -
A^d(...).). Tying this together, one gets
DL = -d(DA ^ G) + DA ^ (J - dG).
The total differential devolves to a boundary term (whose variation is
0, since boundary values are assumed fixed in a variational problem),
and the remainder under a stationary variation problem reduces to the
law
J = dG.
The current conservation is the derived identity is
dJ = d^2 G = 0.
The analogous case for gravity is that the dual fields are given by
the Lagrangian's variation
DL = De ^ P + Dw ^ S - D(tau) ^ T - DR ^ U.
The resulting field equations can be obtained similarly
D(tau)^T = D(de + w^e) ^ T = d(De ^ T) + De ^ dT + Dw ^ e ^ T - De ^ w
^ T
DR^U = D(dw + w^w) ^ U = d(Dw ^ U) + Dw ^ dU + Dw ^ w ^ U + w ^ Dw ^ U
Expressed in terms of the w-"covariant" derivative operator d_w, this
becomes
D(tau)^T = d(De^T) + De ^ d_w T + Dw ^ e ^ T
DR^U = d(Dw^U) + Dw ^ d_w U.
Substituting this into the Lagrangian, you get
DL = -d(De ^ T + Dw ^ U) + De ^ (P - d_w T) + Dw ^ (S - e^T - d_w U).
The analogues here are the "momentum current" P (which underlies the
stress tensor) and "angular momentum current" S (which underlies the
spin tensor). The correction on the last term
Dw ^ (S - e^T - d_w U) --> Dw ^ (J - d_w U)
corresponds to the decomposition of total angular momentum (J) in
terms of an orbital part (that arising from the anti-symmetrization of
e^T) and spin part (S). The resulting equations
d_w U = J & d_w T = P
are analogous to dG = J, while the "conservation laws"
d_w J = -R ^ U & d_w P = -R ^ T
(roughly) are analogous to dJ = 0.
However, at this point the analogy breaks down. The Einstein law
(unlike Maxwell or Yang-Mills) operates in the absence of a background
metric (because it *makes* the metric). The "field" terms therefore
only come in to the FIRST order; while those in the Maxwell theory
come in through the second order, leading to a duality relation (G =
*F). Duality is an expression of the metric.
There is no analogous relation (T = *tau, U = *R) here. Instead, since
R appears linearly in the Lagrangian, U reduces to combinations
involving only e! T reduces to nil! Matter may contribute to each. For
pure gravity, S reduces to nil, while P is non-zero.
(Yup, that's right the stress tensor term P is non-zero for pure
gravity. There is a stress tensor in gravity! Surprise. Even bigger
surprise: it's -k G^{mn}, where k is the coupling constant! That fits
Barbour's analysis of the Mach principle in Newtonian Gravity and
Quantum Gravity, where he showed that (Kinetic energy + Potential
energy = 0) yields a gauge for time-parametrization, while (total
momentum = 0, total angular momentum = 0) yields a gauge for the
inertial frames. Here, the analogous relation is (P_{matter} +
P_{gravity} = 0; i.e., the Einstein equation T_{mn} - k G_{mn} = 0).
That's the analogue of the Barbour gauge T + V = 0.)
This means that T and U are NOT the analogues of the dual fields G <->
(H, D). Instead, they are SUPERPOTENTIALS -- i.e., potentials from
which are derived the source terms (P and J).
Einstein gravity is of the same mould as Chern-Simons gauge theory;
not Yang-Mills gauge theory.
Because of the linearity in the analogues of the F field, this drops
down the designations (G <-> dual of F <-> tau & R) downa level. The
analogues of G now reside with the Gamma's.
That is, the analogues to (D, H) are with the connection coefficients
and frame; not the torsion and curvature coefficients. All this is
while, at the same time, the analogues of (E, B) remain with the
torsion and curvature.
This is already seen in my previous article, discussing the 3+1 quasi-
Galilean frame. What I called the E and H fields there come out of the
connection coefficients, rather than the curvature coefficients. These
are (despite the notation I used for them) the analogues of the
Maxwell D and H fields. The analogue of the Gauss law (div D = rho) is
now the Raychaudhuri equation (div E + ... = k (rho + 3p/c^2), where p
is the pressure).
Rock Brentwood
With respect to the 3+1 frame I defined in a previous article, these
are the H^1,H^2,H^3 connection coefficients; all of which are present
in both the Einsteinian and Newton-Cartan gravity. They're non-zero in
the Galilean limit.
> Think about the Lorentz versus the Galilean symmetry. At low speed, time
> cannot rotate into space. The transition from one to the other would
> be smooth, with initially a wee bit of mixing of time and space.
This is also clarified in the decomposition I related. For the time-
like member of the ("quasi-Galilean" tetrad), T, one has the exterior
derivative
dT = B^1 Y ^ Z + B^2 Z ^ X + B^3 X ^ Y + (D_1 X + D_2 Y + D_3 Z) ^ T
which are related to the E and G connection coefficients by
B^1 = 2/c^2 (G_{23} - G_{32}; etc.
D_1 = 1/c^2 E_1,
with the anti-symmetric part of the G coefficients being related to
the "rotation", as it is called in the fluid dynamic interpretation
underlying the Raychaudhuri equation.
These go to 0 in the Galilean limit; though they may acquire non-zero
values in the presence of torsion in Newton-Cartan gravity.
(Again, section 3.1 of the link
http://federation.g3z.com/Physics/index.htm#Solutions
works out the details of the quasi-Galilean tetrad decomposition)
> For the case at hand, with the antisymmetric DelxA for the gravitomagnetic
> field, like charges would repel, exactly has happens for EM. For the
> full theory based on h, like charges attract.
The E field above follows the Raychaudhuri equation (div E + ... = +4
pi G (rho + 3p/c^2), where rho is the mass density. Note the plus.
This occurs because E is the Raychaudhuri "acceleration". This is
defined as the acceleration of the frame *with respect* to geodesic
motion. For the Earth, with a fixed frame (that is, "fixed" with
respect to altitude), E points radially outwards with a strength
inversely proportional to r^2.
Both pressure and mass contribute positively. So like attracts like.
Spin, however, contributes negatively [1] (a kind of localized
centrifugal effect). This, in fact, has been used in cosmological
solutions to offset the singularity at the Big Bang (that is, the
centrifugal effect dominates near the singularity and there is no
singularity). The effect blows up rapidly with increasing matter
density.
This, of course, directly ties into the comment made a few years ago
[2], appended below...
Notes:
[1] According to Einstein-Cartan gravity. There is, yet, no direct
confirmation of the effects of spin on gravity.
[2] http://groups.google.com/group/sci.physics/msg/94e311f375f3867b?dmode=source
Excerpt:
sci.physics
1993 March 22
Remember the kiddie question: why doesn't the sun fall? Remember the
pat answer always given in a science class? Well, guess what? The
question's back!
..
So, in answering the original question you invariably come to one of
four possibilities: Universal Pressure (Cosmological constant),
Universal Contraction and a Beginning & End, Universal Expansion and a
Beginning, or Universal Rotation.
..
Now the latter possibility is interesting in its own right (and it
proves that hindsight is NOT always 20/20). And I pose it as both a
question and prediction. What if, instead of the Big Bang model, what
we actually have is a Universe with a net angular momentum which
periodically expands and contracts, reentering a new period of
expansion just when it's contracted far enough that the angular
momentum counteracts gravity.
The solutions that were first posed to the Einstein Equations for the
Big Bang model correspond (very closely) to the cases of radial motion
under the influence of gravity for velocities less than/equal to/
greater than the escape velocity. What was never considered, on the
other hand, were the cases that correspond to elliptical/parabolic/
hyperbolic orbits -- where there is also a net angular momentum.
Perhaps someone has constructed such a solution to the Einstein
Equations extending the Big Bang model in this manner?