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Science Forum Index » Physics - Research Forum » Query about non-symmetric metric tensor
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| JimJast |
 Posted: Mon Mar 31, 2008 2:30 pm |
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Can someone please explain how the physics of a non-symmetric metric
tensor would be different from that of a symmetric metric tensor.
Especially, with Einstein's idea that we have to make g_ik <> g_ki
expressed in "On the Generalized Theory of Gravitation" (Scientific
American, April 1950). How would one interpret g_ik versus g_ki and
the physics associated with each?
Thanks,
-- Jim |
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| Oh No |
| Posted: Tue Apr 01, 2008 5:23 am |
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Thus spake JimJast <jim_jastrzebski@yahoo.com>
Quote: Can someone please explain how the physics of a non-symmetric metric
tensor would be different from that of a symmetric metric tensor.
Especially, with Einstein's idea that we have to make g_ik <> g_ki
expressed in "On the Generalized Theory of Gravitation" (Scientific
American, April 1950). How would one interpret g_ik versus g_ki and
the physics associated with each?
Thanks,
There has been a bit of discussion of non-symmetric metrics on s.p.f.
recently and I have scanned a few papers. It seems clear that the phrase
"non-symmetric metric" is something of a misnomer. Einstein is talking
of a generalisation of a field. After this generalisation g is not a
metric (metrics are symmetrical) and, according to the literature and my
own understanding, it is by no means obvious what its relation to the
metric is. I think you should regard this field as consisting of
speculative investigations into unknown physics.
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
http://www.teleconnection.info/rqg/MainIndex |
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| Rock Brentwood |
| Posted: Tue Apr 01, 2008 5:23 am |
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On Mar 31, 7:30 pm, JimJast <jim_jastrzeb...@yahoo.com> wrote:
Quote: Can someone please explain how the physics of a non-symmetric metric
tensor would be different from that of a symmetric metric tensor.
Especially, with Einstein's idea that we have to make g_ik <> g_ki
expressed in "On the Generalized Theory of Gravitation" (Scientific
American, April 1950). How would one interpret g_ik versus g_ki and
the physics associated with each?
In order for a non-symmetric metric to be developed that has geometric
meaning, the 2 indices have to have different roles in whatever is
used to define them.
One way is to treat the gravity field as a gauge field with a
connection given by
omega^a_b = Gamma^a_{mu b} dx^{mu}
omega^a = Gamma^a_{mu} dx^{mu}
omega_b = Gamma_{b mu} dx^{mu}
subject to the generalized Cartan equations
d omega^a_b + omega^a_c ^ omega^c_b = Omega^a_b
d omega^a + omega^a_b ^ omega^b = Omega^a
d omega_b - omega^a_b ^ omega_a = Omega_a
where the indices (a, b, c) range over 1, .., n and represent some
kind of internal indices, while mu represents space time coordinate
indices.
The 2-forms
Omega^a_b = 1/2 R^a_{b mu nu} dx^{mu} ^ dx^{nu}
Omega^a = 1/2 T^a_{mu nu} dx^{mu} ^ dx^{nu}
Omega_b = 1/2 T_{b mu nu} dx^{mu} ^ dx^{nu}
then represent the curvature, torsion and a kind of index-lowered dual
torsion, respectively.
In this geometry there is no metric at the outset to raise and lower
indices with. However, one can be created by
g_{mu nu} = Gamma^a_{mu} Gamma_{a nu}.
The indices are no longet given symmetric treatment. The 2-form
omega^a ^ omega_a is then the "anti-symmetrization" 2-form for the
metric
omega^a ^ omega_a = 1/2 (g_{mu nu} - g_{nu mu}) dx^{mu} ^ dx^{nu}. |
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| Igor |
| Posted: Tue Apr 01, 2008 9:35 pm |
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On Mar 31, 8:30=A0pm, JimJast <jim_jastrzeb...@yahoo.com> wrote:
Quote: Can someone please explain how the physics of a non-symmetric metric
tensor would be different from that of a symmetric metric tensor.
Especially, with Einstein's idea that we have to make g_ik <> g_ki
expressed in "On the Generalized Theory of Gravitation" (Scientific
American, April 1950). How would one interpret g_ik versus g_ki and
the physics associated with each?
Thanks,
-- Jim
Because the Maxwell tensor is antisymmetric, Einstein hoped that the
antisymmetric part of the metric tensor could be related to it. He
ran into difficulties reproducing Maxwell's equations in this
geometry, even under approximations. Despite these setbacks, and his
other varied attempts at unification, he inexplicably and stubbornly
kept returning to the nonsymmetric field, but to no avail. |
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| Guest |
| Posted: Tue Apr 01, 2008 9:35 pm |
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JimJast wrote on Tue, 01 Apr 2008 00:30:20 +0000:
Quote: Can someone please explain how the physics of a non-symmetric metric
tensor would be different from that of a symmetric metric tensor.
Especially, with Einstein's idea that we have to make g_ik <> g_ki
Precisely Einstein was one of the authors who proposed modifications to
General Relativity by introducing a nonsymmetric metric tensor.
Quote: expressed in "On the Generalized Theory of Gravitation" (Scientific
American, April 1950). How would one interpret g_ik versus g_ki and the
physics associated with each?
g_ik = Symmetrical[g_ki] + non-symmetrical[g_ki]
The symmetrical part is interpreted as in General Relativity. The non-
symmetrical part receives different interpretations by different authors.
Take a look to
http://en.wikipedia.org/wiki/Nonsymmetric_gravitational_theory
--
http://canonicalscience.org/en/miscellaneouszone/guidelines.txt |
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| DRLunsford |
| Posted: Fri Apr 04, 2008 9:44 am |
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On Apr 2, 3:35 am, Igor <thoov...@excite.com> wrote:
Quote: Because the Maxwell tensor is antisymmetric, Einstein hoped that the
antisymmetric part of the metric tensor could be related to it. He
ran into difficulties reproducing Maxwell's equations in this
geometry, even under approximations. Despite these setbacks, and his
other varied attempts at unification, he inexplicably and stubbornly
kept returning to the nonsymmetric field, but to no avail.
This is a little too harsh. Einstein well understood the ambiguities
of his final program, which is based on "lambda invariance", a kind of
gauge invariance in the connection, which is taken to be asymmetric.
The main issue of mixing symmetric and anti-symmetric tensors as part
of what is supposed to be a single physical object, is reducibility.
The symmetric and anti-symmetric parts transform independently under
coordinate transformations, and any tensor equations obtained may be
invariantly split into symmetric and antisymmetric parts. Thus, unless
one is going to introduce a new geometry based on something other than
general covariance alone (which is exactly what Einstein was
attempting), the asymmetric object can be split into its symmetric and
antisymmetric parts, which then lead separate lives. Einstein was
forced to this choice because, frankly, the alternatives were
exhausted.
Contrast this with the Weyl geometry, where in addition to general
covariance, one has calibration invariance as a fundamental principle,
both issuing from the single idea of strict locality of the metric.
Here, the Ricci tensor gets an anti-symmetric part that is
proportional to Fmn. Under coordinate transformations, this lives
independently from the symmetric part, Rmn. But the entire geometry
includes scale transformations, and the absolute invariance of the
entire Ricci tensor under both, requires them to live as a unit, with
coupled field equations. Nevertheless, the objects appearing in these
equations are just Rmn and Fmn, independently. This is why the Weyl
theory is still viable as long as one does not demand a 4-d space or
gravitational equations of second order in gmn.
The same remarks apply to any theories with torsion taking a role
inside the energy-momentum tensor. Such theories are based on more
general viewpoints than simple covariance. We still have conservation
of energy and conservation of angular momentum as separate issues,
each embodying a specific aspect of the overall invariance.
-drl |
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| Rock Brentwood |
| Posted: Mon Apr 07, 2008 9:52 pm |
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On Apr 4, 9:44pm, DRLunsford <antimatte...@yahoo.com> wrote:
Quote: The main issue of mixing symmetric and anti-symmetric tensors as part
of what is supposed to be a single physical object, is reducibility.
....
The same remarks apply to any theories with torsion taking a role
inside the energy-momentum tensor.
This is the fallacy of the Conflation of Extensional Equivalents. It's
how people get into trouble with oversights and drawing the wrong
conclusions by applying what's true of A to an "equivalent" B where it
is, in fact, false.
It is never a valid argument to say that A can be reduced to B+C and
completely forgot about, even when they are equivalent in some sense.
The examples you brought up are a perfect case in point where this
kind of unfettered reductionism gets people into trouble.
A theory with torsion is *not* equivalent to one founded on oridinary
Riemannian geometry + torsion added in as a separate field -- even if
the structures are equivalent. Equivalent things can have different
properties. Here, the relevant property is energy positivity. The
criteria for the respective geometries are not equivalent. So, a
solution to the field equations which accords with these or other
extra conditions in one geometry will not necessarily accord with the
other, despite the equivalence of their structures.
Such theories are based on more
Quote: general viewpoints than simple covariance. We still have conservation
of energy and conservation of angular momentum as separate issues,
each embodying a specific aspect of the overall invariance.
-drl |
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