Joseph Voros
Oct12-06, 04:44 AM
Hi folks,
it's been over a decade since I did any physics research; I have recently
managed to get a little time to do some. I have always been interested in
spacetimes with torsion (eg Einstein-Cartan theory, Riemann-Cartan
manifolds, etc). Hence the question:
Does anyone here know of a recent review of work/progress in Einstein-Cartan
theory, which deals with the putative physics as well as the mathematics it
is based upon?
I know of the review article by Hehl et al in 1976, so what I'm looking for
is a more recent one which "fills in" what's been going on since then.
Something which uses both the component and abstract notations would be
quite handy (as I am a bit rusty with both :-)
I am particularly interested in affine connections which are not only
non-symmetric (of course!) but also metrical (in Schouten's terminology; ie
cov div of metric vanishes wrt to that connection). The most interesting
one, I think, is the so-called (according to Schouten) "metrical,
semi-symmetric connection" which looks like (in a coordinate frame):
\Gamma^a_{ij} = CHR^a_{ij} - \delta^a_i S_j + g_{ij}S^a
where latin indicies are spacetime ones, CHR = the usual Christoffel
connection built out of the metric, \delta is the Kronecker delta, and S_i
is a 4-vector field. The torsion thus looks like
S^a_{ij} = S_{[i \delta^a_{j]} = \half S_i \delta^a_j - \half S_j \delta^a_i
which is a particularly interesting special case, from the point of view of
someone who in interested in the interplay of gravity and electromagnetism
(like me :-).
In this geometry, the covariant derivative of the metric vanishes wrt to
both the usual Christoffel connection (naturally) as well as the full
connection defined above.
So, anyone know of any work along these lines?
Cheers,
Joe
it's been over a decade since I did any physics research; I have recently
managed to get a little time to do some. I have always been interested in
spacetimes with torsion (eg Einstein-Cartan theory, Riemann-Cartan
manifolds, etc). Hence the question:
Does anyone here know of a recent review of work/progress in Einstein-Cartan
theory, which deals with the putative physics as well as the mathematics it
is based upon?
I know of the review article by Hehl et al in 1976, so what I'm looking for
is a more recent one which "fills in" what's been going on since then.
Something which uses both the component and abstract notations would be
quite handy (as I am a bit rusty with both :-)
I am particularly interested in affine connections which are not only
non-symmetric (of course!) but also metrical (in Schouten's terminology; ie
cov div of metric vanishes wrt to that connection). The most interesting
one, I think, is the so-called (according to Schouten) "metrical,
semi-symmetric connection" which looks like (in a coordinate frame):
\Gamma^a_{ij} = CHR^a_{ij} - \delta^a_i S_j + g_{ij}S^a
where latin indicies are spacetime ones, CHR = the usual Christoffel
connection built out of the metric, \delta is the Kronecker delta, and S_i
is a 4-vector field. The torsion thus looks like
S^a_{ij} = S_{[i \delta^a_{j]} = \half S_i \delta^a_j - \half S_j \delta^a_i
which is a particularly interesting special case, from the point of view of
someone who in interested in the interplay of gravity and electromagnetism
(like me :-).
In this geometry, the covariant derivative of the metric vanishes wrt to
both the usual Christoffel connection (naturally) as well as the full
connection defined above.
So, anyone know of any work along these lines?
Cheers,
Joe