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karlzr
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I am reading Hayashi's paper titled "http://prd.aps.org/abstract/PRD/v19/i12/p3524_1" " (Phys. Rev. D 19, 3524, 1979).
In Riemann-Cartan geometry, one can choose the Weitzenbock connection rather than the well-known Levi-Civita connection, so that Riemann curvature tensor vanishes while the torsion tensor arises. In this case, the gravitational effect is attributed to the torsion tensor exclusively. This theory is dubbed teleparallel gravity since absolute parallel vectors can be defined in this geometry, because of the vanishing of curvature tensor, as the authors stated.
So I don't quite follow the authors' logic. Does vanishing curvature necessarily guarantee absolute parallelism?
In general(or in general relativity), no absolute parallelism exists because vectors parallel transported along a closed infinesimal parallelogram to the starting position will be different from the original one. The difference is dependent on both curvature and torsion tensor. Quoting Sean Carroll's textbook:
[tex][\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma-T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho[/tex]
If absolute parallelism exists, I believe the second term on the right-hand side of the above equation should vanishes simultaneously. So any comments on this issue?
In Riemann-Cartan geometry, one can choose the Weitzenbock connection rather than the well-known Levi-Civita connection, so that Riemann curvature tensor vanishes while the torsion tensor arises. In this case, the gravitational effect is attributed to the torsion tensor exclusively. This theory is dubbed teleparallel gravity since absolute parallel vectors can be defined in this geometry, because of the vanishing of curvature tensor, as the authors stated.
So I don't quite follow the authors' logic. Does vanishing curvature necessarily guarantee absolute parallelism?
In general(or in general relativity), no absolute parallelism exists because vectors parallel transported along a closed infinesimal parallelogram to the starting position will be different from the original one. The difference is dependent on both curvature and torsion tensor. Quoting Sean Carroll's textbook:
[tex][\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma-T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho[/tex]
If absolute parallelism exists, I believe the second term on the right-hand side of the above equation should vanishes simultaneously. So any comments on this issue?
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