Spacetime and Geometry · Problem 7

Gravitation — Sean Carroll

Physics — General Relativity · parallel-propagator-riemann-tensor

Problem

For this problem you will use the parallel propagator introduced in Appendix I to see how the Riemann tensor arises from parallel transport around an infinitesimal loop. Consider the following loop: a path from

A

B

C

D

and back to

A

, where the loop has

x^1 = 0

x^1 = \delta a

and

x^2 = 0

x^2 = \delta b

. Using the infinite series expression for the parallel propagator, compute to lowest nontrivial order in

\delta a

and

\delta b

the transformation induced on a vector that is parallel transported around this loop from

A

B

C

D

and back to

A

, and show it is proportional to the appropriate components of the Riemann tensor. To make things easy, you can use

x^1

and

x^2

as parameters on the appropriate legs of the journey.

Related theory

Riemann Curvature Tensor

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