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Riemann tensor for metric ds^2 = dt^2 - dx^2 - a(t)^2 dy^2

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1. Suppose we have a 2+1 dimensional spacetime described by the line element
ds^2 = c^2dt^2 - dx^2 - a(t)^2 dy^2
where a(t) is an increasing function of time.

(a) How do you interpret the spatial part of this geometry? In other words, can you describe its shape?
(b) Calculate (by hand) the nonzero components of the curvature tensor R . You may appeal to symmetries of the tensor (and the Christoffel symbols) when appropriate.
(c) Determine the condition on the function a(t) for which the spacetime is flat.

See attached file for full problem description.

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Solution Summary

A detailed solution is given. The Riemann tensor for metric functions are analyzed. Condition functions are determined.

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