The line element of the wormhole geometry is: ds? = –dt? + dr? + (6 + m2) (do? + sino do?). (a) Write down all elements

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The Line Element Of The Wormhole Geometry Is Ds Dt Dr 6 M2 Do Sino Do A Write Down All Elements 1 (53.74 KiB) Viewed 35 times

The line element of the wormhole geometry is: ds? = –dt? + dr? + (6 + m2) (do? + sino do?). (a) Write down all elements of the metric and of the inverse metric. Make a general statement about which elements are zero. (b) What range of r values is necessary for this manifold not to have a boundary at r =0? At a fixed instant and fixed 0 = 7, sketch the 2D surface that this metrie describes. Describe in words what this 2D space looks like. Now for fixed t, but 0 € [0, 1], the resulting 3D space cannot be drawn, but using an analogy with the 2D diagram, describe what this 3D space "looks" like? (c) Find all nonzero partial derivatives of the components of the metric, gab... Make some general statements about which of these partial derivatives are zero. (d) Make some general statements about which Christoffel symbols are immediately seen to be zero from your results in (c). (e) Calculate le and from Eq. [5.75). Write down the result for 1 re

(f) Find the rest of the Christoffel symbols using the fact that the metric is purely diagonal: 13. = 39" (1938.4 +974,3 – 91.7) Bu and also using the symmetry of 1%-