The isotropic-coordinate line element (26.16) describing the spacetime geometry of a Schwarzschild wormhole is independent of the time coordinate t . However, because g_tt = 0 at the wormhole’s throat, r̅ = M/2, the proper time

measured by an observer at rest appears to vanish, which cannot be true. Evidently, the isotropic coordinates are ill behaved at the throat.

(a) Martin Kruskal (1960) and George Szekeres (1960) independently introduced a coordinate system that covers the wormhole’s entire spacetime and elucidates its dynamics in a nonsingular manner. The Kruskal-Szekeres time and radial coordinates v and u are related to the Schwarzschild t and r by

Show that the metric of Schwarzschild spacetime written in these Kruskal- Szekeres coordinates is

where r(u, v) is given by Eq. (26.68).

(b) Draw a spacetime diagram with v increasing upward and u increasing horizontally and rightward. Show that the radial light cones are 45° lines everywhere. Show that there are two r = 0 singularities, one on the past hyperbola,

and the other on the future hyperbola,

Show that the gravitational radius, r = 2M, is at v =±u. Show that our universe, outside the wormhole, is at u ≫ 1, and there is another universe at u ≪ −1.

(c) Draw embedding diagrams for a sequence of spacelike hypersurfaces, the first of which hits the past singularity and the last of which hits the future singularity. Thereby show that the metric (26.69) represents a wormhole that is created in the past, expands to maximum throat circumference 4πM, then pinches off in the future to create a pair of singularities, one in each universe.

(d) Show that nothing can pass through the wormhole from one universe to the other; anything that tries gets crushed in the wormhole’s pinch off.

Transcribed Image Text:

dt = √-ds²