The definition of Komar Mass in GR is associated with one asymptotically flat end. However, a hypersurface may contain more than one end, such as the spacelike Einstein-Rosen bridge in Kruskal Spacetime which has two ends.

Therefore, the total mass is conserved by the Stokes theorem and current conservation, but how is Komar mass associated with each end conserved?

In the example of Kruskal Spacetime with parameter $M$ in the Schwarzschild metric, the total mass integral is certainly zero for the vaccume solution, with the Komar mass associated with two ends are $M$, $ -M$ respectively. So we know that $ M+(-M)=0$ is conserved, but why $M$ or $-M$ is conserved?

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    $\begingroup$ If "Kruzkal spacetime" is just Schwarzschild, why would the Komar mass be $0$? $\endgroup$ – Ryan Unger May 5 '16 at 0:41
  • $\begingroup$ @ocelo7 the komar mass is not 0, but why is it conserved? $\endgroup$ – Shadumu May 5 '16 at 8:52
  • $\begingroup$ So what is $M+(-M)=0$ supposed to mean? It's conserved because the spacetime is static. $\endgroup$ – Ryan Unger May 5 '16 at 13:19
  • $\begingroup$ @0celo7 A constant $t$ slice through Kruskal will have two ends. The Komar masses associated with the ends are $M$ and $-M$. It seems that the usual argument for conservation of Komar mass applies to only the sum of the two, not each individually. $\endgroup$ – knzhou Apr 19 '18 at 14:58

Komar mass is associated to one asymptotically flat end. So a wormhole has two Komar masses, one for each side, and in principle they can even be different! Indeed the simplest wormhole you can imagine is simply two copies Schwarzschild spacetimes glued together at the would-be horizon.

As an aside, even time can run differently on the two sides (often termed universes) of the wormhole. That's why it's very easy to build time machines with wormholes!

  • $\begingroup$ What do you mean by saying that time runs differently on both sides? $\endgroup$ – Khushal Apr 29 '18 at 10:23
  • $\begingroup$ If one universe is denser than the other, time in the denser one will run slower. If you jump into a rotating black hole you can even enter a region where time runs backwards, if you survive the passage through the cauchy horizon. $\endgroup$ – Yukterez Jul 3 '18 at 22:40
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    $\begingroup$ @Khushal , an observer in one universe will see the clock in the other universe running faster/slower. $\endgroup$ – Rexcirus Jul 4 '18 at 19:50

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