Do wormholes have mass that depends on the spacial separation, the temporal separation, the relative motion of the two ends of the wormhole

Question

If there was a wormhole connecting two spatially/temporally separate locations, would it have mass that depends on the spacial separation, the temporal separation, the relative motion of the two ends of the wormhole?

Answer 1

One answer would be: yes, there has to be negative mass there.

The standard way of producing wormholes in general relativity since Misner, Thorne and Yurtsever's classic paper is to define a suitable metric that describes a wormhole (producing the left-hand geometric sides $R_{\mu\nu}-(1/2)Rg_{mu\nu}$ of the GR equations) and then see what kind of mass-energy tensor $T_{\mu\nu}$ that is required to get the right hand side to equal the left hand side. Typically this produces something that has negative mass-energy densities.

This is very hard to avoid, it follows from the Raychaudhuri equation that there has to be a breach of the averaged null energy condition if one wants a light-traversable wormhole (see Visser's excellent book on Lorenzian Wormholes for details). Many think this is a reason to discount traversble wormholes, others point at counterexamples in quantum field theory.

One can be devious and construct a wormhole spacetime where all the negative mass is in a polyhedral framework, leaving no extreme curvature or gravitational fields elsewhere.

To add to the confusion, if a mass or charge passes through a wormhole the original end increases in mass or charge correspondingly, and the other end decreases (as measured by the ADM mass) (p. 111-114 in Visser). So yes, some measures of mass would assign mass to wormholes by looking at curvature at a distance... but this mass may change depending on what passes through. Indeed, it might be that sending too much stuff one way depletes the wormhole with bad effects.

So another answer might be: wormholes could have any mass, depending on (1) how we decide to make the lefthand and righthand side of the GR equations equal each other, (2) how we define mass in general relativity, and (3) their history. This is not very satisfying, but wormholes are really underconstrained.

(Would this mass depend on the distance in space, time and velocity between the wormhole mouths? Yes... depending on how you define the wormhole metric.)