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Suppose we have a spacetime containing a traversible two-sheet wormhole. In other words, there are two otherwise independent asymptotically flat spacetimes glued together at the wormhole. The ADM mass of the spacetime is the sum of the ADM masses that would be measured by a distant observer on sheet 1 and a distant observer on sheet 2.

I guess there also has to be exotic matter to make the wormhole traversible, and I'm not sure if this violation of the energy conditions means that a lot of my normal mental assumptions are false.

Now suppose that a mass moves from sheet 1, through the wormhole, to sheet 2. The distant observer on sheet 2 simply sees the mass of the universe as having grown. This seems to them like a violation of the conservation of mass, but I guess they say, "Oh, there must be a wormhole in my universe." But in addition to this, if this effect propagates outward from the mouth of the wormhole at the speed of light, then it would have to look like monopole radiation. This seems wrong, because monopole radiation isn't a solution of the vacuum field equations.

So what's wrong with this analysis?

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  • $\begingroup$ I'm interested to see what answers this gets :) For now (simplistically): energy is not conserved in GR, so there is nothing to apparently violate to infer a wormhole exists. The mouth of a wormhole doesn't need to be spherical (see. g. Visser's cubic wormhole mouths), so it could in principle radiate, and I believe that a non-spherically symmetric mass distribution on the "other side" of a wormhole should be expected to lead to non-sphericity of the other mouth. You're right: no monopole radiation though [BTW I never fully grasped my answers re ADM mass & what's "in"/"out" of an "end" :| ] $\endgroup$ Commented Feb 2, 2021 at 10:36

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The ADM mass of the spacetime is the sum of the ADM masses that would be measured by a distant observer on sheet 1 and a distant observer on sheet 2.

This is wrong. For such spacetime there are two ADM masses, one for each of asymptotic regions, and they are conserved separately. This could be seen as a consequence of independence of asymptotic timelike Killing vector fields for each of the boundaries.

Now suppose that a mass moves from sheet 1, through the wormhole, to sheet 2. The distant observer on sheet 2 simply sees the mass of the universe as having grown.

No. After the mass entered the wormhole in sheet 1, wormhole's mass as measures by sheet 1 observer increases and stays that way. Also, the mouth of the wormhole in the second “sheet” would become lighter after “emitting” this mass, so that the total mass on the second sheet also will remain unchanged. The distant observer on sheet 2 would not notice the change in the mass of the whole system.

To make this more intuitive, consider electric charge $q$ in such a spacetime initially placed at sheet 1, without any other charges present. Field lines of electric field originate on this charge and escape to infinity of sheet 1. After the charge passes through the wormhole into the sheet 2 the field lines originating on the charge instead of escaping to infinity of sheet 2 all go through the wormhole and still escape to infinity of sheet 1. To an observer in sheet 1 the wormhole appears to have the charge $q$, while to observer in sheet 2, after emitting the charge $q$, wormhole now has charge $-q$. John Wheeler called this property of wormholes “charge without charge”.

Similarly to conservation of charge, which is visualized by electric field lines, conservation of ADM mass could be visualized via corresponding flux lines escaping to spacelike infinity, so that the fluxes at each of the sheets remain constant.

Let me offer a quote from a book

  • Visser, Matt, Lorentzian wormholes. From Einstein to Hawking, Woodbury, N.Y.: American Institute of Physics, 1995.

pages 111–112:

Consider the effect of a mass $m_i$, that is initially far away from the wormhole mouth in the “$+$” universe. Suppose now that this object traverses the wormhole and eventually settles down far from the wormhole in the “$-$” universe. Then the total ADM masses on both sides of the wormhole satisfy \begin{eqnarray} M^{+}_\text{total}&=&M^{+}_i + m_i = M^{+}_f; \tag{11.66}\\ M^{-}_\text{total}&=& M^{-}_i=M^{-}_f + m_f. \tag{11.67} \end{eqnarray} Notation: Here $M^\pm_i$ denote the initial masses and $M^\pm_f$, denote the final masses of the two wormhole mouths in the “$+$”and “$-$” universes, respectively.

That is, when massive objects traverse a wormhole they alter the mass of the wormhole mouths they pass through. The mouth that “absorbs” the object gains mass ($M^{+}_f > M^{+}_i$), while the mouth that “emits” the object loses mass ($M^{-}_i < M^{-}_f$). For generality I have permitted $m_i\ne m_f$; this allows for the possibility that the object traversing the wormhole may gain or loose some (kinetic) energy in the process.

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This suggests (but emphatically does not prove) the possibility of a fundamental limit on the total mass that can traverse a wormhole. For a sufficiently large net transfer of mass the final mass of the “emitting” wormhole mouth becomes negative. Under normal circumstances this would be considered a complete disaster. However, we shall soon see that violations of the average null energy condition are required just to hold the wormhole open in the first place. Because of this, the hypotheses used to derive the usual versions of the positive mass theorem do not apply to traversable wormhole spacetimes. There does not seem to be any guarantee that would prevent the total mass of a wormhole mouth from going negative, though such behavior would certainly make one feel very queasy [194].

If the mass of the “emitting” mouth of the wormhole does become negative one has the possibility of a runaway reaction: its mass now being negative the “emitting” mouth will now gravitationally repel the ambient medium. The “absorbing” mouth will continue to accrete matter, becoming ever more massive and attracting more of the surrounding material to itself. On the other hand, the “emitting” mouth will continue to lose mass, and its mass will become more and more negative, thereby reinforcing its gravitational repulsion of the ambient medium. For an example of the peculiarities this leads to, see [49].

The reference [149] is this paper, and [49] is this.

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