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Mar
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@Timaeus - noting "pls avoid extended discussion in chat" should we not communicate further there, with conclusions to be incorporated here?
Mar
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@Timaeus Re moving mass around. Yes, it is a spatial characteristic, and I'm well aware of the problem of energy conservation in GR. I could speak instead of different spatial manifolds with the nominal same matter content distributed differently rather than the same manifold at different times but that seems not to help much. Of course if the metric is not a function of t one does get energy conservation (there is a timelike Killing field) but then one is considering a special case. After digging into what ADM mass is etc. the "how it is used/useful" question might be next. –
Mar
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@Timaeus. Re connectedness. Limiting discussion to purely Riemannian spatial (sub)manifolds would be of little value when the larger context is the application of ADM mass in issues of GR, such as in the possibility of creating closed timelike curves by joining two such spacetimes as described at their asymptotic infinities; in which context absence of spatial topology change in time is important, per Geroch's theorem. The larger context had been unstated to avoid confusing the issue. HTH
Mar
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@Timaeus. Disjoint surfaces - agreed & fixed, thanks
Mar
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@Timaeus Physical meaning - I don't follow the argument. Can you elaborate? As I read it, the meaningfulness of "mass of an end" seems to be assumed, and if you meant "compare the integrated value of one end with..." I don't see the meaning of the difference.
Mar
31
revised Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Accepted Timaeus "disjoint" comment
Mar
30
revised Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Fixed punctuation in notation section to improve readability
Mar
30
revised Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Added to conclusion: individual end integrals not necessarily equal
Mar
30
revised Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Fixed typos (on Minkowski metric, ...)
Mar
30
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
New answer below; comments welcome
Mar
30
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
New answer below; comments welcome
Mar
30
answered Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Mar
18
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
"@Timaeus" (how do you make that actually appear?) forgot to namecheck you on the last comment... PS Re: "If you think geodesic incompleteness means causal influence" did you mean "geodesic completeness means causal influence" or "geodesic incompleteness doesn't mean lack of causal influence". I'm afraid the original wording doesn't make sense to me. Typo?
Mar
18
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Re: surface vs volume. From scholarpedia (easier to copy TeX) $ P^0 \equiv E = -\int d^3r \nabla^2 g^T =- \oint dS_i g^T \!_{,i} =\oint dS_i (g_{ij,j} - g_{jj,i}) $ Then in ADM's original paper [1] see 1st Eq 2.1 (surface integral) and then Eq 3.2 on p1002 where it says "all these expressions then reduce to Eq. (2.1). The latter can be converted to a volume integral [Eq. 3.2] ... and the last member is correctly the energy [i.e a volume integral]" [1]R. Arnowitt, S. Deser, and C. W. Misner, Phys. Rev. 122, 997 (1961). Seems explicit to me. Can you explain why am I "100% wrong"?
Mar
17
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
I appreciate the time but... ADM mass is supposed to be an integral over a surface that encloses a volume - that's its utility; geodesic completeness just affects what can causally count as inside that volume and hence what can affect the surface values (NB I didn't say anything about trying to add masses of atoms as such). With multiple ends this enclosure could be thought of as enclosing other ends, possibly "compactified" (I thought eg by mapping down to a point). I can't agree with your reading of Bartnik, which seems explicit - if he had meant "of an end" I think he would have said so.
Feb
14
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Thanks; I'll dig up the references later - already have the talk open. So... why does everybody else state that each end has independent mass?
Feb
11
awarded  Notable Question
Feb
9
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Thanks! I'm flattered you seem to think I know more than I do, but that was a little over-compressed for me. However, if I do follow: a standard inter universe wormhole spacetime isn't discontinuous, so the complete $ \gamma_{ij} $ etc. on either asymptotic region should yield the whole manifold so there IS a unique ADM? But if there's a discontinuity bets are off. Is that right? Which specific theorems are being applied in your argument?
Feb
1
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Thanks for taking extra time to think about it. I don't think standard wormhole spacetimes have any such problems; I do think however that some people construct joins and deliberately build in discontinuities (e.g. Visser, a wormhole from joining two Schwarzschild geometries) but their existence doesn't in general seem to me necessary (& makes me wonder if their discontinuities are legit). I look forward to hearing you thoughts on the matter... and I have to say it is reassuring that a PhD relativist needs to think about it - no wonder I've been struggling.
Jan
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
(continued) How is this reconciled with Bartnik who says ADM mass is a manifold invariant? If it is, then it shouldn't matter over which infinity the ADM is calculated. The joined universe is a perfectly good manifold... isn't it?