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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?
Jan
31
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Thks Jerry; think I can advance things now. Agreed, Hamiltonian~energy/mass & is conserved. Maybe my use of "components" (Schwarzschild MAE as contra-ex) confused things? Think of 2 universes, each with an asymptotically flat infinity; as two disconnected "components" obviously the ADM sum is conserved. Now excise a spatial 3sphere from each and identify the 2sphere surfaces; there is now one manifold with two "ends". Each infinity seems to contain the other, so surely integration over either should give the same result for the total content? Consensus says no, my Q is why?
Jan
30
comment Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
@JerrySchirmer Thx for the interest. If I understand that correctly, I'm not sure we would disagree: the Hamiltonian would conserve the mass. The question is why that mass is the sum. The integration can be performed over any boundary so why are they not all equal? I didn't say it that boundaries had to be causally connected (the MAE Schwarzschild example is a counter example), the question is really about those cases in which they are causally connected and there seems to be no physical justification for ADM differences. Interesting that no one has jumped in with an obvious "answer" yet
Jan
29
asked Uniqueness or multiplicity of ADM masses for spacetime manifolds with more than one “end”?
Jan
29
revised Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Added caution re non-standard coordinate numbering in referenced notebook "Christoffel.nb"
Jan
22
revised Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Hmmm... that g was a function of x got lost in (6) somehow; fixed now. Looks like a Lagrangian now!
Jan
22
revised Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Forgot to add the dtau for integration in (6)
Jan
22
comment Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
@KyleKanos Thank you; I am now digesting the edits. NB I had tried using Mathematica's "copy as LaTeX" but that didn't work well for me :(
Jan
22
answered Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Jan
20
awarded  Supporter
Jan
20
comment Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Thank you very much. It's a good place to start; on the one hand I had hoped it would be simpler, on the other hand I don't feel bad that I had overlooked something very simple! I had perused that paper but hadn't got as far as p11 since I had hoped that eq. 1.1.17 on p7 might have been the key. Maybe I can get Mathematica to do the hard work with the Christoffel symbols etc. :)
Jan
20
asked Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric
Jul
19
awarded  Nice Question