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May
23
awarded  Yearling
May
22
awarded  Nice Answer
May
22
awarded  Yearling
May
22
comment What would be the rate of acceleration from gravity in a hollow sphere?
@WetSavannaAnimalakaRodVance Thanks - esp. since I finally disentangled many confusions about "flat" inside (Ricci flat and Riemann flat) and at infinity (Ricci flat but not Riemann flat) etc. thanks to other answers here to more technical related questions.
May
21
answered What would be the rate of acceleration from gravity in a hollow sphere?
May
13
comment Wormholes & Time Machines - for *experts* in GR/maths
Quick acknowledgement: I'm sorry I missed this earlier; that is very extensive and much appreciated. More later. NB since the main discussion was 4 years ago things have moved on, following further research etc.: I have written it up in Phys Rev D style. I will recheck what I have written in the light of your comments - if there is still a gap to bridge, would you be willing to take a look at said "paper"?
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"?