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I know for asymptotically flat spacetimes, one can define the Arnowitt-Deser-Misner (ADM) mass of the spacetime

$$M_{ADM} = \frac{1}{16\pi} \lim_{r\to\infty} \int d^2\sigma r^a \gamma^{cd}\left( \partial_{c}\gamma_{ad}-\partial_{a}\gamma_{cd} \right).$$

I also know that for spherically symmetric spacetimes one can define the Misner-Sharp mass, which is quasi-local and can be defined in terms of gradients of the radial distance from the origin:

$$ M_{MS} = \frac{r}{2}\left(1-\sigma^{ab}\partial_ar \partial_br\right) $$

(for a recent discussion of both concepts, see e.g. https://arxiv.org/abs/2010.00069. See https://arxiv.org/abs/1004.1456 for a more in-depth discussion of the Misner-Sharp mass.)

My question is: in the limit of $r\to\infty$, do the two notions of mass agree for static, spherically symmetric spacetimes? The main difficulty I see is that the ADM mass is only well-defined for asymptotically flat coordinates, while the Misner-Sharp mass is defined regardless of the properties of the coordinates one is using, as long as the coordinate are spherically symmetric (and the spacetime is spherically symmetric). References would be useful as well as explanations!

EDIT: I should clarify that I am only thinking about spacetimes that could be transformed to being asymptotically flat. So, for example Schwarzschild spacetimes are "asymptotically flat", but there are coordinates (such as Painleve-Gullstrand coordinates) for that spacetime that are not asymptotically flat

$$ ds^2 = - \left(1-\frac{2m}{r}\right)dt^2 + 2\sqrt{\frac{2m}{r}}dtdr + dr^2 + r^2 d\Omega^2. $$

In this case, we cannot define an ADM mass, but we also "know" that if we could transform coordinates to, e.g. Schwarzchild coordinates, we could then determine "the" ADM mass of the spacetime, which would be $m$. We can also compute the asymptotic Misner-Sharp mass in Painleve-Gullstrand coordinates, and get $m$. So in this case the two coincide. Is this generically the case for spacetimes like these? I am pretty much convinced the answer is "yes", as asymptotically flat spacetimes basically need to asymptote to asymptotically Schwarzchild-like spacetimes in when the spacetime is spherically symmetric, but I would like some references about this to read as well, if they exist.

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  • $\begingroup$ @physics_researcher Can you please give me a reference for the formula of misner sharp mass for spherically symmetric spacetime, M$_{MS}$ in your question? $\endgroup$
    – apk
    Commented Sep 5, 2023 at 9:58
  • $\begingroup$ @apk This reference discusses the MS mass more arxiv.org/abs/1004.1456 (and includes citations to older literature). I've added that citation to the main text as well $\endgroup$ Commented Sep 6, 2023 at 13:17

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I found this nice paper by S. Hayward which answers my question. Long story short: the Misner-Sharp mass and ADM mass do reduce to one another at spatial infinity in asymptotically flat (and spherically symmetric, which we need to even define the Misner-Sharp mass) spacetimes.

In case the link ever goes bad, the article citation

S. Hayward, Phys.Rev.D 53 (1996) 1938-1949, arXiv:gr-qc/9408002

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