# Do the Misner-Sharp mass and ADM mass asymptote to one another for static spacetimes

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)

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.

• You can have static spherically symmetric metrics that are not asymptotically flat (and therefore do not have a well-defined ADM mass), e.g. Schwarzschild-de Sitter. Would that be the type of counter example you are looking for, or did you want to exclude that case? Apr 15, 2021 at 6:39
• Not quite--I've clarified my question with an edit Apr 15, 2021 at 8:43
• Hi PHY314. Consider writing an answer to your question if you have found one. Aug 10, 2021 at 12:38
• Good idea @Nihar Karve; I've now removed my answer in the question in placed it as an official answer Aug 10, 2021 at 12:41