Komar mass for vacuum solutions I'm a little confused about the integral definition of the Komar mass, and how it gives the correct result for vacuum solutions.
As given by Poisson, the Komar mass is
$$
M_{Komar} = \int \left(2T_{\alpha\beta} - T g_{\alpha\beta}\right)n^\alpha\xi^\beta_{(t)}\sqrt{h}d^3y,
$$
where $n^a$ and $\xi^\beta_{(t)}$ are the normal vectors and timelike killing vectors, respectively, and $\sqrt{h}d^3y$ is the volume element of the hypersurface induced from the chosen slicing.
The thing I don't get is that in a vacuum solution, like Schwarzschild, $T_{\alpha\beta} = 0$, right?  So I don't see how the volume integral could ever give you $M_{Komar} = m$.
 A: Komar integrals are based on the concept of covariant conservation laws associated with every infinitesimal coordinate transformation. If we have a current as $J_i^\mu  = \xi _\nu ^{(i)}{R^{\mu \nu }}$ associated with the Killing vector field $\xi _\nu ^{(i)}$ (which is conserved, ${\nabla _\mu }J_{(i)}^\mu  = 0$), we can find a conserved charge associated with that current as (see Carroll, pages 248-254)
$$
Q[{\xi _{(i)}}] = \frac{1}{{4\pi G}}\int {{d^3}\vec y\sqrt h {{\hat n}_\mu }J_{(i)}^\mu } .
$$
Using the Einstein field equations (${R_{\mu \nu }} - \frac{1}{2}{g_{\mu \nu }}R = \kappa {T_{\mu \nu }}$), the above formula may be written as
$$
Q[{\xi _i}] = \int {{d^3}\vec y\sqrt h {{\hat n}_\mu }\xi _\nu ^{(i)}\left( {2{T^{\mu \nu }} - {g^{\mu \nu }}T} \right)},
$$
which is the volume-integral formulation that you mentioned. But, there is a singularity at $r=0$ which renders the both integrals difficult to evaluate. Clearly, this is ill-defined for the Schwarzschild metric at $r=0$, so one cannot calculate the volume integral. In addition, outside of the singularity we have $R=T=0$. This is a common question and, despite of several proposals in order to resolve this problem (see Carroll, pages 248-254), we don't know quite how to deal with it in a complete, satisfactory way. But, we can still convert the volume integral to a surface integral without singularity using the Stokes' theorem, yielding
$$
Q[{\xi _i}] = \frac{1}{{4\pi G}}\int {{d^2}\vec y\sqrt h {{\hat n}_\mu }{{\hat u}_\nu }{\nabla ^\mu }\xi _{(i)}^\nu }.
$$
Seemingly, in the case of Schwarzschild spacetime, one has to use this form of Komar integral mass. However, some authors, including Wald in pages 289-291 of his great book, consider this definition (surface-integral formulation) as a fully satisfactory notion of the total mass in stationary, asymptotically flat spacetimes.Having this relation, you can easily find the correct answer as
$$
M[{\xi _t}] = \frac{1}{{4\pi G}}\int {{d^2}\vec y\sqrt h {{\hat n}_\mu }{{\hat u}_\nu }{\nabla ^\mu }\xi _{(t)}^\nu }
= \frac{1}{{4\pi G}}\int {{d^2}\vec y({r^2}\sin \theta ){g^{00}}\Gamma _{00}^1\xi _{(t)}^0}
= M\,
$$
We know this definition is true since it can be proved by use of the other well-known methods such as ADM approach, evaluating Euclidean partition function and also Brown-York formalism. Carroll has an interesting idea about the validity of the surface-Komar integral:
''The point is that, since 6.38 [i.e., the above relation in this note] involves contributions only at spatial infinity, it should be a valid expression for the energy no matter what happens in the interior. ''
So, if there exists an asymptotically timelike Killing vector, the total energy of a stationary spacetime (with flat or AdS asymptote) is obtained via surface-Komar integral.
A: 
I love this question and there's a really interesting answer.
If you have an actual planet, like earth, then $T_{\mu \nu}$ won't be $0$ because of the matter of the planet. So in that case, it makes sense.
But in Schwarzschild, there is no $T_{\mu \nu}$. People sometimes say that it blows up at the singularity. But this does not resolve the paradox, and is actually irrelevant. Look at a Penrose diagram for Schwarzschild spacetime. The singularities are in the past/future. They are not on a 3 dimensional time slice you would integrate over to calculate Komar mass.
In fact, the Schwarzschild spacetime really describes a wormhole between two different universes, and this resolve the paradox. You see, the time vector $\partial_t$ points "up" in our universe and "down" in the other universe! So if you really integrate over the full time slice, they cancel out!
A: You have simply used a wrong definition of Komar mass, please read Poisson carefully, sec.5.5.3.

*

*The Komar mass is an integral over a closed 2-surface at infinity;

*It can be separated into two parts, one is a surface integral over the horizon; the other is a volume integral, which is written in your question, see eq.(5.102).

*You simply miss the first part $M_H$ in eq.(5.102), which gives exactly the black hole mass.

*The part, written by you, has no contribution to Komar mass, because energy-momentum tensor is zero for vacuum.

