# Mass and energy definitions

1. Nov 18, 2013

### atyy

Can I check whether these are right? Here let's define the ADM mass as length of the ADM energy-momentum vector.

In the Schwarzschild spacetime
ADM energy = Schwarzschild mass parameter

In a spacetime in which the ADM energy and the Komar mass are both defined

2. Nov 18, 2013

### WannabeNewton

The second statement is indeed true so as long as you take the space-like hypersurface associated with the ADM energy integral to be orthogonal to the stationary killing field at spatial infinity. The first statement is a consequence of the second since is easy to show that the Komar integral in Schwarzschild space-time yields the Schwarzschild mass parameter. The proof is given in this paper: http://scitation.aip.org/content/aip/journal/jmp/20/5/10.1063/1.524151

3. Nov 18, 2013

### atyy

The Ashtekar and Magnon-Asktekar paper says in Lemma 2.3 that the Komar integral (presumably that is the same thing as the Komar mass?) is the ADM mass (length of the ADM 4-vector). Presumably this is because the Komar integral is defined in stationary spacetimes, and the ADM momentum is zero in a stationary spacetime?

4. Nov 18, 2013

### WannabeNewton

Well you can see from Theorem 2 that the ADM 4-momentum is parallel to the time-like killing vector at spatial infinity with the proportionality factor containing the Komar mass itself, so indeed the ADM 3-momentum vanishes. What this means is that the rest frame defined by the ADM 4-momentum agrees with the rest frame defined by the time-like killing vector at spatial infinity, a result which is by no means obvious to me from the definition of the ADM 4-momentum as evaluated in a stationary space-time; in particular, the fact that the twist $\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d$ of the time-like killing field $\xi^a$ satisfies $\lim_{\rightarrow i^0}\hat{\omega}_a = 0$, which was a crucial part of the proof, is not obvious to me just from physical intuition (the hat indicates the twist associated with $\hat{\nabla}_a$, $\hat{\epsilon}_{abcd}$, and $\hat{\xi^a}$). Indeed this result (that is, including the part about the proportionality factor containing the Komar mass) is only obtained in retrospect after the proof of Lemma 2.3.

Of course even before the proof of Lemma 2.3, one could argue through physical intuition that for a stationary space-time, the ADM 3-momentum must clearly vanish but just because the ADM 3-momentum vanishes doesn't mean a priori that the ADM energy must equal the Komar energy; the conclusion is non-trivial as Lemma 2.3 shows.

Last edited: Nov 18, 2013