Is ADM Energy Equivalent to Komar Mass in All Spacetimes?

[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
ADM energy = Komar mass

 

[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

 

[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?

 

[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.

 

[sardar]

I cannot confirm the accuracy of these definitions without further context or information. However, I can provide some clarification and potential considerations.

Firstly, the ADM mass and energy are concepts used in general relativity to describe the total mass/energy of a spacetime. This includes both the mass/energy contained within a region of space, as well as the energy associated with the curvature of spacetime itself.

The ADM energy-momentum vector is a mathematical object that describes the total energy and momentum of a spacetime. It is defined as the integral of the energy-momentum density over a three-dimensional spatial slice of the spacetime. The length of this vector represents the total energy of the spacetime.

In the Schwarzschild spacetime, which describes the curvature of space around a non-rotating, spherically symmetric mass, the ADM energy is equal to the Schwarzschild mass parameter. This makes sense intuitively, as the Schwarzschild mass parameter is a measure of the total mass of the object causing the curvature.

In a more general spacetime where both the ADM energy and the Komar mass are defined, it is possible for the two to be equal. The Komar mass is a measure of the total energy contained within a region of space, similar to the ADM energy. However, it is defined using a different mathematical approach, involving the use of Killing vectors to calculate the energy associated with the spacetime's symmetries. Therefore, in some cases, the ADM energy and the Komar mass may be equal.

However, it is important to note that these are just two possible definitions of mass and energy in general relativity. Other definitions, such as the Bondi mass or the Hawking mass, may be more appropriate in different situations. Additionally, it is important to consider the limitations and assumptions of any given definition, as well as the mathematical and physical implications of using one over the other.

In summary, while these definitions may be correct in certain cases, it is important to consider the broader context and potential limitations when using them in scientific research.