Partitioning Mass in GR: A Cautionary Tale

[pervect]

In the case of Komar mass, we can express the mass as an integral of $\rho + 3P$, so we can meaningful divide the total mass (or energy) of a system into the contribution due to each part, just by integrating over that spatial part of the system.

What happens if we try to do this with other definitions of mass, say the ADM or Bondi mass? My overall impression is that it can't be done, but I don't have a specific reference for this, so I want to be cautious about saying it can't be done.

I suppose I'm open to general ways of partitioning the mass, and not just my suggested approach of integrating some (pseudo) tensor of some sort over a spatial region.

One obstacle that comes to mind with psuedotensors is the obvious issue of the gauge degree of freedom affecting the subdivision process. But this seems lacking as a proof of impossibility, at least without an example illustrating different "partitioning".

 

[WannabeNewton]

Are you referring to the formula for the Komar mass given by $M = \int _{\Sigma}(\rho + 3P)e^{\phi}dV$ where $\phi = \frac{1}{2}\ln(-\xi^{a}\xi_{a})$ is the general relativistic Newtonian potential? This has a very nice physical interpretation when we can define such a potential (i.e. when we have a time-like killing vector field) but considering the ADM energy-momentum can be defined at spatial infinity for non-stationary asymptotically flat space-times as well, I'm not sure how the above would be meaningfully extended to the non-stationary case. Let me see if I can find papers by Ashtekar, Geroch, or Winicour on the matter because they have quite a few papers on the asymptotic structure of space-time at infinity.

EDIT: well more generally, for a stationary asymptotically flat space-time it is easy to show that the definition of the Komar mass given by $M = -\frac{1}{8\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\xi^{d}$ leads to the expression $M = 2\int _{\Sigma}(T_{ab} - \frac{1}{2}Tg_{ab})n^{a}\xi^{b}$ where $n^{a}$ is the normal to the hypersurface $\Sigma$ and $\xi^{a}$ is the time-like killing field. So if the energy-momentum tensor is that of a fluid, we can make sense of it separating the Komar mass into pressures and mass density as per the above (the formula with the $\rho + 3P$ above will come out if we consider for example the Schwarzschild spacetime for a fluid star). The Bondi mass definition is not too different from that of the Komar mass but the ADM energy-momentum is quite different, definition wise, from the Komar mass so I'm not sure if such a calculation can be given for non-stationary space-times.

 

[pervect]

I was thinking Wald's (11.2.10) which I won't retype here - it's basically the same idea though.

One thing that has influenced my thinking was mentioned in Wald, http://link.springer.com/article/10.1007/BF00762133

which says that giving certain conditions, the LL psuedotensor approach gives the same mass as the Bondi mass. I've never seen the LL psuedotensor mass given its own name, I've been assuming that it's just the Bondi mass in disguise (I could be wrong).

 

[WannabeNewton]

I was thinking Wald's (11.2.10) which I won't retype here - it's basically the same idea though.

Ok awesome, that is exactly what I wrote in my recent edit so we are looking at the same thing.

One thing that has influenced my thinking was mentioned in Wald, http://link.springer.com/article/10.1007/BF00762133

which says that giving certain conditions, the LL psuedotensor approach gives the same mass as the Bondi mass. I've never seen the LL psuedotensor mass given its own name, I've been assuming that it's just the Bondi mass in disguise (I could be wrong).

Interesting, I'll have to take a look at that (I'm currently reading Ashtekar's 1980 paper on the asymptotic structure of the gravitational field at spatial infinity to see if there is anything of relevance).

 

[sardar]

Thank you for sharing your thoughts on partitioning mass in general relativity. I agree that it is important to approach this topic with caution and open-mindedness. While the Komar mass definition allows for a straightforward partitioning of mass into different parts of a system, it is not clear if this can be done with other definitions such as the ADM or Bondi mass.

One possible reason for this difficulty could be the gauge degree of freedom, which can affect the subdivision process and make it challenging to define a unique partitioning of mass. However, I believe that it is still worth exploring other ways to partition mass, as you have suggested, and considering how they may differ from the Komar approach.

Overall, I appreciate your cautious approach to this topic and your openness to alternative ways of partitioning mass in general relativity. It is through careful consideration and exploration that we can continue to advance our understanding of this complex subject.