Graduate ADM mass of a spacetime and mass of the associated matter

[spaghetti3451]

The ADM formalism gives a definition for the energy (Hamiltonian) of a static, asymptotically flat spacetime. This energy can be equated to the mass of the matter (for example, a black hole) which resides in this spacetime.

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What is the physical mechanism which allows us to equate the energy of a spacetime with the mass of the matter which resides in the spacetime?

 

[PeterDonis]

The ADM formalism gives a definition for the energy (Hamiltonian) of a static, asymptotically flat spacetime.

The spacetime doesn't actually need to be static. It just needs to be asymptotically flat. A static, asymptotically flat spacetime has an ADM mass that is constant in time. If the spacetime isn't static, the ADM mass can change with time; but it can still be defined.

This energy can be equated to the mass of the matter (for example, a black hole) which resides in this spacetime.

In the case of a black hole, there is no matter; a black hole is a vacuum solution. So thinking of the ADM mass as "the mass of the matter" is not really correct.

What is the physical mechanism which allows us to equate the energy of a spacetime with the mass of the matter which resides in the spacetime?

The ADM mass is not a reflection of a "physical mechanism" of this sort. It's a convenient way of defining a "mass" that, for certain special cases, matches our intuitions, without requiring one to have a detailed picture of how that mass "arises" from the "stuff" that is present (if any is in fact present--as above, there isn't any in a black hole) in the spacetime.

The physical "mechanism" that the ADM mass reflects is actually the effect of the spacetime geometry on the orbits of test particles. Basically, it embodies the intuitive definition of "mass" as the quantity that appears in Kepler's laws and determines orbital parameters. These can be measured without knowing anything about the internal composition of whatever-it-is that is producing the spacetime geometry.

If you are interested in a definition of mass that corresponds to "adding up all the stuff" that is present, that is the Komar mass, not the ADM mass.

 

[spaghetti3451]

In the case of a black hole, there is no matter; a black hole is a vacuum solution. So thinking of the ADM mass as "the mass of the matter" is not really correct.

I understand that the Schwarzschild metric, for example, is obtained by solving Einstein's equations in the vacuum, assuming that the solution is static and spherically symmetric.

However, this contradicts with the popular notion that matter curves spacetime and black holes are created by the curvature of spacetime by matter of infinite density.

Would you be able to clarify the confusion?

 

[PeterDonis]

this contradicts with the popular notion that matter curves spacetime and black holes are created by the curvature of spacetime by matter of infinite density.

Would you be able to clarify the confusion?

Sure: the "popular notion" is wrong as you state it.

Instead of considering an idealized black hole spacetime which is vacuum everywhere, we could consider a more realistic solution such as the Oppenheimer-Snyder model of gravitational collapse to a black hole. In this model, the spacetime is not vacuum everywhere: a spherically symmetric object collapses, forms an event horizon, and then forms a singularity when the matter reaches zero radius and infinite density. However, the region of spacetime containing the matter has a boundary, and to the future of that boundary, everything is vacuum. And we can compute the ADM mass using just that vacuum region, so the popular notion is still wrong as you state it if you are trying to use it to interpret what the ADM mass is telling you.

Once again, I suggest that you look into the Komar mass; it is a much better match for the intuition about "mass" and "matter" that you appear to be using.

 

[PAllen]

The spacetime doesn't actually need to be static. It just needs to be asymptotically flat. A static, asymptotically flat spacetime has an ADM mass that is constant in time. If the spacetime isn't static, the ADM mass can change with time; but it can still be defined.

ADM mass is conserved if it is defined, and can only be computed at infinity. Bondi mass can change, but not ADM mass.

 

[PeterDonis]

ADM mass is conserved if it is defined, and can only be computed at infinity. Bondi mass can change, but not ADM mass.

Ah, that's right, good catch.

 

[pervect]

The ADM formalism gives a definition for the energy (Hamiltonian) of a static, asymptotically flat spacetime.

Yes.

This energy can be equated to the mass of the matter (for example, a black hole) which resides in this spacetime.

It's really the energy of the entire system. If one could localize the energy in the gravitational field, one could split up the total energy of the system into a part that was due to the field, and a part that was due to the matter, as one does in Newtonian theory. For instance, in Newtonian theory, a spherical mass has a gravitational self-binding energy that one can calculate, the amount of work that would need to be done to disassemble the spherical mass . For instance, one could imagine a mass made up of rigid blocks, that were hoisted to infinity with a crane. The crane needs to do work to lift the blocks.

Bu in general, one can't localize the gravitational field energy. The issue is that different observers don't necessarily split up the energy in the same way, so the splitting process is observer-dependent, it's not in general covariant. A specific observer in a specific coordinate system can do a plausible-looking split from their point of view, but it won't necessarily give the same answers as another observer using a different coordinate system - they might have an equally plauslbile split that is not the same as the first observers. This makes talking about such a split rather suspect.

However, all observers do agree on the ADM mass of the system, regardless of their coordinate choices. So we can safely talk about the mass of the system, but we run into covariance issues if we try to divide up the system mass into a part due to matter and a part due to the gravitational field.

The ADM mass is calculated from the metric, so it's really best to regard it as the mass of the space-time, as that's what it's calculated from.