Can Merging Black Holes Have Infinite Kinetic Energy?

[PeterDonis]

In our hypothetical BH power station, the use of tethers to secure the masses to extract energy would have a tremendous load on them.

Yes, this is one of the practical limitations on this scenario; any real tether will have a finite tensile strength, and the tension required to hold an object at a constant altitude increases without bound as the horizon is approached. So any real tether will break at some finite altitude above the horizon.

I believe this tension would become relevant in GR.

If you mean, would it have to be treated as having a non-negligible stress-energy tensor and therefore affect the spacetime geometry, no, that would not be necessary; the tether could have a very large tension in ordinary terms and still have a negligible stress-energy tensor as far as the spacetime geometry was concerned. Remember that we are only needing to drop small masses if we're talking about ordinary power requirements (1 GW power output means roughly 10 micrograms per second of mass dropped in).

 

[kimbyd]

I don't think this is correct; the mass, angular momentum, and charge of a black hole are not internal degrees of freedom. They are externally measured conserved quantities.

You've misread me a bit here. The mass is the total energy in the internal degrees of freedom. Similarly, the angular momentum will be the total angular momentum of the internal degrees of freedom. These statements are true for any object, even if we don't know what the precise internal degrees of freedom are (as with black holes). I wasn't attempting to make any claim about what the internal degrees of freedom are or how we go from those internal degrees of freedom to the externally-measurable quantities.

The reason why the mass is the energy in the internal degrees of freedom is simply that it is the total energy of the object in that object's rest frame. This description is especially useful when thinking about the masses of baryons like protons and neutrons (where most of the energy is the binding energy between the quarks).

 

[PeterDonis]

The mass is the total energy in the internal degrees of freedom.

What internal degrees of freedom? As far as classical GR is concerned, a black hole doesn't have any. It just has external conserved quantities which are properties of the global geometry: mass, angular momentum, and charge.

Also see below.

These statements are true for any object, even if we don't know what the precise internal degrees of freedom are

I don't think this is valid. Before we can say a black hole's mass is the total energy in its internal degrees of freedom, I think we need, at the very least, to have some valid theory that tells us what those degrees of freedom might be. We don't have such a theory, because we don't have a valid (i.e., with at least some experimental confirmation) theory of quantum gravity. So I think the best we can say is that we would like, if possible, to find such a theory, but we don't currently have one.

Also, in GR it is not the case that the externally measured mass of an object is, in general, the total energy in its internal degrees of freedom. Unless the object is in a stationary spacetime, there is no invariant way to even specify the total energy in its internal degrees of freedom. Mass, angular momentum, and charge, as above, are external conserved quantities that are interpreted as global properties of the geometry.

The reason why the mass is the energy in the internal degrees of freedom is simply that it is the total energy of the object in that object's rest frame.

Unless that rest frame is stationary (which it isn't for any real object), this is not an invariant quantity. As an approximation, it works for a wider class of cases, but that's just an approximation.

This description is especially useful when thinking about the masses of baryons like protons and neutrons (where most of the energy is the binding energy between the quarks).

This has nothing to do with GR; all of these models are modeling the objects using QFT in flat spacetime. By construction, such a model cannot capture the interaction between the energy in the object and the spacetime geometry. But such interaction is essential for any GR model. So as far as GR is concerned, all of these models are approximations, and I don't think they come anywhere close to justifying a blanket claim about all objects, including black holes.

 

[kimbyd]

What internal degrees of freedom? As far as classical GR is concerned, a black hole doesn't have any. It just has external conserved quantities which are properties of the global geometry: mass, angular momentum, and charge.

Also see below.

The notion of "internal degrees of freedom" is a fundamentally quantum notion.

As for a black hole, the reason it doesn't have any is because the mass is not located on the manifold at all (as the singularity cannot be part of the manifold). So it's not that they don't exist: it's that General Relativity doesn't tell us what they are.

I don't think they come anywhere close to justifying a blanket claim about all objects, including black holes.

I don't see why not. It's what we mean by the concept of mass. Mass is a coordinate-invariant quantity associated with an object. Because mass is an energy, and because it's a coordiante-invariant quantity, it's the energy which is associated with the object itself independent of reference frame.

The difficulty in extending this description to General Relativity has more to do with the fact that you can't uniquely define a mass for all objects in General Relativity, period, than it does with what we mean when we say the word, "mass". When mass is identifiable, as it is with special relativity, it is the energy in the internal degrees of freedom.

Even with QFT this remains true because even though you can write down a "fundamental" mass of a particle, making that fundamental mass non-zero leads to inconsistencies, suggesting that all masses in QFT must be a result of binding energies of some kind, even though we don't know what those binding energies are.

So yes, I will absolutely claim that mass is the energy in the internal degrees of freedom of a system. This is a difficult statement to make in GR more because it's hard to define mass in general in GR more than it suggests that the concept of mass itself as internal energy is worthless. I think you'll be hard-pressed to present an argument for why the concept is misleading for black holes in particular, where we have a parameter that behaves a lot like a classical mass, and which is exactly the classical mass in the limit of large distance from the event horizon.

 

[PeterDonis]

it's not that they don't exist: it's that General Relativity doesn't tell us what they are.

I don't think that's valid, because as far as GR is concerned, what isn't part of the manifold doesn't exist. The mass, as far as GR is concerned, is, as I've said, a global property of the spacetime geometry (as are the charge and angular momentum for holes that have them).

To put this another way: what do we expect from a quantum gravity model of a black hole? That it will keep the entire GR spacetime, but then adjoin to it some quantum thing in place of the singularity that contains all the internal degrees of freedom? I don't know of any physicist who expects any such thing. Everything I've seen is that physicists expect a quantum gravity model of a black hole to replace a significant portion of the GR spacetime with some other spacetime geometry, which contains the quantum internal degrees of freedom. But so far, such a model does not exist. And until it does, the best we have is the GR model, and the GR model simply does not have the internal degrees of freedom at all. It just has the global properties.

It's what we mean by the concept of mass.

It may be what you mean by the concept of mass. I don't think you are justified in claiming that it is the "true" concept of mass. That term simply does not have a single "true" referent in physics. And as far as the GR model of black holes is concerned, the referent of "mass" is a global property, not something involving internal degrees of freedom.

 

[PeterDonis]

I will absolutely claim that mass is the energy in the internal degrees of freedom of a system.

Aside from my reservations about this, I would note that for this particular discussion, we don't need to agree on it, since answering the question under discussion does not require any model of the internal structure of the objects involved; all we need to know is their externally measured quantities.