Exploring the Kinetic Energy of Merging Black Holes

[Herbascious J]

Imagine two black holes at great distance. They are both spatially separate and both completely collapsed to a singularity. Gravity begins to pull them together. According to the equation for the gravitation potential energy of two objects at distance…

Ug = -GMm/r

…These two objects begin to lose gravitational potential energy as a system. This causes kinetic energy (KE) to increase between the two objects as they approach. If these two black holes are singularities, would it be true that their spatial separation (r) would begin to approach 0, and in so doing, their KE would begin to approach infinity? This seems strange to me, because I can imagine a scenario where the KE among these objects is so high, that their energy contribution from KE is higher than their energy contribution from their matter. And wouldn’t this have an effect on the total energy of the resultant merged black hole and it’s gravitational field?

 

[Drakkith]

If these two black holes are singularities, would it be true that their spatial separation (r) would begin to approach 0, and in so doing, their KE would begin to approach infinity?

Good question. I think you might have to use relativistic equations here, so I'm not sure how the velocity acts as the two singularities move closer together. But, since we're dealing with singularities, even if it tends to infinity that's not really saying much since a singularity is already a breakdown in the theory/math.

And wouldn’t this have an effect on the total energy of the resultant merged black hole and it’s gravitational field?

Nope. The conversion between potential and kinetic energy is conserved, so that no energy is lot or gained that wasn't already there as potential energy. The total gravitational strength of the system is the same before and after. At least until the gravitational waves from the merger move outside the system.

 

[Herbascious J]

In retrospect, I believe I am misinterpreting the math. I attempted to perform a calculation that showed the distance (r) where the KE would approach the mass energy of two colliding 1kg bodies. I expected this diameter to be extremely small (only singularities could reach), but the resultant math gave a distance approximately the size of the universe. I now realize that that would be the distance I would have to spatially separate these objects for them to have enough KE (in a non-expanding universe) to collide at energies equivalent to their masses. I guess I should retract the original question and formulate a better quesiton.

 

[PeterDonis]

According to the equation for the gravitation potential energy of two objects at distance

Which doesn't even apply to two black holes, since two black holes is not a stationary system and gravitational potential energy is only a meaningful concept in a stationary system.

If the two black holes are an isolated system, the system as a whole has a finite mass as viewed from very far away. This mass will just be the sum of the individual masses of the holes, at least if we assume they start out at some instant at rest relative to one another. Since the system remains isolated, its mass as viewed from far away stays constant, no matter what happens to the system internally. Eventually the holes will merge and the system will just be one big hole with the same mass (if we exclude emission of gravitational waves during the merger).

The detailed dynamics of the merger would have to be simulated numerically, but they don't affect any of the above.

 

[Herbascious J]

Nope. The conversion between potential and kinetic energy is conserved, so that no energy is lot or gained that wasn't already there as potential energy. The total gravitational strength of the system is the same before and after. At least until the gravitational waves from the merger move outside the system.

Is there a conventional explanation that describes how spatially separated bodies can have a gravitational field equal to those same bodies moving very fast and in close proximity? I hope my question is clear, I'm trying to imagine how PE contributes to a gravitational field.

 

[PeterDonis]

I'm trying to imagine how PE contributes to a gravitational field.

You have it backwards. "Gravitational potential energy" is not a source of gravity or a contribution to the "gravitational field". It's an effect of a particular kind of gravitational field (a stationary one) that you can use to analyze the motion of objects in the field in a way similar to the usual Newtonian analysis. But you have to already know the entire field, and that it's stationary, before you can even define "gravitational potential energy".

 

[Herbascious J]

You have it backwards. "Gravitational potential energy" is not a source of gravity or a contribution to the "gravitational field". It's an effect of a particular kind of gravitational field (a stationary one) that you can use to analyze the motion of objects in the field in a way similar to the usual Newtonian analysis. But you have to already know the entire field, and that it's stationary, before you can even define "gravitational potential energy".

Ok, so. I imagine the two stationary black holes at distance, and at rest. I measure their masses and then at a distance, I can measure their combined gravitational field and it is the result of the system mass, which is just the sum of the two masses. I get that. Now, as the black holes fall together they gain KE. From a distance I see this change in momentum, but that the momentum of each body cancels the other. Is that also true for KE. Do I ignore the increase in velocity between the two objects as a form of energy, like with momentum? Or, do I say that the system has some KE(mass), and now I measure the individual masses to be less than they were before? Is there something I'm missing?

 

[PeterDonis]

Is there something I'm missing?

Yes. Read my post #4 (especially the second paragraph) and think about how it affects your analysis.

 

[Drakkith]

From a distance I see this change in momentum, but that the momentum of each body cancels the other. Is that also true for KE.

No, kinetic energy is a scalar while momentum is a vector, so you can't cancel kinetic energy out just by having objects move in different directions.

Imagine a comet and a star. When the comet is very far away from the star, it moves very slowly and has lots of potential energy. But when the comet falls in towards its periapsis (closest point to the star in its orbit) it gains a great deal of kinetic energy but loses most of its potential energy. However, the gravitational field of the system doesn't change when viewed from a great distance. The total energy of the system (and thus the mass) remains constant.

Replace the comet and the star with any two objects of any size and you still get the same results.

That's the simple answer. Peter's answers above are a bit more complicated.

 

[metastable]

Is there any collision scenario between 2 black holes that could result in matter exiting one or both of the event horizons? (offset relativistic head-on collision, etc)

 

[PeterDonis]

Is there any collision scenario between 2 black holes that could result in matter exiting one or both of the event horizons?

No. By definition nothing can escape from behind an event horizon.

 

[Herbascious J]

Yes. Read my post #4 (especially the second paragraph) and think about how it affects your analysis.

My hangup is watching these two black holes approach at high speed (velocity they didn't have before, relative to each other). I'm assuming this velocity is a form of energy and would contribute to the mass of each object as measured by me. Just to clarify, in GR energy is the source of gravity, not just the rest-mass, is this correct? The reason rest-mass contributes to a gravitational field is simply because it is a form of energy (E=MC^2)? Two objects at rest would have a lower gravitational field than those same two objects about to collide at high velocity?

 

[PeterDonis]

I'm assuming this velocity is a form of energy and would contribute to the mass of each object as measured by me.

It's not that simple. See below.

in GR energy is the source of gravity, not just the rest-mass, is this correct?

No. The stress-energy tensor is the source of gravity. Kinetic energy does appear in the stress-energy tensor, but it's a tensor, not a scalar, so it's not just a matter of it kinetic energy "adding" to rest mass.

Also, your post asked about black holes, which are vacuum--there is no stress-energy anywhere. So their "mass" is just a property of the spacetime geometry. So even trying to think of them as having "kinetic energy" is problematic, since they're not made of matter to begin with. I would personally advise you to change your scenario to use ordinary massive objects that fall together and collide inelastically, ending up as one larger object at rest in the center of mass frame, since that will eliminate the whole "black hole is vacuum" issue.

Two objects at rest would have a lower gravitational field than those same two objects about to collide at high velocity?

You're leaving things out of the comparison.

Suppose we have four objects, A, B, C, and D. All of them have the same rest mass. A and B are separated by a distance L and, at some instant, are at rest relative to each other. C and D, at the same instant, are separated by the same distance L and are moving towards each other at high speed. All of these statements are relative to the same frame of reference.

In the above scenario, if we consider A + B as an isolated system, it will have a smaller mass than C + D considered as an isolated system. It is tempting to say that the kinetic energy of the C + D system will make an additional contribution to its overall mass, but we should resist the temptation for reasons that will appear below. (Btw, we're assuming we can measure all these masses of isolated systems, but we haven't talked about how we would do that. You might want to think about that aspect as well.)

However, now suppose we decide we want to make A + B move towards each other at the same high speed as C + D. To do that, we would have to apply a huge impulse to A and B, and that would take energy, and that energy has mass. So the additional mass doesn't appear out of nowhere; we added it to the system.

Now suppose further that C + D got that way by starting out much, much farther apart, and at rest relative to each other, and then fell towards each other under their mutual gravity. Then ask: what was the mass of the C + D system far in the past when C + D were at rest relative to each other? Was it the same as the A + B system then? The answer is no. Per my post #4, if the C + D system is an isolated system, its overall mass is constant. So its mass far in the past when C + D were at rest relative to each other is the same as the mass it has at the instant described above, when C + D and A + B were both at the same separation L.

We can check our logic by asking what we would have to do to the A + B system to make it like the C + D system was in the far past when C + D were at rest relative to each other, as A + B are at the instant described above. The answer, again, is that we would have to add energy to the A + B system, enough to push them apart, against their mutual gravity, to the same separation that C + D had far in the past when they were at rest relative to each other. And this would give the A + B system the same overall mass as the C + D system. And, again, this additional mass woudn't come out of nowhere; we would add it to the system.

Pushing A + B apart against their mutual gravity is often described as adding "potential energy" to the system, and as an ordinary language heuristic description, this is fine. But since the spacetime is not stationary, there is no precise mathematical object that corresponds to this "potential energy", at least not an invariant one. That means there is no invariant way to characterize the contribution of "kinetic energy" to the overall mass either, because doing that would require separating "kinetic energy" and "potential energy" and trading off one against the other, and we can only do that in an invariant way in a stationary spacetime. The only invariant concept in our example is just "adding energy", or equivalently "adding mass". As the two examples of what we could do to make the A + B system match the C + D system illustrate, there are different ways to add mass, but they all end up adding the same mass.

 

[Herbascious J]

That was great, thank you so much. I see now how energy is being added to these systems so that A-B can look like C-D and vice-versa. That makes perfect sense. I'm not even going to touch the BHs being a vacuum :) :D Cheers.

Ok, I'm going to try one last inquiry, the reason being is that I would like to steer the thread back to the original question in a way that makes sense. Just typing this, I can already hear the jeers from the GR seats in the audience, but alas... Imagine we have two EXTREMELY DENSE objects, not vacuum black holes, just impossibly dense objects that have not quite become a singularity, but they are matter non-the-less and they are real. If I place their centers of gravity at a position that is EXTREMELY CLOSE (in fact closer that would be physically possible and probably violates all kinds of physics), they would be bound by a very strong (approaching infinity) gravitational field, as 'r' approaches 0. If I attempt to move these objects apart, even just a little, I can imagine there is some hypothetical distance where the energy required to do so would be more than the energy of their rest-energy/mass. This is why I originally used BHs. Does the gravitaional field increase? Does this even come close to describing a theorhetical reality?

 

[PeterDonis]

impossibly dense objects that have not quite become a singularity

You can't make them arbitrarily dense. Buchdahl's Theorem says that an object that is stable against its own gravity (i.e., it won't collapse to a black hole) must have a surface area that is at least 9/8 of the area of the horizon of a black hole of the same mass. So two such objects can't get arbitrarily close together without colliding, merging, etc.

they would be bound by a very strong (approaching infinity) gravitational field

Once more: a two-body system is not stationary. That means you cannot use the kind of reasoning you are using. Saying "bound by a very strong gravitational field" is just another way of saying "has a very large negative gravitational potential energy", and as we've already seen, the concept of gravitational potential energy is not valid for a non-stationary system.

If I attempt to move these objects apart, even just a little, I can imagine there is some hypothetical distance where the energy required to do so would be more than the energy of their rest-energy/mass

Note that this question can be formulated without having to use any invalid concepts like "gravitational potential energy" or "bound by a gravitational field". You can just ask: suppose we have a system of two massive objects (which are not black holes and are stable against collapse against their own gravity) that, at some instant, are at rest relative to each other and separated by some distance that is not very much larger than their sizes. (We can put aside for this discussion the question of how the objects got that way.) How much energy would we have to add to the system to move the objects apart by some specified distance?

This question is well-defined; the problem is that we have no exact solution of the Einstein Field Equation that we can use to answer it. We can only simulate the quantitative behavior of such systems numerically. However, we can answer the related question you ask: Could the energy required to separate the objects by some distance be larger than the combined rest masses of the objects? The answer to this question is no.

The argument for this answer is: suppose the two objects were very, very far apart, so their mutual gravity was negligible. Then the mass of the two-body system as a whole would be just the sum of the two rest masses. Now, to take the system as you describe it, where the two bodies are very close together, and move the bodies very, very far apart, would take energy, so the energy of the two-body system when the bodies are very close together (and, at least momentarily, at rest relative to each other) must be smaller than the sum of the two rest masses. But how much smaller?

We know that there is a way, at least in principle, to recover all of the rest mass of an object as energy, by slowly lowering the object into a black hole--just make sure the object comes to rest just above the horizon, at which point all of its rest mass will have been withdrawn as energy during the lowering process, and then release it to fall into the hole. However, this is a limiting case and is of course not possible if no black hole is present. And we stipulated that neither of the massive objects in our scenario were black holes, or could collapse into black holes. So there can't be any way to take these two objects starting very, very far apart, and somehow withdraw all of their combined rest mass as energy while lowering them together to the point specified in our scenario (where they are separated by some small distance and momentarily at rest relative to each other). And by just reversing this reasoning, we can see that moving the two objects apart, even to a large enough separation that their mutual gravity is negligible, cannot possibly require adding more energy to the system than the combined rest masses of the objects.

We can also see this another way: if it were possible for it to require more energy than the combined rest masses to move the objects apart, then the externally measured mass of the two-object system in the state we specified--separated by a small distance--would have to be zero or negative, since we know that the mass of the two-object system when the two objects are very, very far apart is the sum of the two rest masses, and the energy we would have to add to get to that point from our starting point must be the same as the energy we would have to take away to get from the masses being separated very, very far apart to our starting point. And it's not possible for the mass of the system to be zero or negative; the mass of the system must be positive.

 

[Herbascious J]

Thank you, that answers my question beautifully. Much appreciated. This thread has elevated my understandng of some deep issues I was having surrounding the nature of these things. BTW - I left a very positive review of PF on google Earth today. Thanks for the awesome admin work. It doesn't go unnoticed.

 

[PeterDonis]

Thanks for the awesome admin work. It doesn't go unnoticed.

Glad I could help, and thanks for the kudos!

 

[metastable]

Is there any collision scenario between 2 black holes that could result in matter exiting one or both of the event horizons? (offset relativistic head-on collision, etc)

No. By definition nothing can escape from behind an event horizon.

That makes sense. Strangely I’m still confused about how would the expected outcomes in these 2 slightly different scenarios be different:

1) a smaller black hole approaches a much larger black hole, and the smaller black hole has significantly more kinetic energy than it requires for having escape velocity from the larger black hole, and their event horizons maintain at least 1 meter separation distance at close approach.

2) same scenario as above but the event horizons overlap by 1 meter at close approach

Will the 2 black holes in scenario 2 merge into a larger black hole?

 

[PeterDonis]

I’m still confused about how would the expected outcomes in these 2 slightly different scenarios be different

As you've described the scenarios the difference is obvious: the event horizons don't overlap in #1, but they do in #2. Which means the holes merge in #2, since event horizons can't overlap--you either have two holes that merge into one, or you don't. If the holes merge, there is only one event horizon--it's just shaped like a pair of trousers (with, in this case, legs whose sizes are very different) instead of a cylinder, heuristically speaking. If the holes don't merge, we have two horizons shaped like cylinders.

 

[metastable]

As you've described the scenarios the difference is obvious: the event horizons don't overlap in #1, but they do in #2. Which means the holes merge in #2, since event horizons can't overlap--you either have two holes that merge into one, or you don't. If the holes merge, there is only one event horizon--it's just shaped like a pair of trousers (with, in this case, legs whose sizes are very different) instead of a cylinder, heuristically speaking. If the holes don't merge, we have two horizons shaped like cylinders.

So if I understand you correctly, the singularity of the smaller black hole does not have to penetrate the event horizon of the larger black hole in order to "guarantee" a merger...

 

[PeterDonis]

the singularity of the smaller black hole does not have to penetrate the event horizon of the larger black hole in order to "guarantee" a merger...

The singularity is not a thing that moves in space. It's a moment of time that is to the future of all events inside the horizon. If two black holes merge, there is only one singularity, which is to the future of all events inside the pair of trousers shaped horizon.

 

[metastable]

The singularity is not a thing that moves in space. It's a moment of time that is to the future of all events inside the horizon.

I'm confused because I thought some authors write that the singularity is shaped like a ring:

 

[PeterDonis]

I'm confused because I thought some authors write that the singularity is shaped like a ring

That's for a Kerr black hole, i.e., a rotating one. I have assumed that we were talking about a Schwarzschild black hole, i.e., a non-rotating one, since those are simpler to deal with.

But making the holes rotating doesn't make your description correct: Kerr spacetime is considered unphysical at the inner horizon, before you even get to the singularity (whereas there is no issue, at least at the classical level, with Schwarzschild spacetime all the way to the singularity).

 

[Herbascious J]

We know that there is a way, at least in principle, to recover all of the rest mass of an object as energy, by slowly lowering the object into a black hole--just make sure the object comes to rest just above the horizon, at which point all of its rest mass will have been withdrawn as energy during the lowering process, and then release it to fall into the hole.

So, can we imagine an (admittedly unrealistic) power station that drops weights on tethers into the event horizon (EH) of a black hole? Perhaps done in pairs on each side to keep the BH stationary. The weights could be brought to rest just above the EH, extracting energy, and then dropped, recovering the tethers. I imagine it requires energy to recover the tethers, and we probably won't get all of the rest-mass as energy out of the object before it is discarded, but in theory, if we neglect these last two points; could we repeat this process indefinitely without increasing the mass of the BH?

I imagine, that even though we have extracted the objects mass energy, relative to an observer at the object, the object still has the same matter it always had and therefore has the same mass. The object still feels quite an intense gravitational tug. Do we observe a loss in mass by a loss in the force of gravity weakening between the weights and the BH as observed at a safe distance? I'm imagining all of this energy being extracted to the point where the rest mass is fundamentaly missing from the object but it is gravitating like mad at the EH. Does this become highly relative at this point and I wonder what non-variable quantaties are conserved if any (like charge)?

 

[PeterDonis]

can we imagine an (admittedly unrealistic) power station that drops weights on tethers into the event horizon (EH) of a black hole?

Yes, in principle you could do that. But with limitations--see below.

could we repeat this process indefinitely without increasing the mass of the BH?

In the limit where you can drop the objects just at the horizon, yes. Of course this limit is not achievable--not in practice, and not even in principle, since it is impossible to "hover" exactly at the horizon, even for an instant. So the object has to be released somewhere above the horizon, and there will be some nonzero amount of energy that cannot be extracted from it before it drops, and which will add to the mass of the BH (but the addition will be too small to affect the spacetime geometry--see next paragraph).

But note that this whole scheme implicitly assumes that none of the objects we drop into the hole have enough stress-energy to affect the spacetime geometry. So we're not talking about dropping planets or stars in. We're talking about dropping in items of ordinary size. Note that, in ordinary terms, that still allows us to extract a lot of energy in principle, since in principle we can recover the entire rest energy of the objects we drop in. So, for example, if we wanted a 1 GW power station (typical for a plant that powers a city), and we were 100% efficient at extracting energy, we would only need to drop in about 10 micrograms of matter every second.

relative to an observer at the object, the object still has the same matter it always had and therefore has the same mass

Yes, but, as above, that mass has to be negligible in terms of stress-energy.

The object still feels quite an intense gravitational tug

No, it doesn't, because the gravity of the object we drop in has to be negligible, as above. If it weren't negligible, the object would affect the spacetime geometry, and then all bets are off; we don't have an exact solution that describes what will happen, and we certainly can't model things as simply lowering objects on tethers and then dropping them at the horizon.

I'm imagining all of this energy being extracted to the point where the rest mass is fundamentaly missing from the object

No, the rest mass is still there, and could be measured by an observer falling into the hole along with the object. (Note that, as above, we're talking about the case where the object, and of course the observer too, have negligible stress-energy.)

 

[Herbascious J]

We're talking about dropping in items of ordinary size. Note that, in ordinary terms, that still allows us to extract a lot of energy in principle,

Just for clarification; is it possible to extract slightly more energy than the rest mass, or is this exactly bound to the rest mass and directly related to it? I'm asking so that I can see that this energy is in fact the rest-mass as it's source, or if it's simply proportional to it (maybe my question isn't stated correctly).

EDIT: I think I'm saying that wrong. I don't believe that the source of the energy is the rest-mass exactly. I guess I'm curious if the limit of the energy that we can extract is in fact equal to the rest mass energy. Is it at that point that the object would have to fall beyond the EH and no more energy could be extracted?

 

[kimbyd]

Imagine two black holes at great distance. They are both spatially separate and both completely collapsed to a singularity. Gravity begins to pull them together. According to the equation for the gravitation potential energy of two objects at distance…

Ug = -GMm/r

…These two objects begin to lose gravitational potential energy as a system. This causes kinetic energy (KE) to increase between the two objects as they approach. If these two black holes are singularities, would it be true that their spatial separation (r) would begin to approach 0, and in so doing, their KE would begin to approach infinity? This seems strange to me, because I can imagine a scenario where the KE among these objects is so high, that their energy contribution from KE is higher than their energy contribution from their matter. And wouldn’t this have an effect on the total energy of the resultant merged black hole and it’s gravitational field?

First, bear in mind that the description above of the gravitational potential is a Newtonian concept that doesn't really exist in General Relativity. You can build something similar through pseudotensors (see the Hamiltonian formalism of General Relativity if you're curious, but it's pretty difficult to find good, readable resources). But potential energy just isn't something that is used much in GR in general.

However, you can sort of kinda do something similar in another way.

In Newtonian gravity, there is a sense in which you can take one configuration of mass (e.g. two spheres separated by some distance) and compare the potential energy difference of that configuration with another configuration (one sphere of mass equal to the total mass of the two spheres). You can do that calculation because the masses are distributed across a volume of space, and because you're assuming that the energy released in the merger is negligible.

With this picture, the potential energy of the system can be split into two parts: the potential energy due to the positions of the two masses and the potential energy inside each mass.

You can do something similar with black holes. Before they merge, you can simply use the Newtonian gravitational potential above. It won't be quite correct (with GR corrections you'd have to add some terms, e.g. a term proportional to $1/r^2$ in the potential), but it's a place to start.

What about the potential energy of the black hole itself? Well, that's pretty easy, actually: the mass is the energy in the internal degrees of freedom of an object. Thus for a Schwarzschild black hole, since its only internal degrees of freedom are its mass, the potential energy is its mass energy. For rotating or charged black holes you'd have to add terms related to their rotation and charge, but the picture is largely the same.

 

[PeterDonis]

is it possible to extract slightly more energy than the rest mass

No. Even in the limiting case (not actually achievable) where you extract all possible energy and lower the object to the horizon and release it there, you can't extract more energy than the object's rest mass.

 

[PeterDonis]

bear in mind that the description above of the gravitational potential is a Newtonian concept that doesn't really exist in General Relativity

That's not quite true. In a stationary spacetime you can define such a potential in an invariant way (using the timelike Killing vector field of the spacetime). But any spacetime with more than one gravitating body present will not be stationary.

the potential energy of the system can be split into two parts: the potential energy due to the positions of the two masses and the potential energy inside each mass.

I would put these two things somewhat differently. It's easiest to take them in reverse order.

For the first, I would say that, taking each mass individually as given (i.e., without asking how it got into its current configuration with its current rest mass--see the second item below for that), there is some binding energy associated with the two-mass system: this is the amount of energy that would need to be extracted from the system to bring the masses from infinite separation (or at least a very, very large separation, such that their gravitational interaction was negligible) to whatever finite separation they are at.

For the second, I would say that, considering each mass separately, there is some binding energy associated with the mass: this is the amount of energy that would need to be extracted from the system in order to take, say, $10^{50}$ atoms of various types, all very, very far apart from each other, and bring them together to form the mass in its usual state.

The point is that the concept of "binding energy" has a well-defined meaning (the meaning I just described in these two examples) regardless of whether the spacetime is stationary or not; whereas the term "potential energy" is only really well-defined in a stationary spacetime. So I think it's better not to use "potential energy" even in a heuristic way in a spacetime where it isn't well-defined.

You can do something similar with black holes.

For the first of the two items above, yes, this is true; a two-black-hole system has a binding energy just as a system of any two massive objects does, determined as I described it above.

However, for the second of the two items, things are more problematic. See below.

the mass is the energy in the internal degrees of freedom of an object.

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.

As far as somehow deriving these conserved quantities from some internal degrees of freedom, we don't currently have any way of doing that since we don't have any theory in which the spacetime geometry of a black hole emerges from something more fundamental. That's one of the things physicists expect to get from a theory of quantum gravity, if we can find it.

 

[Herbascious J]

Yes, but, as above, that mass has to be negligible in terms of stress-energy.

An after thought; In our hypothetical BH power station, the use of tethers to secure the masses to extract energy would have a tremendous load on them. I believe this tension would become relevant in GR. Should we interpret the masses and their tethers to have some kind of extreme tension/pressure associated with it that would contribute to the interpretation in General Relativity? Can this simply be ignored to keep things simple and not complicate the basic questions trying to be understood?

 

[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.