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rjbeery

kinetic energy and black hole formation

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Hi there!

"Black hole creation" is obviously an absolute event at a given point in space-time, and black holes are created when a particular energy density threshold is reached. My problem is that kinetic energy is a relative calculation. If a mass is sufficiently dense to be close to the Schwarzschild radius, and we accelerate it, we can obviously surpass the required density threshold according to our frame.

How does general relativity reconcile this?

 

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Kinetic energy is frame dependent ( relative ).
IOW it may be there in one frame and not there in another; BHs form in ALL frames ( absolute ).

18 minutes ago, rjbeery said:

Black hole creation" is obviously an absolute event

 

19 minutes ago, rjbeery said:

kinetic energy is a relative calculation

You state this yourself, but then jump to an unsupported conclusion.

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27 minutes ago, rjbeery said:

Hi there!

"Black hole creation" is obviously an absolute event

Yes

27 minutes ago, rjbeery said:

at a given point in space-time

No. 

Position and time are relative.

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19 minutes ago, swansont said:

Position and time are relative.

Maybe I'm misunderstanding you.
You are correct space and time are relative, but

49 minutes ago, rjbeery said:

"Black hole creation" is obviously an absolute event at a given point in space-time

a point, or event, in space-time is absolute, and the foundation for the block universe model.

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48 minutes ago, rjbeery said:

If a mass is sufficiently dense to be close to the Schwarzschild radius, and we accelerate it, we can obviously surpass the required density threshold according to our frame.

How does general relativity reconcile this?

 

The source term in the gravitational field equations isn’t density or kinetic energy, but the full stress-energy-momentum tensor. This being a tensorial quantity, it is covariant under changes in reference frame, and thus the same for all observers, regardless of their state of relative motion with respect to the gravitational source. So there isn’t anything that needs to be reconciled - both observers are part of the same spacetime with the same dynamics, they just labels events differently.

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1 hour ago, rjbeery said:

Hi there!

"Black hole creation" is obviously an absolute event at a given point in space-time, and black holes are created when a particular energy density threshold is reached. My problem is that kinetic energy is a relative calculation. If a mass is sufficiently dense to be close to the Schwarzschild radius, and we accelerate it, we can obviously surpass the required density threshold according to our frame.

How does general relativity reconcile this?

A black hole is caused by the (rest) mass of the object in its own frame of reference.

After all, you are moving near the speed of light relative to something, but you are not a black hole!

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5 hours ago, MigL said:

Maybe I'm misunderstanding you.
You are correct space and time are relative, but

a point, or event, in space-time is absolute, and the foundation for the block universe model.

That the event happened is absolute - it happened in all frames. Where and when are relative.

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Posted (edited)
On 4/29/2020 at 7:29 PM, rjbeery said:

Hi there!

"Black hole creation" is obviously an absolute event at a given point in space-time, and black holes are created when a particular energy density threshold is reached. My problem is that kinetic energy is a relative calculation. If a mass is sufficiently dense to be close to the Schwarzschild radius, and we accelerate it, we can obviously surpass the required density threshold according to our frame.

How does general relativity reconcile this?

 

Very good question. It reminds me a lot of the twins paradox, but for GTR. The solution to the puzzle may go along these lines:

If I understand you correctly, you've got yourself a bunch of matter that's stopped compressing just before it becomes a black hole, but it hasn't. (The Fermi degeneracy pressure is just enough to hold it partially outside of its Schwarzschild horizon.) Then you push it just enough from one side so that its energy density reaches the threshold, and it turns into a black hole. How could that be, if kinetic energy is --allow me to rephrase-- a frame-dependent concept?

That means that if you go to a different inertial frame the kinetic energy, that in the first inertial frame looks like

\[\frac{mc^{2}}{\sqrt{1-v^{2}/c^{2}}}\]

would look like its rest energy

\[mc^{2}\]

Yes, but you've said that you're accelerating it, so there is no inertial frame you can go to where you can see it with constant zero velocity.

In somewhat more technical words, you must include the fields that are accelerating your black-hole wannabe and either include them on the right-hand side of Einstein's equations (if they're other than gravitational) or in the Einstein tensor (if they are more gravitational fields). That's not frame-dependent, but covariant under general coordinate transformations.

\[G μ ν =8 π G( T μ ν + δ T μ ν )\]

Whether the system collapses or not would, I suppose, depend on the details of the dynamics. Maybe what you would do is save it from collapse.

Many paradoxes arise in relativity (both special and general) when you forget that your reasoning requires non-inertial frames in order to make sense.

19 minutes ago, joigus said:

 

but covariant under general coordinate transformations.

\[G μ ν =8 π G( T μ ν + δ T μ ν )\]

 

Covariant means it is a tensor, and if a tensor is zero in one reference frame at a point, it is zero in every reference frame. So your force is not zero in any reference frame.

Edited by joigus

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Hadn't even noticed the part that involved acceleration; I just assumed it meant a higher speed.

So, assume you have a mass on the verge of gravitational collapse to a BH.
And you accelerate it such that the fields and/or dynamics of the system lead to gravitational collapse.

What happens if you remove the accelerated condition/field geometries/system dynamics ?
Does the BH drop its event horizon, and pop ( very big pop ) back into 'normal' space-time ?
Or does it continue as a BH ?
 

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Posted (edited)
58 minutes ago, MigL said:

Hadn't even noticed the part that involved acceleration; I just assumed it meant a higher speed.

So, assume you have a mass on the verge of gravitational collapse to a BH.
And you accelerate it such that the fields and/or dynamics of the system lead to gravitational collapse.

What happens if you remove the accelerated condition/field geometries/system dynamics ?
Does the BH drop its event horizon, and pop ( very big pop ) back into 'normal' space-time ?
Or does it continue as a BH ?
 

Once it's fallen, there's no way back.That's what the theory says (Hawking radiation aside.) In fact, there is a very, simple, very nice Gedanken experiment explained by Lenny Susskind of a sphere of photons converging to a point, in such a way that a BH is bound to form, but even though you have the illusion that something could be done about it, there is a point past which the photons are doomed. I don't remember the details, but maybe I could find it. But I wouldn't bet my life on what a BH is actually going to do.

On a rather more speculative note, I think gravitational horizons have something very deep to do with the problem of the arrow of time that we haven't understood very well at all.

Edited by joigus
mistyped

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Posted (edited)

I wouldn't think a BH could remain a BH if its mass wasn't equivalent, or greater than    (Rs*c^2)/(2*G).
It doesn't make sense, then again nothing about BHs is common sense.

I had read about J A Wheeler's Geons ( gravitationally constrained ( non-singular ) EM radiation, but can't seem to find anything by L Susskind on gravitationally collapsed photons with a simple google search.

Edited by MigL

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Posted (edited)
33 minutes ago, MigL said:

I wouldn't think a BH could remain a BH if its mass wasn't equivalent, or greater than    (Rs*c^2)/(2*G).
It doesn't make sense, then again nothing about BHs is common sense.

There are so many speculations about them. You're right. There's nothing common-sense about BHs. I think angular momentum may play a key part and, although Schwarzschild's solution is an exact one, only the Kerr-Newman maybe makes sense, the rotating one.

33 minutes ago, MigL said:

I had read about J A Wheeler's Geons ( gravitationally constrained ( non-singular ) EM radiation, but can't seem to find anything by L Susskind on gravitationally collapsed photons with a simple google search.

It's in one of his video lectures. Don't know where it is though. I'll look it up. The thing about Susskind is he's so intuitive and pictorial. Even in the most abstract and difficult topics. Although he always discusses the Schwazschild solution.

His lectures on supersymmetry, even though he's clearly unhappy at the end, are amazing. SS I think must be correct in some sense we haven't understood. A basic exposition like Susskind's I think is perfect for anybody young, without prejudice, that would like to have a go at a possible re-interpretation. But that's off-topic.

Edited by joigus
misworded

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8 hours ago, MigL said:

I had read about J A Wheeler's Geons ( gravitationally constrained ( non-singular ) EM radiation, but can't seem to find anything by L Susskind on gravitationally collapsed photons with a simple google search.

Thank you for this pointer. I didn't know about geons. I'm very interested in self-consistent classical solutions.

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11 hours ago, MigL said:

So, assume you have a mass on the verge of gravitational collapse to a BH.
And you accelerate it such that the fields and/or dynamics of the system lead to gravitational collapse.

One cannot simply assume that accelerating such a mass will automatically lead to collapse; this is actually a very complicated problem, and would have to be treated mathematically to see what would happen. Even accelerating a fully-formed black hole leads to some non-trivial results (see here for example), and a system just about to undergo collapse is far less trivial still (remember we would need to work with interior metrics here).

My intuition is that nothing would happen actually, but I might well be wrong.

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1 hour ago, Markus Hanke said:

One cannot simply assume that accelerating such a mass will automatically lead to collapse; this is actually a very complicated problem, and would have to be treated mathematically to see what would happen. Even accelerating a fully-formed black hole leads to some non-trivial results (see here for example), and a system just about to undergo collapse is far less trivial still (remember we would need to work with interior metrics here).

My intuition is that nothing would happen actually, but I might well be wrong.

Completely agree. That problem calls for numerical GTR.

I was thinking along same lines when I said,

13 hours ago, joigus said:

Whether the system collapses or not would, I suppose, depend on the details of the dynamics. Maybe what you would do is save it from collapse.

On the other hand, linear accelerations for stellar objects are very rare phenomenologically speaking, I surmise. Plus any linear acceleration field would lead to accretion rather that inducing threshold trespassing, my intuition tells me, concurring with you, perhaps.

But there could be an experimental/astrophysical context to do the trick perhaps. A neutron star approached by a heavy stellar interloper which induces strong tidal forces in it, as well as very intense centrifugal potentials. See if any such event of BH formation can be detected. But the acceleration field we're talking about would be very different, of course. 

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I was thinking of more 'local' phenomena, like trying to induce micro BH formation at the LHC...

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4 hours ago, MigL said:

I was thinking of more 'local' phenomena, like trying to induce micro BH formation at the LHC...

During the first months of LHC running its first tests, I got wind that micro BH formation tested negative. It was an informal conversation, but the source was reasonably reliable. We seem to be stuck at Higgs...

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I think I distracted the thread when I said "and we accelerate it [the object]." We could alternatively observe the mass, which is just on the brink of black hole collapse, after we have gone under acceleration ourselves such that the mass now apparently has sufficient energy to collapse.

The answer has already apparently been given, which is that the stress-energy-momentum tensor is covariant in inertial frames. To me, this generates more questions. We know that a rotating mass requires less rest mass than a non-rotating one for gravitational collapse. This could be explained by observing that angular momentum is absolute, and that the mass is under acceleration due to rotation, therefore all inertial frames will acknowledge it...

But what is a rotating mass, exactly? What about a binary star system rotating at speeds sufficient to predict black hole creation? The stars, A and B, would each claim to be free-falling and not under acceleration. They would each calculate the other body to be orbiting them at extraordinary speeds. A remote observer C could predict that the A-B system should collapse to form a black hole, but does that analysis work for either orbiting body?

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2 hours ago, rjbeery said:

The answer has already apparently been given, 

I cannot be completely sure that the answer has been given, as you seem to falter quite a bit in your premises:

2 hours ago, rjbeery said:

I think I distracted the thread when I said "and we accelerate it [the object]."

Naturally. You must really think what you say, say what you think, and may I add, think what you think. And,

2 hours ago, rjbeery said:

which is that the stress-energy-momentum tensor is covariant in inertial frames.

No:

On 5/7/2020 at 12:19 AM, joigus said:

That's not frame-dependent, but covariant under general coordinate transformations.

Covariant under general coordinate transformations. You must really read what you read.

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37 minutes ago, joigus said:

Covariant under general coordinate transformations. You must really read what you read.

I'll try my best, but I do not appreciate the difference between being "covariant under general coordinate transformations" and being "covariant under inertial frames". Isn't the latter just special case of the former, and doesn't the latter fit the scenario we're discussing?

In any event, I'd like to keep acceleration out of this, for simplicity. MigL read my intention correctly in that the original mass was "given" momentum energy by simply changing frames. Sorry for the confusion.

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13 minutes ago, rjbeery said:

I'll try my best, but I do not appreciate the difference between being "covariant under general coordinate transformations" and being "covariant under inertial frames". Isn't the latter just special case of the former, and doesn't the latter fit the scenario we're discussing?

In any event, I'd like to keep acceleration out of this, for simplicity. MigL read my intention correctly in that the original mass was "given" momentum energy by simply changing frames. Sorry for the confusion.

T's ok. I just tire easy around morphing arguments. Maybe in other life.

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14 hours ago, rjbeery said:

We could alternatively observe the mass, which is just on the brink of black hole collapse, after we have gone under acceleration ourselves such that the mass now apparently has sufficient energy to collapse.

These scenarios are not physically equivalent - only “us” would measure proper acceleration in our frame, but not the mass in its own rest frame. There is no symmetry between frames once proper acceleration is involved. 

14 hours ago, rjbeery said:

We know that a rotating mass requires less rest mass than a non-rotating one for gravitational collapse.

Do we really know this? There is no known closed analytic interior counterpart to the exterior Kerr metric (unlike is the case for Schwarzschild), so correctly modelling the gravitational collapse of a rotating body is mathematically a difficult problem. Also, the concept of “mass” is not straightforward here, since it is now one parameter in a 2-parameter family of metrics, and hence a global property of the entire spacetime.

14 hours ago, rjbeery said:

But what is a rotating mass, exactly?

It’s a gravitational source the exterior vacuum of which can be approximately described by the Kerr metric.

14 hours ago, rjbeery said:

What about a binary star system rotating at speeds sufficient to predict black hole creation?

It is unlikely that such a thing is feasible, since the kinetics of that system would simply separate the two bodies, long before the gravitational effects would approach criticality. In terms of mathematical analysis, this is a relativistic 2-body problem, which is much more complex than a single rotating mass; such a spacetime is not described by the Kerr metric. I’m sure the analysis can be done, but only numerically.

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I'm simply going to assume that any system rotating fast enough to have a stress-energy-momentum increase non-trivial enough to cause gravitational collapse, would have lost gravitational cohesion, and flown apart, long before it got to that rotation speed.
( I'll leave the math to you Markus )

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3 hours ago, Markus Hanke said:

Do we really know this? There is no known closed analytic interior counterpart to the exterior Kerr metric (unlike is the case for Schwarzschild), so correctly modelling the gravitational collapse of a rotating body is mathematically a difficult problem. Also, the concept of “mass” is not straightforward here, since it is now one parameter in a 2-parameter family of metrics, and hence a global property of the entire spacetime.

I don't personally know this but that was my impression. If we don't believe that it's true then my confusion is actually resolved...kind of. My layman impression of energy in GR is that its form is irrelevant (mass or otherwise), and that energy density contributes to the local gravitational field. The crux of my confusion is that momentum energy is frame-dependent, whereas black hole creation is obviously an absolute event. If we allow angular momentum energy to contribute to the creation of a black hole, but really scrutinize the difference between angular momentum energy and linear momentum energy, then I'm left with questions.

 

 

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16 hours ago, rjbeery said:

If we don't believe that it's true

Just to clarify my earlier comment...it could be true of course, but my point was that we can’t just assume that it is without running the maths first. If I have learned one thing about GR in all my many years of engaging with it, then that would be that it is always good for a surprise or two, and not to trust my “common sense”!

16 hours ago, rjbeery said:

The crux of my confusion is that momentum energy is frame-dependent, whereas black hole creation is obviously an absolute event. If we allow angular momentum energy to contribute to the creation of a black hole, but really scrutinize the difference between angular momentum energy and linear momentum energy, then I'm left with questions.

Yes, this is a common (and very valid!) source of confusion. The answer to this one lies in realising that within the field equations, the source term for gravity is not energy, but the full stress-energy-momentum tensor. This entity describes the densities, momenta and fluxes of all forms of energy-momentum in spacetime (except nonlinear self-gravitation, which is encoded in the structure of the field equations themselves), and as such it contains contributions of various different kinds, also including angular momentum. To put it very simply, gravitational collapse is governed not just by energy, but by the full energy-momentum tensor; and since it is a tensor, all observers agree on it, and no contradictions arise in the first place. 

Actually writing down the energy-momentum tensor for a given system (never even mind inserting it into the field equations and solving those) can be a very non-trivial task, however. In the case of a spinning mass on the brink of collapse, I am not aware of any closed analytical treatment; one would need to approach this with numerical methods. Note also that while all observers agree on whether a collapse takes place (or not), they will disagree on when the collapse happens, where it happens (if there is relative motion), and how long it takes.

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