How do gravitons escape a black hole?

[BigBuffet]

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One way to think of the way gravity curves spacetime is to picture a bedsheet on your bed with different sized ball bearings on it representing stars. Now put a bowling ball representing a massive black hole in the middle. Well... the smaller bearings are going to roll towards it.

You see, there is no comminication between the bowling ball and the ball bearings at all, but they *are* affected by each other. The key concept here is that *gravity warps spacetime* just like that bowling ball warps your bedsheet.

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Wouldn't the bigger ball bearings roll toward and then hit the bowling ball first thus causing the smaller ball bearings to ricochet off them back in the direction they came from?

[Metric]

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The event horizon *is* the singularity / where the singularity occurs.

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No, it's not.

[Skidoo]

Space aside, the question remains as to just how there could be a causal relationship of any sort between something, even a quantity of mass per se, within the critical radius and the outside world.

[Metric]

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Let's assume the mass (and maybe charge and angular momentum) of a black hole is concentrated at the `singularity'?

Suppose it is a big black hole, so the event horizon is a long way from the `singularity'.

Then in the Hawking evaporation of the black hole (assume nothing is falling in, and don't worry about final stages for now) it seems to me that the mass (and maybe charge and angular momentum) must come not just from around the event horizon, but instead must makes its way out all the way from the `singularity', meaning that there is a flow of mass (and maybe charge and angular momentum) from the `singularity' all the way out to the event horizon and beyond.

QUESTION: Is something like this what is actually happening?

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The standard understanding of the Hawking effect relies only on the properties of fields to the exterior of the BH. It doesn't depend on a flux of particles moving from the singularity to the horizon. In fact, if you try to make the "flux of particles" idea concrete, you quickly run into problems.

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Moreover the strength of this flow should be calculable from standard formulas relating strength of Hawking/Unruh radiation to acceleration (gravitational or otherwise).
So the Hawking/Unruh radiation is stronger nearer the `singularity'. And it has mass. Maybe all of the mass is in this form leaving no singularity!

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Mass with respect to what? Outside the BH, "mass" makes sense with respect to a timelike Killing vector that defines symmetry with respect to time translations. Inside the BH, there is no such symmetry with which to define "mass" unambiguously.

[Upstairs]

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The event horizon *is* the singularity / where the singularity occurs.

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No, it's not.

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I stand corrected.

[thylacine]

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Let's assume the mass (and maybe charge and angular momentum) of a black hole is concentrated at the `singularity'?

Suppose it is a big black hole, so the event horizon is a long way from the `singularity'.

Then in the Hawking evaporation of the black hole (assume nothing is falling in, and don't worry about final stages for now) it seems to me that the mass (and maybe charge and angular momentum) must come not just from around the event horizon, but instead must makes its way out all the way from the `singularity', meaning that there is a flow of mass (and maybe charge and angular momentum) from the `singularity' all the way out to the event horizon and beyond.

QUESTION: Is something like this what is actually happening?

[/ QUOTE ]
The standard understanding of the Hawking effect relies only on the properties of fields to the exterior of the BH. It doesn't depend on a flux of particles moving from the singularity to the horizon. In fact, if you try to make the "flux of particles" idea concrete, you quickly run into problems.

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Moreover the strength of this flow should be calculable from standard formulas relating strength of Hawking/Unruh radiation to acceleration (gravitational or otherwise).
So the Hawking/Unruh radiation is stronger nearer the `singularity'. And it has mass. Maybe all of the mass is in this form leaving no singularity!

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Mass with respect to what? Outside the BH, "mass" makes sense with respect to a timelike Killing vector that defines symmetry with respect to time translations. Inside the BH, there is no such symmetry with which to define "mass" unambiguously.

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Is this question `how does the mass get from the "singularity" to the outside?' something that, at least if formulated correctly (which could be the main difficulty) would be a serious and interesting physics question? Certainly there are many questions associated with the overall process.

Or is it as fundamentally wrongheaded as asking, say, `which slit did the photon go through?' (in the double-slit experiment)?

Or somewhere in between?

[Metric]

I would say that your question lies somewhere between "already answered" (energy leaves the BH because "negative energy" modes are falling in) and "not too well defined" (if you're wanting to do something like calculating how much of the BH mass lies in a particular extended region of the BH).

But you could probably morph your question into something very interesting indeed with a bit of work -- as you said, there are lots of puzzles here.

[thylacine]

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I would say that your question lies somewhere between "already answered" (energy leaves the BH because "negative energy" modes are falling in) and "not too well defined" (if you're wanting to do something like calculating how much of the BH mass lies in a particular extended region of the BH).

But you could probably morph your question into something very interesting indeed with a bit of work -- as you said, there are lots of puzzles here.

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I tried reading something on QFT in curved spacetime once (and I don't really understand QFT to start with) and I vaguely remember there was some issue with defining "negative energy" modes to start with, whereas it is not a problem to canonically make a choice in standard QFT. Is this issue related to making what you said above work.

I vaguely see what the problems are with defining mass, (something along the lines of: Newton would integrate mass density over a 3D region, or do an equivalent surface integral, but there are some problems doing this in GR even though its worth a try) but I'll have to do some reading. My attempt to understand GR comes from reading "Spacetime and geometry : an introduction to general relativity" by Sean Carroll, which I found very good. I did not know what a differentiable manifold was before reading it.

Can you recommend a QFT book for a mathematician?

As for `"negative energy" modes are falling in' (? to the singularity) that's probably good enough for me in place of energy flowing out. I would just expect some kind of causal connectivity, which would be `local' in some sense. That's really what would prompt me to wonder how to bridge what I might perceive to be a (causal) gap between the singularity and the event-horizon.

[Metric]

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tried reading something on QFT in curved spacetime once (and I don't really understand QFT to start with) and I vaguely remember there was some issue with defining "negative energy" modes to start with, whereas it is not a problem to canonically make a choice in standard QFT. Is this issue related to making what you said above work.

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This is related to that whole Killing vector issue I mentioned before. In flat spacetime, there's a canonical choice of positive energy modes (particles) due to the existence of a timelike Killing vector. This nice property goes away in general spacetimes. Or you may have a timelike Killing vector in one region of spacetime (outside the BH) and not in another (inside the BH).

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I vaguely see what the problems are with defining mass, (something along the lines of: Newton would integrate mass density over a 3D region, or do an equivalent surface integral, but there are some problems doing this in GR even though its worth a try) but I'll have to do some reading. My attempt to understand GR comes from reading "Spacetime and geometry : an introduction to general relativity" by Sean Carroll, which I found very good. I did not know what a differentiable manifold was before reading it.

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If that's the relatively new book by Carroll based on his online GR notes, I agree that it's excellent. I would also recommend "A relativist's toolkit: the mathematics of black hole mechanics" by Eric Poisson if you want a great book to reference.

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Can you recommend a QFT book for a mathematician?

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For what we're talking about here, "Quantum fields in curved space" by Birrell and Davies is the right place to go. You get the field theory without all the heavy formalism developed to do big calculations with the standard model, etc. (but on the other hand it's much heavier in the GR department of course) A great deal of the stuff (including Hawking's work) people have studied in the curved spacetime setting has just been done with super-simple, non-interacting scalar fields -- you don't need QCD to get at the general principles of what is going on.

For a more standard treatment of QFT, but with some good tidbits thrown in (like slick, three line derivations of the Hawking effect, etc.) get "Quantum Field Theory in a Nutshell" by Zee.

[Charon]

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A couple of good responses so far. Let's expand the list of questions.

1. How do gravitons escape a black hole?


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A "graviton producing machine" inside the BH will not be able to send real gravitons out past the event horizon. (provided we remain in the limit where the BH solution can be thought of as a reasonable background)


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Isn't it true that virtual particles can travel faster than light as long as they don't violate the uncertainty principle? So if that is true, then gravitons can escape the event horizon as long as their velocity is high enough.

And has a "graviton producing machine" any meaning?
From my knowledge, virtal particles (like photon, Z-boson, W-boson) are obtained by quantizing the appropriate field, where every gauge transformation of the field corresponds to a different type of particle. The excitations of the gauge fields represent particles. So the field (in this case the gravitational field) can be described in terms of excitations (particles) which can be created and annihilated. It's not like the mass in the BH is constantly sending out gravitons. Am I correct?

Thanks.