A Dustball Collapse & Evaporation: No Black Hole Singularity

[OJ Bernander]

TL;DR Summary

Dustball collapse is transformed to standard coordinates. The shrinkage speed of the Schwarzschild radius is much larger than the internal coordinate speed of light.

This old nut is often dismissed as flawed thinking by a naive student. It’s been discussed here too, I know.
However, when you do the math in standard (Schwarzschild-like) coordinates:

Dustball collapse and evaporation in standard coordinates (Arxiv)

A. the (coordinate) speed of light inside a collapsing star is much smaller than outside,
B. the speed at which the Schwarzschild radius shrinks (by evaporation) is intermediate between the two.

Therefore if we perturb the classical solution with an evaporation process at the (shrinking) Schwarzschild radius:
1. the perturbation only spreads to the exterior, not the interior, limited by c,
2. the singularity is not in the part of the (interior) solution that survives,
3. infalling particles don’t cross the Schwarzschild radius,
4. outgoing photons are temporarily frozen, but eventually emerge (no horizon forms).

The analytic work shows A and B. Then 1-4 follow.

I’ve seen others speculate on 1-4, but not back it up with A and B.
Such speculation is often dismissed, using intuition from non-evaporating black holes.
A and B should correct the flawed intuition where it doesn’t apply to evaporating collapse.

Thoughts?
-Öjvind
(I have read the discussion here on PF)

 

[PeterDonis]

if we perturb the classical solution with an evaporation process at the (shrinking) Schwarzschild radius

Your model of this seems to me to be leaving out a crucial component: if the hole is evaporating, outgoing radiation is being emitted. But there is no outgoing radiation in your model. Your model has the spacetime being entirely Minkowski after evaporation is complete; but that can't be correct. The radiation has to be somewhere. The radiation could be in an outgoing region that leaves a Minkowski region behind inside it once evaporation is complete, but it still has to be somewhere.

In the absence of radiation, what you are modeling is a Schwarzschild black hole with a mass that decreases with time, with nothing else present. I'm not even sure this is a consistent solution of the Einstein Field Equation.

 

[PeterDonis]

I'm not even sure this is a consistent solution of the Einstein Field Equation.

In fact, now that I've done some checking, I'm sure it's not a consistent solution of the Einstein Field Equation. In Schwarzschild coordinates, the metric of a spherically symmetric metric can always be written in this form:

$$
ds^2 = - J(r, t) + \frac{1}{1 - \frac{2m(r, t)}{r}} + r^2 d\Omega^2
$$

Here we have two undetermined functions, $J$ and $m$, and in the general case, both could be functions of $r$ and $t$. If we compute the Einstein tensor of this metric, we see that if $m$, the mass, is a function of either $r$ or $t$, the spacetime is not vacuum (the Einstein tensor does not vanish). So there is no such thing as a Schwarzschild black hole with non-constant $m$ and nothing else present; if $m$ is changing, stress-energy has to be present. (The simplest known solutions describing this are, of course, the Vaidya solutions; the outgoing one would be the appropriate one to model outgoing radiation due to black hole evaporation.)

 

[OJ Bernander]

In the absence of radiation, what you are modeling is a Schwarzschild black hole with a mass that decreases with time, with nothing else present. I'm not even sure this is a consistent solution of the Einstein Field Equation.

Thanks for your reply.
You may not have gotten to read the Discussion section yet, which acknowledges that there will be a modification to the exterior Schwarzschild metric, for example something like Vaidya's.
The point of the paper is to put constraints on what the modification to classical collapse would be, and that such modifications would only be in the exterior, not the interior.

 

[PeterDonis]

The point of the paper is to put constraints on what the modification to classical collapse would be, and that such modifications would only be in the exterior, not the interior.

But you can't do that if you're not using a valid solution of the Einstein Field Equation. And you're not. It's good that you recognize that the actual metric won't be vacuum, but then you need to use that fact throughout your entire analysis. The analysis is simply not valid without it.

 

[PeterDonis]

there will be a modification to the exterior Schwarzschild metric, for example something like Vaidya's

The Vaidya metric doesn't just modify the "exterior". The Einstein Field Equation is local: what I said in post #3 applies to any event in the spacetime at which the mass is not constant (i.e., at which its derivative with respect to either $r$ or $t$ is nonzero).

 

[PAllen]

For context, has your (OP paper) gotten any peer review yet? There is no indication of publication or submission for peer review that I can find using arxiv tools.

 

[OJ Bernander]

For context, has your (OP paper) gotten any peer review yet? There is no indication of publication or submission for peer review that I can find using arxiv tools.

PAllen, no, there is no peer review yet. I didn't mean to hide that fact.

But you can't do that if you're not using a valid solution of the Einstein Field Equation.

So the paper offers one actual solution, for classical collapse. It's the Oppenheimer-Snyder dustball collapse transformed from comoving coordinates to standard coordinates. This solution should be correct

Then the paper offers pointers for how that solutions should be modified by evaporation, without trying to actually give a full solution. The pointers are based on two generally accepted notions.
1. Many of the cited papers talk of the process as occurring at or near the Schwarzschild radius R_S (even though the derivation of Hawking radiation involves detecting radiation far away and comparing vacua before and after).
2. The mass, and thus R_S, shrinks with time.

This then suggest a perturbation at the the shrinking R_S, moving along with R_S.
What are limits on how this perturbation can modify the classical solution? Well, a perturbation's effect propagates at most with the (coordinate) speed of light, c. That c is vastly smaller inside than the speed at which R_S (and hence the perturbation) moves. Thus the moving perturbation will overtake any of its earlier effects interior to it. Meanwhile, effects of the perturbation exterior to R_S can propagate away, modifying the exterior metric in some way (the details of which is not proposed).

That's all the paper tries to say (poorly?).

 

[PeterDonis]

the paper offers one actual solution, for classical collapse. It's the Oppenheimer-Snyder dustball collapse transformed from comoving coordinates to standard coordinates. This solution should be correct

Transforming from one coordinate chart to another cannot change the physics, and the physics of the Oppenheimer-Snyder model is already known: there is an event horizon, a black hole, and a singularity.

the paper offers pointers for how that solutions should be modified by evaporation

Your "pointers" do not include any outgoing radiation or any other stress-energy. But a simple computation using the Einstein Field Equation for a spherically symmetric spacetime will show you that, as I have already said, in any region of spacetime where the mass $m$ is not constant (which in your perturbed model is everywhere before $t_\text{Life}$), there must be nonzero stress-energy; the Einstein tensor, and therefore the stress-energy tensor, does not vanish. Your "pointers" do not include this and therefore cannot be correct.

 

[PeterDonis]

there is no peer review yet

For your awareness, PF is not intended for discussion of research in progress. That's what your paper is since it has not been peer-reviewed or published. We have given some feedback here, but PF is not the place for an extended discussion or critique. That is what the peer review process is for.

Accordingly, this thread is now closed.