Collapse and unitary evolution

[Demystifier]

No, I'll ask if there is such a result about the approximation, or is it just expected. What about stubility?

It's expected by my physical intuition, but maybe I should add that I expect it for $r>2M(t)$. I'm sure that someone made explicit calculations and haven't obtained anything very surprising, because otherwise I would already heard about that. After all, the mass of the Sun is not constant, yet the Schwarzschild metric describes the motion of planets around Sun very well.

 

[PeterDonis]

It's the usual radial coordinate in the Schwarzshild metric.

Or, to make clear that this $r$ labels a geometric invariant, it is the "areal radius" of the 2-sphere on which a given event lies, i.e., if the 2-sphere's area is $A$, then $A = 4 \pi r^2$, so $r = \sqrt{A / 4 \pi}$.

 

[PeterDonis]

It's about uncharged nonrotating black hole, with a possibly non-constant mass. If the mass changes sufficiently slowly, then the metric can be approximated by Schwarztschild metric with $M\rightarrow M(t)$.

I'll ask if there is such a result about the approximation

The Vaidya metric is an exact solution describing a non-rotating, uncharged black hole either emitting or absorbing null dust (basically incoherent EM radiation emitted or absorbed isotropically, equally in all directions). See here:

https://en.wikipedia.org/wiki/Vaidya_metric

Note that the usual way of writing this metric is in the equivalent of Eddington-Finkelstein coordinates, where there is a null coordinate $u$ or $v$ (depending on whether you are looking at the outgoing--emitting radiation--or ingoing--absorbing radiation--case) instead of the Schwarzschild $t$. For this case, $M$ is a function of $u$ or $v$ only, as shown in the article, and this is true regardless of the rate of change of $M$ with respect to $u$ or $v$.

I believe what @Demystifier is referring to is an approximation in which (for the ingoing case) $dM / dv$ is small enough that you can transform into standard Schwarzschild coordinates and still have $M$ be a function of only $t$ (instead of both $t$ and $r$, as it would be in the general case with unrestricted rate of change) for the duration of the process he is analyzing.

 

[PeterDonis]

the mass of the Sun is not constant, yet the Schwarzschild metric describes the motion of planets around Sun very well

The usual treatment of the Solar System is basically the order by order PPN expansion of the Schwarzschild metric for the case of small $M / r$, yes. But this works well because the mass loss (from radiation, solar wind, etc.) is so small compared to $M$ that its effects are negligible. It's not because the solution explicitly uses an $M$ that is changing with time (at least, that's my understanding).

 

[martinbn]

It's expected by my physical intuition, but maybe I should add that I expect it for $r>2M(t)$. I'm sure that someone made explicit calculations and haven't obtained anything very surprising, because otherwise I would already heard about that. After all, the mass of the Sun is not constant, yet the Schwarzschild metric describes the motion of planets around Sun very well.

But the Sun is very far from a black hole. I think it requires more justification than intuition to extrapolate. It would be interesting to see a theorem.

Or, to make clear that this $r$ labels a geometric invariant, it is the "areal radius" of the 2-sphere on which a given event lies, i.e., if the 2-sphere's area is $A$, then $A = 4 \pi r^2$, so $r = \sqrt{A / 4 \pi}$.

Which two sphere?

The Vaidya metric is an exact solution describing a non-rotating...

My exclamation wasn't about which solution describes the scenario but about the fact that it is one solution that he had in mind. The statement seems to be about black holes in general, then it turns out that when physicists say "black hole" they mean the Schwartzschild solution or Vaiday, or something else but just one solution. Demystifier said that it should be a good enough approximation, which is probably true, but it would be interesting to see a precise statement.

 

[PeterDonis]

Which two sphere?

Whichever 2-sphere the particular event you are interested in (the one labeled with a given value of $r$) lies on. The entire spacetime is spherically symmetric, which means every event in the spacetime lies on some 2-sphere with a definite area.

 

[martinbn]

Whichever 2-sphere the particular event you are interested in (the one labeled with a given value of $r$) lies on. The entire spacetime is spherically symmetric, which means every event in the spacetime lies on some 2-sphere with a definite area.

Ah, ok, so there is an assumption that the space-time is spherecally symmetric.

 

[Demystifier]

By the way, I have proposed a version in which Hawking radiation is completely irrelevant.
https://arxiv.org/abs/1802.10436

The paper is now published in Universe. It's completely open: everyone can see the paper, the names of reviewers, their reports, and my responses to reviewers.
https://www.mdpi.com/2218-1997/9/1/11

 

[physika]

new interactions does emerge as complexity increases, that was physically impossible at lower complexity.

But this is not a viable strategy for one human interacting with other humans. Instead the complex systems develops behaviour that due to chaos can not be inferred from knowledge of interaction of parts.

Example; altruistic attitudes.

 

[physika]

I'm not sure who said people weren't made of particles.

Indeed.