I Is the Schwarzchild Radius Relativistic or Newtonian?

[PeterDonis]

A Kerr black hole for example is not static (it is - however - stationary).

The exterior (outside the horizon) of Kerr spacetime is stationary. The interior (inside the horizon) is not. (Nor is the interior of Schwarzschild spacetime.)

The interior of the black hole can't be material in (hydrostatic) equilibrium because all world lines for all physical particles must reach the singularity in finite proper time

"All worldlines must hit the singularity in finite proper time" is only true for Schwarzschild spacetime. It is not true for Kerr spacetime; there are worldlines in maximally extended Kerr spacetime that never hit the singularity at all (they eventually emerge into another stationary exterior region that is disconnected from the original one that the material fell in from). Also, Kerr spacetime, unlike Schwarzschild spacetime, has an inner Cauchy horizon that makes any kind of extrapolation beyond it problematic. And also again, the interior regions of both Schwarzschild and Kerr spacetime are unstable against small perturbations, which means they probably don't actually describe the interiors of physically realistic black holes, at least not well inside the horizon. (Look up "BKL singularity" for an example of a more realistic possibility.)

None of this changes the key points for this discussion: black hole interiors are not static, and they must be almost all vacuum because material falling in doesn't stay in place.

 

[Matterwave]

The exterior (outside the horizon) of Kerr spacetime is stationary. The interior (inside the horizon) is not. (Nor is the interior of Schwarzschild spacetime.)

There is no time-like killing field for the interior of a Schwarzschild space time? Somehow this point has been glossed over in my readings on the subject. I assumed that since the "static" and "vacuum" assumptions are built into finding the Schwarzschild solution itself, that it would be valid everywhere. My mistake.

 

[Dale]

How do we know the inside of a black hole is a vacuum? How do we know the inside of a black hole is static?

We don’t have to go inside a black hole. The analogy fails well outside of the horizon, where we have plenty of experimental evidence.

If you still think it is a valid analogy despite the contradictions indicated above then it is up to you to provide some support for your claim.

 

[PeterDonis]

There is no time-like killing field for the interior of a Schwarzschild space time?

No, there is not. Only in the exterior. Inside the horizon, the Killing field is still there, but it's spacelike, not timelike. (On the horizon, it's null; the horizon is actually generated by the null integral curves of the Killing field at $r = 2M$.)

I assumed that since the "static" and "vacuum" assumptions are built into finding the Schwarzschild solution itself, that it would be valid everywhere.

As it happens, I wrote an Insights article a while back about the derivation of Birkhoff's theorem which discusses this.

https://www.physicsforums.com/insights/short-proof-birkhoffs-theorem/

The key is that "static" is not an assumption; it's a result of the derivation, but the result isn't actually "static" (though it's often misstated that way, even in textbooks, unfortunately), it's "there's a fourth Killing field in addition to the three that are there because of spherical symmetry". The theorem does not prove, or require, that the fourth Killing field is timelike everywhere. And indeed, you can see from the line element that it is only timelike for $r > 2M$; it's null at $r = 2M$ and spacelike for $r < 2M$.

 

[Matterwave]

No, there is not. Only in the exterior. Inside the horizon, the Killing field is still there, but it's spacelike, not timelike. (On the horizon, it's null; the horizon is actually generated by the null integral curves of the Killing field at $r = 2M$.)
As it happens, I wrote an Insights article a while back about the derivation of Birkhoff's theorem which discusses this.

https://www.physicsforums.com/insights/short-proof-birkhoffs-theorem/

The key is that "static" is not an assumption; it's a result of the derivation, but the result isn't actually "static" (though it's often misstated that way, even in textbooks, unfortunately), it's "there's a fourth Killing field in addition to the three that are there because of spherical symmetry". The theorem does not prove, or require, that the fourth Killing field is timelike everywhere. And indeed, you can see from the line element that it is only timelike for $r > 2M$; it's null at $r = 2M$ and spacelike for $r < 2M$.

Thanks for this! I will read it when I get the time.

 

[PAllen]

There is no time-like killing field for the interior of a Schwarzschild space time? Somehow this point has been glossed over in my readings on the subject. I assumed that since the "static" and "vacuum" assumptions are built into finding the Schwarzschild solution itself, that it would be valid everywhere. My mistake.

Precise statement is that vacuum plus spherical symmetry imply an additional killing vector field besides those for spherical symmetry. Outside the horizon, it is timelike. Inside it is spacelike, and may be considered axial. If you allow nonanalytic manifolds, there are many different topologies than Kruskal that are possible. I believe some have no timelike killing vector field at all. I will try to dig up a paper I’ve linked in the past that explores this.

Addendum: I found the papers:

https://arxiv.org/abs/0908.4110
https://arxiv.org/abs/0910.5194

See top of page 6 of second paper which gives vacuum spherically symmetric manifolds that are nowhere static, nor are they asymptotically flat. They are also not extensible, thus not a cheat by just chopping a manifold at some boundary, e.g. a piece of the interior.

 

[bbbl67]

We don’t have to go inside a black hole. The analogy fails well outside of the horizon, where we have plenty of experimental evidence.

If you still think it is a valid analogy despite the contradictions indicated above then it is up to you to provide some support for your claim.

What experimental evidence is that?

This paper, by Carlo Rovelli & Marios Christodoulou, shows that the inside of a black hole is bigger than the outside, if you use certain coordinate systems! The volume grows bigger with time, and even with a stellar mass black hole, the interior volume can become bigger than the current observable universe. This sounds pretty much like a universe to me (also sounds like a Tardis)!

Paper: [1411.2854] How big is a black hole?

 

[Dale]

This paper, by Carlo Rovelli & Marios Christodoulou, shows that the inside of a black hole is bigger than the outside, if you use certain coordinate systems! The volume grows bigger with time, and even with a stellar mass black hole, the interior volume can become bigger than the current observable universe.

First, to use your own words, how do you know the inside of a black hole is bigger than the outside? Second, how is that any kind of analogy with the universe since the Big Bang (FLRW) universe has no outside? Third, that does not address the vacuum objection. Fourth, that does not address the homogeneity objection. Fifth, that does not address the asymptotic flatness objection.

Note that at no point in that paper does Rovelli make any claim that a Schwarzschild spacetime is analogous to a FLRW spacetime, as you did. While being interesting, the paper does not support your specific objectionable claim.

Regarding the evidence that we have outside of the EH you can consider all of the solar system tests of the PPN that show the PPN parameters match those of GR. All of the solar system tests are based on the Schwarzschild spacetime.

 

[martinbn]

Just a pedantic comment about Schwartschild being static. To show that the interior is not even stationary one needs to prove that there aren't any timelike Killing fields there. The fact that the Killing field that is timelike outside becomes spacelike inside, by itself is not enough, there still could be another timelike Killing field there. Of course that is not true, but it needs to be shown.

 

[PeterDonis]

To show that the interior is not even stationary one needs to prove that there aren't any timelike Killing fields there. The fact that the Killing field that is timelike outside becomes spacelike inside, by itself is not enough, there still could be another timelike Killing field there.

No, there couldn't, because the metric inside is still Schwarzschild (by Birkhoff's theorem), and we already know Schwarzschild has no other Killing fields.

 

[martinbn]

No, there couldn't, because the metric inside is still Schwarzschild (by Birkhoff's theorem), and we already know Schwarzschild has no other Killing fields.

Of course, may be I wasn't clear. My remark simply says that to say $\partial_t$ becomes spacelike int the interior is not enough, one needs to show that no other Killing field is timelike there. What you say is a perfectly good proof why there aren't any other.

 

[bbbl67]

Just a pedantic comment about Schwartschild being static. To show that the interior is not even stationary one needs to prove that there aren't any timelike Killing fields there. The fact that the Killing field that is timelike outside becomes spacelike inside, by itself is not enough, there still could be another timelike Killing field there. Of course that is not true, but it needs to be shown.

What exactly is a Killing Field?

 

[Nugatory]

What exactly is a Killing Field?

Google will be your friend - if you add the words "general relativity" to your search for "killing field".

 

[stevendaryl]

What exactly is a Killing Field?

It's a gruesome phrase, isn't it? It sounds like it should mean some valley where a mass murder took place.

 

[bbbl67]

It's a gruesome phrase, isn't it? It sounds like it should mean some valley where a mass murder took place.

Yeah, that's what I was thinking, that it was some place where mathematical parameters stop working or something, like an upper limit. But it's just named after a guy named Killing. So reading the Wikipedia article about Killing vector fields does me absolutely no good in understanding it. So how exactly does it apply to General Relativity?

 

[Ibix]

What exactly is a Killing Field?

A vector field that satisfies Killing's equation. A Killing vector defines a direction in which spacetime looks the same. For example, if a spacetime is rotationally symmetric, there's a Killing vector field whose vectors point in the tangential direction.

A timelike Killing field defines a direction in which you can travel so that space is the same. It means that one can have a notion of time such that space isn't changing with time. There is a timelike Killing vector field outside a black hole, and sure enough you can hover at constant altitude and nothing changes. There isn't one inside the horizon, which is because all paths lead to the singularity in finite time, so their notion of space must have a changing "shape".