Black hole event horizon radius = Schwarzschild radius?

[arindamsinha]

I am under the impression that the event horizon radius of a non-rotating black hole is equal to its Schwarzschild radius. Is this correct?

If yes, then I have a mixed bag of questions:

• Is the event horizon radius always calculated using the Schwarzschild metric, no matter what model we are talking about (including O-S model etc.)?
• Or, are there different methods of arriving at event horizon radius via different models?
• Do all models (which I think of as different interpretations of GR) end up with the same radius for the event horizon, or are there possible differences in size, including possible absence of event horizons in some models?

I am assuming that the central singularity does exist in all models, but am ready to stand corrected on this.

 

[PeterDonis]

I am under the impression that the event horizon radius of a non-rotating black hole is equal to its Schwarzschild radius. Is this correct?

Yes.

Is the event horizon radius always calculated using the Schwarzschild metric, no matter what model we are talking about (including O-S model etc.)?

I'm not sure what you mean. The event horizon radius is 2M (or 2GM/c^2 in conventional units). "M" is the mass of the hole; it's the same regardless of which "model" we are using (at least, I think it is, but I'm not sure what you mean by "model", so I'm not sure). You don't need to decide on any particular "metric" (by which you appear to mean "coordinate chart", but I'm not sure); you just need to know the externally measured mass of the hole (or the matter that collapsed to form the hole).

Do all models (which I think of as different interpretations of GR)...

Why do you think this? I think it would help if you gave some specific examples, since as I said above, I'm not sure what you mean by "model".

 

[Dale]

I am also not sure what you mean by "model" and "interpretation" here. In particular, the OS spacetime is a physically different spacetime than the Schwarzschild spacetime, and the EH is not stationary in the OS spacetime.

 

[Nugatory]

Do all models (which I think of as different interpretations of GR) end up with the same radius for the event horizon

I'm guessing here, but I think the bolded part tells me what you're really trying to ask.

You can use different coordinate systems (for example, Schwarzschild or KS) to describe the same solution of the Einstein field equation (for example, the Schwarzschild spacetime) but the properties of that spacetime don't change.

The Schwarzschild metric (more precisely, the spacetime described by the Schwarzschild metric) is the solution of the field equations for the vacuum outside of a spherically symmetric stationary mass distribution. In that spacetime, there is an event horizon at a particular surface in that spacetime; there are no timelike or lightlike outgoing paths through that surface. All of this is true no matter what coordinates I use. KS and SW coordinates may look very different, but they are describing the exact same reality, just as $A^2=x^2+y^2+z^2$ and $r=A$ describe the exact same object (a sphere of radius A) in different coordinate systems. Either way the sphere exists in the same place, and I can talk about it using either set of coordinates or no coordinates at all ("a sphere of radius A"). Llikewise, in KS coordinates I'd say the horizon is at $U=V$; in Schwarzschild coordinates I'd say it's at $r=2M$; but it's the same thing at the same place in spacetime.

Now, the O-S solution is a solution of the Einstein field equations in a completely different situation (collapsing ball of dust), so describes a different spacetime with a different metric. Again, I can use various coordinates to describe the O-S spacetime, and depending on which I choose, the expression for the metric may look very different - but it'll still be describing the same physical reality, and it won't be the one described by the Schwarzschild solution.

 

[Mike Holland]

As I understand it, the two are the same for a non-rotating black hole. But when it is rotating, and dragging space around it, there are effectively two event horizons. Nothing can escape the outer one, but photons can orbit around between the two without being sucked into the singularity.

I hope some expert will tune in and verify this.

Mike

 

[PeterDonis]

As I understand it, the two are the same for a non-rotating black hole. But when it is rotating, and dragging space around it, there are effectively two event horizons. Nothing can escape the outer one, but photons can orbit around between the two without being sucked into the singularity.

(1) For a non-rotating, uncharged black hole (i.e., in Schwarzschild spacetime), photons can orbit the hole at r = 3M (i.e,. at 1.5 times the horizon radius), and there is only one horizon, at r = 2M.

(2) For a rotating, uncharged black hole (i.e., in Kerr spacetime), there is an outer horizon (somewhere between r = M and r = 2M) and an inner horizon (at some r < M), but AFAIK photons can't orbit between them. Photons can orbit at a particular radius outside the outer horizon (I can't remember what it is, but it's similar to the photon orbit at r = 3M outside a non-rotating hole), and they can also orbit (I believe) at some radii inside the *inner* horizon.

(Also, there is a region outside the outer horizon of a rotating hole called the "ergosphere", where there are no static "hovering" observers--everything has to have a nonzero angular velocity. It's called the ergosphere because it is possible to extract energy from the hole by slightly slowing its rotation, if you put objects in the ergosphere into particular orbits.)

(3) For a non-rotating, *charged* black hole (i.e., in Reissner-Nordstrom spacetime), there is also an outer horizon and an inner horizon (the ranges of r where they occur are the same as for a rotating hole, but the exact formulas are different), and photons can orbit at a particular radius outside the outer horizon, and at some radii (I believe) inside the inner horizon, but not in between the two.

 

[arindamsinha]

Yes, I realize that I have probably put some confusing terminology around this (mainly because of my lack of much background in relativity).

So, probably what I am trying to say by different models is two things (a) coordinates as PeterDonis, Nugatory and Mike Holland have correctly guessed and responded to, and (b) the O-S model of star collapse, where the EH does not seem to be stationary as DaleSpam mentioned.

Based on esp. the O-S model, where the EH is changing in size (I assume) I was thinking that there may be different ways in which these are interpretations of GR, and may lead to slightly different conclusions about the EH size.

 

[PeterDonis]

So, probably what I am trying to say by different models is two things (a) coordinates

Changing coordinates doesn't change any of the physics, so thinking of different coordinate charts on the same spacetime as different "models" doesn't strike me as a very fruitful approach. I would call them different descriptions of the same model.

(b) the O-S model of star collapse, where the EH does not seem to be stationary as DaleSpam mentioned.

This is more like what I would call a "model", which could be described by different coordinate charts.

Based on esp. the O-S model, where the EH is changing in size (I assume) I was thinking that there may be different ways in which these are interpretations of GR, and may lead to slightly different conclusions about the EH size.

Now I'm not sure what you mean by "different interpretations". The O-S model is one model, with one set of physical observables associated with it. You can describe it in different coordinate charts, but any chart will give the same values for observables.