TrickyDicky said:
When you talk about "the physics" here, what specifically are you referring to?
This is a good question, and I agree that it's worth some discussion.
TrickyDicky said:
mathematical models that are validated by observations, that is by empirical evidence (and I know this is a can of worms as there seems to be very different opinions amongst physicists about what constitutes evidence)?
This is an important part of it, yes, but not all of it. If we always limited our models to what we already have evidence for, we would never be able to discover anything. We have to extrapolate into the unknown somehow. But *how* we extrapolate is important. See below.
TrickyDicky said:
I mention it because events horizon math is fine too but one is tempted to ask like you do "whether this math has anything to do with the physics" as experiments are absent
One thing to bear in mind is that I wasn't claiming that lukesfn is wrong; I was asking for some kind of argument other than "it seems obvious to me" or "well, the math works". I don't think I've ever used that, by itself, as a justification for believing that event horizons exist.
TrickyDicky said:
and indirect observations are compatible with a number of different things.
Sort of. Our current observations have ruled out a number of proposed models for compact objects without horizons. For example, models in which there is a "slowly collapsing surface" at some ##r > 2M##, where ##r## is slowly decreasing but never quite reaches ##2M##, are ruled out (at least with a good degree of probability) by observations of the spectrum of black hole candidates; if there were an actual surface there, it would reflect radiation back in a way that is not observed. (I'm being very brief here, a more detailed discussion of these models would be for a separate thread--not that there haven't already been quite a few
; my point here is just to illustrate the kinds of arguments I would say are relevant in judging whether a particular bit of math has anything to do with the physics.)
TrickyDicky said:
Recurring to the fact they show up in a solution of our favourite theory so they are "predicted" seems a feeble argument when there are so many unphysical solutions
Yes, but there's a difference between a solution that is "obviously unphysical" in its entirety (such as, IMO, the Godel spacetime; or the maximally extended Schwarzschild spacetime, i.e., vacuum everywhere and including both the black hole and the white hole regions) and a solution that is physically reasonable but with a property that some people find counterintuitive (such as the modern version of the Oppenheimer-Snyder model of a spherically symmetric collapse, which is obviously an idealization but which makes a clear prediction that there *is* an event horizon and a black hole region in the spacetime--more on this in a bit.)
The difference here is not often articulated (most physicists seem to adopt a view similar to the Supreme Court's on pornography, "I can't define it but I know it when I see it"), but I think it comes down to a judgment about whether the model as a whole could reasonably be viewed as "complete" in some sense, or whether it requires postulating some improbable arrangement to bring it into being. For example, the maximally extended Schwarzschild spacetime is vacuum everywhere, yet it has two singularities, which are normally interpreted as indicating the presence of "mass"--i.e., not vacuum. How can that be? Where did the singularities come from if everything is vacuum everywhere? (The same kind of issue arises with the Godel spacetime: how did it come to be "rotating" if there is vacuum everywhere? What started it rotating?)
By contrast, the spherically symmetric collapse model makes it clear where the "mass" comes from: it comes from the object that collapsed. And this model only has one singularity, in the future, which is clearly formed from the collapse of the object that provides the mass. So this model seems much more reasonable physically.
Also, there's the question of how to extrapolate from the domain where we have already verified a model by evidence, e.g., the exterior region of Schwarzschild spacetime, to a domain where we have not, e.g., the future horizon and black hole region of Schwarzschild spacetime. In the case I just named, the extrapolation is simple: all it requires is accepting that (a) the physics is in the invariants, not in any coordinate-dependent quantities; and (b) all invariants are finite and well-behaved at the horizon (or, if one wants to be very careful not to prematurely extrapolate, so to speak
, in the limit as ##r \rightarrow 2M##). Given those assumptions, and given the assumption that the spacetime continues to be vacuum outside the surface of the object that collapsed to form the black hole (more on that in a bit), the prediction that there *is* a black hole is unavoidable.
Of course the model I've just described is classical, and any discussion of Hawking radiation has to take into account quantum effects, and as soon as you include quantum effects, you come up against the issue that in a curved spacetime, "vacuum" is relative. This creates a problem when trying to do the extrapolation I just described. In the classical case, "vacuum" just means setting ##T_{ab} = 0##, i.e., the RHS of the EFE is identically zero. However, when quantum effects are taken into account, the effective ##T_{ab}## in the classical limit is an expectation value of some operator or combination of operators. And as I understand it, it turns out that there is *no* operator whose expectation value is identically zero everywhere in a curved spacetime. One can alternatively phrase this in terms of quantum states, and say that there is no quantum state that has a globally vanishing expectation value in a curved spacetime. The best you can do is to find some state that at least has vanishing expectation value for some class of observers.
The debate over the correct quantum model of gravitational collapse, as I understand it, is therefore basically about which operator to use; or, put another way, which quantum state to assign to the quantum field. The standard derivation of Hawking radiation, as I understand it, uses a state called the "Hartle-Hawking vacuum state", which is a state that, roughly speaking, appears to be a vacuum to observers free-falling into the black hole from infinity. The prediction of Hawking radiation then depends on showing that this state does *not* appear to be a vacuum to observers who are "hovering" at a constant ##r## far from the hole; it appears to be a thermal state with a temperature inversely proportional to the observed mass of the hole.
But of course an observer hovering far away from the hole sees any outgoing radiation to be highly redshifted, compared to an observer hovering very close to the horizon. So if the former observer sees Hawking radiation at the predicted temperature, the latter observer should see radiation at a much *higher* temperature--one that increases without bound as ##r \rightarrow 2M##. *But*, this is still the *same* quantum field state that looks like a vacuum to infalling observers, as above. This is where all the fuss about what actually happens near the horizon comes from; many obvious questions suggest themselves. Is the Hartle-Hawking vacuum state really the "right" state to use here? Is there some other quantum effect that comes into play? Does the derivation work for *any* accelerated observer, or does the observer have to be "hovering" (i.e., following an orbit of the ##\partial_t## KVF)?
I go into all this detail to make it clear why "it just seems obvious to me" is not a good response. There are too many open questions and too many issues involved for any simple line of reasoning to be enough here, or for one to be able to say "well, the math works". The problem is not that we don't have math that works: it's that we have too *much* math, and different pieces of math say different things, and we don't have enough understanding of how the math relates to the physics. We have the classical event horizon math: we have the Hartle-Hawking vacuum state quantum math; we have various other math that has been proposed. Which math is the *right* math, the math that will turn out to match the actual physics? We don't know.
