Counterargument on Graham's Hierarchy of Disagreement
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It would be unreasonable to present the whole of the case for the internal structure of a black hole that is given here without discussing some of the counterarguments. We will look briefly at some of these here.
There is no real Schwarzschild singularity at the event horizon
An object falling into a black hole will cross the event horizon in a finite time
Times and distances depend on your 'frame of reference'. When talking about black holes, there are two commonly used frames of reference: a frame of reference that would be adopted by a distant observer (such a someone here on earth), known as Schwarzschild or bookkeeper coordinates; and a frame of reference that is attached to someone falling into a black hole, known as comoving coordinates. Both are equally valid.
As already stated, time and distance depend on your 'frame of reference', but in the case of Schwarzschild and comoving coordinates applied to a black hole, the differences near the event horizon are uncomfortable. In Schwarzschild coordinates, it takes an infinite time to cross the event horizon whilst for comoving coordinates the time is finite and what is more, having crossed the event horizon, will reach a central singularity only a short time later. But we have to be quite clear as to just what the problem is. We are getting two different time measurements which we would never expect to be the same. But the only real question is how can an event happen in one reference frame but will never ever happen in another? It cannot. But we have to start asking ourselves just what can happen before an infinite time passes? One thing that most of you will have heard of is Hawking radiation. This is a quantum effect and can be shown to make any black hole slowly evaporate. Slow for a stellar-mass black hole will be a very long time. Approximately 1054 times the current life of the Universe for a stellar-mass black hole to completely evaporate! A long time but still nothing compared to infinity. The black hole will be gone long before one could reach the central singularity!
Of course, using comoving coordinates attached to an in-falling observer, the same things must happen but very much more quickly, for Hawking radiation to remain true. Of course, this would mean that from the event horizon onwards, an in-falling traveller would meet a massive wall of radiation - as predicted by proponents of the firewall paradox.
"If you have 2 astronauts above a black hole and one of them gets dragged in, the one outside will see the image of the first astronaut frozen at the event horizon as the light can't escape. Eventually, it would decay to infrared, but this is just an effect on the light, the astronaut himself wasn't really frozen in time, he went straight through the event horizon and died."
This is a variation on it's all an optical illusion or the last photon explanation.
Now, my answer: the strong equivalence principle
In general relativity, any convenient system of coordinates can be used and is valid. I suggest that as far as observational data goes, Schwarzschild coordinates are the most appropriate as these alone can correlate with observations.
Two different coordinate systems -- Schwarzschild and comoving coordinates -- give very different results in the vicinity of a black hole horizon and yet we know that they must describe the same reality for different observers. Understanding the relationship between these two results is therefore crucial to accepting the validity of these results.
Consider twins, one of whom descends towards the event horizon of a black hole. We accept that one, the traveller, will appear to be slowing down compared to their stay at home sibling due to the gravitational effect on the passage of time. However, the traveller sees the opposite: time for the stay at home twin seems to speed up. There is nothing fictitious or illusory about this -- if the traveller returns home, he will certainly be younger than his twin. Depending upon how close to the event horizon he travels, he could be many days or years younger. In principle, he could be even 100,000 years younger and still not have crossed the event horizon. (Apart from the technical difficulties, we are here assuming eternal life.) So just when does the traveller cross the event horizon. By his own watch, it may be just a few hours but for the stay at home twin, it will be an eternity. So the traveller does arrive at the central singularity, but for the stay at home twin, this is after the Universe has ceased to exist. Both are real but only one can produce a measurable outcome as it is only in or close to our earthly home that we can make any measurements at present.
Einstein's equivalence principle requires that space is locally Minkowski flat at the event horizon
This is based on a misunderstanding of the strong equivalence principle. When Einstein came up with this after observing a workman falling from a ladder, he never envisioned that it was still to be applied when he hit the ground! The abrupt change in mass density converts free fall to tragedy! Just as with a black hole.
With a more scientific analysis, the error is simply that on reaching the event horizon, time stands still, and whilst the equivalence principle is maintained, the object can no longer move. Just ask yourself what the equivalence principle means when time stops, or what is Minkowski space without a time axis? One has to tread very carefully whenever a real infinity is encountered in physical systems.
The last photon
This argument is based on a belief that if an object falling towards a black hole emits a light signal that is received by a distant observer, then eventually the last photon will be received before the end of time. It fails to take into account the fact that, as the received photons are red-shifted, they get to have progressively lower and lower energy, and hence there is never a last photon in a finite time.
Quantum mechanics has been an incredibly accurate predictor of behaviour in physics. The accuracy is far greater than that of any other theoretical model. This has led some to suggest that quantum mechanical rules somehow take precedence over those of general relativity. But bear in mind that when a dying star collapses to form a white dwarf, it is held up by electron degeneracy pressure. This fails when the mass is sufficient. The reason? It is because any greater pressure would require the electrons to be travelling at greater than the speed of light. Relativity theory dictates that this is not possible. The next stage is a neutron star but as we increase the mass, a similar point is reached at which the constituent parts of neutrons (quarks) would have to travel at the speed of light. At this point, gravity wins completely. Until we reach a point at which general relativity fails, there will be no greater ultimate force in the Universe. There may still be greater forces; we just have not detected them yet.
Nothing special at the event horizon
This is a common misconception. It is argued that the Einstein equivalence theorem guarantees that nothing special will happen at the event horizon. Einstein argued that locked in a spaceship falling towards the earth - free fall - the occupants would not be able to detect any difference by any local physical experiment, between this and to being in space, far from all gravitational effects. We do of course have to add that this is without looking through the window, and before hitting the ground, and only for a sufficiently large gravitating body so that tidal effects can be ignored.
At the same time, we are expected to accept that once having crossed the event horizon of a black hole, nothing can ever escape.
These two statements are clearly incompatible. We could obviously devise an experiment in which objects are fired back out of the black hole and observing that they fail to leave as they had been doing prior to crossing the event horizon.
The Riemann manifold cannot include an infinity
This is true. The Riemann manifold is made of points drawn from real numbers and infinity is not one of those. To me, this does seem a little like a cop-out. We do not know how to deal with infinity, so we will not include it. However, that is not the sort of answer that I could expect others to accept for anyone who has encountered infinity in mathematics: the answer is not to try to work out what happens at infinity but just to restrict the investigation to what happens when we approach infinity.
- The Schwarzschild solution is pathological
This is another fallacy. The solution is continuous, and infinitely differentiable everywhere as required for a Reimann manifold. In the time axis, it runs from \(-\infty\) to \(+\infty\) like all axes in all coordinate systems.