I have long been fascinated with the mysterious black holes. Over the years I've been following the literature and improved my mathematical skills to better understand what we know about these objects. Over the past several years I followed several heated debates related to numerous paradoxes that our understanding of black holes had caused. Here I'd like to present a few issues I have with our contemporary understanding of the subject. If you are a black hole specialist, I will appreciate feedback.
Classical picture
Existence of black holes is a straightforward result of the theory of general relativity (in fact is conceivable even in the classical Newtonian mechanics). In essence the observation is that an object dense enough would eventually reach the escape velocity equal to the speed of light, at which point in becomes black (since it cannot radiate anything out) and anything that happens to get trapped inside it, has no hope of getting out, or at least has the same hope of getting out as we may have the hope of traveling faster than light. The solution of that particular object was first put forward by Karl Schwarzschild who observed that there is a particular size/radius below which a given mass has to compress in order to become a black hole. That radius is given by:
where G is the constant of gravity, M is the mass of the black hole and c is the speed of light. It is easy to calculate that a solar mass black hole would have a radius of several kilometers and therefore be incredibly dense. Surprisingly however, much larger black holes actually become less dense as they grow eventually reaching quite manageable densities of say 1g/cm^3 for huge black holes. However it is unclear whether such defined density (mass divided by volume inside Schwarzschild radius) has any physical meaning.
As far as things go inside the black hole, all we know from general relativity is that the theory breaks down in a point known as the singularity, where all the hell breaks loose. At least that is the purely relativistic picture.
Aside from that, a few interesting things happen near the horizon, namely there is the photon sphere, a sphere of photons orbiting the black hole in unstable circular orbits, and above everything, there is the time dilation, hence clocks slow down as they approach the horizon (as seen by the external observer). The exact formula for dilation is:
where 
Now one could think that if the observer next to the horizon gets frozen in time, he will experience eternity and see the end to the universe. This view omits however one important detail - if the observer is in free fall, then even though he gets frozen in time from the point of view of the external observer, he himself does not perceive accelerated flow of time behind him. The reason for that, is that when he is in free fall, he gets rapidly accelerated. He is so much accelerated that effectively can only experience the stuff that was in his light cone to begin with and that does not include any of the distant future. Therefore, even though time passes quick for the outside observer, none of that is able to catch up with the free falling observer as he just accelerates too quickly. This can be seen by drawing a black hole in Kruskal–Szekeres coordinates (see also lectures by Leonard Susskind). Consequently, free falling observer technically is not affected by time dilation and can reach singularity in finite amount of time in his frame of reference.
However there are problems in this picture: such defined black hole appears to destroy information, which is a big no-no in quantum mechanics. In fact the most fundamental feature of quantum mechanics is unitary evolution, which basically states that information can never be destroyed. In attempts to resolve that issue, one encounters further paradoxes, such as the AMPS paradox related to quantum entanglement. And above everything there is the singularity itself, a clear indication that something is wrong with this solution. For whatever we know there is something broken with this purely relativistic picture, as attempts of resolving it create really bizarre consequences, particularly once any of that clashes with quantum mechanics which for some reason does not like black holes. So instead of going down that rabbit hole, let us carefully reanalyze what general relativity tells us about black holes.
General relativity once again
Going back to time dilation, it is crucial to note something very important: dilation does not only pertain to a black hole when it is formed. It applies as the process of gravitational collapse happens. In essence when matter gets compressed close to Schwarzschild radius, clocks slow down. From the external observer point of view, the horizon never actually forms, as the infalling matter freezes in time right before it is about to turn into an actual black hole.
In fact the object the external observer sees looks like a black hole for all intents and purposes. Even though the matter stands frozen in time, massive redshift causes the object to quickly darken. Light bends around the object just as it would around a fully developed black hole. So this object really looks like a black hole (hence nothing here contradicts our observations of objects satisfying the definition of black holes or even recent detection of gravitational waves) , but the physics of it is not yet singular in any way. To the external observer all the information is still there (though arguably not easy to extract), there is no singularity (yet) and there is no event horizon, even though not much light comes out of the object (due to extreme redshift). This effectively means that in our frame of reference, or in fact in any frame of reference that is not in free fall onto a black hole, no black hole exists (yet). But there are plenty collapsed stars apparently on their way to become a black hole, but effectively frozen in time.
Now remains the problem of the frame of reference of a free falling observer. Equivalence principle tells us that all frames of reference are equally good, so this should also be a good frame of reference. From that point of view the black hole horizon must exist, since the observer can freely cross it. But we know that this interpretation causes massive problems, therefore let us for the moment assume that something here is wrong.
The question is, can you actually free fall through the horizon? This assumes, that there is nothing unusual on the surface of the horizon, it is just empty space. This seems a reasonable assumption for an already formed black hole, but what about a star that is in the process of collapsing?
Things are becoming a bit more complex. It is assumed that black holes form because we don't know of any force that could stop the collapse. But frankly, a star which is about to form a black hole is a fairly extreme beast and a lot of interesting things could be going on right before a horizon were to emerge.
First thing to note, is that gravitational collapse may likely not look like a free fall. It is reasonable to assume that even after the force holding neutrons apart in a neutron star is finally overpowered by gravity, the matter still resist compression (i.e. the process actually extracts energy from the gravitational pull). Certainly such matter will get denser and hotter.
Let us imagine a star core, which is close to becoming a black hole, but has not crossed the necessary density anywhere yet. If we assume that none of the iso-surfaces of that very dense sphere is actually in free fall, but resists compression and collapses at much slower velocity (likely actually slowing as density increases), the classical free falling frame of reference may not apply. In this case (when the contraction velocity is slow enough), the observer sitting on the surface of such object would actually catch up with his dilated future (we forget for a moment that no human, could survive in such strong gravitational field, unless close to free fall). Hypothetical observer on such slowly collapsing object would begin to see that indeed the outside universe begins to accelerate.
In fact a hypothetical observer standing on the surface of black hole horizon would actually experience eternity, much like typical intuition suggests. This may seem like a movie played in fast forward but is a bit more complex than that - on top of things seen outside accelerating, any radiation coming from there would become blue shifted, hence more energetic. So if any observer could spend a finite amount of time standing (stationary) exactly on the event horizon he would experience photons reaching him from the outside with energies diverging to infinity. Note that a hypothetical observer standing still on the black hole horizon is equivalent to an observer moving at the speed of light - even the 4K background radiation of the big bang would essentially become infinitely energetic.
Here is where things become interesting - the fate of the collapsing star and what it would experience depends heavily on the resistance the compressed matter would exert and hence the velocity of iso-surfaces contracting. If that velocity accelerates or even is constant (in the frame of reference of every iso-surface), then the back hole inevitably forms and we get back to our original picture plagued with paradoxes.
If however the collapse velocity decreases, the iso-surfces are able to catch up with the time dilation, experience enormously energetic radiation from the outside and hence heat up incredibly and perhaps actually never form a black hole, but perhaps radiate that energy back out. Counterintuitively this process could actually resemble a violent explosion from the point of view of the contracting matter, but due to time dilation could take forever as seen from the outside. In fact such an object does not even necessarily need to be black: in some regimes the increasing temperature of the blue shifted radiation will cause the surface to emit high energetic photons which (after they get out of the gravitational well) may actually be quite reasonable energetic photons (say in visible or even ultraviolet range). That does not contradict any of our observations as typically black holes are surrounded by bright in-falling matter which heats up and emits X rays.
In reality we don't really know anything about how matter would perform in such extreme conditions. There have been proposed Quark Stars but it is very difficult to prove their existence. In fact from a distant observer such collapsed quark stars (at the edge of becoming a black hole) could be very hard to distinguish from "actual" theoretical black holes, even though they are actually more like degenerate neutron stars. So we may never know. In fact given the scale of the universe it is actually quite conceivable that we will never find out.
So what about other means of creating a black hole? Maybe it does not have to be a gravitational collapse? Leonard Susskind has the example of a contracting sphere of photons. The total energy contained in all those photons is enough to form a black hole. Eventually the sphere of photons contracts enough that it crosses the Schwarzschild radius and we have a black hole. In fact in this thought experiment we can even see, that a hypothetical observer sitting inside of such contracting sphere would not notice anything unusual until the very last moment, hence he would not even notice he is inside the horizon while he may well be already enveloped.
I have an issue with such thought experiments. Aside from the (im-)practical aspect of it, in reality such highly energetic wave of photons would begin to produce particles and scatter off of them (even in perfect vacuum) way before it would get anywhere close to the Schwarzschild radius (interestingly, this thought experiment is possible in pure relativity but gets killed by quantum mechanics of electromagnetic field and uncertainty principle - quantum mechanics really does not like black holes for some reason!). In the end, it would have generated a cloud of hot gas and we would again need the regular implosion to happen. Stories of several large stars just happening to pass close enough to close a Schwarzschild radius around them are similarly bogus.
Scientists like to jump into the wonderland
To summarize, scientists conclude that black holes exist based on the assumption that once gravity overpowers strong nuclear force, the resulting star core implodes in free fall like manner. From that we conclude a set of disturbing paradoxes and spend the good part of century trying to fix them via all sorts of questionable constructs such as the holographic principle and so on. I suggest we pause for a moment: the "free fall-like collapse" is a huge assumption! Such collapsing star gets into energy/density regimes which we have zero understanding of way before it hits the singular state. It is as if we wrote a huge set of literature about hypothetical particles moving faster than light (so called tachyons)! I would not oppose discussing such things hypothetically, but I don't think anyone serious in physics would really consider those things a possibility. Not at least outside of the realm in which Warp drive (Alcubierre drive) or wormholes are a possibility too. They may eventually, but for now they are way outside of what we can study experimentally. This in my opinion is fringe science and although we may venture into those subjects every once in a while, we probably should not waste huge resources on them. Black holes somehow pass the scrutiny and everybody happily assumes that they absolutely exist, though we have no means whatsoever to experimentally prove that (in fact it would be impossible even if we had such a black-hole like object right next door).
My stance is that nature does not like singularities. Any time our equations suggest a singularity (particularly followed by paradoxes), we likely have the wrong equations. But theoreticians love singularities, because essentially they are like a wildcard - you can put out any sort of crazy theory, it will do just fine if it gets rid of the singularity. Singularity gives you the license to be crazy, since the alternative is even worse! Black holes in that sense are really like the rabbit hole through which Alice falls into the wonderland. All bets are off, everything is possible. But in the long run, did we really learn anything valuable about the Universe?
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