The Problem Of Black Holes Nobody Talks About（原声文稿）

There is one problem regarding black holes that I never truly understood.

And whenever I asked someone about it, he was either also confused or the answer was never satisfactory to me.

So I started to do some research and I eventually found that this issue is still being discussed and it's also a lot more complicated than I originally thought.

So what is this issue?

After the discovery of general theory of relativity in 1915 by Albert Einstein, just Just a year later, first solution to the Einstein's field equations was found by Karl Schwarzschild.

This solution describes the spacetime around a non -rotating, spherically symmetric object of mass m. But this solution has its limits.

For example, there is a singularity if the observer's radial coordinate, which is the value of 2gm over c squared, as this term blows up to infinity, but also this This term goes to zero, which means infinite time dilation.

But normally this never happens, because usually there is a planet or a star that prevents you to reach that point.

And if you wanna go inside the planet, then the Schwarzschild solution is no longer valid, because it's only a vacuum solution.

So the only way this can happen is to squeeze the object below this radius, which was thought Not to be impossible at first, but later it was proven that this can really happen.

And this is a black hole.

This boundary, where Schwarzschild's solution is singular, is called an event horizon.

And from below it, nothing can ever escape. Nothing can ever escape a black hole.

That is probably the first thing you'll learn when you read or watch something about black holes.

But the Schwarzschild solution not only tells us that nothing can ever escape a black hole, but also nothing can ever go into the black hole.

The type of solution goes to zero at the event horizon, and therefore if you saw your friend approach the horizon, his time would start to slow down, and his approach would be slower and slower, and eventually your friend would appear as frozen in time forever, and would never cross the horizon.

But the light your friend is emitting towards you will be infinitely redshifted until it becomes black, which means your friend does not fall into a black hole, but emerges with it.

This is how it would play out in a reference frame of the Observer at large distance from the black hole, where the space time is effectively flat.

And this was kinda boggling my mind, and I was asking the professors, is it true that nothing can really fall into a black hole and everything is kinda stuck on the horizon and the answer to it was usually the same.

Things can totally fall into a black hole and the singularity you see in a Schwarzschild solution is just a coordinate singularity.

A good example of a coordinate singularity can be demonstrated on a 2d plane.

You can either describe it using Cartesian coordinates as x, y or cylindrical coordinates using the distance from the center and the angle.

In the cylindrical coordinates if you go to the origin then the angle becomes undefined, whereas in the Cartesian coordinates the origin is well defined as 0 ,0.

So that means that it's only the coordinates that fail to describe the event of crossing the horizon.

It's not like the origin is some sort of special point in space.

Physically, Usually it's indistinguishable from any other point, it's just the coordinates we have chosen are not good enough.

But if you choose different coordinates, namely the coordinates of the infaling of server, the horizon crossing is regular and it happens at finite time.

And this is usually the argument that you can totally fall into a black hole.

But I was totally unsatisfied with this answer, because one of the most fundamental rules of physics is objectivity.

This means that if a certain event happens for one observer, then it must happen for all observers.

So if in any frame the observer crosses the horizon, then this must happen also for the observer far away.

But according to the Schwarzschild solution, this happens in infinite time due to time dilation, and therefore it never happens.

So what is really going on?

While objectivity must be fulfilled in general relativity, it is not true that all observer must assign a finite coordinate value to one particular event.

So while for the infalling observer, crossing the event horizon happens at finite time coordinate it does not happen at finite time for an observer far away.

But to preserve the objectivity for the infalling observer an infinite amount of time must have passed far away from the black hole at the moment of horizon crossing.

But this raises a lot of questions.

For example, according to the Hawking radiation black holes evaporate and therefore there are finite objects and they are finite for an observer at infinity and therefore for an observer at infinity black hole evaporates at finite time.

So when you use this logic for a faraway observer his friend would not be able to cross the horizon before the black hole evaporates and therefore it must play out the same also for the infalling observer.

So the infalling observer would see the black hole evaporate right before he crosses the horizon and therefore never actually fall in.

Moreover he should also see the entire black hole evaporate in a very short amount of time and therefore enormous energy flux.

Of course as the black hole shrinks the tidal effect grows stronger and stronger eventually disintegrating you into atoms so this wouldn't mean you could survive or anything.

After all we don't expect a black hole to give us everything back after it evaporates. But this is kind of an highway of thinking about it.

Because if we allow evaporation, then Schwarzschild solution is no longer static, but dynamic, since the mass depends on time.

And exactly this was done in the article from 2005, and the result they found is nicely illustrated in these two pictures.

The left picture represents the infalling process from the reference frame of the This line represents the size of the Schwarzschild radius of the evaporating black hole, and therefore they used a black hole with a lifetime of 20 time units.

And this line represents the trajectory of the infalingop server starting at the double the Schwarzschild radius, and as you can see it kind of freezes about the horizon until evaporation is complete.

The right picture depicts the same thing from the reference frame of the infalingop server.

You can see that the overall lifetime is shorter, but the evaporation starts slow at the start but when the observer gets sufficiently close, the evaporation completes nearly instantaneously.

So such observer will basically see the black hole explode in front of him.

Be aware though that these time units are in natural units, where C and G is equal to 1, and therefore these time scales are in nanoseconds.

So they used a very small black hole from the start that is already evaporating very fast, because doing this on a stellar size black hole would be kinda tricky on the numerics.

However, authors in the article say this paper should be used namely for academic purposes, as the problem is not fully understood yet.

And the reason is that just promoting this end to a time -dependent variable is not a consistent way of doing it.

because when you do it, the metric you get is no longer a solution to the Einstein's field equations.

And you can even intuitively see the inconsistency.

If you just promote the mass to be time dependent, then obviously the total energy is not conserved.

The mass is escaping in the form of Hawking radiation, but it should also influence the space -time structure, which is clearly not captured by this metric.

The proper approach is to use stress energy tensor, which allows for Hawking radiation, and then solve Einstein's field equations again.

But the Hawking radiation was always associated with a black hole that was already formed, with already established event horizon.

But this is not consistent with what we wanna study, because if a collapsing mass freezes just about the Schwarzschild radius, then the true event horizon never really forms.

So how can we talk about the Hawking radiation if we don't have the event horizon?

But obviously, it's natural to expect that the Hawking radiation starts sooner than the true event horizon really forms.

So in this article from Bacetti -Man internal, they studied the collapsing shell of mass and assume vadaia metric where they assume pre -hockeying radiation from the start.

And vadaia metric is already self -consistent, which means that the energy is conserved and therefore the radiation causes back -reaction on the geometry. So this is quite realistic model.

You don't start with an eternal black hole that suddenly starts evaporating.

You start with an object that is not a black hole and collapses into one. And let's look at the results.

So, since I use collapsing sphere, which I am depicting in 2D, as it approaches the Schwarzschild radius, the time dilation kicks in, and it freezes right before it.

Since the shell is hollow, there is no formation of any kind of trapped surface, nor the event horizon, and it radiates the pre -hawking radiation.

As the shell evaporates, it loses mass, and its Schwarzschild radius shrinks, and the shell follows, but never reaches it.

Eventually, it will evaporate before the shell reaches the Schwarzschild radius.

So if you did the experiment again, but now with an observer commoving with the shell, he would never fall into a black hole, as it wouldn't evaporate before him.

But now, it is interesting to see what would happen from the reference frame of the observer that is commoving with the shell.

So, according to the article, the observer would register the pre -hawking radiation from the shell, and as the shell approaches the Schwarzschild radius, the flux of energy would increase, until suddenly, right before the horizon, the flux of energy would become so high that the evaporation rate would accelerate tremendously, and the Schwarzschild radius would shrink at the same rate as the speed of the observer.

However, according to the article, there is an almost constant distance, which they call epsilon star between you and the Schwarzschild radius, and you can't go closer because the blue shift will make the energy flux larger, making the shrinking faster, and you can't get further away because the energy flux would decrease and shrinking would slow down.

This distance though is very small, it depends on the mass of the shell but for a stellar mass it will be around 10 to the minus 77 meters, which is deep into subplankian regime.

Since you maintain a roughly constant distance from the horizon, the energy flux you would experience would reach some natural upper bound.

The authors call this a tame firewall.

Towards the end of evaporation, this distance would suddenly start to grow, as the evaporation rate speeds up and the black hole is very small.

But in reality, this would happen really fast, because when you get close to the Schwarzschild radius, you are basically moving with a speed of light, and therefore the time scale is the same as traversing the black hole from the edge to the center at the speed of light.

But you also have to take into account Lorentz's contraction, and the gamma factor at this sub -plankian distance from the Schwarzschild radius will be massive.

But what is interesting to me, if I understood the article correctly, is that this energy flux you would experience is the same for all black holes, independent of their masses.

The reason is that larger black holes evaporate slower than smaller black holes, but larger black holes let you get closer to the Schwarzschild radius than smaller black holes, and you would experience more blue shift.

This clears all the mass dependencies, if you try to calculate the number then you would get something very absurd like 10 to the 116 watts per meter squared, but still a finite number and as authors say it could be regarded as a firewall with a natural upper cutoff.

Since you fell into a black hole you wouldn't cross the horizon just fine and look at the singularity, you would be falling, falling and suddenly you hit a firewall.

But that would happen so quickly there is no chance you would experience it, you would just stop existing.

It's one of the better ways to die. But still, this is not the final answer.

The gap I was talking about is far below the Planck scale and the energy densities are too.

Our knowledge of physics almost certainly breaks at this scale and we need new physics to fully understand what is going on there.

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