"A little book about black holes"

Despite the complexity of the topic, Princeton University Professor Stephen Gabser offers a succinct, accessible and entertaining introduction to this one of the most discussed areas of physics today. Black holes are real objects, not just a thought experiment! Black holes are extremely convenient from the point of view of theory, since mathematically they are much simpler than most astrophysical objects, such as stars. The oddities begin when it turns out that black holes are actually not so black.

What is actually inside them? How can you imagine falling into a black hole? Or maybe we already fall into it and just don’t know about it yet?

In the Kerr geometry, there are geodesic orbits completely enclosed in the ergosphere, with the following property: the particles moving along them have negative potential energies, which outweigh in absolute value the rest mass and the kinetic energies of these particles combined. This means that the total energy of these particles is negative. This circumstance is used in the Penrose process. Being inside the ergosphere, a ship that extracts energy shoots a projectile in such a way that it moves in one of these orbits with negative energy. According to the law of conservation of energy, the ship receives sufficient kinetic energy in order to compensate for the lost rest mass, equivalent to the energy of the projectile, and in addition to obtain a positive equivalent to the net negative energy of the projectile. Since the projectile after a shot should disappear into a black hole, it would be good to make it from some kind of waste. On the one hand, the black hole still eats anything, and on the other, it will give us back more energy than we have invested. So in addition the energy acquired by us will be "green"!

The maximum amount of energy that can be extracted from a Kerr black hole depends on how quickly the hole rotates. In the most extreme case (at the highest possible rotational speed), the share of the rotational energy of space-time accounts for approximately 29% of the total energy of a black hole. It may seem to you that this is not very much, but do not forget that this is the proportion of the total rest mass! For comparison, remember that nuclear reactors powered by radioactive decay energy use less than one tenth of a percent of energy equivalent to the rest mass.
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The space-time geometry inside the horizon of a rotating black hole is very different from the Schwarzschild space-time. Follow our probe and see what happens. At first, everything looks like the Schwarzschild case. As before, space-time begins to collapse, dragging everything along with itself towards the center of the black hole, and tidal forces begin to grow. But in the Kerr case, before the radius vanishes, the collapse slows down and begins to reverse. In a rapidly rotating black hole, this will happen long before the tidal forces are large enough to threaten the integrity of the probe. In order to intuitively understand why this happens, let us recall that in Newtonian mechanics, so-called centrifugal force arises during rotation. This force is not among the fundamental physical forces: it arises due to the joint action of the fundamental forces, which is necessary to ensure the state of rotation. The result can be represented as an effective force directed outward - centrifugal force. You feel it on a sharp turn in a fast moving car. And if you ever ride on the carousel, you know that the faster it turns, the stronger you have to grab the handrails, because if you let them go, you will be thrown out. This analogy for space-time is not ideal, but it conveys the essence correctly. The momentum in the space-time of the Kerr black hole provides an effective centrifugal force that counteracts gravitational attraction. When the collapse inside the horizon tightens the space-time to smaller radii, the centrifugal force increases and eventually becomes capable of first resisting the collapse, and then reversing it.

At the moment when the collapse stops, the probe reaches a level called the inner horizon of the black hole. At this point, the tidal forces are small, and the probe, after it has crossed the event horizon, takes only some finite time to reach it. However, the mere cessation of the collapse of space-time does not mean that our problems are over and that the rotation somehow led to the elimination of the singularity inside the Schwarzschild black hole. Until this far away! Indeed, in the mid-1960s, Roger Penrose and Stephen Hawking proved the system of singularity theorems, from which it followed that if a gravitational collapse had occurred, even if short, then some form of singularity should result. In the Schwarzschild case, this is a comprehensive and all-destructive singularity that subordinates all the space within the horizon. In the Kerr solution, the singularity behaves differently and, I must say, quite unexpectedly. When the probe reaches the inner horizon, the Kerr singularity reveals its presence - but it turns out that this happens in the causal past of the probe's world line. It is as if the singularity was always there, but only now the probe felt how its influence reached it. You will say that it sounds fantastic and that is true. And there are several inconsistencies in the picture of space-time, from which it is also clear that this answer cannot be considered final.

The first problem with a singularity appearing in the past of an observer who reaches the inner horizon is that at this moment the Einstein equations cannot unambiguously predict what will happen to the space-time outside this horizon. That is, in a sense, the presence of a singularity can lead to anything. Perhaps what actually happens can be explained to us by the theory of quantum gravity, but the Einstein equations do not give us any chance to find out. Just out of interest, we describe below what happens if we demand that the intersection of the horizon of space-time is as smooth as mathematically possible (if the functions of the metric are, as mathematicians say, “analytical”), but there are no clear physical reasons for this not. In essence, the second problem with the inner horizon assumes exactly the opposite: in the real Universe, in which matter and energy exist outside black holes, the space-time near the inner horizon becomes very uneven, and a loop-like singularity develops there. It does not act as destructively as the infinite tidal force of the singularity in the Schwarzschild solution, but in any case its presence raises doubts about the consequences that follow from the concept of smooth analytic functions. Perhaps this is good - very strange things entail an assumption of analytical expansion.


In essence, a time machine works in the area of ​​closed timelike curves. Away from the singularity, there are no closed timelike curves, and apart from the repulsive forces in the region of the singularity, space-time looks completely normal. However, there are motion paths (they are not geodesic, so you need a rocket engine) that will take you to the region of closed time-like curves. Once you are there, you will be able to move in any direction along the t coordinate, which shows the time of the remote observer, but in your own time you will always move forward anyway. This means that you can go at any time t, at which you want, and then return to the remote part of the space-time - and even arrive there before you go. Of course, now all the paradoxes associated with the idea of ​​time travel come to life: for example, what if, after taking a walk in time, you convinced your past "I" to abandon it? But can there be such kinds of space-time and how can the associated paradoxes be resolved - issues beyond the scope of this book. However, just as in the case of the “blue singularity” problem on the inner horizon, the general theory of relativity indicates that areas of space-time with closed time-like curves are unstable: as soon as you try to combine with one of these curves the amount of mass or energy these areas may become singular. Moreover, in the rotating black holes that form in our Universe, it is the “blue singularity” itself that can prevent the formation of negative mass regions (and all Kerr other universes into which white holes lead). Nevertheless, the fact that the general theory of relativity allows such strange solutions is intriguing. They are, of course, easy to declare pathology, but let's not forget that Einstein himself and many of his contemporaries said the same thing about black holes.

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