# "A little book about black holes"

Despite the complexity of the topic under consideration, Princeton University professor Stephen Gabser offers a capacious, affordable 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. Oddities begin when it turns out that black holes are actually not so black.

What is really inside them? How can one imagine falling into a black hole? Or maybe we are already falling into it and just do not know about it yet?

In Kerr geometry, there are geodetic orbits completely enclosed in the ergosphere with the following property: particles moving along them have negative potential energies that outweigh the absolute mass of the rest and the kinetic energies of these particles combined. This means that the total energy of these particles is negative. It is this circumstance that is used in the Penrose process. Being inside the ergosphere, a ship that produces energy shoots a shell in such a way that it moves along 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 the positive equivalent of the net negative energy of the projectile. Since the projectile after the shot should disappear into the black hole, it would be nice to make it from some waste. On the one hand, a black hole still eats anything, but on the other, it will return more energy to us than we invested. So in addition, the energy we acquired will be “green”!

The maximum amount of energy that can be extracted from the Kerr black hole depends on how fast the hole rotates. In the most extreme case (at the highest possible rotation speed), the space-time rotation energy accounts for approximately 29% of the total energy of the black hole. You may think that this is not very much, but do not forget that this is a fraction of the total rest mass! For comparison, recall that nuclear reactors operating on radioactive decay energy use less than one tenth of a percent of the energy equivalent to the rest mass.

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 similar to the Schwarzschild case. As before, space-time begins to collapse, dragging everything after 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 become large enough to threaten the integrity of the probe. In order to intuitively understand why this happens, we recall that in Newtonian mechanics during rotation there is a so-called centrifugal force. This force is not one of the fundamental physical forces: it arises due to the joint action of fundamental forces, which is necessary in order to ensure a state of rotation. The result can be represented as an effective force directed outward - a centrifugal force. You feel it in a sharp bend in a fast moving car. And if you ever rode a carousel, you know that the faster it spins, the harder you have to grab onto 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 moment of momentum in the space-time of the Kerr black hole provides an effective centrifugal force that counteracts gravitational attraction. When a collapse within the horizon draws space-time to smaller radii,

At the moment when the collapse stops, the probe reaches a level called the inner horizon of the black hole. At this point, tidal forces are small, and the probe, after it has crossed the event horizon, only needs some finite time to reach it. However, the mere cessation of the collapse of space-time does not yet mean that our problems are behind and that the rotation somehow led to the elimination of the singularity inside the Schwarzschild black hole. So far so far! Indeed, back in the mid-1960s, Roger Penrose and Stephen Hawking proved a system of theorems on singularity, from which it followed that if a gravitational collapse happened, even if it was short, then some form of singularity should form as a result. In the Schwarzschild case, this is a comprehensive and overwhelming singularity, which subjugates all the space within the horizon. In Kerr's decision, the singularity behaves differently and, it must be said, 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 it 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. 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 it 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. 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 it 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 the singularity that appears in the past of the observer who reaches the inner horizon is that at this moment Einstein's equations cannot unambiguously predict what will happen to spacetime outside this horizon. That is, in a sense, the presence of a singularity can lead to anything. Perhaps what really happens can be explained to us by the theory of quantum gravity, but Einstein's equations do not give us any chance to find out. Just out of interest, we will describe below what happens if you require that the intersection of the space-time horizon be as smooth as mathematically possible (if the metric functions are, as mathematicians say, “analytic”), but there are no clear physical grounds for such an assumption not. In fact, 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 at the inner horizon becomes very unstable, 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 makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension. and there a loop-like singularity develops. It does not act as destructively as the infinite tidal force of the singularity in the Schwarzschild solution, but in any case its presence makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension. and there a loop-like singularity develops. It does not act as destructively as the infinite tidal force of the singularity in the Schwarzschild solution, but in any case its presence makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension.

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 singularity region, space-time looks quite ordinary. However, there are motion paths (they are not geodesic, so you will need a rocket engine) that will take you to the region of closed time-like curves. Once you are there, you can move in any direction along the t coordinate, which shows the time of the distant observer, but in your own time, you will always always move forward. And this means that you can go at any moment of time t at which you want, and then return to the remote part of space-time - and even arrive there before you leave. Of course, now all the paradoxes associated with the idea of time travel come to life: for example, what if, taking a walk in time, you convinced your past self to abandon it? But whether such forms of space-time can exist and how the related paradoxes can be resolved are issues that are 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 contains indications that regions of spacetime with closed timelike curves are unstable: as soon as you try to combine some of these curves with amount of mass or energy, these areas can become singular. Moreover, in the rotating black holes that form in our universe, it is the “blue singularity” in itself that can prevent the formation of regions of negative masses (and to all Kerr other universes into which white holes lead). Nevertheless, the fact that the general theory of relativity allows such strange decisions seems intriguing. They, of course, are easily declared pathological, but we will not forget that Einstein himself and many of his contemporaries said the same thing about black holes.

»The book can be found in more detail on the publisher’s website

For Khabrozhiteley 25% discount on the coupon -

Upon payment of the paper version of the book, an electronic version of the book will be sent by e-mail.

What is really inside them? How can one imagine falling into a black hole? Or maybe we are already falling into it and just do not know about it yet?

In Kerr geometry, there are geodetic orbits completely enclosed in the ergosphere with the following property: particles moving along them have negative potential energies that outweigh the absolute mass of the rest and the kinetic energies of these particles combined. This means that the total energy of these particles is negative. It is this circumstance that is used in the Penrose process. Being inside the ergosphere, a ship that produces energy shoots a shell in such a way that it moves along 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 the positive equivalent of the net negative energy of the projectile. Since the projectile after the shot should disappear into the black hole, it would be nice to make it from some waste. On the one hand, a black hole still eats anything, but on the other, it will return more energy to us than we invested. So in addition, the energy we acquired will be “green”!

The maximum amount of energy that can be extracted from the Kerr black hole depends on how fast the hole rotates. In the most extreme case (at the highest possible rotation speed), the space-time rotation energy accounts for approximately 29% of the total energy of the black hole. You may think that this is not very much, but do not forget that this is a fraction of the total rest mass! For comparison, recall that nuclear reactors operating on radioactive decay energy use less than one tenth of a percent of the energy equivalent to the rest mass.

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 similar to the Schwarzschild case. As before, space-time begins to collapse, dragging everything after 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 become large enough to threaten the integrity of the probe. In order to intuitively understand why this happens, we recall that in Newtonian mechanics during rotation there is a so-called centrifugal force. This force is not one of the fundamental physical forces: it arises due to the joint action of fundamental forces, which is necessary in order to ensure a state of rotation. The result can be represented as an effective force directed outward - a centrifugal force. You feel it in a sharp bend in a fast moving car. And if you ever rode a carousel, you know that the faster it spins, the harder you have to grab onto 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 moment of momentum in the space-time of the Kerr black hole provides an effective centrifugal force that counteracts gravitational attraction. When a collapse within the horizon draws space-time to smaller radii,

At the moment when the collapse stops, the probe reaches a level called the inner horizon of the black hole. At this point, tidal forces are small, and the probe, after it has crossed the event horizon, only needs some finite time to reach it. However, the mere cessation of the collapse of space-time does not yet mean that our problems are behind and that the rotation somehow led to the elimination of the singularity inside the Schwarzschild black hole. So far so far! Indeed, back in the mid-1960s, Roger Penrose and Stephen Hawking proved a system of theorems on singularity, from which it followed that if a gravitational collapse happened, even if it was short, then some form of singularity should form as a result. In the Schwarzschild case, this is a comprehensive and overwhelming singularity, which subjugates all the space within the horizon. In Kerr's decision, the singularity behaves differently and, it must be said, 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 it 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. 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 it 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. 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 it 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 the singularity that appears in the past of the observer who reaches the inner horizon is that at this moment Einstein's equations cannot unambiguously predict what will happen to spacetime outside this horizon. That is, in a sense, the presence of a singularity can lead to anything. Perhaps what really happens can be explained to us by the theory of quantum gravity, but Einstein's equations do not give us any chance to find out. Just out of interest, we will describe below what happens if you require that the intersection of the space-time horizon be as smooth as mathematically possible (if the metric functions are, as mathematicians say, “analytic”), but there are no clear physical grounds for such an assumption not. In fact, 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 at the inner horizon becomes very unstable, 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 makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension. and there a loop-like singularity develops. It does not act as destructively as the infinite tidal force of the singularity in the Schwarzschild solution, but in any case its presence makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension. and there a loop-like singularity develops. It does not act as destructively as the infinite tidal force of the singularity in the Schwarzschild solution, but in any case its presence makes one doubt the consequences that follow from the notion of smooth analytic functions. Perhaps this is good - very strange things entail the assumption of an analytical extension.

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 singularity region, space-time looks quite ordinary. However, there are motion paths (they are not geodesic, so you will need a rocket engine) that will take you to the region of closed time-like curves. Once you are there, you can move in any direction along the t coordinate, which shows the time of the distant observer, but in your own time, you will always always move forward. And this means that you can go at any moment of time t at which you want, and then return to the remote part of space-time - and even arrive there before you leave. Of course, now all the paradoxes associated with the idea of time travel come to life: for example, what if, taking a walk in time, you convinced your past self to abandon it? But whether such forms of space-time can exist and how the related paradoxes can be resolved are issues that are 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 contains indications that regions of spacetime with closed timelike curves are unstable: as soon as you try to combine some of these curves with amount of mass or energy, these areas can become singular. Moreover, in the rotating black holes that form in our universe, it is the “blue singularity” in itself that can prevent the formation of regions of negative masses (and to all Kerr other universes into which white holes lead). Nevertheless, the fact that the general theory of relativity allows such strange decisions seems intriguing. They, of course, are easily declared pathological, but we will not forget that Einstein himself and many of his contemporaries said the same thing about black holes.

»The book can be found in more detail on the publisher’s website

For Khabrozhiteley 25% discount on the coupon -

**Black Holes.**Upon payment of the paper version of the book, an electronic version of the book will be sent by e-mail.