Despite the complexity of the topic, Princeton University professor Stephen Gabser offers a concise, 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 they are mathematically much simpler than most astrophysical objects, such as stars. Weirdness begins when it turns out that black holes are not really so black after all.
What is really inside them? How can you imagine falling into a black hole? Or maybe we're already falling into it and just don't know it yet?
In the geometry of Kerr, there are geodesic orbits, completely enclosed in the ergosphere, with the following property: the particles moving along them have negative potential energies, which outweigh the rest masses and kinetic energies of these particles taken together in absolute value. 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, the ship producing energy fires a projectile 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 enough kinetic energy to compensate for the lost rest mass equivalent to the energy of the projectile, and in addition to receive a positive equivalent of the net negative energy of the projectile. Since the projectile must disappear into a black hole after being fired, it would be good to make it from some kind of waste. On the one hand, the black hole will still gobble up anything, and on the other hand, it will return us more energy than we put in. So in addition, the energy we acquire will be "green"!
The maximum amount of energy that can be extracted from a Kerr black hole depends on how fast the hole is spinning. In the most extreme case (at the maximum possible rotation speed), the space-time rotational energy accounts for approximately 29% of the total energy of the black hole. This may not sound like much, but remember that this is a fraction of the total rest mass! By comparison, remember that nuclear reactors fueled by radioactive decay use less than one-tenth of a percent of the rest-mass-equivalent energy.
The space-time geometry inside the horizon of a rotating black hole differs sharply from Schwarzschild's space-time. Let's 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 along with it 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 spinning black hole, this will happen long before the tidal forces are large enough to threaten the integrity of the probe. To intuitively understand why this happens, remember that in Newtonian mechanics, during rotation, the so-called centrifugal force arises. This force is not one of the fundamental physical forces: it arises due to the combined action of the fundamental forces, which is necessary to provide a state of rotation. The result can be thought of as an effective outward force, a centrifugal force. You feel it on a sharp turn in a fast moving car. And if you've ever ridden a merry-go-round, you know that the faster it spins, the tighter you have to grip the rails, because if you let go, you'll be thrown out. This analogy for space-time is not ideal, but it conveys the essence correctly. The angular momentum in the spacetime of a Kerr black hole provides an effective centrifugal force that counteracts the gravitational pull. As the collapse within the horizon shrinks spacetime to smaller radii, the centrifugal force increases and eventually becomes able to first counteract the collapse and then reverse it.
The moment the collapse stops, the probe reaches a level called the black hole's inner horizon. 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, just stopping the collapse of space-time does not mean that our problems are over and that the rotation has somehow led to the elimination of the singularity inside the Schwarzschild black hole. This is far from it! Indeed, back in the mid-1960s, Roger Penrose and Stephen Hawking proved a system of singularity theorems, from which it followed that if a gravitational collapse had already happened, albeit a short one, then some form of singularity should form as a result. In the Schwarzschild case, this is an all-encompassing and all-destroying singularity that subjugates all space inside the horizon. In Kerr's solution, 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 occurs in the causal past of the probe's world line. It is as if the singularity had always been there, but only now did the probe feel its influence reach him. You will say that this 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 a singularity appearing in the past of an observer who reaches the inner horizon is that, at that point, Einstein's equations cannot unambiguously predict what will happen to spacetime outside that horizon. That is, in a sense, the presence of a singularity can lead to anything. Perhaps what will actually happen can be explained to us by the theory of quantum gravity, but Einstein's equations do not give us any chance of knowing this. Just out of curiosity, we describe below what happens if we require that the space-time horizon crossing be as smooth as mathematically possible (if the metric functions are, as mathematicians say, "analytic"), but there is no clear physical basis for such an assumption. No. In fact, the second problem with the inner horizon suggests exactly the opposite: in the real Universe, in which matter and energy exist outside of black holes, space-time at 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 casts doubt on the consequences that follow from the concept of smooth analytic functions. Perhaps this is good - very strange things are entailed by the assumption of an analytic extension.
In essence, a time machine operates in the area of closed timelike curves. Far from the singularity, there are no closed time-like curves, and apart from the repulsive forces around the singularity, space-time looks completely normal. However, there are trajectories (they are not geodesic, so you need a rocket engine) that will take you to the region of closed timelike curves. Once you're there, you can move in any direction on the t-coordinate, which is the time of the distant observer, but in your own time, you'll still always be moving forward. And this means that you can go to any point in time t you want, and then return to a remote part of space-time - and even arrive there before you go. Of course, all the paradoxes surrounding the idea of time travel now come to life: for example, what if, by taking a time walk, you persuaded your past self to give it up? But whether such kinds of space-time can exist, and how the paradoxes associated with it can be resolved, are questions beyond the scope of this book. However, just as with the problem of the "blue singularity" on the inner horizon, general relativity contains indications that regions of spacetime with closed timelike curves are unstable: as soon as you try to match one of these curves with some amount of mass or energy, these areas can become singular. Moreover, in rotating black holes that form in our Universe, it is the “blue singularity” itself that can prevent the formation of a region of negative masses (and all other Kerr universes into which white holes lead). Nevertheless, the fact that general relativity allows such strange solutions is intriguing. Of course, it is easy to declare them pathological, but let's not forget that Einstein himself and many of his contemporaries said the same about black holes.
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Source: habr.com