This is a student text that revises the previous edition from 2009. The main change is the addition of chapter 6 on black holes in more than four spacetime dimensions and a nonzero cosmological constant.

The previous editions had the subtitle "An introduction" and the new one is called "A Student text", which perhaps is a better description. It is written for a public of last year undergraduate, first year graduate students. Some training in physics, tensor calculus, and preferably also in relativity theory is strongly advised. The text is peppered with problems for the reader to solve, but solutions are provided at the end. These problems are often shortcuts not to disrupt the flow of the exposition. The chapters mainly restrict to the study of the metric and how this defines motion in the presence of black holes. Long derivations are avoided and it is not shown how these metrics solve Einstein's field equations.

The book starts by recalling the essential elements from relativity theory (Minowski metric, gravitational field, geodesics, transformations, etc.) and of course the definition of a black hole. Its mass is collapsed into a physical singularity and its boundary is the event horizon, a kind of mathematical one-way membrane represented by a coordinate singularity separating different physics inside and outside. An ousider cannot observe what is happening inside the event horizon.

The simplest case is a spherical black hole with the Schwarzschild metric (which is diagonal) in a 4-dimensional spacetime. Several orbits are described (radial infall and circular orbits for particles and for light). How does a distant observer sees something falling into a black hole? How does it 'feel' to fall into a black hole? How to describe the physics near the event horizon? What happens inside the black hole? Answers to such questions may appear in popular science books or even in science fiction books, but here the answers are read off from the formulas which is something quite different. What happens inside the event horizon can not be observed but the mathematics can be done and the strange physics can be derived. The coordinate systems are important and some phenomena are better understood in different coordinate systems. An observer at infinity and a particle falling into a black hole have quite different experiences.

The next complication is when the black hole rotates, i.e., has an angular momentum which defines an axial symmetry. This momentum can range from zero (spherical Schwarzschild case) to some finite upper bound (an extreme Kerr hole). Kerr refers to the Kerr metric that is used in this case (diagonal plus nonzero elements in the NE and SW corners). The same kind of exploring the orbits can be done as in the previous chapter. A new element is the ergo sphere (actually an ellipsoid bulging near the equator) that encloses the sphere defined by the event horizon. Rotational energy can be extracted in this bulging area.

Black holes do not radiate, so they do not seem to have a temperature. A Kerr hole has several thermodynamic-like properties such as the area of the event horizon cannot decrease, its mass can decrease, but there is a lower bound (the Schwarzschild hole), and hence there is an upper bound for the energy that can be extracted from the ergosphere, but this does not reduce to a physical temperature.

It is shown though that quantum theory does need temperature and there are thermodynamic laws. For example entropy cannot decrease and there is Hawking radiation. It creates some problems that need solution, a solution that could be provided by string theory.

Chapter 5 is about wormholes and time travel, very popular topics in science fiction. Energy requirements are derived for a macroscopic wormhole to exist. If ever this amount of energy would be available, then wormholes would create shortcuts between two points in spacetime and time travel would be possible. Since this would create the well known paradoxes, there is a chance that such wormholes are highly unstable and collapse immediately or some yet unknown physical laws may just forbid them to exist.

The new chapter 6 discusses the existence of more than the classical 3 space dimensions. Allowing a negative cosmological constant relaxes the condition of an asymptotically flat spacetime, which allows the introduction of a new parameter. Einstein's field equations are now solved by an anti-de Sitter (AdS) metric and black holes need not be topologically spherical anymore, but they may have a positive genus. Quantum theory of gravity seems to match nicely with a conforming field theory (CFT) on the boundary of an AdS of higher dimension. Strings are alternative 1-dimensional building blocks whose excitations at low energy correspond to point-like particles, one of which could be a graviton. Strings can be connected to higher dimensional objects: the branes. Supersymmetric string theory requires 10-11 dimensions. The extra dimensions can not be observer because they are curled up so tightly, but we experience their existence in the form of fields. Black strings and membranes are discussed and linked to black hole entropy. This chapter is even more a survey than are the other chapters. It certainly is an invitation for further reading.

The last chapter is about astrophysical black holes. In reality the perfect symmetry studied so far does not exist. So what observations do confirm the existence of black holes? There are the collapsed very massive stars, usually appearing as a dark twin to another star, there might be a massive black hole at the center of the galaxy. However, in principle intermediate or even smaller black holes can exist. The former will still be the result of star evolution, but mini black holes should result from physical conditions in the early universe. Their existence is still hypothetical but of course of great interest to cosmology.

So it will be clear that with this limited number of papers, one should not start reading the book unprepared. Knowledge of relativity theory is indeed something the reader should be somewhat familiar with, and some cosmology does not harm. The mathematics are relatively elementary, but formulas are often dropped saying that you need to take this and that into account and then such equations will result in this formula, or just a reference is given. Thus, if as a mathematician, you want to learn about black holes, you will get a general idea by reading this book, but it will only be a starting point for further exploration.