Introduction to Black Holes

INTRODUCTION

This book presents a broad overview of black holes, starting from the mathematical and physical concepts that indicate their presence in spacetime up to their properties.

Among the most exotic celestial objects, black holes have represented an immense theoretical challenge for general relativity.

In fact, this discipline, born to describe spacetime at every point, without any distinction, must bow to the existence of singularities foreseen by the various metric solutions of its own equations.

For years there have been neither experimental findings nor physical theories capable of understanding the properties of black holes.

However, from the 1970s onwards, the application of quantum field theory to black holes has made it possible to understand some fundamental mechanisms such as Hawking radiation and the thermodynamics of black holes.

In addition, the refinement of some formalisms (Carter-Penrose diagrams for example) has allowed us to describe their main properties.

All of this is far from a comprehensive understanding of such celestial objects.

To date, there are no univocal theories that allow to describe what really happens in the presence of a spacetime singularity, mainly due to the fact that quantum general relativity has not yet been enunciated as a consistent physical theory.

The problems relating to black holes therefore intersect with other fundamental aspects of contemporary physics, such as the unification of forces, a probable theory of everything that explains the physical mechanisms of the Universe and cosmological assumptions such as the shape of the Universe and its origin .

What we are going to explain needs some prerequisites concerning general relativity itself, tensor mathematics and, in general, the physical theories of quantization of the fields.

Therefore, this book has a cut strongly addressed to those who have physical and mathematical knowledge of a specialist university type in these sectors or to those who have a strong passion for astrophysics, intimately knowing its profound mathematical-physical aspects.

I

SCHWARZSCHILD'S BLACK HOLES

Gravitational collapse

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We can consider a star as a sphere of hydrogen atoms supported by a thermal pressure given by the product of the temperature, the density of the atoms and a constant.

At equilibrium, total energy has a minimum.

The total energy can be expressed as the sum of a gravitational part of a kinetic part:

Where the term represents the average kinetic energy of the atoms, while M and R are the mass and radius of the sphere.

We note that, if the temperature of a star were =0, the pressure does not go to zero as there is the mechanism of degeneration of the pressure.

If electrons can