Table of Contents
Table of Contents
"Introduction to Black Holes"
INTRODUCTION
SCHWARZSCHILD'S BLACK HOLES
ELECTRICALLY CHARGED BLACK HOLES
ROTATING BLACK HOLES
BLACK HOLE MECHANICS
HAWKING'S RADIATION
"Introduction to Black Holes"
"Introduction to Black Holes"
SIMONE MALACRIDA
The following topics are presented in this book:
basics of black holes: gravitational collapse, event horizon, geodesics
Schwarzschild, Reissner-Nordstrom and Kerr-Newman metrics
spherically symmetric, rotating and electrically charged black holes
Carter-Penrose diagrams, naked singularities and Kruskal coordinates
mechanics of black holes
thermodynamics of black holes and Hawking radiation
quantum black holes
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – SCHWARZSCHILD'S BLACK HOLES
Gravitational collapse
Geodesics
Schwarzschild metric
Krusk coordinates in spacetime
Carter-Penrose diagrams
Event horizon
naked singularities
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II - ELECTRICALLY CHARGED BLACK HOLES
Reissner-Nords metric t rom
Cauchy horizon
Isotropic coordinates
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III - ROTATING BLACK HOLES
Uniqueness theorem
Kerr solutions
Ergosphere
Penrose process
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IV - BLACK HOLE MECHANICS
Energy and angular momentum
Geodetic congruences
The laws of black hole mechanics
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V - HAWKING'S RADIATION
Quantization of a scalar field
particle production
Hawking radiation
The thermodynamics of black holes
INTRODUCTION
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
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 <E> 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 be considered non-relativistic we have:
And so the total energy will be:
Where the coefficients alpha and beta do not depend on R.
Graphing this expression, we have:
Put another way, the star will not collapse and a white dwarf will be formed supported by the electron degeneracy pressure.
For this to happen, the approximation of non-relativistic electrons must exist, i.e.:
If, on the other hand, this approximation does not hold, i.e. if the mass of the star is greater or comparable to this quantity, the electrons will have to be considered relativistic.
In this case, the dependence of the total energy on the radius of the star will be:
The equilibrium condition exists only for:
For smaller masses, the radius will continue to grow until the electrons become non-relativistic.
For larger masses, the radius will continue to decrease and the electron degeneracy pressure will fail to support the star.
The critical mass for which this occurs is:
This is the so-called Chandrasekhar limit and the critical mass is about 1.4 times that of the Sun.
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The fact that the electrons can be relativistic or not based on the mass of the star, decisively influences its evolution.
In fact, in a white dwarf beta decay is substantially stable due to the presence of "low energy" electrons.
This reaction, representing beta decay, is exactly counterbalanced by its inverse:
If, on the other hand, the electrons are relativistic, i.e. above the Chandrasekhar limit, the neutrino generated by this decay does not remain trapped in the star and therefore the reverse reaction can no longer take place.
Through beta decay, the star will transform electrons and protons into neutrons by creating a neutron degeneracy pressure.
That's why it's called a neutron star .
This will give a further push to the gravitational collapse of the star, significantly increasing its density.
At such densities, the
Imprint
Publisher: BookRix GmbH & Co. KG
Publication Date: 04-19-2023
ISBN: 978-3-7554-3954-7
All Rights Reserved