General Relativity and its Applications

Black Holes, Compact Stars and Gravitational Waves

Valeria Ferrari, Leonardo Gualtieri, Paolo Pani

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9781138589773
• eBook ISBN: 9780429957796
• Edition: 1
• Topic: Biological Sciences
• Subtopic: Astronomy & Astrophysics
• Index: Biological Sciences

Containing the latest, groundbreaking discoveries in the field, this text outlines the basics of Einstein's theory of gravity with a focus on its most important astrophysical consequences, including stellar structures, black holes and the physics of gravitational waves. Blending advanced topics - usually not found in introductory textbooks - with examples, pedagogical boxes, mathematical tools and practical applications of the theory, this textbook maximises learning opportunities and is ideal for master and graduate students in Physics and Astronomy.

Key features:
• Provides a self-contained and consistent treatment of the subject that does not require advanced previous knowledge of the field.
• Explores the subject with a new focus on gravitational waves and astrophysical relativity, unlike current introductory textbooks.
• Fully up-to-date, containing the latest developments and discoveries in the field.

CHAPTER 1

Introduction

The General Theory of Relativity (or “General Relativity” for short) is the theory of gravity formulated by Albert Einstein in 1915. It is based on two fundamental principles: the Equivalence Principle of gravitation and inertia, which establishes a relation, central to the theory, between the gravitational field and the geometry of the spacetime, and the Principle of General Covariance, according to which the laws of physics have the same form in any reference frame. General Relativity has changed dramatically our understanding of space, time, and gravity. In this book we shall investigate its fascinating and far-reaching predictions, such as the existence of black holes, and the emission of gravitational waves in the most violent and energetic phenomena occurring in our universe, as the collision of black holes and of neutron stars.

The language of General Relativity is that of differential geometry. There is no way to understand Einstein’s theory of gravity without mastering the concept of manifold, or of tensor field. Therefore, we shall dedicate a few chapters to the mathematical tools that are essential to describe the theory and its physical consequences. This first chapter, however, will be a pedagogical introduction aimed at answering the following questions:

1. Why did Newtonian theory become inappropriate to describe the gravitational field?
2. Why do we need to introduce geometrical objects, like the metric tensor or the Christoffel symbols, to describe the gravitational field; and what is the role of the Equivalence Principle in this new geometrical framework?

In the next chapters we shall rigorously define manifolds, vectors, tensors, and then, after introducing the principle of general covariance, we will formulate Einstein’s equations. But first of all, since as anticipated there exists a relation between the gravitati...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• Preface
• Notation and conventions
• CHAPTER 1 ◾ Introduction
• CHAPTER 2 ◾ Elements of differential geometry
• CHAPTER 3 ◾ Affine connection and parallel transport
• CHAPTER 4 ◾ The curvature tensor
• CHAPTER 5 ◾ The stress-energy tensor
• CHAPTER 6 ◾ The Einstein equations
• CHAPTER 7 ◾ Einstein's equations and variational principles
• CHAPTER 8 ◾ Symmetries
• CHAPTER 9 ◾ The Schwarzschild solution
• CHAPTER 10 ◾ Geodesic motion in Schwarzschild's spacetime
• CHAPTER 11 ◾ Kinematical tests of General Relativity
• CHAPTER 12 ◾ Gravitational waves
• CHAPTER 13 ◾ Gravitational waves in the quadrupole approximation
• CHAPTER 14 ◾ Gravitational wave sources
• CHAPTER 15 ◾ Gravitational waves from oscillating black holes
• CHAPTER 16 ◾ Compact stars
• CHAPTER 17 ◾ The far-field limit of an isolated, stationary object
• CHAPTER 18 ◾ The Kerr solution
• CHAPTER 19 ◾ Geodesic motion in Kerr's spacetime
• CHAPTER 20 ◾ Black hole thermodynamics
• Bibliography
• Index