Hyperbolic Equations and General Relativity


Series: Mathematics Research Developments
BISAC: MAT027000

This work is divided into three parts. In the first part, the hyperbolic equations’ theory is analysed, the second part concerns the Cauchy problem in General Relativity, whereas the third part gives a modern perspective of General Relativity.

In the first part, the study of systems of partial differential equations allows the introduction of the concept of wave-like propagation and the definition of hyperbolic equation is given. Thus, once the definition of Riemann kernel is given, Riemann’s method to solve a hyperbolic equation in two variables is shown. The discussion moves on the fundamental solutions and its relation to Riemann kernel is pointed out. Therefore, the study of the fundamental solutions concludes by showing how to build them providing some examples of solution with odd and even number of variables. Moreover, the fundamental solution of the scalar wave equation with smooth initial conditions is studied.

In the second part, following the work of Fourès-Bruhat, the problem of finding a solution to the Cauchy problem for Einstein field equations in vacuum with non-analytic initial data is presented by first studying under which assumptions second-order systems of partial differential equations, linear and hyperbolic, with n functions and four variables admit a solution. Hence, it is shown how to turn non-linear systems of partial differential equations into linear systems of the same type for which the previous results hold. These considerations allow us to prove the existence and uniqueness of the solution to the Cauchy problem for Einstein’s vacuum field equations with non-analytic initial data. Eventually, the causal structure of space-time is studied. The definitions of strong causality, stable causality and global hyperbolicity are given and the relation between the property of global hyperbolicity and the existence of Cauchy surfaces is stressed.

In the third part, Riemann’s method is used to study the news function describing the gravitational radiation produced in axisymmetric black hole collisions at the speed of light. More precisely, since the perturbative field equations may be reduced to equations in two independent variables, as was proved by D’Eath and Payne, the Green function can be analysed by studying the corresponding second-order hyperbolic operator with variable coefficients. Thus, an integral representation of the solution in terms of the Riemann kernel function can be given.
(Imprint: Nova)

Table of Contents

Table of Contents


List of figures


Part I: Hyperbolic Equations’ Theory

Chapter 1. Hyperbolic Equations

Chapter 2. Fundamental Solutions

Chapter 3. How to Build the Fundamental Solution

Part II: The Cauchy Problem in General Relativity

Chapter 4. Linear Systems of Normal Hyperbolic Form

Chapter 5. Linear System from a Non-linear Hyperbolic System

Chapter 6. General Relativity and the Causal structure of Space-

Part III: A Modern Perspective

Chapter 7. Riemann’s method in Gravitational Radiation Theory




Keywords: Mathematical Relativity, General Relativity, PDE, Pure Analysis, Black Hole, Black Hole Collisions, Physics, Mathematics, Mathematical Physics, Partial Differential Equations, Riemann Kernel, Hyperbolic Equations, Cauchy problem.

Audience: Math students, Physics Students, Mathematical Relativity Researchers and Professors, General Relativity Professors and students, Pure Analysis students and professors. Eventually, people who have interest in Mathematical Relativity, General Relativity, Partial differential equations and Pure Analysis.


“This is an excellent book that can be used to introduce the reader to the theory of hyperbolic equations and to the mathematical theory of gravitational waves and Einstein equations. Several basic concepts such as wavelike propagation, fundamental solution, Riemann kernel and its existence, world function and its role in the fundamental solution, characteristic conoid, are well presented. The reader is then offered the opportunity to learn in detail how Choquet-Bruhat proved existence and uniqueness of the solution of vacuum Einstein equations with non-analytic Cauchy data. Last, an introduction to some important aspects of high-speed black hole collisions, a masterpiece work of D’Eath and Payne, is presented. The appendices on Sobolev spaces and Kasner spacetime are also very useful, as well as the suggested literature at the end. The author should be congratulated for having written a book of high pedagogical value for future generations of research workers in general relativity.” Giampiero Esposito, INFN Sezione di Napoli, Naples, Italy

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