Mathematical general relativity: a sampler

Lecture notes on selected topics in the theory of gravitation.

It took eight years after Einstein announced the basic physical ideas behind the relativistic gravity theory before the proper mathematical formulation of general relativity was mastered. The efforts of the greatest physicist and of the greatest mathematician of the time were involved and reached a breathtaking concentration during the last month of the work. Recent controversy, raised by a much publicized 1997 reading of Hilbert's proof-sheets of his article of November 19...

This is an English translation of the Italian version of an encyclopedia chapter that appeared in the Italian Encyclopedia of the Physical Sciences, edited by Bruno Bertotti (1994). Following requests from colleagues we have decided to make it available to a more general readership. We present the motivation for constructing General Relativity, provide a short discussion of tensor algebra, and follow the set up of Einstein equations. We discuss briefly the initial value pro...

In this short note, a brief overview with a critical appraisal of the acclaimed singularity theorems, the most genuine post-Einsteinian result of General Relativity, is presented.

We give a brief non-technical introduction to non-regular spacetime geometry. In particular, we discuss how curvature, and hence gravity, can be defined without a smooth (differential geometric) calculus.

The authors present concepts and mathematical developments which give rise to the Hamiltonian formulation of Einstein's general relativity, first introduced by Arnowitt, Deser, and Misner. All the geometrical quantities needed for the construction are explicitly obtained, and examples of some of the application of the formalism are given.

This paper is a collection of lecture notes on modified gravity. Various modified gravity models formulated within the Riemannian formalism are discussed.

This year marks the hundredth anniversary of Einstein's 1915 landmark paper "Die Feldgleichungen der Gravitation" in which the field equations of general relativity were correctly formulated for the first time, thus rendering general relativity a complete theory. Over the subsequent hundred years physicists and astronomers have struggled with uncovering the consequences and applications of these equations. This contribution, which was written as an introduction to six chapter...

This paper gives a self-contained, elementary, and largely pictorial statement of Einstein's equation.

General relativity's successes and limitations are compared to those of special relativity.