Quo Vadis, Mathematical General Relativity?

These lectures notes contain an introduction to General Relativity. They are addressed to a general mathematical audience with no specific background in physics. The goal is to motivate and explain Einstein's theory of gravity and discuss some of the fundamental examples.

This survey paper is divided into two parts. In the first (section 2), I give a brief account of the structure of classical relativity theory. In the second (section 3), I discuss three special topics: (i) the status of the relative simultaneity relation in the context of Minkowski spacetime; (ii) the "geometrized" version of Newtonian gravitation theory (also known as Newton-Cartan theory); and (iii) the possibility of recovering the global geometric structure of spacetime f...

Popular wisdom amongst theoretical physicists says that the continuum structure of spacetime is probably not elementary, but rather emergent. While many arguments to support that view arise from speculative ideas, the argument can also be made by only invoking standard physics. In this manuscript, I shall argue that a novel general theory of relativity might change the deal, while it corresponds to a somewhat minimal extension of the core theory of physics.

This is Chapter 1 in the book General Relativity and Gravitation: A Centennial Perspective, Edited by Abhay Ashtekar (Editor in Chief), Beverly Berger, James Isenberg, Malcolm MacCallum. Publisher: Cambridge University Press (June, 2015). It gives a survey of themes that have been developed during the 100 years of progress in general relativity theory.

In papers on the history of general relativity and in personal remembrances of relativists, keywords like "renaissance" and "golden age" of general relativity have been used. We try to show that the first label rests on a weak empirical basis. The second one, while describing a period of vivid growth in research in general relativity, exaggerates the importance of this particular development.

These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.

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

In this paper a mathematically precise global (i.e. not the usual local) approach is presented to the variational principles of general relativistic classical field theories. Problems of the classic (usual) approaches are also discussed in comparison. The aim of developing a global approach is to provide a possible tool for future efforts on proving global existence theorems of field theoretical solutions.

This article has a dual purpose: i) to provide a flavor of the scientific highlights of the landmark conference, GR3, held in July 1962 at Jablonna, near Warsaw; and, ii) to present a bird's eye view of the tremendous advances that have occurred over the half century that separates GR3 and GR20, which was again held in Warsaw in July 2013.

This is a review devoted to some results of Algebraic Programming (Computer Algebra) used in treating several problems of general relativity, based mainly on already published articles. The article contains the talk given by the author at The Albert Einstein Institut, Max Planck Institut fur Gravitationstheorie, Golm, Germany, september 2000