Analytical and Numerical Approaches to Mathematical Relativity [Paperback]

General relativity ranks among the most accurately tested fundamental theories in all of physics. Deficiencies in mathematical and conceptual understanding still exist, hampering further progress. This book collects surveys by experts in mathematical relativity writing about the current status of, and problems in, their fields. There are four contributions for each of the following mathematical areas: differential geometry and differential topology, analytical methods and differential equations, and numerical methods.

The general theory of relativity, as formulated by Albert Einstein in 1915, provided an astoundingly original perspective on the physical nature of gr- itation, showing that it could be understood as a feature of a curvature in the four-dimensional continuum of space-time. Now, some 90 years later, this extraordinary theory stands in superb agreement with observation, prov- ing a profound accord between the theory and the actual physical behavior of astronomical bodies, which sometimes attains a phenomenal precision (in one case to about one part in one hundred million million, where several d- ferent non-Newtonian e?ects, including the emission of gravitational waves, are convincingly con?rmed). Einsteins tentative introduction, in 1917, of an additional term in his equations, speci?ed by a cosmological constant, - pearsnowtobeobservationallydemanded,andwiththistermincluded,there is no discrepancy known between Einsteins theory and classical dynamical behavior, from meteors to matter distributions at the largest cosmological scales. One of Einsteins famous theoretical predictions that light is bent in a gravitational ?eld (which had been only roughly con?rmed by Eddingtons solareclipsemeasurementsattheIslandofPrincipein1919,butwhichisnow very well established) has become an important tool in observational cosm- ogy, where gravitational lensing now provides a unique and direct means of measuring the mass of very distant objects.l#�