SPECIAL SECTION: 100 YEARS OF GENERAL RELATIVITY CURRENT SCIENCE, VOL. 109, NO. 12, 25 DECEMBER 2015 2230 *e-mail: bala.iyer@icts.res.in Advances in classical general relativity Bala R. Iyer* International Centre for Theoretical Sciences–TIFR, Hessaraghatta, Bengaluru 560 089, India The year 2015 is the centenary of Einstein’s creation of general relativity. Over the century general relati- vity has gradually increased its footprints on main- stream physics and this article highlights advances in the classical aspects of general relativity since its crea- tion. Keywords: Black holes, centenary, cosmology, general relativity, gravitational waves, ISGRG conference. Introduction EINSTEIN arrived at General Relativity (GR), his relativis- tic classical theory of gravitation on 25 November 1915 (ref. 1). The prime motivation was to have a theory of gravity compatible with special relativity (SR) that agreed with Newton’s theory in the appropriate limit. Unlike SR that came in its final form over a year in 1905 and involved only himself, GR went through manifold ‘tinkering phases’ over eight long years and involved collaboration on mathematical aspects with Marcel Grossman and Michele Besso. As was later shown by Lovelock 2 , if one allows only second order equations and a single 4-dimensional ST metric, GR is the unique gravi- tation theory based on Riemannian geometry. Beyond the new mathematical description was also a profound physi- cal insight. The geometry of spacetime (ST) was no longer a fixed backdrop but a dynamical physical entity determined by the matter-energy content and nonlinear equations. The geodesics of ST determined the paths of light and freely falling particles. The geodesic deviation equation determines the curvature and the tidal forces; the curvature is determined by the matter and motion con- tent of ST. The metric in the non-relativistic limit is re- lated to the gravitational potential. Standard physics based on linearity is not adequate in general and the search for exact solutions using tensor calculus, covariant equations, structures in non-Euclidean geometry and co- ordinate-free methods, the way forward. Generalizations of GR do exist. They include scalar tensor theories, theo- ries with higher derivatives or torsion, bimetric theory, unimodular theories and theories in higher dimensions 3 . Not only is GR universally acknowledged to being the epitome of mathematical elegance and conceptual depth but importantly for over a century demonstrated remark- able observational success. Being very nonlinear, it has collaterally led to many developments in analytic and numerical techniques. GR is mathematically a compli- cated nonlinear theory. So it was surprising to Einstein himself that an exact solution could be found so quickly: the Schwarzschild solution 4 that describes the gravita- tional field exterior to a spherically symmetric body as also the Schwarzschild constant density interior solution. Other interesting solutions included the linearized gravity solutions describing gravitational waves in analogy to electromagnetic waves, solutions corresponding to the Einstein static universe 5 , static de Sitter 6 , Friedman 7,8 and Lemaître 9 expanding models 10 describing the gravita- tional field of the whole universe, Vaidya metric 11 repre- senting a spherically symmetric radiating solution and Majumdar–Papapetrou solution representing system of charged black holes in equilibrium under their gravita- tional and electrostatic forces 12 . For a long time this inspired mathematical research in GR to seek exact solu- tions of Einstein’s equations (EE). The complexity of EE made this an interesting mathematical challenge even if the physical interpretation of some of these solutions was not very obvious. It led to interesting developments in al- gebraic computing for long computations in tensor calcu- lus typical of GR and later means to classify solutions of EE and recognize equivalent solutions that appeared new due to a different choice of coordinates. The use of sym- metry groups to simplify the system of equations, the classification of exact solutions on the basis of symme- tries, the search for techniques to generate new solutions from old constituted a major area of research in GR for many years 13 . It naturally led to the use of coordinate free methods, tetrad formalism, use of null tetrads, the New- man Penrose formalism 14 and methods to investigate kinematical properties of null and timelike vector fields describing radiation and matter respectively. In spite of these theoretical developments, in its first half century of its existence, GR was outside of mainstream physics in contrast to the following fifty years with increasing pro- found applications in astronomy, astrophysics and cos- mology. Though classical differential geometry was an adequate starting point for the initial studies, later deve- lopments required a careful understanding of the global structure of spacetime, singularities and asymptotics to interpret these solutions 15 and formalisms to disentangle physical effects from coordinate or gauge-dependent ones.