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Raychaudhuri Equation at the Crossroads Digitized by the Internet Archive in 2018 with funding from Public.Resource.Org https://archive.org/details/raychaudhuriequaOOunse Special issue on Raychaudhuri equation at the crossroads . Special Issue on Raychaudhuri equation at the crossroads Guest Editors Naresh Dadhich Pankaj Joshi Probir Roy Published by Indian Academy of Sciences Bangalore © 2007 Indian Academy of Sciences Reprinted from Pramana - J. Phys. Vol. 69, No. 1, July 2007 ISSN No. 0304-4289 Edited by Naresh Dadhich, Pankaj Joshi and Probir Roy and printed for the Indian Academy of Sciences by Tholasi Prints, Bangalore Contents Preface. 1 A little reminiscence.A K Raychaudhuri 3 A K Raychaudhuri and his equation.JEhlers 1 On the Raychaudhuri equation.George F R Ellis 15 Singularity: Raychaudhuri equation once again.Naresh Dadhich 23 A singularity theorem based on spatial averages.J MM Senovilla 31 The Raychaudhuri equations: A brief review. .Sayan Kar and Soumitra SenGupta 49 Black hole dynamics in general relativity.Abhay Ashteker 77 String theory and cosmological singularities.Sumit R Das 93 Horizons in 2+1-dimensional collapse of particles. .Dieter Brill, Puneet Khetarpal and Vijay Kaul 109 On the genericity of spacetime singularities.Pankaj S Joshi 119 On a Raychaudhuri equation for hot gravitating fluids. .Chandrasekhar Mukku, Swadesh MMahajan and Bindu A Bambah 137 Raychaudhuri equation in quantum gravitational optics. .N Ahmadi and M Nouri-Zonoz 147 ; ' - M 11H111 ■ ■ - | 1 I ..•.. B..I._I ■ ■ Preface In 1953 something extraordinary happened at the Indian Association for the Cultivation of Science in Calcutta. Amal Kumar Raychaudhuri, twenty-seven years of age and employed ungainfuily as a scientific assistant at the Experimen¬ tal X-ray Section, made a startling theoretical discovery in General Relativity (GR). Without assuming any symmetry constraint on the underlying spacetime, he derived an equation which showed the unavoidable occurrence of spacetime singularities in GR under quite general conditions. Nearly a decade later, by global topological arguments utilizing the causal structure of spacetime and Einstein’s equations, this result was given a complete mathematical general¬ ization and proved rigorously in terms of a set of precisely enunciated theorems, now very well-known as singularity theorems, by Hawking, Penrose and Geroch. The equation of Raychaudhuri was the critical starting point for these theorems which held under more general conditions of which Raychaudhuri’s conditions were a subset. Its import and significance were immediately recognized as was evident from the fact that Prof. Charles Misner could obtain a grant from NSF for an year’s visit of AKR to the University of Maryland in 1964. The Raychaudhuri equation has three aspects which are logically sequen¬ tial. First and foremost, it is a geometric statement on the congruence of non-spacelike paths, including geodesics, in an arbitrary spacetime manifold. Second, on introducing the Principle of Equivalence, it becomes a statement on the congruence of the trajectories of material particles and photons in an arbitrary gravitational field. Finally, the use of Einstein’s equations and of the energy conditions leads to the result that in a generally nonflat spacetime manifold there exist trajectories which are necessarily incomplete in the sense that they and their neighbouring trajectories inevitably focus into singulari¬ ties at finite comoving times. The equation describes how trajectories behave during the course of their dynamical evolution, i.e. how they expand, recon¬ verge, get distorted under shearing effects of gravitational fields and rotate under the influence of the energy density and matter fields present. The scope of the Raychaudhuri equation is very wide since it is a geometric statement on the evolution of paths in a general (not necessarily spacetime) manifold. For gravitational dynamics, it encompasses all spacetime singularities from the cosmological big bang to black holes and naked singularities that could arise in astrophysics from collapsing stars. More than fifty years have passed since this powerful equation was writ¬ ten down. In this intervening half-century, it has influenced different types of research, not only in classical GR as well as in still incomplete theories of quantum gravity, but also in string theory and even in hydrodynamics. There is every likelihood that research involving the Raychaudhuri equation will take Preface new directions in future. Just to illustrate this point, let us mention that in the currently fashionable Loop Quantum Cosmology, this equation is needed in a new avatar in the possible avoidance of the cosmological big bang singularity. Standing at the crossroads, we feel this to be an appropriate juncture to view this equation in perspective. To this end, we have invited essays from several experts, working in different areas, whose current research not only derives inspiration from this equation, but in fact makes use of it in some way or the other. We feel that, as a celebration of the golden jubilee of the birth of this amazing equation, this is the best tribute that we can offer to the memory of its deceased creator. We thank all our authors for magnificently responding to our request for their contributions. We are grateful to the editorial board of Pramana for agreeing to publish this volume and we especially thank Prof. Rohini Godbole for facilitating that. The article by Professor Jurgen Ehlers is being produced from the Proceedings of the International Conference on Einstein’s Legacy in the New Millennium published in Int. J. Mod. Phys. D15 (2006). We thank both the author and the publisher, World Scientific, for their permission to reproduce it. Also reproduced is Prof. Raychaudhuri’s reminiscence from the Proceedings of the Symposium on the Fortieth Anniversary of the Raychaudhuri Equation held at IUCAA in December 1995. Naresh Dadhich Inter-University Centre for Astronomy and Astrophysics, Pune 411 007 Pankaj Joshi Tata Institute of Fundamental Research, Mumbai 400 005 Probir Roy Tata Institute of Fundamental Research, Mumbai 400 005 Spring 2007 (Guest Editors) 2 Pramana (c) Indian Academy of Sciences Vol. 69, No. 1 — journal of July 2007 physics pp. 3-5 A little reminiscence* A K RAYCHAUDHURI I feel rather exhilerated by the honour that the organizers of ICGC-III have be¬ stowed on me by arranging a particular session in connection with my works. How¬ ever, I must confess that I am embarassed to speak at this meeting. Firstly, I believe that a characteristic beauty of science is its objectivity but my presence as a speaker apparently brings in a subjective element. Secondly, in a conference like the present one, the discussions are generally on the pressing problems of the present and possible developments that are likely to take place in the foreseeable future. Frankly, I am an old, out-of-date person who is a misfit to take part in such deliberations. In the circumstances, acceding to your request, all I can do is to travel down the memory lane and tell you something about how I came across the equation that bears my name. It was early fifties when words like the black hole, static limit or geodesic in¬ completeness had not entered the scientific vocabulary and quite different types of peculiarities were clubbed under the name ‘singularity’. My first interest was in the so-called Schwarzschild singularity where some metric tensor components vanished or blew up. Nevertheless, the metric determinant remained well behaved and the signature condition was nowhere violated. This was an indication that this so-called singularity was a peculiarity of the particular coordinate system, nevertheless, there were some awkward questions to be answered: (a) What does this peculiarity which has no analogue in Newtonian gravitation signify? (b) Is this singularity physically attainable? Bergmann in his book ‘Introduction to the theory of relativity’ referred to an unpublished work of Robertson on the penetrability of the ‘Schwarzschild singular¬ ity’ and also Einstein’s model of a rotating cluster in equilibrium. On the basis of his model, Einstein conjectured that matter could not be compactified to an extent sufficient for the appearance of the singularity. Bergmann’s concluding remark that the Schwarzschild singularity is only ‘partly singular’ intrigued me. ’Reprinted from Proceedings of the Symposium on the Fortieth Anniversary of the Ray- chaudhuri Equation (IUCAA, 1995). 3 A K Raychaudhuri Bergmann was apparently unaware of the work of Oppenheimer and Snyder on the collapse of a spherical dust distribution and so was I. Thus I took up the study of a collapsing homogeneous dust distribution, which could be imagined to be a portion of a contracting Friedmann universe, in an outside empty space. The problem was solved using a comoving coordinate system and provided a couter- example to Einstein’s conjecture. In view of the earlier work of Oppenheimer and Snyder, the result was not new but my paper correctly considered the boundary conditions which were overlooked by Oppenheimer and Snyder. This work led me to the ultimate occurrence of the collapse singularity - the singularity with which one is familiar with as the big bang origin of the universe. But the then prevailing ideas about the big bang were confusing. Einstein spec¬ ulated that this singularity signalled a failure of the general theory of relativity for high concentrations and intense fields and might be removed in a unified field theory where there will be no separation between field and matter. Others thought that the origin of the singularity lay in the assumptions of homogeneity and isotropy and in a more realistic picture, the singularity will disappear. However, Einstein did not succeed in building up such a unified field theory and Tolman and Omer investigated a non-homogeneous model only to find that the singularity persisted. True, the steady state theory did away with the singularity but a breakdown of the long cherished conservation principle of energy did not find favour with the average physicist.
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