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Astrofísica y Relatividad


Scientific American Bookmarks Summary: As longtime readers of Scientific American probably know, discoveries made over the past century have revolutionized the study of the cosmos. Indeed, as our January special report reveals, the universe seems to get stranger and stranger with each new finding. But if all this talk about the past, present and future of our universe has left you out in the cold, you might want to check out the Cosmology Tutorial created by Edward L. Wright
Scientific American - Cosmology Introducción a al cosmología - Publicaciones de scientific Am. sobre el tema.
Cosmology Tutorial Ned Wright's Cosmology Tutorial UCLA- My cosmology tutorial is an attempt to summarize these discoveries. It will be "under construction" for the foreseeable future as new discoveries are made. I will attempt to keep these pages up-to-date as a resource for the cosmology courses I teach at UCLA
Relativity tutorial Ned Wright's Relativity Tutorial UCLA
Relativity on the World Wide Web Numerosos vinculos a sitios sobre Relatividad general y Cosmología - En Ingles


Graduate Level Course Notes and Tutorial Papers Related to 

General Relativity

Original by Chris Hillmam (Last modified by Chris Hillman, 24 Jan 2001) 

Relativity on the World Wide Web


To really appreciate the beauty and subtleties of general relativity, you must grapple with the mathematics, which lies, unfortunately, just beyond the undergraduate level. Here are some fine graduate level course notes, tutorial papers, and review papers on the following topics: Some of these resources are written at the beginning graduate level; others are more specialized papers and these tend to be more demanding.

Important Note! I don't have a secretary to help me maintain these pages, and to keep the work manageable, I have only attempted to list here some representative review papers, not any individual research papers, however important these might be. I hope the resources gathered here will help graduate students in any area of physics get some idea of the wealth of current research in gtr and closely related areas, and I'd also like to try do what little I can to ``reward'' those experts who have taken the trouble to try to write review papers of their special areas of expertise.


Basic General Relativity

  • A Short Course on GR, by William L. Burke, (Physics, UC Santa Cruz). Topics covered include weak field theory, gravitational waves, radiation damping, cosmology, the Friedmann and Lemaitre dusts, singularities, black holes, the Schwarzschild metric and Kruskal's extension of it. There is an appendix on mathematical notation. This is a single postscript document (about 75 pages).
  • Lecture Notes on General Relativity by Matthias Blau (ICTP, Trieste). A very readable and complete set of course notes, written for advanced undergraduates but also useful to graduate students, particularly since they cover some topics which are slighted in other notes. Topics covered include the equivalence principle, gravitational redshift, tensor analysis, covariant derivatives, Lie derivatives, Killing vectors and conservation laws, geodesics and effective potential analysis, the Riemann, Weyl, Ricci, and Einstein curvature tensors, intrinsic versus extrinsic geometry, the Bianchi identities, the Jacobi equation for geodesic deviation, the principle of minimal coupling, the matter tensor and the Einstein field equation, variational principles, a thorough study of the Schwarzschild solution, weak field theory and gravitational waves (including detector theory), and cosmology (vacuum, radiation, and matter dominated FRW models). The lecture notes end with a very thorough discussion of the Kaluza-Klein theory and and introduction to the notion of a non-abelian gauge theory. This is a single postscript document (about 180 pages).
  • Lecture Notes on General Relativity, by Sean M. Carroll (Physics, University of Chicago). From a course taught at MIT. Topics covered include str, manifolds, covariant derivatives, connections, curvature, Lie derivatives, pullbacks, Killing vectors, the Equivalence Principle, the matter tensor, the field equation of gtr (Einstein's equation), the initial value and variational principle formulations of the field equation, weak field theory, gravitational waves, a complete discussion of the Schwarzschild solution, cosmology and the Friedmann solutions. Carroll's careful discussion of the geometry of the Kerr solution is particularly noteworthy. The lectures are available as either html or postscript documents (about 200 pages total).
  • General Relativity, by Petr Hadrava, (Astronomical Institute, Academy of Sciences of the Czech Republic). Lecture notes (in English) on str and gtr. Topics include the Equivalence Principle, the field equations, weak-field theory, the Schwarzschild exterior (vacuum) and interior (stellar "fluid") solutions, the Friedmann cosmological solutions. Two mathematical appendices sketch the mathematics of tensor algebra, exterior algebra, connection, Lie derivatives, Killing vectors, and variational principles. This is a single postscript document (50 pages).
  • Introduction to General Relativity, by Gerard 't Hooft (Institute for Theoretical Physics, Utrecht University). Lecture notes (in English) on gtr. Written for advanced undergraduates, these notes work through the basics in careful detail, but they would also be good for graduate students. 't Hooft won the Nobel prize in physics for his work on quantum field theory, so it is interesting to see his viewpoint on general relativity. This is a single postscript document (69 pages).


White Dwarfs and Neutron Stars

  • Rotating Stars in Relativity , by Nikolaos Stergioulas (Physics, University of Wisconsin-Milwaukee) offers a very nice introduction to models of rotating stars in hydrostatic equilibrium, as treated in gtr. Stergioulas also gives an introduction to the important topic of CFS instabilities, a phenomenom by which certain types of perturbations in a rotating relativistic star in which frame dragging is significant (e.g. a rapidly rotating neutron star) can actually be "pumped up" rather than "damped" by the emission of gravitational radiation. ("CFS" stands for Chandrasekhar, Friedmann, and Schutz, after the researchers who first established the existence of this intriguing phenomenom.) See also the review paper by Kokkotas and Schmidt listed below, for more information about quasinormal modes in the perturbations of rotating relativistic stars. This is an invited paper in the Living Reviews series.
  • Gravitational-Wave Driven Instability of Rotating Relativistic Stars , by John L. Friedman (Physics, University of Wisconsin-Milwaukee) and Keith H. Lockitch (Physics, Penn. State) offers a brief review to r-mode instabilities, the most frequently studied type of CFS instability.
  • The Properties of Matter in White Dwarfs and Neutron Stars , by Shmuel Balberg and Stuart L. Shapiro (Physics, University of Illinois at Urbana-Champaign). This readable survey provides an introduction to what is currently known about the physical properties of condensed matter at the extreme densities found in white dwarfs and neutron stars, and examines how well existing theories agree with observations.
  • Recent Progress in Neutron Star Theory , by H. Heiselberg (NORDITA) and V. Pandharipande (Physics, University of Illinois at Urbana-Champaign). This is a more advanced survey of the theory of neutron stars, featuring a -huge- bibliography!
  • Superfluidity in Relativistic Neutron Stars , by David Langlois (DARC, Meudon & IAP, Paris) is concise survey of one of the most mysterious aspects of neutron star interiors: there must be microscopic tubes of magnetic flux threading the superfluid interior.
  • Accretion Processes Around Black Holes And Neutron Stars: Advective Disk Paradigm , by Sandip K. Chakrabarti (S. N. Bose National Center for Basic Sciences, Calcutta) offers a brief introduction to the physics of accretion disks, as treated by the most popular model (advection dominated flow).


Gravitational Collapse

  • Gravitational Collapse and Cosmic Censorship , by Robert M. Wald (Physics, University of Chicago) offers a review of the theoretical status of the weak cosmic censorship hypothesis.
  • Gravitational Collapse , by P. S. Joshi (Tata Insitute of Fundamental Research, Bombay) offers a gentle introduction to some of the most commonly studied models of gravitationcal collapse, including
    • the OS (Oppenheimer-Snyder) model of an imploding spherical dust ball,
    • the VP (Vaidya-Papapetrou) model of imploding spherical shells massless radiation,
    • the LTB(Lemaire-Tolman-Bondi) models of inhomogeneous collapsing dust balls,
    • numerical and theoretical studies of collapsing fluid drops,
    • numerical and theoretical studies of collapsing scalar fields.
    Unfortunately, colliding gravitational waves are not considered. See however the review paper by Bicak listed below for more information about exact solutions modeling the collision of gravitational plane waves. Joshi's paper appeared in Singularities, Black Holes and Cosmic Censorship (On the fortieth anniversary of the Raychaudhuri Equation), IUCAA publication, Pune, India.
  • Critical Phenomena in Gravitational Collapse, by Patrick R Brady and Mike J Cai offers a concise and very readable introduction to this important and very surprising recent discovery.
  • Critical Phenomena in Gravitational Collapse, by C. Gundlach (Enrico Fermi Institute, University of Chicago), a very clear invited review paper in the Living Reviews series. It is more detailed than the preceeding paper. (Note: see the review paper by Coley listed below for another way in which modern dynamical systems theory is useful in gtr.)


Classical Black Holes

  • Black Holes, by Paul Townsend (Applied Mechanics and Theoretical Physics, Cambridge). A very thorough introduction, studies the Schwarzschild, Reissner-Nordstrom, and Kerr solutions using a variety of coordinate systems. Additional topics include gravitational collapse, horizons, singularities, Carter-Penrose diagrams (aka conformal compactification), Hawking radiation and black hole thermodynamics. This is a 145 page postscript document.
  • Scattering by Black Holes, by N. Andersson and B.P. Jensen (Mathematics, Univerity of Southampton). This excellent survey is one chapter in the Encyclopedia on Scattering, which will be published by Academic Press. The paper surveys wave propagation in black-hole spacetimes, diffraction effects in wave scattering (including "light bending"), resonances, and quasinormal modes, among other topics.
  • Stringy Black Holes, by Martijn Derix and Jan Pieter van der Schaar (Institute for Theoretical Physics, Rijksuniversiteit, Groningen, The Netherlands). An extensive set of htmlified lecture notes. Includes a review of Schwarzschild and Reissner-Nordstrom holes, elecric-magnetic duality, dilaton holes, axion holes, and much more.


Gravitational Waves

  • The Gravitational Wave Symphony of the Universe, by B.S. Sathyaprakash (Physics and Astronomy, Cardiff) offers a concise overview of the theory of astrophysical sources of gravitational radiation and the theory of laser interferometric detectors..
  • Gravitational Radiation, by Bernard F Schutz (Max Planck Institute for Gravitational Physics, Potsdam), offers a very nice introduction to this subject. This is an article from the Encycopedia of Astronomy and Astrophysics.
  • Gravitational Wave Astronomy, by Bernard F Schutz (Max Planck Institute for Gravitational Physics, Potsdam). An readable and well balanced overview of the theory of the generation and detection of gravitational waves. Appeared in Class.Quant.Grav. 16 (1999) A131-A156.
  • Gravitational Radiation Sources and Signatures, by Lee Samuel Finn (Center for Gravitational Physics and Geometry, Penn State). An excellent introduction to the theory of gravitational wave generation by astrophysical sources, the theory of interferometer detectors such as LIGO, and a survey of possible signals.
  • Gravitational Radiation from Relativistic Sources , by Luc Blanchet (CNRS, France). Another very thorough introduction to the theory of gravitational wave generation by astrophysical sources, more challenging than the previous one.
  • Gravitational Radiation Theory and Light Propagation , by Luc Blanchet (CNRS, France), Sergei Kopeikin (Physics & Astronomy, University of Missouri-Columbia) and Gerhard Shaefer (Theoretical Physics Institute, Friedrich-Schiller University, Jena, Germany) offers an introduction to the propagation of light rays in gravitational fields, including post-Newtonian effects beyond the quadrupole approximation, such as higher order multipoles, spin-spin interactions, and back-reaction. Appeared in the book Gyros, Clocks, and Interferometers: Testing Relativistic Gravity in Space, ed. C. Laemmerzahl, C.W.F. Everitt, F.W. Hehl, Springer-Verlag, 2000.
  • Probing Black Holes and Relativistic Stars with Gravitational Waves, by Kip Thorne (Theoretical Astrophysics, Cal Tech). Focuses on the theory of the LIGO interferometers and similar detectors, and gives a survey of what kinds of signals astrophysicists expect to ``hear'' with this instruments. This paper appeared in the book Black Holes and Relativistic Stars: Proceedings of a Conference in Memory of S. Chandrasekhar, ed. R. M. Wald, University of Chicago Press, 1999. See also the same author's earlier survey.
  • Gravitational Wave Experiments and Early-Universe Cosmology, by Michele Maggiore. This is a review of possible signatures of the so called ``relic waves'' from the very early universe, and the possibilities for detecting them using LIGO and other interferometers. Appeared in Phys.Rept. 331 (2000) 283-367.


Tests of General Relativity

  • Gravitation and Experiment, by Thibault Damour (IHES, DARC) offers a concise overview of this gigantic subject. Appeared in Proceedings of Princeton's 250th Anniversary Conference on Critical Problems in Physics (October 31-November 2, 1996), Princeton University Press, 1997. See also this updated review by the same author, which appears in the year 2000 edition of the Review of Particle Physics.
  • The Confrontation between General Relativity and Experiment: A 2001 Update, by Clifford M. Will (Physics, Washington University, St. Louis) offers a more extensive survey, including the classical solar system tests and binary pulsar data, together with the prospects for direct detection of gravitational waves.
  • Binary-pulsar tests of strong-field gravity, by Gilles Esposito-Farese (Theoretical Physics, CNRS) gives more detail concerning the very stringent tests posed by binary-pulsar timing (and passed by gtr!), including the work of Taylor and Hulse on PSR1913+16.
  • Astrophysical Evidence for the Existence of Black Holes by Annalisa Celotti, John C. Miller, and Dennis W. Sciama (SISSA, Trieste, Italy), offers a short history of how the existence of astrophysical black holes came to be essentially universally accepted by astronomers. The paper focuses on the rather different types of evidence for the two best known classes of astrophysical black holes: supermassive black holes and solar mass black holes. Appeared in the millenium issue of Class.Quant.Grav. 16 No 12A (December 1999), A3.
  • Supermassive Black Holes in Active Galactic Nuclei, by John Kormendy (Univ. Texas at Austin) and Luis C. Ho (Carnegie Observatories). In the past few decades it has become universally accepted that supermassive black holes provide the ``engine'' powering active galatic nuclei (AGN's) such as quasars and Seyfert galaxies. This review article (to appear in The Encyclopedia of Astronomy and Astrophysics (Institute of Physics Publishing), discusses the currently available stellar dynamical evidence for supermassive black holes living at the core of AGN's.


Thermodynamics of Black Holes

I have listed here some expository papers on the laws of black holes mechanics and their reformulation in terms of classical mechanics, using Hawking's astonishing discovery that black holes radiate with a black body spectrum and thus have a well defined temperature like any other black body (which is of course classical a perfect absorber, just like a black hole).

  • The Thermodynamics of Black Holes, by Robert M. Wald. An up-to-date review by the leading expert in this field.
  • Introductory Lectures on Black Hole Thermodynamics, by Ted Jacobson (Insitute for Theoretical Physics, University of Utrecht). Another up-to-date review by the phsyicist who has made some progress toward deriving the Einstein field equation from the laws of black hole thermodynamics, rather than the other way around.
  • An Introduction to Black Hole Evaporation, by Jennie Traschen (Physics, University of Massachusetts at Amherst). This very readable paper offers a tutorial in the actual computation of Unruh, Hawking, Gibbons and other radiation, and examines the end states of evaporation in de Sitter and AdS backgrounds, in the case of charged and rotating holes. The paper has appeared in print in Mathematical Methods of Physics, proceedings of the 1999 Londrina Winter School, ed. by A. Bytsenko and F. Williams, World Scientific, 2000.


Further Special Topics in General Relativity

  • Selected Solutions of Einstein's Field Equations: Their Role in General Relativity and Astrophysics, by Jiri Bicak (Institute of Theoretical Physics, Charles University, Prague). This is a comprehensive introduction to the most important exact solutions in gtr, including the Minkowksi, de Sitter, anti-de Sitter background vacuums, the Schwarzschild solution and charged and rotating generalizations, the Taub-Nut solution, plane waves, colliding plane waves, cylindrical waves, and cosmological models. This is an invited paper from a new book, Einstein Field Equations and Their Physical Implications, edited by Berndt Schmidt, Springer, 2000. (126 pages).
  • The Cauchy Problem for the Einstein Equations , by H. Friedrich and A. D. Rendall (Max Planck Institue for Gravitational Physics, Potsdam). This is a comprehensive introduction to a fundamental but technically rather involved topic in general relativity. This is a single postscript document (98 pages). This paper appeared in print in the book Einstein's Field Equations and their Physical Interpretation, (ed. B. G. Schmidt, Springer-Verlag, 2000.
  • Quasi-Normal Modes of Stars and Black Holes, by Kostas D. Kokkotas and Bernd G. Schmidt. In Newtonian models of spherical stars, perturbations (e.g. due to a small lump) can be decomposed into a sum over normal modes, rather like a multidimensional Fourier series. In gtr, such vibrations are slowly damped out (or sometimes -pumped up-!!) by the emission of gravitational radiation, so they are called quasinormal modes. This important topic is the subject of this invited review paper in the Living Reviews series.
  • Reflections on Gravity, by Norbert Straumann (Institute for Theoretical Physics, University of Zurich). This brief paper offers a nice sketch of an approach to deriving the EFE which was advocated by Feynman. The basic idea is to start with Newtonian gravitostatics, considered to consist of the Poisson equation on -Minkowksi spacetime-, and then try to follow the model of how one passes from electrostatics to Maxwell's theory of electrodynamics (which is Lorentz covariant) and then to quantum electrodynamics, fixing up the approach as needed. In particular, it turns out that one must introduce back reaction of the gravitational field on matter, which leads a kind of infinite series of approximations, which was cleverly "summed" by Deser. The end result is the EFE! However, the original metric of flat spacetime turns out to be unobservable and the original hypothesis of Lorentz covariance becomes moot! Caution!: Straumann inexplicably fails to mention the fact that the approach he is discussing only yields a "local mimic" of gtr; unless one carries the "geometrization" one step further by interpreting the quantum fields as existing on one of many coordinate charts, one excludes all the solutions to the EFE which have nontrivial topology. The following paper covers the ideas Straumann is discussing from a somewhat different persepective (among many other topics).
  • Actions for Gravity, with Generalizations: A Review, by Peter Peldan (Institute of Theoretical Physics, Chalmers Technical Univerisity and Goetteborg University, Sweden) offers a very concise but comprehensive (as of 1993) review of Lagrangian and Hamiltonian formulations of general relativity, including Askhetar's "new variables" Hamiltonian formalism. This is a 61 page postscript document--- the diagram relating the various formalisms is alone worth the price of admission. A word of advice: prospective readers should first study the formulation of classical mechanics in terms of Lagrangians and Hamiltonians first, and then study the appendix in Wald carefully and make sure they understand the distinction between tensors and tensor densities. Without this background, the reader will soon get lost. This paper appeared in print in the premier journal in gravitation physics: Class.Quant.Grav. 11 (1994) 1087.
  • Topological Censorship, by Kristin Schleich and Donald M. Witt (Physics, University of British Columbia, Vancouver) is an expository paper on an important theorem, the Topological Censorship Theorem (proven by the authors and John Friedman) which says in essence that any nontrivial topology of an isolated system such as an individual black hole (possibly with exotic fields, aka "hair") cannot be observed by distant observers. The theorem assumes that the spacetime is globally hyperbolic (see the preceding paper for the definition!) and that the null energy condition holds. Note well: the first assumption fails for some very important exact solutions in gtr, e.g. plane waves, and the second assumption fails for "traversable wormholes" and for at least some solutions in many proposed classical field theories involving scalar fields. Nevertheless, this theorem is an important and fairly general result. This paper appeared in print in Proceedings of the Lake Louise Winter Institute, Particle Physics and Cosmology, Feb. 20-26, 1994, (World Scientific, 1994).
  • New properties of Cauchy and event horizons, by Robert Budzynski (Physics, Univ. of Warsaw), Witold Kondracki and Andrzej Krolak (Institute of Mathematics, Polish Academy of Sciences) discusses the state of our knowledge concerning the smoothness properties of Cauchy and event horizons.
  • Some Recent Progress in Classical General Relativity, by Felix Finster, Joel Smoller, and Shing-Tung Yau. This is an introduction to the positive mass theorem and related results in the global analysis of generic solutions in gtr. The third author won a Fields Medal (the highest award in mathematics) in part for his role in proving the Positive Mass Theorem. This appeared in print: J.Math.Phys. 41 (2000) 3943-3963.
  • Boost-Rotation Symmetric Spacetimes - Review, by V. Pravda and A. Pravdova (Mathematical Institute, Academy of Sciences, Prague) offers a readable introduction to some of the most interesting exact solutions in gtr, including accelerating pairs of black holes and many more examples. This paper appeared in print: Czech.J.Phys. 50 (2000) 333-376.


Quantum Gravity and String/M-Brane Theory

  • Quantum Gravity at the Turn of the Millennium, by Gary Horowitz (Physics, UC Santa Barbara), offers a concise and very readable (but of course very sketchy!) overview of the state of the art in the search for a quantum theory of gravity.
  • Here are three very readable expository papers by Carlo Rovelli (Physics, University of Pittsburgh), one of the leading researchers on quantum gravity:


    As Rovelli points out in one of these papers, this is a very rapidly developing and extremely active field of research, and his viewpoint should not be regarded as being the only one reasonable one, or as the definitive account of what quantum gravity is or should be and how it may be in the process of becoming a reality.

  • Spacetime and the Philosophical Challenge of Quantum Gravity, by J.Butterfield (Oxford) and C.J.Isham (Blackett Laboratory, Imperial College, London) offers a thoughtful discussion of conceptual problems which arise various approaches to quantum gravity and how these might be overcome.
  • Are We at the Dawn of Quantum-Gravity Phenomenology?, by Giovanni Amelino-Camelia (CERN) reviews recent progress in quantum gravity and argues that, contrary to a long-held viewpoint, theories of quantum gravity may be testable.
  • String/M-branes for Relativists, by Donald Marolf (Physics, Syracuse University). This offers a tutorial in string theory and M-brane theory for gtr students.
  • Large N Field Theories, String Theory and Gravity, by O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz. This huge review paper covers the holographic conjecture, connections with conformal field theories and the anti-de-Sitter vacuum, and discusses some possible implications for the theory of black holes. (261 page postscript document, with figures).
  • Superstring Cosmology, by James E. Lidsey (Astronomy, Queen Mary & Westfield, London), David Wands (Computer Science and Mathematics, Portsmouth), and E. J. Copeland (Center for Theoretical Physics, Suffex). This review stresses the role of duality symmetries in superstring theory and their cosmological implications.


Astrophysical Background

  • Advanced Astrophysics, by Neb Duric (Physics and Astronomy, University of New Mexico). A full length on-line set of course notes. Covers the virial theorem, galactic rotation curves, galactic formation, galactic clusters and other large scale structures, dark matter, the Hubble expansion, applications of thermodynamics and statistical mechanics to astrophysics and cosmology, COBE and the CMBR, astrophysics of planets, white dwarfs, supernovaes, X-ray binaries, neutron stars, and black holes, accretion, the Eddington limit, Reaction Rates and Equilibria in Astrophysics Planck, Boltzmann and Saha equations, cosmological nucleosynthesis, solar neutrion problem, EM radiation, magento, relativistic and thermal bremmsstrahlung, inverse Compton emisssion, and more.
  • Nucleosynthesis Basics and Applications to Supernovae, by F.-K. Thielemann, T. Rauscher , C. Freiburghaus (Physik und Astronomie, Universitaet Basel), K. Nomoto, M. Hashimoto (Institute for Theoretical Physics, UC Santa Barbara), B. Pfeiffer and K.-L. Kratz (Institut fuer Kernchemie, Universitaet Mainz). This is a 52 page introduction to the basic equations for thermonuclear reaction rates and nuclear reaction networks, and applications of this theory to nucleosynthesis of heavy elements in aging stars.
  • Topics in Neutrino Astrophysics, by W. C. Haxton (Physics, University of Washington, Seattle), offers a 56 page introduction to solar neutrino problem and its implications.
  • A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects by Fred C. Adams and Gregory Laughlin (Physics, Univ. of Michigan). This is a 57 page postscript document, which appeared in Rev.Mod.Phys. 69 (1997) 337-372.


Mathematical Background

  • Differential Geometry, by Sergei Yakovenko (Weizmann Institute). A complete set of lecture notes. Topics include manifolds, diffeomorphisms, partitions of unity, the Whitney embedding theorem, tangent bundle, algebra of vector fields, Lie derivatives, commutators, points as maximal ideals, derivations, local rings, differentiable forms, etc.
  • The Theory of Caustics and Wavefront Singularities with Physical Applications, by Juergen Ehlers (Max Planck Institute for Gravitation Physics, Potsdam) and Ezra T. Newman (Physics and Astronomy, Pittsburgh). This is a tutorial paper on the important work of V. I. Arnold and his collaborators, which is directly applicable to geometric optics (including the propagation of gravitational waves) in gtr.
  • Noncommutative Geometry for Pedestrians, by J. Madore (Universite de Paris Sud and Max Planck Institute). As part of his far-reaching programme of re-expressing most of physics and a huge chunk of modern mathematics in terms of ``non-commutative geometry'', Alain Connes (a Fields's Medalist) has recently partially reformulated the EFE in terms of operator theory. This tutorial paper attempts to explain the background for this work.



  • Riemannian Geometry and General Relativity, the problem sets (with solutions) from a course taught by Michael Shubin (Mathematics, Northeastern).
  • Exercises in General Relativity, from courses taught at the DEA of Theoretical Physics (Ecole Normale Supérieure de Paris) and at the University of Geneva (1996-1999) by Jean-Philippe Uzan (Theoretical Physics, University of Geneva, Switzerland).

I hope you'll inspired by the on-line resources listed on this page to go off-line and do some outside reading! Here is a list of recommended books at a variety of levels. And be sure to get hold of some symbolic tensor manipulation software--- this will make your life a lot easier!



Cosmología  -  Cosmology


Numerical Relativity



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