General relativity

Key topics

Introduction to... Mathematical formulation of...

Fundamental concepts

Special relativity Equivalence principle World line · Riemannian geometry

Phenomena

Kepler problem · Lenses · Waves Frame-dragging · Geodetic effect Event horizon · Singularity Black hole An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Newton’s conception and quantification of gravitation held until the beginning of the 20th century, when Albert Einstein extended the special relativity to form the general relativity (GR) theory. ... Notational point: General relativity articles using tensors will use the abstract index notation . ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In general relativity, the Kepler problem involves solving for the motion of a particle of negligible mass in the external gravitational field of another body of mass M. This gravitational field is described by the Schwarzschild solution to the vacuum Einstein equations of general relativity, and particle motion is described... This article or section is in need of attention from an expert on the subject. ... For the concept in fluid dynamics and meteorology, see Gravity wave. ... According to Albert Einsteins theory of general relativity, space and time get pulled out of shape near a rotating body in a phenomenon referred to as frame-dragging. ... The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a spinning, moving body. ... For the science fiction film, see Event Horizon (film). ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ... Simulated view of a black hole in front of the Milky Way. ...

Equations

Linearized Gravity Post-Newtonian formalism Einstein field equations

Advanced theories

Kaluza-Klein Quantum gravity

Solutions

Schwarzschild Reissner-Nordström Kerr · Kerr-Newman Kasner · Milne · Robertson-Walker It has been suggested that Weak-field approximation be merged into this article or section. ... The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... This article or section is in need of attention from an expert on the subject. ... Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism. ... This article does not cite any references or sources. ... It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. ... In physics and astronomy, a Reissner-Nordström black hole, discovered by Gunnar Nordström and Hans Reissner, is a black hole that carries electric charge , no angular momentum, and mass . ... In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... The Kerr-Newman metric is a solution of Einsteins general relativity field equation that describes the spacetime geometry around a charged (), rotating () black hole of mass m. ... The Kasner metric is an exact solution to Einsteins theory of general relativity. ... Milnes model follows the description from special relativity of an observable universes spacetime diagram containing past and future light cones along with elsewhere in spacetime. ... // The Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity and which describes a homogeneous, isotropic expanding/contracting universe. ...

Scientists

Einstein · Minkowski · Eddington Lemaître · Schwarzschild Robertson · Kerr · Friedman Chandrasekhar · Hawking · others “Einstein” redirects here. ... Hermann Minkowski. ... One of Sir Arthur Stanley Eddingtons papers announced Einsteins theory of general relativity to the English-speaking world. ... Father Georges-Henri Lemaître (July 17, 1894 – June 20, 1966) was a Belgian Roman Catholic priest, honorary prelate, professor of physics and astronomer. ... Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... Howard Percy Robertson (January 27, 1903 - August 26, 1961) was a scientist known for contributions related to cosmology and the uncertainty principle. ... Roy Patrick Kerr (1934- ) is a New Zealand born mathematician who is best known for discovering the famous Kerr vacuum, an exact solution to the Einstein field equation of general relativity, which models the gravitational field outside an uncharged rotating massive object, or even a rotating black hole. ... Alexander Alexandrovich Friedman or Friedmann (Александр Александрович Фридман) (June 16, 1888 – September 16, 1925) was a Russian cosmologist and mathematician. ... Chandrasekhar redirects here. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... This is a partial list of persons who have made major contributions to the development of standard mainstream general relativity. ...

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Books

Popular

• Geroch, Robert (1981). General Relativity from A to B. Chicago: University of Chicago Press. ISBN 0-226-28864-1.  Leisurely pace, provides superb intuition for Schwarzschild geometry.

• Wald, Robert M. (1992). Space, Time, and Gravity: the Theory of the Big Bang and Black Holes. Chicago: University of Chicago Press. ISBN 0-226-87029-4.  Covers much more ground, while remaining concise and readable.

• Thorne, Kip S. (1995). Black Holes and Time Warps: Einstein's Outrageous Legacy. New York: W. W. Norton. ISBN 0-393-31276-3.  A delightful romp through the physics of black holes. Features many personal anecdotes from the author's distinguished career.

Robert Wald (b. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... Simulated view of a black hole in front of the Milky Way. ...

Textbooks

Beginning undergraduate level

• Rindler, Wolfgang (2001). Relativity: Special, General, and Cosmological. Oxford University Press. ISBN 0-19-850836-0.  This book is an expanded version of an earlier book by the same author, Essential Relativity, but covers more topics in greater detail. The physics of general relativity is developed with great care, followed by an introduction to the usual formalism and some more advanced topics.

• Schutz, Bernard F. (2003). Gravity from the ground up. Cambridge University Press. ISBN 0-521-45506-5.  A more elementary treatment than Rindler's, this book uses no more than a little algebra and trigonometry to explore Einstein's theory of gravity. A good book to develop an intuitive understanding of general relativity, underpinned by helpful back-of-the-envelope type calculations.

Wolfgang Rindler is a leading physicist working in the field of General Relativity where he is well known for introducing the term event horizon, rindler coordinates, and (in collaboration with Roger Penrose) for popularizing the use of spinors in general relativity. ...

Advanced undergraduate level

• Cheng, Ta-Pei (2005). Relativity, Gravitation and Cosmology: a Basic Introduction. Oxford and New York: Oxford University Press. ISBN 0198529570.  Full tensor formulation of GR is postponed till the last of the three parts of the book. Particularly suitable for an introductory GR course with an emphasis on cosmology.

• P.A.M.Dirac, General Theory of Relativity (1975), Princeton University Press, quick presentation of the bare essentials of GTR. ISBN 0-691-01146-X

• Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8662-9.  One feature of this textbook not found in its competitors is a nice discussion of accretion discs.

• Ohanian, Hans C. & Ruffini, Remo (1994). Gravitation and Spacetime (3rd ed.). New York: W. W. Norton. ISBN 0-393-96501-5.  In contrast to other introductions, these authors use an exceptionally clear comparison of linearized general relativity with electromagnetism to motivate Einstein's field equations. Superb treatment of observational tests and of gravitational lensing. Should be useful for students wishing to master the textbook by Weinberg.

• d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0-19-859686-3.  Readable, well illustrated, fairly comprehensive without becoming encyclopedic.

• Hughston, L. & Tod, K. P. (1991). Introduction to General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-33943-X.  Clearly written, short and sweet; covers less ground than the others but much cheaper.

• Schutz, Bernard F. (1985). A First Course in General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-27703-5.  Features an outstanding treatment of tensor calculus and the stress-energy tensor, a key topic which beginners often have trouble grasping. The treatment of linearized gravitational waves and stellar models is also outstanding.

James Hartle is an American physicist. ... Artists conception of a binary star system with one black hole and one main sequence star Unsolved problems in physics: Accretion disc jets: Why do the discs surrounding certain objects, such as the nuclei of active galaxies, emit radiation jets along their polar axes? These jets are invoked by... It has been suggested that Weak-field approximation be merged into this article or section. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... A gravitational lens is formed when the light from a very distant, bright source (such as a quasar) is bent around a massive object (such as a massive galaxy) between the source object and the observer. ... For more technical Wiki articles on tensors, see the section later in this article. ... This article is in need of attention from an expert on the subject. ...

Graduate level

• Carroll, Sean M. (2004). Spacetime and Geometry: An Introduction to General Relativity. San Francisco: Addison-Wesley. ISBN 0-8053-8732-3.  Readable, up-to-date book. Features an outstanding treatment of the mass, charge, and spin of isolated objects, plus an elementary introduction to quantum field theory on curved spacetimes and Hawking radiation. Further essential material is concisely explained in valuable appendices. Book website.

• Stephani, Hans (1990). General Relativity: An Introduction to the Theory of the Gravitational Field,. Cambridge: Cambridge University Press. ISBN 0-521-37941-5.  Clear and very well organized. Features excellent treatment of far-field and weak-field expansions and linearized gravitational waves, including multipole moments. Offers more on solution techniques than other introductory textbooks.

• Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87033-2.  Often cited as the definitive graduate level textbook. Features an outstanding introduction to tensors (with a clear distinction between abstract indices and particular indices, overlooked by most other authors), as well as the basic singularity, stability, and uniqueness theorems, quantum field theory on curved spacetimes, and black hole thermodynamics. Much valuable material is clearly explained in a series of superb appendices. In general, this book focuses more on developing insight into mathematical formalism and techniques than on developing physical insight.

• Landau, Lev D. & Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, ISBN 0-7506-2768-9 A unique textbook straddling the modern and pre-modern eras in general relativity, this offers a dual introduction to Maxwell's theory of electromagnetism and Einstein's theory of gravitation. Noteworthy topics include a good treatment of multipole moments and background material needed for the BKL conjecture.

• Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973), Gravitation, San Francisco: W. H. Freeman, ISBN 0-7167-0344-0 A classic general relativity textbook. Features a unique two-track organization, with numerous boxes, tables, figures, and citations. In general, this book focuses more on developing physical and geometrical intuition than the textbook by Wald. Generally regarded as the first modern textbook on general relativity.

• Weinberg, Steven (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: John Wiley & Sons. ISBN 0-471-92567-5.  Demanding but full of valuable physical insight and techniques. No pictures, in marked contrast to the textbook by Misner, Thorne & Wheeler. Excellent treatment of topics related to PPN formalism, weak field approximations, gravitons, as well as applications of particle physics to cosmology. No exercises.

Sean M. Carroll is a senior research associate in the Department of Physics at the California Institute of Technology. ... This article or section is in need of attention from an expert on the subject. ... In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... Quantum field theory (QFT) is the quantum theory of fields. ... In physics, Hawking radiation (also known as Bekenstein-Hawking radiation) is a thermal radiation thought to be emitted by black holes due to quantum effects. ... The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ... In physics, gravitational radiation is energy that is transmitted through waves in the gravitational field of space-time, according to Albert Einsteins theory of general relativity: The Einstein field equations imply that any accelerated mass radiates energy this way, in the same way as the Maxwell equations that any... Multipole moments in mathematics and mathematical physics are an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. ... Robert Wald (b. ... In physics and especially relativity, General Relativity is a popular textbook on Einsteins theory of general relativity written by Robert Wald. ... This article or section is in need of attention from an expert on the subject. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Evgeny Mikhailovich Lifshitz (Russian: ; February 21, 1915 – October 29, 1985) was a notable Soviet physicist. ... Charles W. Misner is one of the authors of Gravitation. Kip Thorne John Archibald Wheeler http://www. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... John Archibald Wheeler (born July 9, 1911) is an eminent American theoretical physicist. ... A book on gravitation (often considered the Bible by researchers for its prominence) by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Steven Weinberg (born May 3, 1933) is an American physicist. ...

Special topics

• Will, Clifford M. (1993). Theory and experiment in gravitational physics. Cambridge: Cambridge University Press. ISBN 0-521-43973-6.  A systematic introduction to tests of general relativity - from the theoretical underpinnings (Parameterized post-Newtonian formalism) to the actual observations. Compare the same author's article The Confrontation between General Relativity and Experiment on Living reviews in relativity.

• Poisson, Eric (2004). A Relativist's Toolkit: The Mathematics of Black Hole Mechanics. Cambridge: Cambridge University Press. ISBN 0-521-83091-5.  Don't be fooled by the subtitle; this book explains many key concepts and techniques which are needed by all contemporary graduate students, but are not adequately explained elsewhere. Essential topics covered here include congruences (expansion, vorticity, and shear), optical scalars, junction conditions for matching interior solutions to exterior solutions, thin shells (including null shells), spatial hyperslices, and energy conditions.

• Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard (2003). Exact Solutions to Einstein's Field Equations (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-46136-7.  This awesome monograph attempts to provide an up-to-date survey of many of the tens of thousands of known exact solutions, plus solution techniques and essential background such as Newman-Penrose formalism.

• Stewart, John (1993). Advanced General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-44946-4.  Not easy to read, but one of the few textbooks to offer an introduction to the important Newman/Penrose formalism. Also features much material on gravitational waves.

• De Felice, F.; and Clarke, C. J. (1992). Relativity on Curved Manifolds. Cambridge: Cambridge University Press. ISBN 0-521-42908-0.  This book is billed as an introductory textbook, but has no exercises and may be hard to read. Unique features include a chapter on measurement theory for general relativity, plus an introduction to tetrad formalism.

• Lightman, Alan P.; Press, William H.; Price, Richard H. & Teukolsky, Saul A. (1975), Problem Book in Relativity and Gravitation, Princeton: Princeton University Press, ISBN 0-691-08162-X A collection of excellent problems, with sketch solutions in the back. Test your skills!

• Hawking, Stephen & Ellis, G. F. R. (1973), The Large Scale Structure of Space-Time, Cambridge: Cambridge University Press, ISBN 0-521-09906-4 A classic and highly influentical monograph; features excellent motivation of the field equation and careful discussion of some important exact solutions, especially their causal or conformal structure.

The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. ... This article or section is in need of attention from an expert on the subject. ... The energy conditions refer to various constraints which can be imposed on a spacetime that any physically reasonable matter distributions in physics are expected to satisfy. ... The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose[1] for General Relativity. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... Alan Lightman is a physicist, novelist, and essayist born in Memphis, Tennessee in 1948, son of Richard Lightman, a movie theater owner, and Jeanne Garretson, a dancing teacher and volunteer Braille typist. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... George Ellis is the Distinguished Professor of Complex Systems at the University of Cape Town (South Africa), in the Department of Mathematics and Applied Mathematics. ...

External links

Popular

• Einstein Online. Website featuring articles on a variety of aspects of relativistic physics for a general audience, hosted by the Max Planck Institute for Gravitational Physics
• NCSA Spacetime Wrinkles. Website produced by the numerical relativity group at the NCSA, featuring an elementary introduction to general relativity, black holes and gravitational waves

The Max Planck Institute for Gravitational Physics (Albert Einstein Institute) is a Max Planck Institute whose research is aimed at investigating Einstein’s theory of relativity and beyond: Mathematics, quantum gravity, astrophysical relativity, and gravitational wave astronomy. ... This article is in need of attention from an expert on the subject. ... National Center for Supercomputing Applications NCSA Building, 1205 W. Clark St. ... Simulated view of a black hole in front of the Milky Way. ... For the concept in fluid dynamics and meteorology, see Gravity wave. ...

Online tutorials

• Baez, John & Bunn, Ted; The Meaning of Einstein's Equation. Retrieved on January 4, 2006. This superb expository paper explains the meaning of the field equation in terms of the motion of a cloud of free falling test particles.
• Baez, John; The General Relativity Tutorial.
• Carroll, Sean M.; A No-Nonsense Introduction to General Relativity. Retrieved on Nov 26, 2006. A concise but very readable overview.

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Webcourses

• Rappoport, Saul. Relativity (Fall 2003). MIT OpenCourseWare. Retrieved on May 29, 2005. An elementary introduction to relativistic physics, including a smattering of gtr.

• Brown, Kevin. Reflections on relativity. Mathpages.com. Retrieved on May 29, 2005. An online book providing extensive discussion of various aspects of special and general relativity.

• Carroll, Sean M.. Lecture Notes on General Relativity. Retrieved on Nov 26, 2006. Extensive (227 pages) lecture notes for an introductory course on general relativity. Readable, largely self-contained material.

• Waner, Stefan. Introduction to Differential Geometry and General Relativity. Retrieved on Jan 31, 2006. Another set of readable, largely self-contained lecture notes.

• Moor, Rafi. Understanding General Relativity. Retrieved on Jul 11, 2006. - The theory of general relativity in an easily understandable way.

May 29 is the 149th day of the year (150th in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... May 29 is the 149th day of the year (150th in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... November 26 is the 330th day (331st on leap years) of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... January 31 is the 31st day of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... July 11 is the 192nd day (193rd in leap years) of the year in the Gregorian Calendar, with 173 days remaining. ... For the Manfred Mann album, see 2006 (album). ...

Collection of review articles

• Living Reviews in Relativity. Electronic journal which features about 50 review articles (and counting) about various subjects of relativistic physics, which are regularly updated by their authors.

Special topics

• John D Norton;
  What was Einstein's principle of equivalence? [PDF-file, 376 KB] Discussion of the difference between what Einstein introduced and the 'infinitisimal principle of equivalence' (which Einstein objected against) that is encountered in virtually all textbooks.

• John D Norton;
  General Covariance and the Foundations of General Relativity: Eight Decades of Dispute (PDF-file, 460 KB) Discussion of the question whether the General Covariance of GTR is physically significant. John Norton documents that on this issue there is fundamental dissent in the physics community.

• Michel Janssen;
  Einstein's first systematic exposition of general relativity (PDF-file, 187 KB) Discussion of the development of Einstein's views on the foundations of general relativity, such as Einstein's embracing of Mach's principle during development of general relativity, (explicit introduction in 1918), and why Einstein abandoned Mach's principle in the early twenties.

Reading lists

• A Guide to Books on Relativity Theory. Relativity on the World Wide Web. Retrieved on May 29, 2005. An idiosyncratic website providing a much more extensive reading list, on which this page is loosely based.
• Bibliography. The Meaning of Einstein's Equation. Retrieved on January 4, 2006. By John Baez and Emory F. Bunn.