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General relativity (GR) or general relativity theory (GRT) is a fundamental physical theory of gravitation which corrects and extends Newtonian gravitation , especially at the macroscopic level of stars or planets.
General relativity may be regarded as an extension of special relativity, this latter theory correcting Newtonian mechanics at high velocities. General relativity has a unique role amongst physical theories in the sense that it interprets the gravitational field as as a geometric phenomena. More specifically, it assumes that any object possessing mass curves the 'space' in which it exists, this curvature being equated to gravity. It deals with the motion of bodies in such 'curved spaces' and has survived every experimental test performed on it since its formulation by Albert Einstein in 1915.
General relativity forms the basis for modern studies in fields such as astronomy, cosmology and astrophysics. It describes with great accuracy and precision many phenomena where classical physics fails, such as the perihelion motion of planets (classical physics cannot fully account for the perihelion shift of Mercury, for example), the bending of starlight by the Sun (again, classical physics can only account for half the experimentally observed bending), the existence of black holes and the expansion of the universe. In fact, even Einstein himself initially believed that the universe cannot be expanding, but experimental observations of distant galaxies by Edwin Hubble finally forced Einstein to concede.
Unlike the other revolutionary physical theory, quantum mechanics, general relativity was essentially formulated by one man - Albert Einstein. However, Einstein required the help of one of his friends, Marcel Grossmann, to help him with the mathematics of general relativity.
Physical Description of the Theory
In relativity theory, physical phenomena are described by observers in reference frames making measurements. In general relativity, these reference frames are arbitrarily moving relative to each other (unlike in special relativity, where the reference frames are assumed to be inertial). Consider two such reference frames, for example, one situated on Earth (the 'Earth-frame'), and another in orbit around the Earth (the 'orbit-frame'). An observer (O) in the orbit-frame will feel weightless as they 'fall' towards the Earth. In Newtonian gravitation, O's motion is explained by the action at a distance formulation of gravity, where it is assumed that a force between the Earth and O causes O to move around the Earth. General relativity views the situation in a different manner, namely, by demonstrating that the Earth modifies ('warps') the geometry in its vicinity and O will naturally follow the curves (geodesics) in this geometry unless he applies accelerative force (e.g. rockets). More precisely, the presence of matter determines the geometry of spacetime, the physical arena in which all events take place. This is a profound innovation in physics, all other physical theories assuming the structure of the spacetime in advance. It is important to note that, given a matter distribution, the spacetime is fixed once and for all. There are a couple of caveats here: (1) the spacetime within which the matter is distributed can't be properly defined without the matter, so most solutions require special assumptions, such as symmetries, to allow the relativist to concoct a candidate spacetime, then see where the matter must lie, then require its properties be "reasonable" and so on. (2) Initial and boundary conditions can also be a problem, so that gravitational waves may violate the idea of the spacetime being "fixed once and for all." The motion of the observer "O" in orbit is rather like a ping-pong ball being forced to follow the 'dent' or depression created in a trampoline by a relatively massive object like a medicine ball. The geometry is determined by the medicine ball, the relatively light ping-pong ball causing no significant change in the local geometry. Thus, general relativity provides a simpler and more natural description of gravity than Newton's action at a distance formulation. An oft-quoted analogy used in visualising spacetime curvature is to imagine a universe of one-dimensional beings living in one dimension of space and one dimension of time. Each piece of matter is not a point on any imaginable curved surface, but a world line showing where that point moves as it goes from the past to the future. The precise means of calculating the geometry of spacetime given the matter distribution is encapsulated in Einstein's field equation.
The Principle of Equivalence
We distinguish inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, from non-inertial frames in which freely moving bodies have an acceleration deriving from the reference frame itself. In non-inertial frames there is a perceived force which is accounted for by the acceleration of the frame, not by the direct influence of other matter. Thus we feel acceleration when cornering on the roads when we use a car as the physical base of our reference frame. Similarly there are coriolis and centrifugal forces when we define reference frames based on rotating matter (such as the Earth or a child's roundabout). In Newtonian mechanics, the coriolis and centrifugal forces are regarded as non-physical ones, arising from the use of a rotating reference frame. In General Relativity there is no way, locally, to define these "forces" as distinct from those arising through the use of any non-inertial reference frame. The principle of equivalence in general relativity states that there is no local experiment to distinguish non-rotating free fall in a gravitational field from uniform motion in the absence of a gravitational field. In short there is no gravity in a reference frame in free fall. From this perspective the observed gravity at the surface of the Earth is the force observed in a reference frame defined from matter at the surface which is not free, but is acted on from below by the matter within the Earth, and is analogous to the acceleration felt in a car.
General relativity's mathematical foundations go back to the axioms of Euclidean geometry and the many attempts over the centuries to prove Euclid's fifth postulate, that parallel lines remain always equidistant, culminating with the realisation by Lobachevsky, Bolyai and Gauss that this postulate need not be true. It is an eternal monument to Euclid's genius that he classified this principle as a postulate and not as an axiom. The general mathematics of non-Euclidean geometries was developed by Gauss' student, Riemann, but these were thought to be mostly inapplicable to the real world until Einstein developed his theory of relativity. The existing applications were restricted to the geometry of curved surfaces in Euclidean space, as if one lived and moved in such a surface, and to the mechanics of deformable bodies. While such applications seem trivial compared to the calculations in the four dimensional spacetimes of general relativity, they provided a minimal development and test environment for some of the equations.
Gauss had realised that there is no a priori reason that the geometry of space should be Euclidean. What this means is that if a physicist holds up a stick, and a cartographer stands some distance away and measures its length by a triangulation technique based on Euclidean geometry, then he is not guaranteed to get the same answer as if the physicist brings the stick to him and he measures its length directly. Of course, for a stick he could not in practice measure the difference between the two measurements, but there are equivalent measurements which do detect the non-Euclidean geometry of space-time directly; for example the Pound-Rebka experiment (1959) detected the change in wavelength of light from a cobalt source rising 22.5 meters against gravity in a shaft in the Jefferson Physical Laboratory at Harvard, and the rate of atomic clocks in GPS satellites orbiting the Earth has to be corrected for the effect of gravity.
Newton's theory of gravity had assumed that objects had absolute velocities: that some things really were at rest while others really were in motion. He realized, and made clear, that there was no way these absolutes could be measured. All the measurements one can make provide only velocities relative to one's own velocity (positions relative to one's own position, and so forth), and all the laws of mechanics would appear to operate identically no matter how one was moving. Newton believed, however, that the theory could not be made sense of without presupposing that there are absolute values, even if they cannot be determined. In fact, Newtonian mechanics can be made to work without this assumption: the outcome is rather innocuous, and should not be confused with Einstein's relativity which further requires the constancy of the speed of light.
Physical assumptions of GR
The mathematical result of the general principle of relativity is the principle of general covariance which states that the laws of physics must be tensor equations (as then they will take the same form in all coordinate systems).
In the process of discovering GR, Einstein used a fact that was known since the time of Galileo, namely, that the inertial and gravitational masses of an object happen to be the same. He used this as the basis for the principle of equivalence, which describes the effects of gravitation and acceleration as different perspectives of the same thing (at least locally), and which he stated in 1907 as:
- We shall therefore assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.
In other words, he postulated that no experiment can locally distinguish between a uniform gravitational field and a uniform acceleration. The meaning of the Principle of Equivalence has gradually broadened, in consonance with Einstein's further writings, to include the concept that no physical measurement within a given unaccelerated reference system can determine its state of motion. This implies that it is impossible to measure, and therefore virtually meaningless to discuss, changes in fundamental physical constants, such as the rest masses or electrical charges of elementary particles in different states of relative motion. Any measured change in such a constant would represent either experimental error or a demonstration that the theory of relativity was wrong or incomplete.
The equivalence principle explains the experimental observation that inertial and gravitational mass are equivalent. Moreover, the principle implies that some frames of reference must obey a non-Euclidean geometry: that spacetime is curved (by matter and energy), and gravity can be seen purely as a result of this geometry. This yields many predictions such as gravitational redshifts and light bending around stars, black holes, time slowed by gravitational fields, and slightly modified laws of gravitation even in weak gravitational fields. However, it should be noted that the equivalence principle does not uniquely determine the field equations of curved spacetime, and there is a parameter known as the cosmological constant which can be adjusted.
The modifications to Isaac Newton's law of universal gravitation produced the first great theoretical success of general relativity: the correct prediction of the precession of the perihelion of Mercury's orbit. Many other quantitative predictions of general relativity have since been confirmed by astronomical observations. However, because of the difficulty in making these observations, theories which are similar but not identical to general relativity, such as the Brans-Dicke theory and the Rosen bi-metric theory cannot be ruled out completely, and current experimental tests can be viewed as reducing the deviation from GR which is allowable. However, the discovery in 2003 of PSR J0737-3039, a binary neutron star in which one component is a pulsar and where the perihelion precesses 16.88° per year (or about 140,000 times faster than the precession of Mercury's perihelion), enabled the most precise experimental verification yet of effects predicted by general relativity.  
There are no known experimental results that suggest that a theory of gravity radically different from general relativity is necessary. For example, the Allais effect was initially speculated to demonstrate "gravitational shielding," but was subsequently explained by conventional phenomena.
Nevertheless, there are good theoretical reasons for considering general relativity to be incomplete. General relativity does not include quantum mechanics, and this causes the theory to break down at sufficiently high energies. A continuing unsolved challenge of modern physics is the question of how to correctly combine general relativity with quantum mechanics, thus applying it also to the smallest scales of time and space.
Mathematics of GR
The idea of curvature can be clarified by the following considerations. While it can be helpful for visualization to imagine a curved surface sitting in a space of higher dimension, this model is not very useful for the real universe; although a two dimensional surface can be embedded in three, and thus visualized well, a curved four dimensional spacetime such as our universe cannot be imbedded in a flat space of even five dimensions, but many more are required. Curvature can be measured entirely within a surface, and similarly within a higher-dimensional manifold such as space or spacetime. On Earth, if you start at the North Pole, walk south for about 10,000 km (to the Equator), turn left by 90 degrees, walk for 10,000 more km, and then do the same again (walk for 10,000 more km, turn left by 90 degrees, walk for 10,000 more km), you will be back where you started. Such a triangle with three right angles is only possible because the surface of the earth is curved. The curvature of spacetime can be evaluated, and indeed given meaning, in a similar way. Curvature may be quantified by the Riemann tensor, essentially a matrix of numbers which describes how a vector that is moved along a curve parallel to itself changes when a round trip is made. In flat space, the vector returns to the same orientation, but in a curved space it generally does not. In spaces of two dimensions, the Riemann tensor is a matrix (i.e., just a number) called the Gaussian or scalar curvature.
To formulate the description of gravity as a geometric phenomena, the idea of a spacetime is used, as it incorporates a very useful mathematical structure known as a manifold. As in special relativity, general relativity views physical phenomena as events in a four-dimensional spacetime, the four dimensions being the usual three spatial dimensions and one time dimension. The paths of particles and light beams are described by world lines, as in the special theory. However, whereas the spacetime in special relativity is flat, the spacetime of general relativity is, in general, curved and so the paths of light beams are curved. The description of events takes place by observers in reference frames, represented mathematically by the coordinate charts on the spacetime. The idea of coordinate charts is an important feature of physics in general, as in reality we can only physically describe (or measure) events in our immediate vicinity, i.e., it highlights the local nature of (non-quantum) physics (see quantum mechanics, for non-local physical effects). It is also assumed that test particles and light beams travel on the geodesics of the given spacetime.
Mathematically, the physical spacetime is represented by a 4-dimensional, connected pseudo-Riemannian manifold. The paths of point particles are represented by timelike geodesics on this manifold, whereas the paths of light beams are represented by null geodesics. The principle of general covariance stipulates that the equations of physics should be written in tensor form. Accordingly, the Einstein field equation should be a tensor equation.
Einstein field equation
The field equation reads, in components, as follows:
where Rab are the Ricci curvature tensor components, R is the scalar curvature, gab are the metric tensor components, Λ is the cosmological constant, Tab are the stress-energy tensor components describing the non-gravitational matter, energy and forces at any given point in space-time, π is pi, c is the speed of light in a vacuum and G is the gravitational constant which also occurs in Newton's law of gravity.
The Ricci tensor is defined by contracting the Riemann tensor over the first and third indices:
Rab = Rcacb
Some important identities in general relativity are the Bianchi identities:
Ra[bcd] = 0 (algebraic Bianchi identity)
Rab[cd;e] = 0 (differential Bianchi identity)
The field equation is not uniquely proven (it is only an assumption of GR) and there is room for other models, provided that they do not contradict observation. General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law. Few physicists doubt that such a theory of everything will give general relativity in the appropriate limit, just as general relativity predicts Newton's law of gravity in the non-relativistic limit.
Solutions of the field equations
The study of exact solutions of Einstein's field equation is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe. In relation to this, note that Einstein's field equation contains a parameter called the "cosmological constant" Λ, which was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations distant galaxies by Hubble a decade later confirmed that our universe is in fact not static but expanding. So Λ was abandoned, with Einstein calling it the "biggest blunder [I] ever made". However, quite recently, improved astronomical techniques have found that a non-zero value of Λ is needed to explain some observations. Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor, and then interpreted as a form of dark energy whose density is constant in space-time.
There is a useful way to study exact solutions by using the tetrad formulation. This involves using four reference vector fields, called a vierbein or tetrad. The four vector fields are denoted by ea, a = 1, 2, 3, 4 and satisfy g(ea, eb) = ηab where
is the Minkowski metric (see sign convention). One thing to note is that we can perform an independent proper, orthochronous Lorentz transformation at each point (subject to smoothness, of course) and still get a valid tetrad. So, the tetrad formulation of GR is a gauge theory, but with a noncompact gauge group SO(3,1). It is also invariant under diffeomorphisms.
See vierbein and Palatini action for more details. See Einstein-Cartan theory for an extension of general relativity to include torsion. See teleparallelism for another theory which predicts the same results as general relativity but with FLAT spacetime (no curvature).
Relationship to other physics theories
The special theory of relativity (1905) modified the equations used in comparing the measurements made by differently moving bodies, in view of the constant value of the speed of light, i.e. its observed invariance in reference frames moving uniformly relative to each other. This had the consequence that physics could no longer treat space and time separately, but only as a single four-dimensional system, "space-time," which was divided into "time-like" and "space-like" directions differently depending on the observer's motion. The general theory added to this that the presence of matter "warped" the local space-time environment, so that apparently "straight" lines through space and time have the properties we think of "curved" lines as having.
In relativity theory, all events are referred to a reference frame. A reference frame is defined by choosing particular matter as the basis for its definition. Thus, all motion is defined and quantified relative to other matter. In the special theory of relativity it is assumed that reference frames can be extended indefinitely in all directions in space and time. The theory of special relativity concerns itself with refernce frames that move at a constant velocity with respect to each other (i.e. inertial reference frames), whereas general relativity deals with all frames of reference. In the general theory it is recognised that we can only define local frames to given accuracy for finite time periods and finite regions of space (similarly we can draw flat maps of regions of the surface of the earth but we cannot extend them to cover the whole surface without distortion). In general relativity Newton's laws are assumed to hold in locally inertial reference frames . Thus Newton's first law is replaced by the law of geodesic motion.
The notion of curvature in relativity arises by considering the non-inertial relationship between two reference frames. With special relativity, Einstein assumed that the description of physical phenomena should be the same for any two inertial observers - the so-called special principle of relativity. He then extended this principle to state the general principle of relativity, namely, that all reference frames (inertial or otherwise) should be equivalent for the description of physical phenomena. By considering a thought experiment in which a man falls of the roof of his house, Einstein intuited that the 'man would not feel his own weight'. This led to the equivalence principle.
- The theory appeared to me then, and still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art, to be enjoyed and admired from a distance. —Max Born
- Carroll, Sean M., Spacetime and Geometry: An introduction to general relativity, Addison Wesley, San Francisco (2004). ISBN 0-8053-8732-3. A modern graduate level textbook.
- D'Inverno, Ray, Introducing Einstein's Relativity, Oxford University Ass Press (1993). A modern undergraduate level text.
- Misner, Charles, Kip Thorne, and John Wheeler, Gravitation, Freeman (1973). ISBN 0716703440. A classic graduate level text book, which, if somewhat long winded, pays more attention to the geometrical basis and the development of ideas in general relativity than some other approaches.
Online notes and courses
- Baez, Bunn, 2001, The Meaning of Einstein's Equation, intuitive explanation of Einstein-Hilbert equations - requires familiarity with special relativity.
- Carroll, Sean M., A No-Nonsense Introduction to General Relativity. Also see the notes from an earlier version of his above textbook: arXiv:gr-qc/9712019.
- MIT 8.962 Course Notes Notes and handouts from the MIT 8.962 course on General Relativity
- MIT OCW Site Notes and resources from the MIT open Courseware website
- Reflections on Relativity A complete online course on Relativity
- Bondi, Herman, Relativity and Common Sense, Heinemann (1964). A school level introduction to the principle of relativity by a renowned scientist.
- Einstein, Albert, Relativity: The special and general theory. ISBN 0517884410. The special and general relativity theories in their original form.
- Epstein, Lewis Caroll, Relativity Visualized. ISBN 093521805X. Requires no mathematical background. Actually explains general relativity, rather than merely hinting at it with a few metaphors.
- Perret, W. and G.B. Jeffrey, trans.: The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, New York Dover (1923).
- Thorne, Kip, and Stephen Hawking, Black Holes and Time Warps, Papermac (1995). A recent popular account by leading experts.
- J. J. O'Connor and E. F. Robertson, History of General Relativity at the MacTutor History of Mathematics archive.
- The original 1915 article by David Hilbert containing the gravitational field equation.
- Malcolm MacCallum's GR News service for current research in relativity.
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