General relativity

Two-dimensional visualisation of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.

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 a geometric phenomenon. More specifically, it assumes that any object possessing mass curves the 'space' in which it exists, this curvature being equated to gravity. To conceptualize this equivalence, it is helpful to think, as several author-physicists have suggested, in terms of gravity not causing or being caused by spacetime curvature, but rather that gravity is spacetime curvature. 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) and the bending of starlight by the Sun (again, classical physics can only account for half the experimentally observed bending). It also predicts phenomena such as the existence of gravitational waves, 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.

It is believed that Einstein developed the general relativity from the simple but elegant consideration that no "action at a distance", like the effect of Newtonian gravitation, can propagate through space-time instantaneously. The speed of propagation is limited by the velocity of light, as required by his Special relativity.

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 curved manifolds. Template:Cosmology

Physical Description of the Theory

In relativity theory, physical phenomena are described by observers making measurements in reference frames. 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 O 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 a given matter distribution will fix the spacetime once and for all. There are a few caveats here: (1) the spacetime within which the matter is distributed cannot 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.

More specifically, let us ask how the nearly circular path on which the Earth travels can be a geodesic, which we always thought looked more like a straight line. But in the four dimensions of relativity, the principal motion of the Earth is into the future. Consider the situation in four dimensions, but for simplicity assume the Earth's velocity is perpendicular to the Z axis. Considering the time axis vertical, the Earth's path is a spiral (helix) about the t-axis, and not a tightly wound one at that. In one complete turn of the spiral, one year has elapsed, so the coordinate ct has increased one light year, but the Earth is moving in the x-y plane much more slowly, having gone only 2 <math> \pi <math> astronomical units in a year; i.e. the slope of the helix is c divided by the orbital velocity, or about ten thousand.

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 Equivalence Principle

(For more detailed information about the equivalence principle, see equivalence principle)

Inertial reference frames, in which bodies maintain a uniform state of motion unless acted upon by another body, are distinguished 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 force in a reference frame in free fall other than tidal gravity forces, which can deform objects but not accelerate them. Indeed, attempts to detect gravitational waves depend on just those tidal forces. 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 prevented from falling by the matter below (on and within the Earth, including the continents, furniture, etc.,) and is analogous to the acceleration felt in a car.

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 Covariance Principle

(For more detailed information about the covariance principle, see the article principle of general covariance)

Following on from the spirit of special relativity, the principle of general covariance states that all coordinate systems are equivalent for the formulation of the general laws of nature. Mathematically, this suggests that the laws of physics should be tensor equations.

Geometric Foundations

For a long time, it was believed that the universe obeyed the axioms of Euclidean geometry, including Euclid's parallel postulate. In crude terms, 'space is Euclidean' seemed to be the general rule. Although the development of non-Euclidean geometries by Lobachevsky, Bolyai, Gauss and others, opened up a new field of research, the general consensus was still that space is Euclidean. Early on, Gauss decided to test this assumption and found (with experiments using the crude equipment of that age) that the sum of the angles of a triangle was 180 degrees, affirming that to available precision, physical space obeyed the parallel postulate and was Euclidean. Modern experiments are capable of detecting 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, in order to synchronize these clocks with earth-bound ones. In so "correcting" a clock, of course, one tweaks it so as to be an imperfect or nonstandard clock from the standpoint of the equivalence principle. In other words, in order to establish a more or less global time standard, one has to adjust or modify clocks originally of standard construction and operation so as to, in a sense, violate the equivalence principle.

In 1854, Gauss' student Riemann gave a famous lecture in which he developed the general mathematics of non-Euclidean geometries. In the lecture, he defined what is nowadays called an n-dimensional Riemannian space and defined the curvature tensor, a fundamental mathematical object in GR. He also inquired as to the dimension of the space of reality (the dimension of our world's space), as well as wondering about the actual geometry of the world. In retrospect, Riemann's lecture was ahead of its time, it finding fruition when Einstein developed GR. In fact, Einstein began with physical concepts to develop GR and knew that he needed the mathematics of curved spaces to formulate his theory. The required mathematics was precisely that developed by Riemann, the modern designation called manifold theory.

Predictions of GR

(For more detailed information about tests and predictions of general relativity, see Tests of general relativity)

Like any good scientific theory, general relativity makes predictions which can be tested. Some of the predictions of general relativity include the perihelion shifts of planetary orbits (particularly that of Mercury), bending of light by massive objects, and the existence of gravitational waves. The first two of these tests have been verified to a high degree of accuracy and precision. Most researchers believe in the existence of gravitational waves, but more accurate experiments are needed to raise this prediction to the status of the other two, if one demands direct detection of the waves. Nevertheless, indirect effects of gravitational wave emission have been observed for a binary system of orbiting neutron stars, as described in Tests of general relativity.

Other predictions include the expansion of the universe, the existence of black holes and possibly the existence of wormholes. The existence of black holes is generally accepted, but the existence of wormholes is still very controversial, many researchers believing that wormholes may exist only in the presence of exotic matter. The existence of white holes is very speculative, as they appear to contradict the second law of thermodynamics.

Many other quantitative predictions of general relativity have since been confirmed by astronomical observations. One of the most recent, 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 the effects predicted by general relativity. [1] (http://skyandtelescope.com/news/article_1124_1.asp) [2] (http://skyandtelescope.com/news/article_1473_1.asp).

Mathematics of GR

(For more detailed information about the mathematics of general relativity, see mathematics of general relativity)

The mathematics of general relativity involves heavy use of tensor calculus. The use of tensors in relativity greatly simplifies many calculations and serves to reflect the fact that all observers are equivalent for the description of physical laws.

An important tensor in relativity is the Riemann tensor, which is a matrix of numbers that essentially measures the deviation of a vector that is moved along a curve parallel to itself when a round trip is made. In flat space, the vector returns to the same orientation (the Riemann tensor is zero), but in a curved space it generally does not (in general, a non-zero Riemann tensor). In spaces of two dimensions, the Riemann tensor is a <math>1 \times 1<math> matrix (i.e. just a real number) called the Gaussian or scalar curvature. Curvature can be measured entirely within a surface, and similarly within higher-dimensional manifolds such as space or spacetime.

The dynamics of general relativity are incorporated in the Einstein field equation, a tensor equation that describes how matter affects the geometry of spacetime, and the geodesic equation, which describes how objects move in the resulting geometry. Often, approximations are made in working with both these equations.

An important feature of the Einstein field equations is that they are a set of nonlinear partial differential equations for the metric. As such, this distinguishes the field equations of general relativity from some of the other important field equations in physics, such as Maxwell's equations (which are linear in the electric and magnetic fields) and Schrodinger's equation (which is linear in the wavefunction). This constitutes another major difference between general relativity and other physical theories.

Relationship to other physical theories

Special and general relativity

In relativity theory, all events are referred to one or more reference frames. 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 reference 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).

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.

Thus Newton's first law is replaced by the law of geodesic motion.

There are no known experimental results that suggest that a non-quantum 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.

Quantum mechanics and general relativity

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. Most scientists consider this unifying theory's leading candidates to be M-theory and loop quantum gravity. This unification would achieve Einstein's dream of a grand unification theory, combining the strong, electroweak, and gravitational forces into one force, as well as successfully creating one set of equations that do not break down under any conditions.

Other theories

The Brans-Dicke theory and the Rosen bi-metric theory are modifications of general relativity and cannot be ruled out by current experiments.

See Einstein-Cartan theory for an extension of general relativity to include torsion.

There have been attempts to formulate consistent theories which combine gravity and electromagnetism, some of the first being the Kaluza-Klein theory and Weyl's gauge theory.

History

Full article: History of general relativity
See also: Tests of general relativity

General relativity was developed by Einstein in a process that began in 1907 with the publication of an article by Einstein on the influence of gravity and acceleration on the behaviour of light in special relativity. Most of this work was done in the years 1911–1915, beginning with the publication a second article of the effect of gravitation on light. By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. In 1915, these efforts culminated in the publication of the Einstein field equations, which are a set of differential equations.

Since 1915, the development of general relativity has focused on solving the field equations for various cases. This generally means finding metrics which correspond to realistic physical scenarios. The interpretation of the solutions and their possible experimental and observational testing also constitutes a large part of research in GR.

The expansion of the universe created an interesting episode for general relativity. In 1922, Alexander Friedmann found a solution in which the universe may expand or contract, and later Georges Lemaître derived a solution for an expanding universe. Einstein did not believe in an expanding universe, and so he added a cosmological constant to the field equations to permit the creation of static universe solutions. In 1929, Edwin Hubble found evidence that the universe is expanding. This resulted in Einstein dropping the cosmological constant, referring to it as "the biggest blunder in my career".

Progress in solving the field equations and understanding the solutions has been ongoing. Notable solutions have included the Schwarzschild solution (1916), the Reissner-Nordström solution and the Kerr solution.

Observationally, general relativity has a history too. The perihelion precession of Mercury was the first evidence that general relativity is correct. Eddington's 1919 expedition in which he confirmed Einstein's prediction for the deflection of light by the Sun helped to cement the status of general relativity as a likely true theory. Since then, many observations have confirmed the predictions of general relativity. These include studies of binary pulsars, observations of radio signals passing the limb of the Sun, and even the GPS system. For more information, see the Tests of general relativity article.

Quotes

Spacetime grips mass, telling it how to move, and mass grips spacetime, telling it how to curve - John Archibald Wheeler.

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

References

This reading list is loosely based on Template:Web reference

Popular Books

• Template:Book reference Leisurely pace, provides superb intuition for Schwarzschild geometry.

• Template:Book reference Covers much more ground, while remaining concise and readable.

• Template:Book reference A delightful romp through the physics of black holes. Features many personal anecdotes from the author's distinguished career.

Textbooks

Introductory

• Template:Book reference Clearly written, short and sweet; covers less ground than the others but much cheaper.

• Template:Book reference Readable, well illustrated, fairly comprehensive without becoming encyclopedic--- what's not to love?

• Template:Book reference Features an outstanding treatment of tensor calculus and the matter tensor, a key topic which beginners often have trouble grasping. The treatment of linearized gravitational waves and stellar models is also outstanding.

• Template:Book reference 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.

• Template:Book reference 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.

Advanced

• Template:Book reference Readable, up-to-date. 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 (http://pancake.uchicago.edu/~carroll/grbook/).

• Template:Book reference 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.

• Template:Book reference 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.

• Template:Book reference 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.

• Template:Book reference 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.

Special Topics

• Template:Book reference A collection of excellent problems, with sketch solutions in the back. Text your skills!

• Template:Book reference 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.

• Template:Book reference 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.

• Template:Book reference 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.

External Links

Online Tutorials

• Baez, John & Bunn, Ted; Template:Web reference This superb expository paper explains the meaning of the field equation in terms of the motion of a cloud of free falling test particles.

• Carroll, Sean M.; Template:Web reference A concise but very readable overview.

Webcourses

• Rappoport, Saul; Template:Web reference An elementary introduction to relativistic physics, including a smattering of gtr.

• Bertschinger, Edmund; Template:Web reference An introduction to general relativity at the level of Misner, Thorne & Wheeler.

• Brown,Kevin; Template:Web reference An idiosyncratic work, providing in-depth discussions of various aspects of special and general relativity. The subjects are treated exceptionally thoroughly. The book is written for people who already have a firm grasp of relativity.

• van Putten, Maurice; Template:Web reference An topics course on gravitational wave detectors, featuring the draft of the instructor's forthcoming textbook.

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