General relativity
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General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16.[1][2] It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational acceleration can be described by the curvature of space and time. General relativity further calls for the curvature of space-time to be produced by the mass-energy and momentum content of the matter in space-time. General relativity is distinguished from other metric theories of gravitation by its use of the Einstein field equations to relate space-time content and space-time curvature.[2][3]
In the mathematical formalism of general relativity, the Einstein field equations are a system of partial differential equations whose solution represents the metric tensor (or the metric) of space-time, describing its "shape". Some important solutions of the Einstein field equations are the Schwarzschild solution (for the space-time surrounding a spherically symmetric uncharged and non-rotating massive object), the Reissner-Nordström solution (for a charged spherically symmetric massive object), and the Kerr metric (for a rotating massive object). An object moving inertially in a gravitational field follows a geodesic path that may be found using the Christoffel symbol of the metric.
General relativity is currently the most successful gravitational theory, being almost universally accepted and well-supported by observations. The first success of general relativity was in explaining the anomalous perihelion precession of Mercury. Then in 1919, Sir Arthur Stanley Eddington announced that observations of stars near the eclipsed Sun confirmed general relativity's prediction that massive objects bend light. Since then, other observations and experiments have confirmed many of the predictions of general relativity, including gravitational time dilation, the gravitational redshift of light, signal delay, and gravitational radiation. In addition, numerous observations are interpreted as confirming one of general relativity's most mysterious and exotic predictions, the existence of black holes. However, the current framework is incompatible with contemporary theories of quantum mechanics and the singularities of black holes which, in turn, has given rise to alternative theories such as the Brans-Dicke theory.
| 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 |
| Equations |
| Linearized Gravity Post-Newtonian formalism Einstein field equations |
| Advanced theories |
| Kaluza-Klein Quantum gravity |
| Solutions |
| Schwarzschild Reissner-Nordström · Gödel |
| Scientists |
| Einstein · Minkowski · Eddington |
| |
Contents
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[edit] Justification
The justification for creating general relativity came from the equivalence principle, which dictates that free-falling observers are the ones in inertial motion. Roughly speaking, the principle states that the most obvious effect of gravity – things falling down – can be eliminated by making the transition to a reference frame that is in free fall, and that in such a reference frame, the laws of physics will be approximately the same as in special relativity.[4] A consequence of this insight is that inertial observers can accelerate with respect to each other. For example, a person in free fall in an elevator whose cable has been cut will experience weightlessness: objects will either float alongside him or her, or drift at constant speed. In this way, the experiences of an observer in free fall will be very similar to those of an observer in deep space, far away from any source of gravity, and in fact to those of the privileged ("inertial") observers in Einstein's theory of special relativity.[5] Albert Einstein realized that the close connection between weightlessness and special relativity represented a fundamental property of gravity.
Einstein's key insight was that there is no fundamental difference between the constant pull of gravity we know from everyday experience and the fictitious forces felt by an accelerating observer (in the language of physics: an observer in a non-inertial reference frame).[6][7] So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration which could just as easily be imitated by placing an observer within a rocket accelerating at the same rate as gravity (9.81 m/s²).
This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself:
| “ | If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them." [8] | ” |
Thus the equivalence principle led Einstein to develop a gravitational theory which involves curved space-times. Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.[9]
Another motivating factor was the realization that relativity calls for the gravitational potential to be expressed as a symmetric rank-two tensor, and not just a scalar as was the case in Newtonian physics (An analogy is the electromagnetic four-potential of special relativity). Thus, Einstein sought a rank-two tensor means of describing curved space-times surrounding massive objects.[10] This effort came to fruition with the discovery of the Einstein field equations in 1915.[1]
[edit] Fundamental principles
General relativity is a metric theory of gravitation. For this class of theory, the main defining feature is the concept of gravitational 'force' being replaced by spacetime geometry. Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion within a curved geometry of spacetime.
General relativity (and all other metric theories of gravitation) are predicated upon several underlying assumptions. The general principle of relativity states that the laws of physics must be the same for all observers (accelerated or not). The principle of general covariance states the laws of physics must take the same form in all coordinate systems. General relativity also requires equivalence between inertial and geodesic motion because the world lines of particles unaffected by physical forces are timelike or null geodesics of spacetime. The principle of local Lorentz invariance requires that the laws of special relativity apply locally for all inertial observers. Finally there is the principle that the curvature and of spacetime and its energy-momentum content are related. (As mentioned above, this relationship between curvature and spacetime content is specifically dictated by the Einstein field equations in general relativity.)
The equivalence principle, which was the starting point for the development of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.
[edit] Mathematical framework
- See also: Physical theories modified by general relativity
The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian. In addition, the principle of general covariance requires that mathematics to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.
[edit] Geometry
Due to the expectation that spacetime is curved, non-Euclidean geometry must be used. (In particular, the geometry is described by a pseudo-Riemannian metric, or more specifically still, a Lorentzian metric.) In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime (meaning that their velocities do not differ to first order in their separation) come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. Similarly, two balls initially at rest with respect to and above the surface of the Earth (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent free-fall.
The curvature of spacetime (caused by the presence of stress-energy) can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the top of this article. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at some suitable combination of direction and speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example.
Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime. Notice that the most important part of the curvature near a massive object is in the plane defined by the time and radial directions, although there is also some purely spatial curvature.
[edit] Coordinate vs. physical acceleration
One of the greatest sources of confusion about general relativity comes from the need to distinguish between coordinate and physical accelerations.
In classical mechanics, space is preferentially mapped with a Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a constant coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronization procedure. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary.
In general relativity, we abandon the unwarranted assumption that nature has provided us with a preferred set of coordinates. Instead an observer may choose a set of coordinates at his own convenience, and we only require that coordinates of a point in different coordinate systems can be expressed into each other through some smooth functional dependence. Only statements that do not depend on the arbitrary choice of a coordinate system by the observer (i.e. the description of a physical phenomenon) can be considered of physical relevance. This is the principle of general covariance of physical laws. It means for example that a quantity like acceleration, cannot simply be described as the second derivative of the coordinate functions of a velocity, because a non zero "coordinate acceleration" may merely be an artifact of the choice of coordinates. In fact such an artifact already occurs for the description in polar coordinates of a uniformly moving particle not passing through the (chosen!) origin. In fact the definition of acceleration of a particle requires that we know how to subtract velocities measured at two different points along its track in space time. Equivalently we must be able to define in an observer and coordinate invariant way which velocity vectors are constant along such a path in space time. This is called parallel transport. It does not come for free but requires additional structure of space-time, a connection. It so happens that if there is defined a "length" of all velocity vectors (which may be negative), a Lorentzian metric , there is a natural connection , the Levi Civita connection, uniquely determined by requiring that parallel velocity vectors have constant "lenghth" and the technical assumption of zero torsion. It was one of Einsteins great insights that this description can be applied to describe the influence of gravity. In general relativity, gravity is seen as a consequence of the fact that parallel transport of a velocity vector may depend on the path through space time, not only on its endpoints. How metric and thereby connection and parallel transport, are determined by the transport of energy-momentum in space time by matter and radiation described with the so called stress energy momentum tensor is the content of the theory of general relativity.
[edit] Einstein field equations
The Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in tensor form (using abstract index notation) as
where Gab is the Einstein tensor, Tab is the stress-energy tensor and κ is a constant. The Einstein tensor is related to the curvature of space-time and is a function only of the metric tensor and its first and second derivatives. The stress energy tensor, which is the source of the gravitational field, includes stress (pressure and shear), the density of momentum, and the density of energy including the energy of mass (the source for Newtonian gravity). The tensors Gab and Tab are both rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains 10 independent terms.
The EFE reduce to Newton's law of gravity in the limiting cases of a weak gravitational field and slow speed relative to the speed of light. In fact, the value of κ in the EFE is determined to be 
by making these two approximations.[2]
The solutions of the EFE are metrics of spacetime. These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. Being non-linear differential equations, the EFE often defy attempts to obtain an exact solution; however, many such solutions are known.
The EFE are the identifying feature of general relativity. Other theories built out of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen's bimetric theory, and Einstein-Cartan theory).
[edit] Consequences of Einstein's theory
General relativity, as laid out in the previous section, has a number of consequences; some follow directly from the theory's axioms, others have only become clear in the course of the ninety years of research that followed Einstein's initial publication.
[edit] Gravitational waves
There are several analogies between weak-field gravity and electromagnetism. One is that, for electromagnetic waves, there are corresponding gravitational waves: ripples in spacetime that propagate at the speed of light.[11]
The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right).[12] Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by 10 − 21 or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.[13] It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there will no be linear superposition for gravitational waves. Describing such more general waves is not an easy task. There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty space[14] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,[15] while, when it comes to describing the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical methods are the only way to construct appropriate models.[16]
[edit] Orbital effects and the relativity of direction
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The most striking one are the relativistic apside shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction.
[edit] Precession of apsides
In general relativity, the apsides of orbits (the points of an orbiting body closest approach to the system's center of mass) will precess – the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rosette-like shape (see image). Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle[17]; the result can also be obtained by either using the exact Schwarzschild metric (describing spacetime around a spherical mass)[18] or the much more general post-Newtonian formalism.[19] The effect is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body's gravity (in other words, the special kind of nonlinearity exhibited by Einstein's theory).[20]
One of the earliest successes of general relativity (and one of the criteria Einstein used in his search for the final form of his field equations) was that this effect offered a straightforward explanation for a previously known[21] anomalous perihelion shift of the planet Mercury. Later and more accurate measurements show that general relativity does indeed predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).[22] More recently, binary pulsar systems have been found in which the observed effects are larger by five orders of magnitude.[23]
[edit] Orbital decay and gravitational waves
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period; since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.[24]
The first observation of a decrease in orbital period due to the emission of gravitational waves was made byHulse and Taylor using binary pulsar PSR1913+16 they had discovered in 1974; it amounts to first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993.[25] Since then, several other binary pulsars have been found; the most spectacular find was the double pulsar PSR J0737-3039 in which both stars are pulsars.[26]
[edit] Geodetic precession and frame-dragging
Several relativistic effects are directly related to the relativity of direction.[27] One is geodetic precession: for a gyroscope in free fall in curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars – even though its motion comes closest to keeping its axis direction constant ("parallel transport").[28] For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging;[29] more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.[30]
Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass gets "dragged around"; this is most extreme for rotating black holes where, for an object entering a zone known as the ergosphere, rotation is inevitable.[31] Such effects can again be tested through their influence on the orientation of a gyroscope in free fall:[32] somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction;[33] a precision measurement is the main aim of the Gravity Probe B mission, whose results are due in late 2007.[34]
[edit] Gravitational time dilation and frequency shift
In general relativity (and, in fact, in any theory in which the equivalence principle holds[35]), gravity has an immediate influence on the passage of time. Imagine two observers Alice and Bob, both of which are at rest in a stationary gravitational field, with Alice closer to the source of gravity ("deeper in the gravity well") and Bob at a greater distance. Then for light sent from Alice to Bob or vice versa, Bob will measure a lower frequency than Alice: light sent down into a gravity well is blue-shifted, light climbing out of a gravity well is redshifted. Also, Alice's clocks tick more slowly than Bob's: whenever the two are compared (either by sending light signals back and forth, or by slowly transporting clocks from one location to the other), the result will be that Bob's clocks are running faster. This effect is not restricted to clocks, but applies to all processes (the rate at which Alice and Bob age, cook five-minute eggs, or play Chopin's Minute Waltz); it is known as gravitational time dilation.[36].
The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound and Rebka[37] and later confirmed by astronomical observations.[38] There are numerous direct measurements of gravitational time dilation using atomic clocks[39] while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).[40] Tests in stronger gravitational fields are provided by the observation of binary pulsars.[41] All results are in agreement with general relativity;[42] however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.[43]
[edit] Light deflection and gravitational time delay
In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics – a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity.[44] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),[45] several effects of gravity on the propagation of light emerge.
The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also derived by extending the universality of free fall to light,[46] the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity; from the standpoint of Einstein's theory they take into account the effect of gravity on time, but not its consequences for the warping of space.[47] An important example of this is starlight being deflected as it passes the Sun; in consequence, the positions of stars observed in the Sun's vicinity during a solar eclipse appear shifted by up to 1.75 arc seconds. This effect was first measured by a British expedition directed by Arthur Eddington, and confirmed with significantly higher accuracy by subsequent measurements.[48]
Closely related to the bending of light is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of radar signals sent from Earth to planets such as Venus or Mercury and thence reflected back;[49] later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.[50] In both the case of the planets and the probes, what was measured was the propagation of signals in the Sun's gravitational field. More recent measurements have detected the Shapiro effect in signals sent by a pulsar that is part of a binary system; in that case, the gravitational field causing the time delay is that of the other pulsar.[51] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called γ that reflects the influence of gravity on the geometry of space.[52]
[edit] Causal structure and global geometry
In general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using so-called Penrose-Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.[53]
Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are nowadays known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations; rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) to derive general results.[54]
[edit] Cosmic segregation: Horizons
One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called horizons, which segregate one spacetime region from the rest of the world. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space, one can define a surface that separates the inside from the outside world. No light from the inside can escape to the outside, and since, in general relativity, no object can overtake a light pulse, all inside matter is imprisoned, as well. The resulting object is known as a black hole, and the surface in question as the black hole's horizon.[55] The hoop conjecture states when a black hole is expected to form: With every mass M, one can associate a length known as the Schwarzschild radius,
where G is the gravitational constant and c the speed of light. Imagine a circular hoop with the circumference 
. A mass small enough to fit through that hoop regardless of their relative orientation, then it is compact enough to form a black hole.[56]
The first studies of black holes relied on simplified model universes (namely explicit solutions of Einstein's equation, in particular the spherically-symmetric Schwarzschild solution, which turns out to describe a static black hole, and the axisymmetric Kerr solution which describes a rotating stationary black hole). Subsequent studies using global geometry have revealed more general properties of black holes. In the long run, they are rather simple objects, characterized by eleven parameters specifying: energy, linear momentum, angular momentum, location at a specified time, and electric charge. This is the result of what are called the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks akin to the differing hairstyles of humans. However complex an object that might collapse to form a black hole; in the long term (having emitted gravitational waves), the resulting object is very simple.[57]
Even more remarkable, there is a general set of laws known as black hole mechanics, analogous to the laws of thermodynamics. For example, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, just as the entropy of a thermodynamic system; among other things, this law sets a limit to the energy that can be extracted from a rotating black hole (e.g. by the Penrose process).[58] In fact, there is strong evidence that the laws of black hole mechanics are indeed a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy:[59] semi-classical calculations indicate that black holes do emit thermal radiation, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation, and we will come back to it in the section on general relativity and quantum theory, below.[60]
Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("particle horizon"), and some regions of the future cannot by any means be influenced (event horizon); in both cases, the location of the horizon in spacetime depends on the event under study.[61] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons[62] (associated with a semi-classical radiation known as Unruh radiation).[63]
[edit] Singularities
Another general – and quite disturbing – feature of general relativity is the appearance of space-time boundaries known as singularities. Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all time-like and light-like geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges" – regions where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. These, by definition, are spacetime singularities; in many of the more interesting cases, the geometrical quantities characterizing spacetime curvature (e.g. the Ricci scalar take on infinite values at such "curvature singularities").[64] Well-known examples of spacetimes with future singularities – where worldlines end – are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,[65] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.[66] The Friedmann-Lemaître-Robertson-Walker solutions and related spacetimes, which describe universes, feature examples for past singularities on which worldlines begin, namely big bang singularities.[67]
Given just these examples, which are all highly symmetric and thus simplified, one might think the occurrence of singularities to be an artifact of idealization. The famous singularity theorems, proved using the methods of global geometry, prove otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage[68] and also at the beginning of a wide class of expanding universes.[69] However, these theorems say very little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the so-called BKL conjecture).[70] As problematic as singularities are, there are indications that all realistic future singularities (where no symmetry is perfect, and matter has realistic properties) are safely hidden away behind a horizon, and thus invisible for all distant observers. This is postulated by the cosmic censorship conjecture (Penrose 1969); while no formal proof of this conjecture exists, numerical simulations offer supporting evidence that it is, indeed, valid.[71]
[edit] Evolution equations
Each solution of Einstein's equation encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields, for example, they are self-interacting (that is, non-linear even in the absence of other fields, and they have no fixed background structure – the stage itself evolves as the cosmic drama is played out.[72]
Nevertheless, in order to understand Einstein's equations in their more general mathematical context (as partial differential equations), it is crucial to re-formulate them in a way that describes the evolution of the universe over time. This is achieved by so-called "3+1" formulations (splitting spacetime into three space dimensions and one time dimension) such as the ADM formalism;[73] with their help, it can also be shown that the spacetime evolution equations following from Einstein's theory are indeed well-behaved (solutions always exist and, once suitable initial conditions are specified, are defined uniquely).[74] Formulations like this are also the basis of numerical relativity: attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers.[75]
[edit] Global and quasi-local quantities
The notion of evolution equations is intimately tied in with another aspect of generally relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason for this is that, in a certain sense, the gravitational field (just as any other physical field) must be ascribed a certain energy; on the other hand, it is fundamentally impossible to localize that energy.[76]
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)[77] or suitable symmetries (Komar mass)[78] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity.[79] It can be shown that, just as in classical physics, these masses are positive.[80] Analogous global definitions exist for momentum and angular momentum.[81]In addition, there have been a number of attempts to define quasi-local quantities, such as the mass of an isolated system defined using only quantities defined within a finite region of space containing that system; the hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.[82]
[edit] Astrophysical applications
[edit] Gravitational lensing
The deflection of light by gravity can have an intriguing side effect: if there is a massive object between the observer and a distant target object, it is possible for the observer to see multiple distorted images of the target! This and similar effects are known as gravitational lensing[83] and, depending on the configuration, scale, and mass distribution, it can result in two images, a bright ring known as an Einstein ring, or partial rings called arcs.[84] The earliest example was discovered in 1979;[85] since then, more than a hundred gravitational lenses have been observed.[86] Images too close to be resolved can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object; a number of such "microlensing events" has been observed, as well.[87]
Gravitational lensing has developed into a tool of observational astronomy. Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data are also used to understand the structural evolution of galaxies.[88]
[edit] Gravitational wave astronomy
From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational wave (see the section Orbital decay and gravitational waves, above). However, as of yet, gravitational waves reaching us from the depths of the cosmos have not been detected directly – this is one of the major goals of current relativity-related research.[89] To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.[90] A joint US-European mission to launch a space-based detector, LISA, is currently under development,[91] with a precursor mission (LISA Pathfinder) due for launch in late 2009.[92]
Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation:[93] Terrestrial detectors are expected to yield new information about inspiral phase and mergers of binary stellar mass black holes and binaries consisting of one such black hole and a neutron star (of interest as a candidate mechanism for gamma ray bursts); they could also detect signals from core-collapse supernovae and from periodic sources such as rotating neutron stars with small deformation. If there is truth to speculation about certain kinds of phase transitions or kink bursts from long cosmic strings in the very early universe (at cosmic times around 10 − 25 seconds) these could also be detectable.[94] Space-based detectors like LISA should detect objects such as binaries consisting of two White Dwarfs, and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star), and also observe the mergers of supermassive black holes and the inspiral of smaller objects (between one and a thousand solar masses) into such black holes. LISA should also be able to listen to the same kind of sources from the early universe as ground-based detectors, but at even lower frequencies and with greatly increased sensitivity.[95]
[edit] Black holes and other compact objects
In the currently accepted models of stellar evolution, neutron stars with around 1.4 solar mass and so-called stellar black holes with a few to a few dozen solar masses are thought to be the final state for the evolution of massive stars.[96] Supermassive black holes with between a few million and a few billion solar masses are now thought to be the rule rather than the exception in the centers of galaxies,[97] and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.[98]
From an astronomical point of view, the most important property of compact objects such as black holes is that they provide a superbly efficient mechanism for converting gravitational into radiation energy.[99] Accretion, that is, the falling of material such as gas or dust onto stellar or supermassive black holes is thought to be responsible for some of the most spectacularly luminous astronomical objects, notably diverse kinds of Active Galactic Nuclei on galactic scales, and stellar-size objects such as Microquasars;[100] in particular, it can lead to relativistic jets: focused beams of highly energetic particles that are being flung into space at almost the speed of light.[101] For modelling all these phenomena, general relativity plays a central role,[102] and relativistic lensing effects are thought to play a role for the signals we receive from X-ray pulsars.[103]
Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity")[104] observations of stellar dynamics in the center of our own Milky Way galaxy,[105] and indications that at least some of the compact objects in question appear to have no solid surface[106] are strong indirect evidence for the existence of black holes; more direct evidence such as observing the "shadow" of the horizon of the Milky Way galaxy's central black hole.[107] is eagerly sought for.
Black holes are also sought-after targets in the search for gravitational waves (see the section Gravitational waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity;[108] the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances;[109] the gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole's geometry.[110]
[edit] Cosmology
Each solution of Einstein's equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. The current models are based on an extension of the original form of Einstein's equations which include the cosmological constant Λ, an additional term that has an important influence on the large-scale dynamics of the cosmos,
where gab is the spacetime metric.[111]
On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker solutions,[112] are built the models of modern cosmology in which the universe has evolved over the past 14 billion years from a hot, early Big bang phase.[113] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,[114] further observational data can be used to put the models to the test: successful predictions include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,[115] which is in good agreement with astronomical observations;[116] the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation,[117] and the large-scale distribution of galaxies.[118]
The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,[119] and they have proven a sound basis to explaining the evolution of the universe's large-scale structure.[120] On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observations[121]) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly); there is currently no generally accepted description of this new kind of matter within the framework of particle physics[122] or otherwise.[123] A similar open question is that of dark energy. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, namely dark energy;[124] the nature of this new form of energy remains unclear.[125]
A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")[126] have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around 10 − seconds, known as an inflationary phase.[127] While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,[128] problems remain. The re is a bewildering variety of possible inflationary scenarios not restricted by current observations.[129] Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity; an authoritative answer would require a complete theory of quantum gravity, which does not exist at the moment[130] (cf. the section Quantum gravity, below).
[edit] Relationship with quantum mechanics
Quantum mechanics is viewed as a fundamental theory of physics along with general relativity, but combining the two theories has presented many difficulties.
[edit] Quantum field theory in curved spacetime
Normally, quantum field theory models are considered in flat Minkowski space (or Euclidean space), which is an excellent approximation for weak gravitational fields like those on Earth. In the presence of strong gravitational fields, the principles of quantum field theory have to be modified. The spacetime is static so the theory is not fully relativistic in the sense of general relativity; it is neither background independent nor generally covariant under the diffeomorphism group. The interpretation of excitations of quantum fields as particles becomes frame dependent. Hawking radiation is a prediction of this semiclassical approximation.
[edit] Einstein gravity is nonrenormalizable
It is often said that general relativity is incompatible with quantum mechanics. This means that if one attempts to treat the gravitational field using the ordinary rules of quantum field theory, one finds that physical quantities are divergent. Such divergences are common in quantum field theories, and can be cured by adding parameters to the theory known as counterterms. These counterterms are infinities which are equal in magnitude and opposite in sign to the divergent terms. When they are added, the infinities cancel, leaving only finite terms, but modifying the meaning of terms in the equation such as "mass" and "charge" [citations needed].
Many of the best understood quantum field theories, such as quantum electrodynamics, contain divergences which are canceled by counterterms that have been effectively measured. One needs to say effectively because the counterterms are formally infinite, however it suffices to measure observable quantities, such as physical particle masses and coupling constants, which depend on the counterterms in such a way that the various infinities cancel.
A problem arises, however, when the cancellation of all infinities requires the inclusion of an infinite number of counterterms. In this case the theory is said to be nonrenormalizable. While nonrenormalizable theories are sometimes seen as problematic, the framework of effective field theories presents a way to get low-energy predictions out of non-renormalizable theories. The result is a theory that works correctly at low energies, though such a theory cannot be considered to be a theory of everything because it cannot be self-consistently extended to the high-energy realm.
[edit] Proposed quantum gravity theories
General relativity fits nicely into the effective field theory formalism and makes sensible predictions at low energies.[131] However, high enough energies will "break" the theory.
It is generally held that one of the most important unsolved problems in modern physics is the problem of obtaining the true quantum theory of gravitation, that is, the theory chosen by nature, one that will work at all energies. Discarded attempts at obtaining such theories include supergravity, a field theory which unifies general relativity with supersymmetry. In the second superstring revolution, supergravity has come back into fashion, with its quantum completion rebranded with a new name: M-theory.
A very different approach to that described above is employed by loop quantum gravity. In this approach, one does not try to quantize the gravitational field as one quantizes other fields in quantum field theories. Thus the theory is not plagued with divergences and one does not need counterterms. However it has not been demonstrated that the classical limit of loop quantum gravity does in fact contain flat space Einsteinian gravity. This being said, the universe has only one spacetime and it is not flat at all scales.
Of these two proposals, M-theory is significantly more ambitious in that it also attempts to incorporate the other known fundamental forces of Nature, whereas loop quantum gravity "merely" attempts to provide a viable quantum theory of gravitation with a well-defined classical limit which agrees with general relativity.
[edit] Alternative theories
Well known classical theories of gravitation other than general relativity include:
- Nordström's theory of gravitation (1913) was one of the earliest metric theories (an aspect brought out by Einstein and Fokker in 1914). Nordström soon abandoned his theory in favor of general relativity on theoretical grounds, but this theory, which is a scalar theory, and which features a notion of prior geometry, does not predict any light bending, so it is solidly incompatible with observation.
- Alfred North Whitehead formulated an alternative theory of gravity that was regarded as a viable contender for several decades, until Clifford Will noticed in 1971 that it predicts grossly incorrect behavior for the ocean tides.
- George David Birkhoff's (1943) yields the same predictions for the classical four solar system tests as general relativity, but unfortunately requires sound waves to travel at the speed of light. Thus, like Whitehead's theory, it was never a viable theory after all, despite making an initially good impression on many experts.
- Like Nordström's theory, the gravitation theory of Wei-Tou Ni (1971) features a notion of prior geometry, but Will soon showed that it is not fully compatible with observation and experiment.
- The Brans-Dicke theory and the Rosen bimetric theory are two alternatives to general relativity which have been around for a very long time and which have also withstood many tests. However, they are less elegant and more complicated than general relativity, in several senses.
- There have been many attempts to formulate consistent theories which combine gravity and electromagnetism. The first of these, Weyl's gauge theory of gravitation, was immediately shot down (on a postcard) by Einstein himself,[citation needed] who pointed out to Hermann Weyl that in his theory, hydrogen atoms would have variable size, which they do not. Another early attempt, the original Kaluza-Klein theory, at first seemed to unify general relativity with classical electromagnetism, but is no longer regarded as successful for that purpose. Both these theories have turned out to be historically important for other reasons: Weyl's idea of gauge invariance survived and in fact is omnipresent in modern physics, while Kaluza's idea of compact extra dimensions has been resurrected in the modern notion of a braneworld.
- The Fierz-Pauli spin-two theory was an optimistic attempt to quantize general relativity, but it turns out to be internally inconsistent. Pascual Jordan's work toward fixing these problems eventually motivated the Brans-Dicke theory, and also influenced Richard Feynman's unsuccessful attempts to quantize gravity.
- Einstein-Cartan theory includes torsion terms, so it is not a metric theory in the strict sense.
- Teleparallel gravity goes further and replaces connections with nonzero curvature (but vanishing torsion) by ones with nonzero torsion (but vanishing curvature).
- The Nonsymmetric Gravitational Theory (NGT) of John W. Moffat is a dark horse in the race.
Even for "weak field" observations confined to our Solar system, various alternative theories of gravity predict quantitatively distinct deviations from Newtonian gravity. In the weak-field, slow-motion limit, it is possible to define 10 experimentally measurable parameters which completely characterize predictions of any such theory. This system of these parameters, which can be roughly thought of as describing a kind of ten dimensional "superspace" made from a certain class of classical gravitation theories, is known as PPN formalism (Parametric Post-Newtonian formalism). [1] Current bounds on the PPN parameters [2] are compatible with GR.
See in particular The confrontation between Theory and Experiment in Gravitational Physics, a review paper by Clifford Will.
[edit] History
General relativity was developed by Einstein in a process that began in 1907 with the publication of an article on the influence of gravity and acceleration on the behavior of light in special relativity. Soon after, he began to think about how to incorporate gravity into his new relativistic framework. His considerations led him from a simple thought experiment involving an observer in free fall to a fully geometric theory of gravity.[132] Most of this work was done in the years 1911–1915, beginning with the publication of a second article on the effect of gravitation on light. By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. In December of 1915, these efforts culminated in Einstein's submission of a paper presenting the Einstein field equations, which are a set of differential equations.[2] Since 1915, the development of general relativity has focused on solving the field equations for various cases. The interpretation of the solutions and their possible experimental and observational testing also constitutes a large part of research in GR.
Starting in 1922, researchers found that cosmological solutions of the Einstein field equations call 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, the Friedmann-Robertson-Walker solution and the Kerr solution.
Observationally, general relativity has accounted for the discrepancy between the Newtonian prediction and observed perihelion precession of Mercury. In 1919, Eddington's announcement that his observations of stars near the eclipsed Sun 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 observations of gravitational red shift, studies of binary pulsars, observations of radio signals passing the limb of the Sun, and the GPS system.
The Golden age of general relativity was a period between approximately 1960 and 1975 when study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics with concepts such as the big bang, black holes, quasars, and pulsars. While there were many contributors to general relativity, the "golden age" is generally regarded as having ended in 1980 when Stephen Hawking proposed that black holes could radiate energy.
[edit] Status
The status of general relativity is decidedly mixed[citation needed].
On the one hand, general relativity is a highly successful model of gravitation and cosmology. It has passed every unambiguous test to which it has been subjected so far, both observationally and experimentally. It is therefore almost universally accepted by the scientific community.
On the other hand, general relativity is inconsistent with quantum mechanics, and the singularities of black holes also raise some disconcerting issues[citation needed]. So while it is accepted, there is also a sense[citation needed] that something beyond general relativity may yet be found.
Currently, better tests of general relativity are needed. Even the most recent binary pulsar discoveries only test general relativity to the first order of deviation from Newtonian projections in the post-Newtonian parameterizations[citation needed]. Some way of testing second and higher order terms is needed, and may shed light on how reality differs from general relativity (if it does).
Any Lorentzian manifold is a solution of the Einstein field equation for some conceivable stress-energy tensor. Thus one must add auxiliary assumptions about the kinds of energy, momentum, and stress in the universe to make any inferences from GTR, e.g. about cosmology.
[edit] 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.