From classical mechanics to general relativity

The structure of general relativity, as well as the way the theory is formulated, are best understood by examining its similarities with, and departures from, classical physics.

The geometry of Newtonian gravity

At the basis of classical mechanics, there is the notion that in describing a body's motion, we can differentiate between a special type of motion commonly known as free (or inertial) motion and deviations from such free motion; such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion: the force acting on a body is equal to that body's (inertial) mass times its acceleration. There is a direct connection between the preferred inertial motions and the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed; in modern parlance, their paths are geodesics, or straight world lines in space-time.

Conversely, it would seem that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space and a time coordinate. However, there is an ambiguity once gravity comes into play. Following from Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors, there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket.

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time – in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry; however, as can be shown using simple thought experiments, the Newtonian connection is not integrable – space-time is curved. The result is a geometric formulation of Newtonian gravity in geometrical terms. The geometric concepts used are all covariant, in other words: this description can be formulated using any desired coordinate system.

In this geometric formulation, tidal effects – the relative acceleration of bodies in free fall – are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.

Relativistic generalization

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics; in the language of symmetry: where gravity can be neglected, physics is Lorentz invariant, not Galilei invariant – and the differences between the two become significant when we are dealing with fast motions or high-energy phenomena.

Lorentz symmetry introduces an additional conformal structure, namely the set of lightcones (see the image on the left). For each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.

Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. As gravity comes into play, assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles; translated into the language of space-time: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.

A priori, it is not clear whether the new local frames in free fall are indeed those in which the laws of special relativity hold – that theory, is based on the propagation of light, and thus on electromagnetism, and its preferred frames might not be the same as the local free-falling inertial frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift; the actual measurements show that free-falling frames in which light propagates as it does in special relativity. The generalization of this statement, namely that the laws of special relativity hold, to good approximation, in freely falling (and non-rotating) reference frames, is known as the Einstein [equivalence principle], and is one of the guiding principles when it comes to generalizing special relativistic physics to include gravity.

However, again with reference to experimental data, it becomes clear that, for instance, proper time for clocks in a gravitational field, is not measured by the Minkowski metric of special relativity. As in the Newtonian case, this is suggestive of a more general geometry: where all reference frames in free fall are equivalent, and approximately Minkowskian, we are dealing with a curved generalization of Minkowski space: instead of Minkowskian, assume the metric tensor to be, more generally, semi-Riemannian Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, a Levi-Civita connection; assume this to be the connection implied by the universality of free fall.

Einstein's equations

While the preceding section shows the relativistic, geometric version of the effects of gravity, there is still the question of the source of gravity. In Newtonian gravity, the source is mass; in special relativity, mass is equivalent to energy, and in fact turns out to be part of a more inclusive quantity called energy-momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear). Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this and the tensor that embodies tidal effects, the Ricci tensor. Adding a suitable geometric form for the conservation of energy-momentum, the simplest set of equations are what are called Einstein's (field) equations, which equate the energy-momentum tensor and a specific combination of the Ricci tensor and the metric known as the Einstein tensor:

$G\_\{ab\}\; =\; kappa,\; T\_\{ab\},$

where G_ab is the Einstein tensor, T_ab is the energy-momentum tensor (both written in abstract index notation). Matching the theory's prediction to observational results for planetary orbits, the proportionality constant has the value $kappa\; =\; 8pi\; G/c^4,$ with $G$ the gravitational constant and $c$ the speed of light. The tensors G_ab and T_ab are both rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations; the fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations.

While the metric description of gravity follows rather straightforwardly from special relativity and the universality of free fall, it is worth mentioning that there are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans-Dicke theory, teleparallelism, and Einstein-Cartan theory.

General relativity: definition and basic applications

As a result of the derivation sketched in the previous section, we now have all the information needed to define and characterize general relativity.

Definition and basic properties

General relativity is a metric theory of gravitation. Its core are Einstein's equations, which link the geometry of a four-dimensional, semi-Riemannian manifold representing space-time with the energy-momentum contained in that space-time. Phenomena that, in classical mechanics, are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity. The curvature is, in turn, caused by the energy-momentum of matter; paraphrasing the relativist John Archibald Wheeler, space-time tells matter how to move; matter tells space-time how to curve.

While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases: for weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of gravity.

As it is constructed using tensors, general relativity exhibits general covariance, that is, its laws – and further laws formulated within the general relativistic framework – take on the same form in all coordinate systems. Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers. Locally, as expressed in the equivalence principle, space-time is Minkowskian, and the laws of physics have local Lorentz invariance.

Model-building

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equation and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates) on which are defined specific matter fields, in such a way that matter and geometry satisfy Einstein's equations, and that the matter satisfies whatever equations have been imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.

Einstein's equations are non-linear partial differential equations and, as such, very difficult to solve. Nevertheless, a number of exact solutions are known, although only a few of them have direct physical applications. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe, and the Friedmann-Lemaître-Robertson-Walker and de Sitter universes, each describing an expanding cosmos. Exact solutions of great theoretical interest include the Gödel universe, the Taub-NUT solution, and Anti-de Sitter space.

In addition, significant efforts are being made in the field of numerical relativity, where the goal is to find interesting numerical solutions describing, say, two black holes orbiting each other, with the help of powerful computers.

Also, there are different methods for finding approximate solutions in the context of perturbation theory. The best-known of these are linearized gravity and its generalization, the Post-Newtonian expansion, which represents a systematic way of describing a space-time containing matter which is not particularly compact and moves but slowly compared with the speed of light; the description starts with Newtonian gravity and, in a systematic sequence, takes into account smaller and smaller effects arising from the difference between Newton's theory and general relativity. An extension of this expansion is the Parametrized Post-Newtonian (PPN) formalism, a framework of testing general relativity against alternative theories in a way that allows quantitative comparisons.

Consequences of Einstein's theory

General relativity has a number of consequences, some following directly from the theory's axioms, others having become clear only in the course of the ninety years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift

In general relativity (and, in fact, in any theory in which the equivalence principle holds), 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..

The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound and Rebka and later confirmed by astronomical observations. There are numerous direct measurements of gravitational time dilation using atomic clocks while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS). Tests in stronger gravitational fields are provided by the observation of binary pulsars. All results are in agreement with general relativity; however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.

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. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion), several effects of gravity on light propagation 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 be derived by extending the universality of free fall to light, 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. 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.

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; later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders. 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. 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 $gamma$ that reflects the influence of gravity on the geometry of space.

The angular deflection of a beam of light may be represented by the following equation: $A.D.=4GM/c^2R$ where A.D. is the angular deflection of the light beam, G is the gravitational constant, M is the mass of the object causing the deflection, c² is the speed of light squared, and R is the radius of the object (such as a star) that is causing the deflection. This is equivalent to saying $A.D=CM/R$ where C is 1.75 arc seconds (the amount of deflection caused by the sun), M is the mass of the star divided by the mass of the Sun, and R is the radius of the star divided by the radius of the Sun.

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.

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). 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.

It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there is no 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 or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves, 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 presently the only way to construct appropriate models.

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 concerns the relativistic apside shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction.

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; the result can also be obtained by using either the exact Schwarzschild metric (describing spacetime around a spherical mass) or the much more general post-Newtonian formalism. 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).

An early success of general relativity was that the theory offered a straightforward explanation for an anomalous perihelion shift of the planet Mercury, which had been discovered by Urbain Le Verrier in 1859 but had remained mysterious. This agreement between theory and experiment confirmed for Einstein that he had at last identified the correct form of the gravitational field equations. More recent observations have shown that the field equations predict the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth). The effect has also been checked in binary pulsar systems where it is larger by five orders of magnitude.

Orbital decay

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.

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor using binary pulsar PSR1913+16 they had discovered in 1974; it amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993. Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars.

Geodetic precession and frame-dragging

Several relativistic effects are directly related to the relativity of direction. 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"). For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging; more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.

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. Such effects can again be tested through their influence on the orientation of a gyroscope in free fall: somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction; a precision measurement is the main aim of the Gravity Probe B mission, whose final results are expected in May 2008.

Astrophysical applications

Gravitational lensing

The deflection of light by gravity can have an intriguing side effect: a massive object between the observer and a distant target object makes it possible for the observer to see multiple distorted images of the target. This and similar effects are known as gravitational lensing 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. The earliest example was discovered in 1979; since then, more than a hundred gravitational lenses have been observed. 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.

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.

Gravitational wave astronomy

From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational waves (see the section on Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, this being one of the major goals of current relativity-related research. 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. A joint US-European mission to launch a space-based detector, LISA, is currently under development, with a precursor mission (LISA Pathfinder) due for launch in late 2009.

Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation: 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. 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.

Black holes and other compact objects

Whenever an object becomes sufficiently compact, general relativity predicts the formation of a black hole: a region of space from which nothing, not even light, can escape. 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. 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, and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.

Astronomically, the most important property of compact objects is that they provide a superbly efficient mechanism for converting gravitational into radiation energy. Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as Microquasars. In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed. General relativity plays a central role in modelling all these phenomena, relativistic lensing effects being thought to play a role for the signals received from X-ray pulsars.

Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"), observations of stellar dynamics in the center of our own Milky Way galaxy, and indications that at least some of the compact objects in question appear to have no solid surface provide strong indirect evidence for the existence of black holes. Direct evidence, such as observing the "shadow" of the Milky Way galaxy's central black hole horizon, 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; 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; 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.

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 $Lambda$, an additional term that has an important influence on the large-scale dynamics of the cosmos,

$G\_\{ab\}\; +\; Lambda\; g\_\{ab\}\; =\; kappa,\; T\_\{ab\}$

where g_ab is the spacetime metric.

On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker solutions, 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. Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation, 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, which is in good agreement with astronomical observations; the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation, and the large-scale distribution of galaxies.

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, and they have proven a sound basis to explaining the evolution of the universe's large-scale structure. On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observations) 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 or otherwise. 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; the nature of this new form of energy remains unclear.

A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous") 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. While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario, problems remain. There is a bewildering variety of possible inflationary scenarios not restricted by current observations. 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 (cf. the section Quantum gravity, below).

Advanced concepts

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 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.

Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are now termed 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.

Cosmic partitions: horizons

One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called horizons, which demarcate one spacetime region from the rest of the spacetime. 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. However, matter and radiation may cross the horizon into the black hole - this illustrates the idea that horizons are not physical barriers that act as 'blockers'. The resulting object is known as a black hole, and the surface in question as the black hole's horizon. The hoop conjecture states when a black hole is expected to form: every mass $M$ determines a length known as the Schwarzschild radius,

$r\_s=frac\{2GM\}\{c^2\},$

where $G$ is the gravitational constant and $c$ the speed of light. Imagine a circular hoop with the circumference $2pi\; r\_s$. A mass $M$ small enough to fit through that hoop, regardless of their relative orientation, is compact enough to form a black hole.

Initial black hole studies relied on simplified models obtained from explicit solutions of Einstein's equation, especially the spherically-symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe 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 like hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.

Even more remarkably, 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. This law sets a limit to the energy that can be extracted from a rotating black hole (e.g. by the Penrose process). 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: 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.

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 be influenced (event horizon); in both cases, the location of the horizon in spacetime depends on the event in question. Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons (associated with a semi-classical radiation known as Unruh radiation).

Singularities

Another general – and quite disturbing – feature of general relativity is the appearance of spacetime 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 timelike and lightlike 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. By definition, these are spacetime singularities. In more interesting cases, the geometrical quantities characterizing spacetime curvature (e.g. the Ricci scalar) take on infinite values at such "curvature singularities". 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, or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole. The Friedmann-Lemaître-Robertson-Walker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely big bang singularities.

Given just these examples, which are all highly symmetric and thus simplified, one might think the occurrence of singularities to be an idealization. The famous singularity theorems proved using the methods of global geometry suggest 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 and also at the beginning of a wide class of expanding universes. 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). 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 hypothesis (Penrose 1969); while no formal proof of this conjecture exists, numerical simulations offer supporting evidence of its validity.

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).

Nevertheless, in order to understand Einstein's equations 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, where spacetime is split into three space dimensions and one time dimension, such as the ADM formalism. These decompositions show that the spacetime evolution equations of general relativity are indeed well-behaved, meaning that solutions always exist and are uniquely defined (once suitable initial conditions are specified). 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.

Global and quasi-local quantities

The notion of evolution equations is intimately tied in with another aspect of general 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 the gravitational field - like any physical field - must be ascribed a certain energy. However, it is fundamentally impossible to localize that energy.

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass) or suitable symmetries (Komar mass) 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. Just as in classical physics, it can be shown that these masses are positive. Analogous global definitions exist for momentum and angular momentum. In addition, there have been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated 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.

Relationship with quantum theory

Along with general relativity, quantum theory, the basis of our understanding of matter from elementary particles to solid state physics is considered one of the two pillars of modern physics. However, it is still an open question of how the concepts of quantum theory can be reconciled with those of general relativity.

Quantum field theory in curved spacetime

The unification of quantum theory and special relativity has led to the highly successful quantum field theories which form the basis of modern elementary particle physics. These theories are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.

Short of constructing a theory of quantum gravity, in which all interactions, including general relativity's description of gravity, are formulated within the framework of quantum theory, there is a way to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself: use classical general relativity to describe a curved background space-time, and define a generalized quantum field theory to describe the behavior of quantum matter within that space-time. The corresponding models have led to highly interesting results. Most notably, they indicate that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that black holes evaporate over time. As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.

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.

Einstein gravity is nonrenormalizable

General relativity is presently known to be incompatible with quantum mechanics, meaning that if the gravitational field is analysed using the ordinary rules of quantum field theory, physical quantities are found to be 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".

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 nonrenormalizable theories. The result is a theory that works correctly at low energies, though such a theory cannot be considered a theory of everything as it cannot be self-consistently extended to the high-energy realm.

Proposed quantum gravity theories

General relativity fits nicely into the effective field theory formalism and makes sensible predictions at low energies. 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 as yet undefined 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, the M-theory approach 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.

History

Soon after publishing his theory of special relativity in 1905, Einstein 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 the equivalence principle and thence to a fully geometric theory of gravity: from explorations of some consequences of the equivalence principle such as the influence of gravity and acceleration on the propagation of light published in 1907 to the main work in the years 1911 to 1915 with the realization of the role of differential geometry (with help from Marcel Grossmann on the intricacies of that field of mathematics) and a long search, including detours and false starts, for the field equations relating geometry and the mass-energy content of spacetime. In December of 1915, these efforts culminated in Einstein's presentation to the Prussian Academy of Science of the Einstein field equations.

Already in 1916, Schwarzschild found the eponymous solution to the Einstein field equations, laying the groundwork for the description of gravitational collapse and, eventually, black holes. The same year saw the first steps of generalization to electrically charged objects that would result in the Reissner-Nordström solution. In 1917, Einstein initiated the field of relativistic cosmology. However, in line with contemporary thinking, he tried to describe a static universe, adding the cosmological constant to his original field equations for that purpose. When it became clear in 1929 with the work of Hubble and others that our universe is indeed expanding (and thus better described by expanding cosmological solutions found by Friedmann in 1922), Lemaître formulated the earliest version of the big bang models.

During all that time, general relativity remained something of a curiosity among physical theories. There was evidence that it was indeed to be preferred to Newton's description: Einstein himself had shown in 1915 how it explained the anomalous perihelion advance of the planet Mercury, and a 1919 expedition led by Eddington had announced confirmation of general relativity's prediction for the deflection of the light of distant stars by the Sun (instantly catapulting Einstein to world fame). Yet it was only with the developments between approximately 1960 and 1975, now known as the Golden age of general relativity, that the theory entered the mainstream of theoretical physics and astrophysics, as both the theoretical basis of black holes as well as their astrophysical applications (quasars) became clear, ever more precise solar system tests confirmed the theory's predictive power, and relativistic cosmology, too, became amenable to direct observational tests.

Status

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.

Although it is widely accepted by the scientific community, general relativity is inconsistent with quantum mechanics. Furthermore, the nature of the Gravitational singularity remains an important open question in general relativity.

Currently, better tests of general relativity are needed. Even the most recent binary pulsar discoveries only test general relativity to the third post-Newtonian (3PN) approximation in the post-Newtonian parameterizations with testing of higher order approximations under way that may shed light on how reality differs from general relativity (if it does).

See also

• Classical theories of gravitation
• Contributors to general relativity
• David Hilbert
• Einstein-Hilbert action
• General relativity resources, an annotated reading list giving bibliographic information on some of the most cited resources.
• Golden age of general relativity
• History of general relativity
• Introduction to mathematics of general relativity
• Mathematics of general relativity

Notes

References