GENERAL RELATIVITY

(Redirected from General theory of relativity)

'General relativity' (GR) (aka 'general theory of relativity' (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. Die Feldgleichungun der Gravitation, , Albert, Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, The Foundation of the General Theory of Relativity, , Albert, Einstein, Annalen der Physik, 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 by its use of the Einstein field equations to relate space-time content and space-time curvature.
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 Eddington announced that observations of stars near the eclipsed Sun confirmed general relativity's prediction that massive objects bend light. Since then, many 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.
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.
In spite of its overwhelming success, there is discomfort with general relativity in the scientific community due to its being incompatible with quantum mechanics and the reachable singularities of black holes (at which the mathematics of general relativity breaks down). Because of this, numerous other theories have been proposed as alternatives to general relativity. An early and still-popular class of modifications is Brans-Dicke theory, which, although not solving the problems of singularities and quantum gravity, appeared to have observational support in the 1960s. However, those observations have since been refuted and modern measurements indicate that any Brans-Dicke type of deviation from general relativity must be very small if it exists at all.
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Justification

Main articles: Equivalence principle

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.^[1] 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.^[2] 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).^[3]^[4] 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:
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.^[5]
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.^[6] This effort came to fruition with the discovery of the Einstein field equations in 1915.

Fundamental principles

Two-dimensional analogy of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity. Note that the white lines do not represent the curvature of space, but instead represent the coordinate system imposed on the curved spacetime which would be rectilinear in a flat spacetime

One of the defining features of general relativity is the idea that gravitational 'force' is replaced by 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.
In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy and momentum (or stress-energy) within it. The relationship between stress-energy and the curvature of spacetime is described by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime (inertial motion) occurs along special paths called timelike and null geodesics of spacetime.^[7]
General relativity is 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 local Lorentz invariance requires that the laws of special relativity apply locally for all inertial observers. The curvature of spacetime is created by stress-energy within the spacetime as described by the Einstein field equations. Moreover, this curvature permits gravitational effects to be described as a form of inertial motion. 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.

Mathematical framework

Main articles: Mathematics of 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 forces 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.

Geometry

Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole

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.

Two bodies falling towards the center of the Earth accelerate towards each other as they 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.

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, the elegance of a flat spacetime and the ability to use a preferred coordinate system are lost (due to stress-energy curving spacetime and the principle of general covariance). Consequently, coordinate and physical accelerations become sundered. In the case of someone standing on the Earth, where they are at rest with respect to the surface coordinates for the Earth (latitude, longitude, and elevation) but are undergoing a continuous physical acceleration because the mechanical resistance of the Earth's surface keeps them from free-falling.

Einstein field equations

Main articles: 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
:

G_{ab} = kappa, T_{ab}

where ''G_ab'' is the Einstein tensor, ''T_ab'' is the stress-energy tensor and

kappa

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 ''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 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

kappa

in the EFE is determined to be

kappa = 8 pi G / c^4

by making these two approximations.
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).

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.

Gravitational waves

Main articles: Gravitational waves

Ring of test particles floating in space

Ring of test particles whose relative distances are changed by a passing gravitational wave

There are several analogies between gravity and electromagnetism. For weak fields, gravity is governed by Newton's law, which has the same form as Coulomb's law of electrostatics; magnetic fields have their analogy in gravitomagnetism. As such, it comes as no surprise that electromagnetic waves have their analogue, as well: gravitational waves; ripples in spacetime that propagate at the speed of light.^[8]
The simplest way to describe gravitational waves (and the one in which the analogy is best apparent) is in a weak field approximation, in which deviations from Newtonian gravity are assumed to be small, and minute perturbations of a flat spacetime background are studied. The resulting gravitational waves correspond to small perturbations of the metric, which can be visualized using 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). Each such simple wave is completely characterized by its frequency, amplitude (the maximal relative distortion factor for lengths) and polarization (the orientation of the directions of maximal distortion).^[9]
There is a limited analogy of linearized gravitational waves with electromagnetic waves: in both cases, the monopole moment is conserved, so there are no spherical waves; in both cases, the waves are transversal. There are, however, also important differences: the time derivative of the dipole moment for gravitational waves is the total momentum, a conserved quantity. Hence, dipole waves, dominant in electrodynamics, are absent; the dominant contribution is made by quadrupole waves, and in computing the quadrupole moment of a given mass distribution, one can calculate to good approximation the gravitational waves it emits.^[10]
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.^[11] 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^[12] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,^[13] 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.^[14]

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 (geodesic precession and, in the presence of rotating bodies, frame-dragging).

Precession of apsides

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star

The first difference is in the prediction that apsides of orbits (that is, the points at which an orbiting body most closely approaches the system's center of mass) will precess on their own – such an 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 a corresponding solution of the geodetic equation;^[15] similar calculations can be performed using the exact Schwarzschild metric, which describes a model spacetime containing a single, spherically symmetric mass.^[10] More precise calculations, in which the smaller body is not treated as a test particle but as a mass with its own gravitational influence can be performed using the post-Newtonian formalism; this calculation can also be generalized to include additional bodies.^[10]
In Newton's theory of gravity, a precession of this kind will generically occur whenever the motion of more than one planet around a central mass is described; however, as early as 1859, astronomers realized that for the planet Mercury, there was an unexplained excess periapside shift of 43 arcseconds per century, an "anomalous perihelion shift".^[18] One of the earliest successes of general relativity (in fact, one of the criteria used by Einstein in his search for the final form of his field equations) was the correct prediction of this additional precession, since known as the anomalous perihelion shift . The most accurate results for Mercury and for other planets to date are based on measurements using radio telescopes that were undertaken between 1966 and 1990.^[19] General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).^[20]
More recently, binary pulsar systems have been found in which the observed effects are much larger – where Mercury takes millions of years to complete one rosetta (i.e. for the apsis to precess 360 degrees), it takes these star systems mere tens of years.^[10]In the parameterized post-Newtonian formalism (PPN), a framework for testing both general relativity and related theories of gravity, measurements of this effect can be used to determine a linear combination of the terms

eta

and

gamma

, indicating that their magnitude depends on the influence of gravity on the geometry of space and on the way that self energy contributes to a body's gravity (in other words, the special kind of nonlinearity exhibited by Einstein's theory).^[18]

Orbital decay and gravitational waves

Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.

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 increases, and so does their orbital period. Under ordinary circumstances (such as within the solar system or for ordinary double stars), the effect is impossible to observe. The situation changed with the discovery in 1974 of the first binary pulsar, which has been given the designation PSR1913+16, by Hulse and Taylor. Pulsars are neutron stars that emit a narrow beam of electromagnetic radiation from their poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses is useful as a highly accurate "clock" that reports on the activity in its neighborhood.^[23] In a binary pulsar (a system with two neutron stars orbiting each other), the compactness of the system makes for an appreciable loss of energy through gravitational wave emission, while the pulsar provides a highly accurate means of keeping track of the system's orbital period. The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor; it amounts to first indirect detection of gravitational waves, and the discoverers were awarded the Nobel Prize in physics in 1993.^[24] 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.^[25]

Geodetic precession and frame-dragging

Main articles: Geodetic precession, Frame dragging

Reflector placed on the Moon by the Apollo 11 mission to enable lunar laser ranging measurements

Another class of relativistic effects is directly related to the relativity of direction – with spacetime distortions due to mass and other sources of gravity, it becomes impossible to define uniquely when two directions at two different spacetime points are the same and when they are different.^[26] This leads to interesting relativistic effects.
The first such effect is called geodetic precession: as a gyroscope moves through curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars – even though the gyroscope has been in free fall, moving in the way that comes closest to keeping its axis direction constant (in technical terms, the gyroscope has been parallel transported).^[13]
The effects of geodetic precession on the Moon-Earth-system – a gigantic gyroscope in itself – have been measured with the help of lunar laser ranging experiments.^[28] More recently, the prediction of geodetic precession was tested and verified as an auxiliary result of the Gravity Probe B experiment to a precision of better than 1 percent.^[10]
The situation is somewhat different close to a mass that is rotating. For a distant observer, it will seem as if any object close to the mass gets "dragged around" with the rotation; this is most extreme for rotating black holes: once an object approaches closely enough (once it enters the zone known as the ergosphere), rotation is inevitable.^[30] The related effects are collectively known as gravitomagnetic or frame-dragging effects; in particular, it is once more the variation in the axis direction of a gyroscope in free fall that can be used to test the influence of a rotating mass on its surroundings.^[31] There have been (somewhat controversial) tests of these effect using the LAGEOS satellites leading to a confirmation of the relativistic prediction;^[32] a precision measurement is the main aim of the Gravity Probe B mission, whose results are due in late 2007.^[33]

Time delay and gravitational frequency shift

Main articles: Gravitational time dilation

Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body

In general relativity, electromagnetic radiation travels at the speed of light along the straightest possible worldlines in spacetime (in other words, along
light-like geodesics); also, time as measured by a clock is determined only by that clock's motion and its position in spacetime (more precisely, what a clock shows is the proper time associated with its world line). These two assumptions have two immediate consequences. 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. In other words, 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 delay.^[34].
Both effects have been tested by observation or direct experiment, with all the results in agreement with general relativity:^[35]
A laboratory measurement of the gravitational redshift was performed as early as 1959 by Pound and Rebka^[36] Astronomical observations involving the Sun and White Dwarf stars (notably Sirius B) followed suit.^[37] After the advent of atomic clocks, direct measurements of the gravitational time delay became possible, starting with the Hafele-Keating experiment and culminating in the Gravity Probe A experiment.^[38] Ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS), which involves the comparison of atomic clocks on the ground and aboard orbiting satellites.^[18] Tests in stronger gravitational fields are provided by the observation of binary pulsars.^[40]
It is important to note that, for weak and approximately stationary gravitational fields, gravitational redshift and time delay can be derived without employing the full machinery of general relativity, using only the weak equivalence principle and Newton's description of gravity. In fact, Einstein derived them along these lines as early as 1907. Hence, the current observational tests are tests of the (weak) equivalence principle, not of the whole of general relativity.^[41]

Propagation of light

The most famous early test of the general theory of relativity was made possible by observation of the 1919 solar eclipse. According to Sir Arthur Stanley Eddington, starlight could be seen to bend around the sun as it made its way to the observer on earth.

In contrast with Newton's theory, general relativity sets up strict rules for the propagation of light and other types of electromagnetic radiation, which are a generalization of earlier such laws: even in classical physics, light travels at constant speed along straight lines; in curved space-time, it travels along the straightest-possible world lines, namely along geodesics. In special relativity, the speed of light is invariant; in general relativity, this special property is embodied in a special condition for the tangent vectors of light geodesics. The resulting world-lines for light are called light-like or null geodesics.^[42]
There are several ways to model spacetime containing separate massive objects. Where only a single, sufficiently spherical object is important, the (exterior) Schwarzschild solution provides a good approximation; for more richly populated systems such as a Solar system, the so-called Post-Newtonian expansion provides a framework for systematically adding relativistic correction terms to a Newtonian universe.^[43] Examination of light propagation in such model universes reveals several interesting relativistic effects.

Bending of light

The best-known is the bending of light in a gravitational field. Heuristic derivations using Newtonian gravity and a corpuscular theory of light date back to 1801;^[44] the same result can be obtained by assuming that the geometry of space is Euclidean and that the equivalence principle holds.^[45] Einstein published one such derivation in 1907;^[44] however, such heuristic derivations give a maximal angle of deflection with only half the value given by general relativity. From the standpoint of general relativity, 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 by similar subsequent measurements.^[48] More modern observations of the deflection of the light of distant quasars by the Sun, which utilize highly accurate techniques of radio astronomy, have confirmed Eddington's results with significantly higher accuracy.^[49]

Gravitational time delay

Main articles: Shapiro delay

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;^[50] later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.^[51] 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.^[25]
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.^[12]

Causal structure and global geometry

Main articles: Causal spacetime structure

Penrose diagram of an infinite Minkowski universe

In special relativity, light gives spacetime a causal structure. No material body can travel faster than light, and neither can any influence one object can have on another. In general relativity, this is still true. No material body can catch up with or overtake a light signal; no influence from an event A can reach any other location before light sent out at A does so. Thus, an exploration of all light worldlines within a given spacetime – the spacetime's null geodesics – yields key information about the spacetime's structure. So-called Penrose-Carter diagrams are a way of displaying this causal structure. In the creation of these maps, spacetime is compactified – infinitely large regions of space and infinite time intervals are shrunk so as to fit onto a finite map while preserving structure; light still travels along diagonals as in standard spacetime diagrams.^[54]
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.^[55]

Horizons

Main articles: Horizon (general relativity)

One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called horizons, which serve to segregate one region of spacetime from the rest of the world. There are several ways of definining horizons. The two most important ones are the event horizon, the boundary from inside which no light and no particles can escape to infinity, and the apparent horizon, the boundary from inside which no light and no particles can escape to the outside; both are defined globally for a whole spacetime.^[56] In addition, there are ways to define horizons in a more intuitive way as isolated horizons, namely surfaces that can be defined for isolated systems without knowing spacetime properties at infinity.^[25]
The best-known examples for regions bounded by a horizon 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. Since, in general relativity, no object can overtake a light beam, all inside matter is imprisoned, as well. Such surfaces are known as (black hole) horizons, and the resulting object is known as a black hole.^[58] More precisely, the horizon is a null hypersurface, and for a black hole in asymptotically flat spacetime, neither null nor timelike geodesics from the inside will never reach infinity. The so-called hoop conjecture makes more precise when a black hole is expected to form: With every mass

M

, one can associate a length known as the Schwarzschild radius,

mathcal{R}=rac{2GM}{c^2},

where

G

is the gravitational constant and

c

the speed of light. Imagine a circular hoop with the circumference

2pimathcal{R}

. If the mass is small enough to fit through that hoop regardless of their relative orientation, then it is compact enough to form a black hole.^[59]
The first studies of black holes relied on 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 making use of global geometric methods have revealed more general properties of black holes: In the long run, black holes are rather simple objects. Stationary black holes are characterized by a minimal set of parameters: their mass, angular momentum, and electric charge. This is the result of what are called the black hole uniqueness theorems; as the statement of these theorems is sometimes phrased more whimsically as "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.^[60]
Even more remarkable, there is a general set of laws known as black hole mechanics which are analogous to the laws of thermodynamics. The ''zeroth law'' states that the surface gravity of a stationary black hole has a constant value on its horizon, just as the temperature is constant throughout a system in thermodynamic equilibrium. The ''first law'' states how small changes in the mass of a stationary black hole are related in a specific way to small changes in its area, its angular momentum and its charge, just as changes in the internal energy of a system are related to changes in its generalized volume, temperature, and particle number; among other things, this law sets a limit to the energy that can be extracted from a rotating black hole by means of the so-called Penrose process.^[10] The ''second law'' states that the area of the event horizon of a general black hole will never decrease with time, just as the entropy of a thermodynamic system. By the ''third law'', it is impossible to reduce the surface gravity to zero in a finite number of steps, in analogy with Nernst's theorem of thermodynamics.^[62] 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:^[13] 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.^[64]
Horizons also play a role for other kinds of solutions. In an expanding universe, there can be particle horizons, the limit to which parts of the past can be observed, and event horizons limiting which parts of the universe can be influenced by a given event. In both cases, the location of the horizon in spacetime depends on the events under study.^[25] Even in flat Minkowski space, as described by an accelerated observer (Rindler space), there will be horizons;^[66] these come with their own semi-classical radiation known as Unruh radiation.^[67]

Singularities

Main articles: Spacetime singularity

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 time-like and light-like geodesics – all potential worldlines of particles in free fall, and of electromagnetic radiation. In a way, this amounts to defining space as everywhere that particles and light can go, given time. But in some spacetimes that are solutions of Einstein's equations, there are regions where time-like and light-like singularities come to an abrupt end, and spacetime geometry becomes ill-defined. This, by definition, are spacetime singularities.^[13] 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 (a space-like singularity shielded by a horizon),^[69] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole (a time-like singularity, also shielded by a horizon).^[70] 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.^[71]
Given these examples, one might think the singularities related to the unrealistically perfect symmetries exhibited by the exact solution, and thus an artefact of idealization. Proving otherwise is one of the remarkable triumphs of globally geometric methods: the singularity theorems prove conclusively that singularities are a generic feature of general relativity; they are unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage ("trapped surfaces"),^[10] and conversely at the beginning of a wide class of expanding universes.^[10]
While the singularity theorems make wide-ranging statements about the conditions under which singularities occur, they say very little about their properties – are we dealing with a ''conical singularity'', similar to the tip of the cone, where tangent vectors simply jump? or, more interesting for physics, with a ''curvature singularity'', where quantities associated with curvature (such as the Ricci scalar or other invariants) take on infinite values? Much of current research on singularities is devoted to finding more general information about their properties, e.g. to testing the so-called BKL conjecture that posits a rather simple structure for singularities, namely that the neighbouring spacetime is dominated by nonlinear gravitational effects (so whatever matter there may be can be neglected), and that the gravitational field's variation from location to location can be neglected, and local geometry described locally to good approximation studying time-dependence only.^[74]
As problematic as singularities are, there are indications that all but very few of them are safely hidden away. This is postulated by the cosmic censorship conjecture (Penrose 1969): in all physically reasonable spacetimes (namely those in which matter has realistic properties, and where no symmetry is perfect), any singularity that is formed will be hidden behind a horizon, and thus invisible for all distant observers (naturally, this does not extend to initial singularities such as the big bang singularity). While no formal proof of this conjecture exists, numerical simulations offer supporting evidence that it is, indeed, valid.^[75]

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: they are self-interacting (that is, non-linear even in the absence of other fields; they are diffeomorphism invariant, so two obtain a unique solution, a fixed background metric and gauge conditions need to be introduced; finally, the metric determines the spacetime structure, and thus the domain of dependence for any set of initial data, so the region on which a specific solution will be defined is not, a priori, defined.^[25]
There is, however, a way to re-formulate Einstein's equations that overcomes these problems. First of all, there are ways of re-writing spacetime as the evolution of "space" in time; an earlier version of this is due to Paul_Dirac, while a simpler way is known after its inventors Arnowitt, Deser and Misner as ADM formalism. In these formulations, also known as "3+1" approaches, spacetime is split into a three-dimensional hypersurface with interior metric and an embedding into spacetime with exterior curvature; these two quantities are the dynamical variables in a Hamiltonian formulation tracing the hypersurface's evolution over time.^[77] With such a split, it is possible to state the initial value formulation of general relativity. It involves initial data which cannot be specified arbitrarily but needs to satisfy specific constraint equations, and which is defined on some suitably smooth three-manifold

Sigma

; just as for other differential equations, it is then possible to prove existence and uniqueness theorems, namely that there exists a unique spacetime which is a solution of Einstein equations, which is globally hyperbolic, for which

Sigma

is a Cauchy surface (i.e. all past events influence what happens on

Sigma

, and all future events are influenced by what happens on it), and has the specified internal metric and extrinsic curvature; all spacetimes that satisfy these conditions are related by isometries.^[78]
The initial value formulation with its 3+1 split is the basis of numerical relativity; attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers.^[10] However, there are significant differences to the simulation of other physical evolution equations which make numerical relativity especially challenging, notably the fact that the dynamical objects that are evolving include space and time itself (so there is no fixed background against which to evaluate, for instance, perturbations representing gravitational waves) and the occurrence of singularities (which, when they are allowed to occur within the simulated portion of spacetime, lead to arbitrarily large numbers that would have to be represented in the computer model).^[80]

Global and quasi-local quantities

Main articles: Mass in general relativity

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 "gravitational field energy" is not a part of the energy-momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situation it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the Stress-energy-momentum pseudotensor, this separation is not true for all observers, and there is no general (in other words, covariant) definition for obtaining it.^[25]
How, then, does one define a concept as a system's total mass – which is easily defined in classical mechanics? As it turns out, at least for spacetimes which are asymptotically flat (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space), the ADM 3+1 split mentioned in the previous section leads to a solution: as in the usual Hamiltonian formalism, the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the ADM mass (or, equivalently, ADM energy).^[12] Alternatively, there is a possibility to define mass for a spacetime that is stationary, in other words, one that has a time-like Killing vector field (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called Komar mass^[83] Although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes.^[84] The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity. In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity.^[85] Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of positivity: if there were no lower limit, than no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that Minkowski space (for which both are zero) is indeed stable.^[86] While the focus here has been on energy, it should be noted that analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.^[47]
The disadvantage of all the definitions mentioned so far is that they are defined only at (null or spatial) infinity; since the 1970s, physicists an mathematicians have worked on the more ambitious endeavor of defining suitable ''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. However, while there is a variety of proposed definitions such as the Hawking energy, the Geroch energy or Penrose's quasi-local energy-momentum based on twistor methods, the field is still in flux. Eventually, the hope is to use a suitable defined quasi-local mass to give a more precise formulation of the hoop conjecture, prove the so-called Penrose inequality for black holes (relating the black hole's mass to the horizon area) and find a quasi-local version of the laws of black hole mechanics.^[88]

Astrophysical applications

Gravitational lensing

Main articles: Gravitational lensing

Einstein cross: four images of the same astronomical object, produced by a gravitational lens

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, then it is possible that the observer might see multiple distorted images of the target! This and similar effects are known as gravitational lensing. The earliest example, two images of one and the same quasar, was discovered in 1979.^[10] Since then, many other examples of distant galaxies and quasars being affected by gravitational lensing have been found.^[90]
An image that has been lensed by a point-like mass will typically result in two images or, in the case where the observer, the lensed object and the lens mass are exactly lined up, a bright ring known as an Einstein ring. If the more distant object is slightly offset to one side and/or the gravitational field is not uniform, partial rings (called arcs) will appear instead.^[91] To date, more than a hundred gravitational lenses have been observed.^[92] A distorted image that cannot be resolved by an observer on Earth can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object. This effect is known as microlensing, and such events are now regularly observed.^[8]
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.^[10]

Gravitational wave astronomy

Main articles: Gravitational waves

Artist's impression of the space-borne gravitational wave detector LISA

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.^[95] 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.^[96]
A joint US-European mission to launch a space-based detector, LISA, is currently under development,^[10] with a precursor mission (LISA Pathfinder) due for launch in late 2009.^[10]
Gravitational waves have a number of features that promise to make them an important astronomical tool. In many ways, they are complementary to the electromagnetic waves that are the astronomers' main source of information: electromagnetic waves have small wavelengths compared to the linear extent of the systems under consideration – they can be used to make detailed pictures; gravitational waves, in contrast, carry coherent information about overall properties. Electromagnetic waves are easily scattered, while gravitational waves interact with matter only very weakly, and thus promise access to otherwise inaccessible regions (such as the core of a supernova or the earliest phases of the universe).^[25]
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); depending on deviations from symmetry, they could detect signals from core-collapse supernovae; they could detect periodic sources such as rotating neutron stars with small deformations; 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.^[10] For a space-based detectors like LISA, there are well-known sources in binaries consisting of two White Dwarfs and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star); detecting these sources will be an important check that the detector in question is working as expected. Also, LISA should observe the mergers of supermassive black holes as well as 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 with greatly increased sensitivity.^[10]

Black holes and other compact objects

Main articles: Black holes

Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves

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.^[10] 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,^[47] and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.^[104]
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.^[105] 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.^[106] Typically, matter will collect in an accretion disks whence it gradually spirals into the black hole; however, complex interactions between the accreting matter and magnetic fields can lead to spectacular phenomena such as relativistic jets, focused beams of highly energetic particles that are being flung into space at almost the speed of light.^[107] For modelling all these phenomena, general relativity plays a central role, be it for predicting stellar end-states,^[108] describing the supernovae in which they are formed,^[109]or simulating accretion and the formation of jets.^[25] Also, relativistic lensing effects are thought to play a role for the signals we receive from X-ray pulsars.^[25]
While observations of accretion-driven phenomena, and the associated estimates for the extent and mass of the central objects (e.g. via the limiting Eddington luminosity) are an important part of current evidence for the existence of black holes,^[10] there are other types of evidence as well. Notably, observations of stellar dynamics in the center of our own Milky Way galaxy – stars orbiting an almost entirely invisible central mass – give estimates for mass and compactness that exclude all other reasonable candidate objects safe for a supermassive black hole.^[25] It should be noted, however, that for all evidence that relies on a mass and size estimate, part of the argument is that, using currently accepted physics, the only astronomical body beyond a certain compactness is a black hole. In this context, indirect evidence for the existence of black hole horizons provided by X-ray bursts is of great interest; roughly speaking, the occurrence of such bursts depends on the accretion of material onto solid surfaces, and available data is consistent with the current classification of X-ray sources into those driven by a central neutron star of up to 1.4 solar masses, which has a surface, and those driven by black holes, which are more massive and have no solid surface;^[20] there is also the possibility of future observation of the "shadow" of the horizon of the Milky Way galaxy's central black hole.^[25]
In the context of the current search for gravitational waves (see the section Gravitational waves, above), black holes are of interest as sources of such waves. For instance, 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 and the resulting gravitational waves are one of the main goals of current research in numerical relativity;^[25] for the phase immediately preceeding the merger ("chirp"), there is hope that it might 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;^[25] 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.^[47]

Cosmology

Main articles: Physical 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.^[119]

Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope WMAP

On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker solutions,^[10] cosmologists, astronomers and particle physicists have built the models of modern cosmology in which the current state of the universe has evolved over roughly 14 billion years from a hot, early Big bang phase;^[44] this is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous.^[25] The Friedmann-Lemaître-Robertson-Walker solutions are a whole family of solutions. Their exact form depends on a number of parameters; one possible set is the age of the universe, the amount of matter it contains (given, for instance, by the mean matter density at the present time) and the value of the cosmological constant. A complete description of the universe requires a number of other parameters to be fixed, as well, notably the number ratio of photons to baryonic matter particles. Part of the astronomical data can be used to fix the parameters values;^[123] further data can then be used to put the cosmological model to the test. Predictions for cosmic times between a few seconds and a few hundred thousand years include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,^[124] which is in good agreement with astronomical observations;^[125] the existence of a "thermal echo" from the early cosmos, namely the cosmic background radiation^[83], whose properties have since been measured to great precision by satellite observatories such as COBE or WMAP.^[127] Further observational data that can either be used to fix some of the parameters, or else to test the model, comes from galaxy statistics (how many how far away, at which redshifts?) based on wide-ranging surveys.^[128]
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,^[129] and they have proven a sound basis to explaining the evolution of the universe's large-scale structure (such as galaxy clusters, super-clusters and their spatial distribution).^[130] On the other hand, there are a number of important open questions. The first concerns the nature of matter found in the universe. According to the determination of the cosmological parameters, 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); moreover, all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter").^[25] While this is in accord with other astronomical observations, such as the dynamics of galaxies and galaxy clusters,^[132] evidence from gravitational lensing,^[25] and simulations of large-scale structure formation,^[10] there is currently no generally accepted description of this new kind of matter within the framework of particle physics;^[135] some physicists have even questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity.^[8]
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.^[137] Here, too, no conventional explanation of this energy (say, in terms of the vacuum energy of conventional or new quantum fields) has yet been found; here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models.^[138]
The classical cosmological models face a number of further problems, such as the exact flatness of space following from observations (flatness problem), problems in explaining the high degree of homogeneity of the cosmic background radiation (horizon problem) and the observed rarity of certain species of exotic particles that should have been produced in the early universe (monopole problem).^[139] These have been the motivation of introducing an additional phase of strongly accelerated expansion at cosmic times of around

10^{-}

seconds, known as an inflationary phase.^[140] While recent measurements of the cosmic background radiation have resulted in first evidence for the inflationary scenario,^[10] problems remain. Present inflationary scenarios are phenomenological; the potential function that is crucial in determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory. Also, even with an inflationary phase, there remains the question of 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^[10] (cf. the section Quantum gravity, below).

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.

Quantum field theory in curved spacetime

:Main articles: 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.

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" .
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.

Proposed quantum gravity theories

General relativity fits nicely into the effective field theory formalism and makes sensible predictions at low energies.^[143] 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.
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.

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, 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,

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