Introduction to general relativity
2008/9 Schools Wikipedia Selection. Related subjects: Mathematics
General relativity (GR) is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915. According to general relativity, the observed gravitational attraction between masses results from those masses warping nearby space and time. Previously, Newton's law of universal gravitation (1686) had described gravity as a force between masses, but experiments have shown that Einstein's description is more accurate. What is more, general relativity predicts interesting new phenomena such as gravitational waves.
General relativity accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the orbits of Mercury and other planets, and it makes numerous predictions – since confirmed – for novel effects of gravity, such as the bending of light and the slowing of time. Although general relativity is not the only relativistic theory of gravity, it is the simplest such theory that is consistent with the experimental data. However, a number of open questions remain: the most fundamental is how general relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity.
The theory has developed into an essential tool for modern astrophysics. It provides the foundation for our current understanding of black holes; these are regions of space where gravitational attraction is so strong that not even light can escape. Their strong gravity is thought to be responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars).
The bending of light by gravity can lead to the curious phenomenon of multiple images of one and the same astronomical object being visible in the sky. This effect is called gravitational lensing, and its study is an active branch of astronomy. Direct evidence of gravitational waves is being sought by several teams of scientists, as in the LIGO and GEO 600 projects; success should allow scientists to study a variety of interesting phenomena, from black holes to the early universe, by analyzing the gravitational waves they produce. General relativity is also the basis of the standard Big Bang model of cosmology.
General relativity  
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From special to general relativity
In 1905, Einstein published his theory of special relativity, which reconciles Newton's laws of motion with electrodynamics (the interaction between objects with electric charge). Special relativity provided a new framework for all of physics by introducing radically new concepts of space and time. However, some thenaccepted physical theories were inconsistent with that framework; a key example was Newton's theory of gravity, which describes the mutual attraction experienced by bodies due to their mass.
Several physicists, including Albert Einstein, attempted to find a theory that would reconcile Newton's law of gravity and special relativity; however, only Einstein's theory ultimately proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow the trajectory of Einstein's thinking between 1907 and 1915, from his simple thought experiment involving an observer in free fall (the "Equivalence Principle") to his fully geometric theory of gravity.
Equivalence principle
A person in a free falling elevator will experience weightlessness during their fall: objects will either float alongside them, or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed. Thus, the experiences of an observer in free fall will be similar to those of an observer in deep space, far from any source of gravity. Such observers are the privileged ("inertial") observers Einstein described in his theory of special relativity: observers for whom light travels along straight lines at constant speed. Such observers feel no acceleration, and they need not introduce what physicists call fictitious forces (such as the force pressing the driver of an accelerating car into his or her seat) to explain what happens around them.
Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and made it the cornerstone of his theory of general relativity. Einstein formalized this idea in his equivalence principle; roughly speaking, this principle states that a person in a freefalling elevator cannot tell that they are in free fall. Every experiment in such a freefalling environment has the same results as it would if the observer were at rest (or moving uniformly) in deep space, far from all sources of gravity.
Gravity and acceleration
Just as most effects of gravity can be made to vanish by observing them in free fall, the same effects can be produced by observing objects in an accelerated frame of reference. An observer in a closed room cannot tell which of the following is true:
 Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
 Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s^{2}. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into his seat.
Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed Einstein to predict several novel effects of gravity in 1907, as explained in the next section.
Einstein's key insight was that the constant pull of gravity we know from everyday experience is basically the same as the fictitious forces felt by an accelerating observer. Since fictitious forces are always proportional to the mass of the object on which they act, an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in Newton's law of gravitation.
Physical consequences
In 1907, having formulated the equivalence principle (and unaware that it would take him another eight years to arrive at the final theory of general relativity), Einstein was already able to derive a number of interesting observable consequences from the equivalence of gravitation and accelerated reference frames.
The first new effect is the gravitational frequency shift of light. Assume that there are two observers, both of them at rest in an accelerating rocketship. Assume also that one observer is higher than the other, where "higher" and "lower" are defined relative to the acceleration: the acceleration drives the "higher" observer "downwards" towards the "lower" observer. If the lower observer sends a light signal to the higher observer, the acceleration causes the light to be redshifted, as may be calculated from special relativity; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the higher observer to the lower is blueshifted (shifted towards higher frequencies). Einstein argued that such frequency shifts must be also observed in a gravitational field, as illustrated in the Figure at left. A light wave is gradually redshifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described below.
This frequency shift corresponds to gravitational time dilation. The observers can use the same light wave to compare their clock rates: since the second observer measures a lower frequency than the first, the second observer's clocks are running faster than those of the first observer. More generally, clocks that are lower in a gravitational field tick more slowly than those that are higher.
It is important to stress that nothing changes locally; for each observer, time flows as normal (Or it can be said, time is relative). Fiveminuteeggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. However, it is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher. This effect is minute, but it too has been confirmed experimentally, as described below.
In a similar way, Einstein also predicted another effect, the gravitational deflection of light: in a gravitational field, light is deflected downward. This can be understood as follows. Suppose that a lightwave passes through a gravitational field so that its left side is "lower" than its right side. By the time dilation effect, the left side will move more slowly than the right side, causing the whole wave to veer to the left (downwards). However, Einstein's numerical predictions in 1907 for this deflection were off by a factor of two; it requires the complete theory of general relativity to get this prediction correct.
Tidal effects
As useful as the equivalence between gravitational and inertial effect might be, it does not constitute a complete theory of gravity. Notably, it cannot answer the following simple question: what keeps the people on the other side of the world from falling off? We might be able to explain gravity near our location on the Earth's surface as a fictitious force – as due to the fact that we have chosen a reference frame that is not in free fall. But a freely falling reference frame on our side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.
A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly – but not completely so: after all, if you look more closely, these bodies are not falling in the same direction, but towards the same point in space: the Earth's centre of gravity. Because of this, there is a component of the motion which accelerates the two bodies towards each other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minute, while for skydivers at the opposite side of the Earth, the effect is much larger.
Whenever bodies fall in different directions or at different rates due to differences in the strength and direction of gravitational forces, the term " tidal effect" is often used, since such differences in force are also responsible for the tides in the Earth's oceans. The equivalence between inertia and gravity cannot explain these tidal effects – it cannot explain the variation of the gravitational field from location to location. For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.
From acceleration to geometry
In exploring the equivalence of gravity and acceleration as well as the role of tidal forces, Einstein had discovered several interesting analogies with the geometry of surfaces. One example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speed) to a rotating reference frame (in which extra terms corresponding to fictitious forces have to be introduced in order to explain particle motion). It is analogous to the transition from a Cartesian coordinate system (in which the coordinate lines are straight lines) to a curved coordinate system.
A deeper analogy relates tidal forces with a property of surfaces called curvature. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature on a surface determines whether or not it is equivalent ( isometric) to a plane. In the summer of 1912, and inspired by these analogies, Einstein went in search of a geometric formulation of gravity.
The elementary objects of geometry – points, lines, triangles – are traditionally defined in threedimensional space or on twodimensional surfaces. In 1907, however, the mathematician Hermann Minkowski introduced a geometric formulation of Einstein's special theory of relativity in which the geometry included not only space, but also time. In this extended geometry, space is replaced by a four dimensional entity called spacetime. The orbits of moving bodies – bodies changing their position in space over time – are lines in spacetime; the orbits of bodies moving at constant speed without changing direction then correspond to straight lines.
For surfaces, the generalization from the geometry of a plane – a flat surface – to that of a general curved surface had been described in the early nineteenth century by Carl Friedrich Gauss, and this had been generalized to higher dimensional spaces in a mathematical formalism introduced by Bernhard Riemann in the 1850s. With the help of Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces.
Even after he had realized the validity of this geometric analogy, it took Einstein three years to find the missing cornerstone of his theory: the equations describing how the matter in a spacetime influences its curvature. Having formulated what are now known as Einstein's equations (or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the Prussian Academy of Sciences in late 1915.
Geometry and gravitation
Paraphrasing the doyen of American relativity research, John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve. What this means requires the understanding of three things: first, the motion of particles which are so small that their effect on the gravitional field they move in is negligible; second, the nature of matter as a source for gravity; third, Einstein's equation, which shows how this matter source is related to the curvature of spacetime.
Probing the gravitational field
In order to map a body's gravitational influence, it is useful to look at what physicists call probe or test particles – particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of spacetime, this is equivalent to saying that such test particles move along straight world lines in spacetime. In the presence of gravity, however, spacetime is nonEuclidean, or curved in an analogous way to a twodimensional surface such as a sphere. In such a spacetime, straight world lines may not exist; instead test particles move along lines called geodesics, which are "as straight as possible". The term geodesic comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely a segment of a great circle, such as a line of longitude or the equator. These paths are not straight because they must follow the curvature of the Earth's surface, but they are as straight as possible subject to this constraint.
The properties of geodesics differ from those of straight lines. For example, in a plane, straight lines that start out in parallel directions will remain at a constant distance from each other. This is not the case for geodesics on the surface of the Earth: for example lines of longitude are parallel at the equator, but intersect at the pole. The world lines of test particles in free fall are spacetime geodesics – they are the straightest possible lines in spacetime – but there are analogous important differences between them and the truly straight lines in the gravityfree spacetime of special relativity. In special relativity, parallel geodesics remain parallel, whereas in a gravitational field with tidal effects, this need not be the case. For example, if two bodies, initially at rest relative to each other, are dropped in the Earth's gravitational field, they will move towards each other as they fall towards the centre of the Earth.
In going from test particles to real matter objects, the laws of motion become somewhat more complicated, but it remains true that spacetime tells matter how to move. Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have comparatively little mass. Where such objects are concerned, the laws governing the behaviour of test particles are perfectly sufficient to describe what happens. Notably, in order to deflect a test particle from its geodesic path, an external force must be applied. A person sitting on a chair is trying to follow a geodesic ( free fall towards the centre of the Earth), but the chair applies an external upwards force preventing the person from falling. In this way, general relativity explains the daily experience of gravity on the surface of the Earth not as the downwards pull of a gravitational force, but as the upwards push of external forces which deflect bodies on the Earth's surface from the geodesics they would otherwise follow.
Einstein's equations
Einstein's equations are the centerpiece of general relativity. They provide a precise formulation, using the language of mathematics, of the relationship between spacetime geometry and the properties of matter.
These equations are formulated using the language of Riemannian geometry, in which the geometric properties of a space (or a spacetime) are described by a quantity called a metric. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).
A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic latitude and longitude. However, unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,051 mi); for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,181 mi). Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime: additional information is needed to convert coordinate differences to real distances. This is precisely the information encoded in the metric: it is a function defined at each point of the surface (or space, or spacetime) which describes how much the space is stretched near that point. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet can be computed from this metric function.
The amount of stretching there is at each point of a space (or spacetime) determines how curved it is. More precisely, the metric function and the rate at which it changes from point to point can be used to define a geometrical quantity called the Riemann curvature tensor, which describes exactly how the space (or spacetime) is curved at each point. In general relativity, the metric and the Riemann curvature tensor are quantities defined at each point in spacetime. The matter content of the spacetime defines another quantity, the Energymomentum tensor T, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related in some way. Einstein found this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity G, now called the Einstein tensor, which describes some aspects of the way spacetime is curved. Einstein's equation then states that
i.e., up to a constant multiple, the quantity G (which measures curvature) is equated with the quantity T (which measures matter content). The constants involved in this equation reflect the different theories that went into its making: G is the gravitational constant that is already present in Newtonian gravity; c is the speed of light, the key constant in special relativity; and π is one of the basic constants of geometry. The appearance of π in the equation is associated with the area (4π) of the unit sphere, whereas the constants c and G are needed to convert the quantity T (which has physical units) into purely geometrical units.
This equation is often referred to in the plural as Einstein's equations, since the quantities G and T are each determined by ten functions of the coordinates of spacetime, and the equations equate each of these component functions. However, because of the freedom to change the four coordinates in which the equations are expressed, there are actually only six physical equations that must be satisfied at every point in spacetime. A solution of these equations describes a particular geometry of space and time; for example, one solution describes the geometry around a spherical, nonrotating mass such as a star or a black hole, whereas another describes a rotating black hole. Still other solutions can describe a gravitational wave or the expansion of the universe. The simplest solution is the uncurved Minkowski spacetime, the spacetime described by special relativity.
Experimental tests
No scientific theory is apodictically true; each is a model that must be checked by experiment. Newton's law of gravity was accepted because it accounted for the motion of planets and moons in the solar system with exquisite accuracy. However, as the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed. These discrepancies were accounted for in the general theory of relativity, but the predictions of that theory must also be checked with experiment. Three experimental tests were devised by Einstein himself and are now known as the classical tests of the theory:
 Newtonian gravity predicts that the orbit of a single planet orbiting a perfectly spherical star should be an ellipse. Einstein's theory predicts a more complicated orbit in which the planet behaves as if it were travelling around an ellipse, but the ellipse itself is slowly rotating around the star at the same time. In the diagram on the right, the ellipse of Newtonian gravity is shown in red, the orbit predicted by Einstein in blue. For a planet orbiting the Sun, this deviation from Newton's orbits is known as the anomalous perihelion shift. The first measurement for Mercury dates back to 1859; 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. General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).
 Einstein's theory predicts that light rays do not follow straight lines in a gravitational field; there is a gravitational deflection. In particular, starlight is deflected as it passes the Sun so that the positions of stars appear shifted by up to 1.75 arc seconds. In contrast, since the photon is a massless particle, a direct application of Newtonian gravity predicts it should not be deflected at all; a more subtle application of Newtonian gravity, discovered in 1804 by J.G. von Soldner using particles of infinitesimal mass, can be used to predict a deflection, but this deflection is half the amount of that predicted by Einstein. These predictions can be tested by observing stars in the Sun's vicinity during a solar eclipse. During the total eclipse of 1919, a British expedition to Brazil and West Africa, directed by Arthur Eddington, confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong. Eddington's results were not very accurate; subsequent 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 better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).
 Gravitational redshift has been measured in a laboratory setting ( Pound and Rebka, 1959) as well as in astrophysics (for instance on Sirius B, with increasingly accurate measurements between the 1950s and 2005); the related gravitational time dilation effect has been measured by transporting atomic clocks to altitudes of up to tens of thousands of kilometers (first by Hafele and Keating in 1971; most accurately to date by Gravity Probe A launched in 1976).
Of these tests, only the perihelion advance of Mercury was known prior to Einstein's final publication of general relativity in 1916. The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted him to international stardom. These three experimental tests justified adopting general relativity over Newton's theory and, incidentally, over a number of alternatives to general relativity that had been proposed.
Further tests of general relativity include precision measurements of the Shapiro effect or gravitational time delay for light (most recently in 2002 by the Cassini space probe) and measurements of effects predicted by general relativity for the behaviour of gyroscopes travelling through space. For example geodetic precession has been tested with Lunar laser ranging experiments (high precision measurements of the orbit of the Moon), while framedragging will be tested by the Gravity Probe B satellite experiment launched in 2004 (with results expected in late 2007).
By cosmic standards, gravity throughout the solar system is comparatively weak. Since the differences between the predictions of Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively strong gravitational fields. This has become possible thanks to precision observations of binary pulsars. In such a star system, two highly compact neutron stars orbit each other, of which one is a pulsar – an astronomical object that sends radio pulses that strike the Earth at very regular intervals. General relativity predicts specific deviations from the regularity of these radio pulses; for instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to the relativistic predictions.
One particular set of observations is part of what is presently the only practical application of general relativity: satellite navigation systems such as the Global Positioning System that are used both for precise positioning and timekeeping. Such systems rely on two sets of atomic clocks: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, the satellite clocks are either slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of universal coordinated time) are testament to the validity of the relativistic predictions.
A number of other tests have probed the validity of various versions of the equivalence principle (strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself). So far, general relativity has passed all observational tests.
Astrophysical applications
Models of the astronomical phenomena based on general relativity play an important role in astrophysics, and the success of these models is further testament to the theory's validity.
Gravitational waves
Gravitational waves, a direct consequence of Einstein's theory, have been detected indirectly in binary stars. Such pairs of stars orbit each other, and gradually lose energy by emitting gravitational waves. This energy loss is normally so gradual that it is difficult to detect. However, in 1974, this energy loss was observed in a binary pulsar called PSR1913+16; the discoverers, Hulse and Taylor, were awarded the Nobel prize in physics in 1993. Since then, several other binary pulsars have been found; the most useful are those in which both stars are pulsars, since they provide the most accurate tests of general relativity. 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 neighbourhood.
Currently, one major goal of research in relativity is the direct detection of gravitational waves. To this end, a number of landbased gravitational wave detectors are in operation, and a mission to launch a spacebased detector, LISA, is currently under development, with a precursor mission ( LISA Pathfinder) due for launch in late 2009. If gravitational waves are detected, they could be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early universe fractions of a second after the Big Bang.
Gravitational lensing
Since light is deflected by mass, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a quasar can pass along one side of a massive galaxy and be deflected slightly so as to reach an observer on Earth, but light passing along the opposite side of that same galaxy could be deflected as well, reaching the same observer, but from a slightly different direction. Similar phenomena are wellknown for optical lenses, which can focus different light rays onto a single point, and so the corresponding gravitational effect is called gravitational lensing. The result is that an observer will see two or more different images of the same astronomical object in the night sky.
Observational astronomy uses lensing effects as an important tool to infer properties of the lensing object: the shape of a lensed image, even if the lensing object is not directly visible, provides information about the mass distribution of the lensing object. In particular, gravitational lensing provides one way to detect dark matter: such lensing configurations are often very large, spanning a significant fraction of the extent of the observable universe. In consequence, they can be used to obtain information about the largescale properties of our cosmos, notably about the Hubble constant, a measure of the universe's ongoing expansion.
Black holes
When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational attraction so strong that not even light can escape. Certain types of black holes are thought to be the final state in the evolution of massive stars. Supermassive black holes with the mass of millions or billions of Suns are thought to be present in the cores of most galaxies. They play a key role in current models of how galaxies have formed over the past billions of years.
Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are quasars and other types of active galactic nuclei. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of jets, in which focused beams of matter are flung away into space at speeds near that of light.
Black holes are promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong. Another reason follows from what are called black hole uniqueness theorems: over time, black holes retain only a minimal set of distinguishing features (since different hair styles are a crucial part of what gives different people their different appearances, these theorems have become known as "no hair" theorems). For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cubeshaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass, but with one important difference: in its transition to a spherical shape, the black hole formed by the collapse of a cube will emit gravitational waves.
Cosmology
One of the most important aspects of general relativity is that it can be applied to the universe as a whole. All current observations suggest that the structure of the cosmos appears to be approximately the same (on average) from every point in space and in every direction of observation; in other words, the universe is approximately homogeneous and isotropic. Such comparatively simple (homogeneous and isotropic) universes can be described by simple solutions of Einstein's equations, and the current cosmological models of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's matter content, namely thermodynamics, nuclear and particle physics. According to these models, our present universe emerged from an extremely dense hightemperature state (the Big Bang) roughly 14 billion years ago, and has been expanding ever since.
Einstein's equations can be generalized by adding a term called the cosmological constant. When this term is present, empty space itself acts as a source of attractive or, unusually, repulsive gravity. This term was originally introduced in Einstein's first paper on cosmology in 1917 to construct a static model of the universe in line with contemporary cosmological thought. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term for aesthetic reasons. However, his reaction proved to be premature. A steadily accumulating body of astronomical evidence, starting from about 1998, has shown that the expansion of the universe is accelerating in a way that suggests the presence of a cosmological constant or, equivalently, of a dark energy with specific properties that pervades all of space.
Modern research: general relativity and beyond
General relativity is very successful in providing an accurate model for an impressive array of physical phenomena, but there are many interesting open questions; in particular, the theory as a whole is almost certainly incomplete.
Notably, in contrast to all other modern theories of fundamental interactions, general relativity is a classical theory which does not include the effects of quantum physics. The question of what a quantum version of general relativity looks like (a theory of quantum gravity) is one of the most exciting and active open problems in physics. While there are promising candidates such as string theory and loop quantum gravity, there is at present no consistent and complete theory of quantum gravity which reconciles Einstein's geometric picture of gravity with the laws of the quantum world. There is a longstanding hope that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of spacetime singularities. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes illdefined and so general relativity itself loses its predictive power. Furthermore, there are singularity theorems which predict that such singularities must exist within the universe if the laws of general relativity were to hold without any quantum modifications. The bestknown examples are the singularities that are believed to exist inside black holes and at the beginning of the universe.
Other attempts to modify general relativity have been made in the context of cosmology: in the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. There have been several (controversial) proposals to obviate the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of cosmic expansion (for example, modified Newtonian dynamics first proposed in 1981, and attempts to eliminate dark energy since the reemergence of the cosmological constant in 1998).
It is possible that another reason to modify Einstein's theory can be found much closer to home: the Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from their expected positions according to known effects (gravitational or otherwise). The possibility of new physics has not been ruled out, despite thorough attempts to find more conventional explanations.
Beyond the challenges of quantum effects and cosmology, the area of general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties Einstein's equations, ever more comprehensive computer simulations of specific spacetimes (such as those describing merging black holes) are run, and the race for the first direct detection of gravitational waves continues apace. More than ninety years after the theory was first published, research is more active than ever.