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General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the fourmomentum (massenergy and linear momentum) of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
Many predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, the gravitational redshift of light, and the gravitational time delay. General relativity's predictions have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and selfconsistent theory of quantum gravity.
Einstein's theory has important astrophysical implications. It points towards the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an endstate for massive stars. There is evidence that such stellar black holes as well as more massive varieties of black hole are 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 phenomenon of gravitational lensing, where multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been measured indirectly; a direct measurement is the aim of projects such as LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
History
First page from Einstein's manuscript explaining general relativity
Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eightyear search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the November, 1915 presentation to the Prussian Academy of Science of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter is present, and form the core of Einstein's general theory of relativity.^{[1]}
The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found the first nontrivial exact solution to the Einstein field equations, the socalled Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which eventually resulted in the ReissnerNordström solution, now associated with charged black holes.^{[2]} In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to reproduce that "observation".^{[3]} By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the big bang models, in which our universe has evolved from an extremely hot and dense earlier state.^{[4]} Einstein later declared the cosmological constant the biggest blunder of his life.^{[5]}
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors").^{[6]} Similarly, a 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919,^{[7]} making Einstein instantly famous.^{[8]} Yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between approximately 1960 and 1975, now known as the Golden age of general relativity. Physicists began to understand the concept of a black hole, and to identify these objects' astrophysical manifestation as quasars.^{[9]} Ever more precise solar system tests confirmed the theory's predictive power,^{[10]} and relativistic cosmology, too, became amenable to direct observational tests.^{[11]}
From classical mechanics to general relativity
General relativity is best understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit of a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.^{[12]}
Geometry of Newtonian gravity
At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration.^{[13]} The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in spacetime.^{[14]}
Ball falling to the floor in an accelerating rocket (left), and on
Earth (right)
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passivegravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.^{[15]} A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket.^{[16]}
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the freefall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (timelike vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The result is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system.^{[17]} In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.^{[18]}
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.^{[19]} In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.) The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with highenergy phenomena.^{[20]}
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see the image on the left). The lightcones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observerindependent.^{[21]} In conjunction with the worldlines of freely falling particles, the lightcones can be used to reconstruct the spacetime's semiRiemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure.^{[22]}
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight timelike lines that define a gravityfree inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.^{[23]}
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the specialrelativistic frames (such as their being earthfixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that freefalling frames are the ones in which light propagates as it does in special relativity.^{[24]} The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and nonrotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing specialrelativistic physics to include gravity.^{[25]}
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi or pseudoRiemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the LeviCivita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).^{[26]}
Einstein's equations
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energymomentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear).^{[27]} Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energymomentum corresponds to the statement that the energymomentum tensor is divergencefree. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curvedmanifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energymomentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations:
On the lefthand side is the Einstein tensor, a specific divergencefree combination of the Ricci tensor R_{ab} and the metric. In particular,
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
On the righthand side, T_{ab} is the energymomentum tensor. All tensors are written in abstract index notation.^{[28]} Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weakgravity, lowspeed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c^{4}, with G the gravitational constant and c the speed of light.^{[29]} When there is no matter present, so that the energymomentum tensor vanishes, the result are the vacuum Einstein equations,
There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are BransDicke theory, teleparallelism, and EinsteinCartan theory.^{[30]}
Definition and basic applications
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for modelbuilding.
Definition and basic properties
General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a fourdimensional, semiRiemannian manifold representing spacetime on the one hand, and the energymomentum contained in that spacetime on the other.^{[31]} Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as freefall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightestpossible paths that objects will naturally follow.^{[32]} The curvature is, in turn, caused by the energymomentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.^{[33]}
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric ranktwo tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.^{[34]}
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.^{[35]} Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.^{[36]} Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.^{[37]}
Modelbuilding
The core concept of generalrelativistic modelbuilding is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semiRiemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energymomentum tensor must be divergencefree. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.^{[38]}
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.^{[39]} Nevertheless, a number of exact solutions are known, although only a few have direct physical applications.^{[40]} The bestknown exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the ReissnerNordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,^{[41]} and the FriedmannLemaîtreRobertsonWalker and de Sitter universes, each describing an expanding cosmos.^{[42]} Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the TaubNUT solution (a model universe that is homogeneous, but anisotropic), and Antide Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).^{[43]}
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.^{[44]} In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity^{[45]} and its generalization, the postNewtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.^{[46]} An extension of this expansion is the parametrized postNewtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.^{[47]}
Consequences of Einstein's theory
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of the ninety years of research that followed Einstein's initial publication.
Gravitational time dilation and frequency shift
Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body
Assuming that the equivalence principle holds,^{[48]} gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.^{[49]}
Gravitational redshift has been measured in the laboratory^{[50]} and using astronomical observations.^{[51]} Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,^{[52]} while ongoing validation is provided as a sideeffect of the operation of the Global Positioning System (GPS).^{[53]} Tests in stronger gravitational fields are provided by the observation of binary pulsars.^{[54]} All results are in agreement with general relativity.^{[55]} However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.^{[56]}
Light deflection and gravitational time delay
General relativity predicts that the path of light is bent in a gravitational field; light passing a massive body is deflected towards that body. This effect has been confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.^{[57]}
Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray)
This and related predictions follow from the fact that light follows what is called a lightlike or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.^{[58]} As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the postNewtonian expansion),^{[59]} several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,^{[60]} the angle of deflection resulting from such calculations is only half the value given by general relativity.^{[61]}
Closely related to light deflection is the gravitational time delay (or Shapiro effect), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.^{[62]} In the parameterized postNewtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.^{[63]}
Gravitational waves
Ring of test particles floating in space
Ring of test particles influenced by gravitational wave
One of several analogies between weakfield gravity and electromagnetism is that, analogous to electromagnetic waves, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light.^{[64]} The simplest type of such a wave can be visualized by its action on a ring of freely floating particles (upper image to the right). A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (lower, animated image to the right).^{[65]} Since Einstein's equations are nonlinear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, for weak fields, a linear approximation can be made. Such linearized gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from faroff cosmic events, which typically result in relative distances increasing and decreasing by 10 ^{− 21} or less. Dataanalysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.^{[66]}
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space^{[67]} or socalled Gowdy universes, varieties of an expanding cosmos filled with gravitational waves.^{[68]} But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.^{[69]}
Orbital effects and the relativity of direction
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.
Precession of apsides
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star
In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess—the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curvelike shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of the anomalous perihelion shift of the planet Mercury, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.^{[70]}
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)^{[71]} or the much more general postNewtonian formalism.^{[72]} It is due to the influence of gravity on the geometry of space and to the contribution of selfenergy to a body's gravity (encoded in the nonlinearity of Einstein's equations).^{[73]} Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus and the Earth),^{[74]} as well as in binary pulsar systems, where it is larger by five orders of magnitude.^{[75]}
Orbital decay
Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.
^{[76]}
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.^{[77]}
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.^{[78]} Since then, several other binary pulsars have been found, in particular the double pulsar PSR J07373039, in which both stars are pulsars.^{[79]}
Geodetic precession and framedragging
Several relativistic effects are directly related to the relativity of direction.^{[80]} One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").^{[81]} For the MoonEarthsystem, this effect has been measured with the help of lunar laser ranging.^{[82]} More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.^{[83]}
Near a rotating mass, there are socalled gravitomagnetic or framedragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.^{[84]} Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.^{[85]} Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.^{[86]}. Also the Mars Global Surveyor probe around Mars has been used ^{[87]} ^{[88]}; see the entry framedragging for an account of the debate. A precision measurement is the main aim of the Gravity Probe B mission, with the results expected in September 2008.^{[89]}
Astrophysical applications
Gravitational lensing
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.^{[90]} Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.^{[91]} The earliest example was discovered in 1979;^{[92]} since then, more than a hundred gravitational lenses have been observed.^{[93]} Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.^{[94]}
Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.^{[95]}
Gravitational wave astronomy
Artist's impression of the spaceborne gravitational wave detector
LISA
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, which is a major goal of current relativityrelated research.^{[96]} Several landbased gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.^{[97]} A joint USEuropean spacebased detector, LISA, is currently under development,^{[98]} with a precursor mission (LISA Pathfinder) due for launch in 2012.^{[99]}
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.^{[100]} They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.^{[101]}
Black holes and other compact objects
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and 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.^{[102]} Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,^{[103]} and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.^{[104]}
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.^{[105]} Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellarsize objects such as microquasars.^{[106]} In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.^{[107]} General relativity plays a central role in modelling all these phenomena,^{[108]} and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.^{[109]}
Black holes are also soughtafter targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.^{[110]} The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about supermassive black hole's geometry.^{[111]}
Cosmology
The current models of cosmology are based on Einstein's equations including cosmological constant Λ, which has important influence on the largescale dynamics of the cosmos,
where g_{ab} is the spacetime metric.^{[112]} Isotropic and homogeneous solutions of these enhanced equations, the FriedmannLemaîtreRobertsonWalker solutions,^{[113]} allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase.^{[114]} Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,^{[115]} further observational data can be used to put the models to the test.^{[116]} Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,^{[117]} the largescale structure of the universe,^{[118]} and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.^{[119]}
Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope
WMAP
Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90 percent of all matter appears to be socalled dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.^{[120]} There is no generally accepted description of this new kind of matter, within the framework of known particle physics^{[121]} or otherwise.^{[122]} Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also 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, known as dark energy, the nature of which remains unclear.^{[123]}
A socalled inflationary phase,^{[124]} an additional phase of strongly accelerated expansion at cosmic times of around 10 ^{− 33} seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.^{[125]} Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.^{[126]} However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.^{[127]} An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed^{[128]} (cf. the section on quantum gravity, below).
Advanced concepts
Causal structure and global geometry
Penrose diagram of an infinite
Minkowski universe
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using PenroseCarter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.^{[129]}
Aware of the importance of causal structure, Roger Penrose and others developed what is 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, and additional nonspecific assumptions about the nature of matter (usually in the form of socalled energy conditions) are used to derive general results.^{[130]}
Horizons
Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The bestknown examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius^{[131]}), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.^{[132]}
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. In the long run, they are rather simple objects characterized by eleven parameters specifying energy, linear momentum, angular momentum, location at a specified time and electric charge. This is stated by the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.^{[133]}
Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process).^{[134]} There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.^{[135]} This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with nonzero temperature, black holes should emit thermal radiation. Semiclassical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).^{[136]}
There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon).^{[137]} Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semiclassical radiation known as Unruh radiation.^{[138]}
Singularities
Another general—and quite disturbing—feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes illdefined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.^{[139]} Wellknown examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,^{[140]} or the Kerr solution with its ringshaped singularity inside an eternal rotating black hole.^{[141]} The FriedmannLemaîtreRobertsonWalker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely big bang singularities, and some have future singularities (big crunch) as well.^{[142]}
Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artefact of idealization. The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage^{[143]} and also at the beginning of a wide class of expanding universes.^{[144]} However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the socalled BKL conjecture).^{[145]} The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.^{[146]}
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, possibly matterfilled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.^{[147]}
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in socalled "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The bestknown example is the ADM formalism.^{[148]} These decompositions show that the spacetime evolution equations of general relativity are wellbehaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.^{[149]} Such formulations of Einstein's field equations are the basis of numerical relativity.^{[150]}
Global and quasilocal quantities
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.^{[151]}
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)^{[152]} or suitable symmetries (Komar mass).^{[153]} If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the socalled Bondi mass at null infinity.^{[154]} Just as in classical physics, it can be shown that these masses are positive.^{[155]} Corresponding global definitions exist for momentum and angular momentum.^{[156]} There have also been a number of attempts to define quasilocal quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.^{[157]}
Relationship with quantum theory
If general relativity is considered one of the two pillars of modern physics, quantum theory, the basis of understanding matter from elementary particles to solid state physics, is the other.^{[158]} However, it is still an open question as to how the concepts of quantum theory can be reconciled with those of general relativity.
Quantum field theory in curved spacetime

Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.^{[159]} In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on classical general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.^{[160]} Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that they evaporate over time.^{[161]} As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.^{[162]}
Quantum gravity
The demand for consistency between a quantum description of matter and a geometric description of spacetime,^{[163]} as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.^{[164]} Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.^{[165]}
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.^{[166]} At very high energies, however, the result are models devoid of all predictive power ("nonrenormalizability").^{[167]}
Simple
spin network of the type used in loop quantum gravity
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute onedimensional extended objects.^{[168]} The theory promises to be a unified description of all particles and interactions, including gravity;^{[169]} the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.^{[170]} In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity^{[171]} form part of a hypothesized elevendimensional model known as Mtheory, which would constitute a uniquely defined and consistent theory of quantum gravity.^{[172]}
Another approach starts with the canonical quantization procedures of quantum theory. Using the initialvalueformulation of general relativity (cf. the section on evolution equations, above), the result is the WheelerdeWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be illdefined.^{[173]} However, with the introduction of what are now known as Ashtekar variables,^{[174]} this leads to a promising model known as loop quantum gravity. Space is represented by a weblike structure called a spin network, evolving over time in discrete steps.^{[175]}
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,^{[176]} there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being dynamical triangulations,^{[177]} causal sets,^{[178]} twistor models^{[179]} or the pathintegral based models of quantum cosmology.^{[180]}
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.^{[181]}
Current status
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed every unambiguous observational and experimental test. Even so, there are strong indications the theory is incomplete.^{[182]} The problem of quantum gravity and the question of the reality of spacetime singularities remain open.^{[183]} Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics,^{[184]} and while the socalled Pioneer anomaly might yet admit of a conventional explanation, it, too, could be a harbinger of new physics.^{[185]} Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,^{[186]} and increasingly powerful computer simulations (such as those describing merging black holes) are run.^{[187]} The race for the first direct detection of gravitational waves continues apace,^{[188]} in the hope of creating opportunities to test the theory's validity for much stronger gravitational fields than has been possible to date.^{[189]} More than ninety years after its publication, general relativity remains a highly active area of research.^{[190]}
See also
Notes
 ^ This development is traced in chapters 9 through 15 of Pais 1982 and in Janssen 2005; an uptodate collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, p. 110ff.. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15.
 ^ See Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918).
 ^ Einstein 1917, cf. Pais 1982, ch. 15e.
 ^ Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4.
 ^ As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the sectionCosmology, below.
 ^ Cf. Pais 1982, p. 253–254.
 ^ Cf. Kennefick 2005 and Kennefick 2007.
 ^ Cf. Pais 1982, ch. 16.
 ^ Cf. Israel 1987, ch. 7.8–7.10 and Thorne 1994, ch. 3–9.
 ^ Cf. the sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein.
 ^ Cf. the section Cosmology and references therein; the historical development is traced in Overbye 1999.
 ^ The following exposition retraces that of Ehlers 1973, section 1.
 ^ See, for instance, Arnold 1989, chapter 1.
 ^ See Ehlers 1973, pp. 5f..
 ^ See Will 1993, section 2.4 or Will 2006, section 2.
 ^ Cf. Wheeler 1990, chapter 2; similar accounts can be found in most other popularscience books on general relativity.
 ^ See Ehlers 1973, section 1.2, Havas 1964, and Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959.
 ^ See Ehlers 1973, pp. 10f..
 ^ Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006.
 ^ An indepth comparison between the two symmetry groups can be found in Giulini 2006a.
 ^ For instance Rindler 1991, section 22; a thorough treatment can be found in Synge 1972, ch. 1 and 2.
 ^ E.g. Ehlers 1973, sec. 2.3.
 ^ Cf. Ehlers 1973, sec. 1.4. and Schutz 1985, sec. 5.1.
 ^ See Ehlers 1973, p. 17ff.; a derivation can be found e.g. in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below.
 ^ Cf. Rindler 2001, sec. 1.13; for an elementary account, see chapter 2 of Wheeler 1990; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985.
 ^ Ehlers 1973, sec. 1.4. for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with nonzero torsion leads to a modified theory known as EinsteinCartan theory.
 ^ Cf. Ehlers 1973, p. 16; Kenyon 1990, sec. 7.2; Weinberg 1972, sec. 2.8.
 ^ See Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergencefree tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3.
 ^ E.g. Kenyon 1990, sec. 7.4.
 ^ Cf. Brans & Dicke 1961 and section 3 in ch. 7 of Weinberg 1972, Goenner 2004, sec. 7.2, and Trautman 2006, respectively.
 ^ E.g. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other textbook on general relativity.
 ^ At least approximately, cf. Poisson 2004.
 ^ E.g. p. xi in Wheeler 1990.
 ^ E.g. Wald 1984, sec. 4.4.
 ^ E.g. in Wald 1984, sec. 4.1.
 ^ For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b.
 ^ E.g. section 5 in ch. 12 of Weinberg 1972.
 ^ Cf. the introductory chapters of Stephani et al. 2003.
 ^ A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996.
 ^ For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006.
 ^ E.g. chapters 3, 5, and 6 of Chandrasekhar 1983.
 ^ E.g. ch. 4 and sec. 3.3. in Narlikar 1993.
 ^ Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5.
 ^ See Lehner 2002 for an overview.
 ^ For instance Wald 1984, sec. 4.4.
 ^ E.g. Will 1993, sec. 4.1 and 4.2.
 ^ Cf. section 3.2 of Will 2006 as well as Will 1993, ch. 4.
 ^ Cf. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198.
 ^ Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5.
 ^ PoundRebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186.
 ^ E.g. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond & Holberg 2005.
 ^ Starting with the HafeleKeating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186.
 ^ GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003.
 ^ Reviews are given in Stairs 2003 and Kramer 2004.
 ^ General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, section 4.2.
 ^ Cf. Ohanian & Ruffini 1994, pp. 164–172.
 ^ Cf. Kennefick 2005 for the classic early measurements by the Eddington expeditions; for an overview of more recent measurements, see Ohanian & Ruffini 1994, chapter 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004.
 ^ This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, section 5.
 ^ A brief descriptions and pointers to the literature can be found in Blanchet 2006, section 1.3.
 ^ See Rindler 2001, section 1.16; for the historical examples, Israel 1987, p. 202–204.; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997.
 ^ From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11.
 ^ For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, with a pedagogical introduction to be found in Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, section 4.4.
 ^ Will 1993, sec. 7.1 and 7.2.
 ^ These have been indirectly observed through the loss of energy in binary pulsar systems such as the HulseTaylor binary, the subject of the 1993 Nobel Prize in physics. A number of projects are underway to attempt to observe directly the effects of gravitational waves. For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Unlike electromagnetic waves, the dominant contribution for gravitational waves is not the dipole, but the quadrupole; see Schutz 2001.
 ^ Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9.
 ^ For example Jaranowski & Królak 2005.
 ^ Rindler 2001, ch. 13.
 ^ See Gowdy 1971, Gowdy 1974.
 ^ See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy.
 ^ See Schutz 2003, pp. 48–49 and Pais 1982, pp. 253–254.
 ^ See Rindler 2001, sec. 11.9.
 ^ See Will 1993, pp. 177–181.
 ^ In consequence, in the parameterized postNewtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3.
 ^ The most precise measurements are VLBI measurements of planetary positions; see Will 1993, chapter 5, Will 2006, section 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407.
 ^ See Kramer et al. 2006.
 ^ A figure that includes error bars is figure 7, in section 5.1, of Will 2006.
 ^ See Stairs 2003 and Schutz 2003, pp. 317–321; an accessible account can be found in Bartusiak 2000, pp. 70–86.
 ^ An overview can be found in Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994.
 ^ Cf. Kramer 2004.
 ^ See e.g. Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, sec. §11.4.
 ^ See Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8.
 ^ See Bertotti, Ciufolini & Bender 1987 and, for a more recent review, Nordtvedt 2003.
 ^ See Kahn 2007.
 ^ E.g. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471.
 ^ E.g. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004.
 ^ E.g. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009
 ^ Iorio L. (2006). "COMMENTS, REPLIES AND NOTES: A note on the evidence of the gravitomagnetic field of Mars". Classical Quantum Gravity 23 (17): 5451–5454. doi:10.1088/02649381/23/17/N01.
 ^ Iorio L. (2009). "On the LenseThirring test with the Mars Global Surveyor in the gravitational field of Mars". Central European Journal of Physics. doi:10.2478/s1153400901176.
 ^ A mission description can be found in Everitt et al. 2001; a first postflight evaluation is given in Everitt et al. 2007; further updates will be available on the mission website Kahn 1996–2008.
 ^ For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998.
 ^ For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3.
 ^ See Walsh, Carswell & Weymann 1979.
 ^ Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007.
 ^ For an overview, see Roulet & Mollerach 1997.
 ^ See Narayan & Bartelmann 1997, sec. 3.7.
 ^ For an overview, Barish 2005; accessible accounts can be found in Bartusiak 2000 and Blair & McNamara 1997.
 ^ An overview is given in Hough & Rowan 2000.
 ^ See Danzmann & Rüdiger 2003.
 ^ See Landgraf, Hechler & Kemble 2005.
 ^ Cf. Thorne 1995.
 ^ Cf. Cutler & Thorne 2002.
 ^ See Miller 2002, lectures 19 and 21.
 ^ E.g. Celotti, Miller & Sciama 1999, sec. 3.
 ^ Cf. Springel et al. 2005 and the accompanying summary Gnedin 2005.
 ^ Cf. Blandford 1987, section 8.2.4
 ^ For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996.
 ^ For a review, see Begelman, Blandford & Rees 1984. Interestingly, to a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see Rees 1966.
 ^ For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from Xray pulsars, cf. Kraus 1998.
 ^ The evidence includes limits on compactness from the observation of accretiondriven phenomena ("Eddington luminosity"), see Celotti, Miller & Sciama 1999, observations of stellar dynamics in the center of our own Milky Way galaxy, cf. Schödel et al. 2003, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of Xray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000.
 ^ Cf. Dalal et al. 2006.
 ^ E.g. Barack & Cutler 2004.
 ^ Originally Einstein 1917; cf. the description in Pais 1982, pp. 285–288.
 ^ See Carroll 2001, ch. 2.
 ^ See Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million lightyears and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991.
 ^ E.g. with WMAP data, see Spergel et al. 2003.
 ^ These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003.
 ^ See Peebles 1966; for a recent account of predictions, see Coc et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005.
 ^ A review can be found in Lahav & Suto 2004 and Bertschinger 1998; for more recent results, see Springel et al. 2005.
 ^ Cf. Alpher & Herman 1948 and, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997.
 ^ Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. chapter 18 of Peebles 1993, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of largescale structure formation, see Springel et al. 2005.
 ^ See Peacock 1999, ch. 12, and Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("nonbaryonic matter"), cf. Peacock 1999, ch. 12.
 ^ Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9.
 ^ See Carroll 2001; an accessible overview is given in Caldwell 2004. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2007.
 ^ A good introduction is Linde 1990; for a more recent review, see Linde 2005.
 ^ More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1.
 ^ See Spergel et al. 2007, sec. 5 & 6.
 ^ More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory.
 ^ See Brandenberger 2007, sec. 2.
 ^ See Frauendiener 2004, Wald 1984, section 11.1, and Hawking & Ellis 1973, section 6.8 & 6.9
 ^ E.g. Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6.
 ^ See Thorne 1972; for an account of more recent numerical studies, see Berger 2002, sec. 2.1.
 ^ For an account of the evolution of this concept, see Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004.
 ^ For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results.
 ^ The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see chapter 2 of Wald 2001. A thorough, booklength introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969.
 ^ See Bekenstein 1973, Bekenstein 1974.
 ^ The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in chapter 3 of Wald 2001.
 ^ Cf. Narlikar 1993, sec. 4.4.4 and 4.4.5.
 ^ Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, chapter 3.
 ^ See Hawking & Ellis 1973, section 8.1, Wald 1984, section 9.1.
 ^ See Townsend 1997, chapter 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, chapter 3.
 ^ See Townsend 1997, chapter 4; for a more extensive treatment, cf. Chandrasekhar 1983, chapter 6.
 ^ See Ellis & van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
 ^ Namely when there are trapped null surfaces, cf. Penrose 1965.
 ^ See Hawking 1966.
 ^ The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007.
 ^ The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a textbook level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1.
 ^ Cf. Hawking & Ellis 1973, sec. 7.1.
 ^ Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, §21.4–§21.7.
 ^ FourèsBruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998.
 ^ See Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001.
 ^ Cf. Misner, Thorne & Wheeler 1973, §20.4.
 ^ Arnowitt, Deser & Misner 1962.
 ^ Cf. Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & MagnonAshtekar 1979.
 ^ For a pedagogical introduction, see Wald 1984, sec. 11.2.
 ^ See the various references given on p. 295 of Wald 1984; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states.
 ^ E.g. Townsend 1997, ch. 5.
 ^ Such quasilocal massenergy definitions are the Hawking energy, Geroch energy, or Penrose's quasilocal energymomentum based on twistor methods; cf. the review article Szabados 2004.
 ^ An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003.
 ^ Cf. textbooks such as Ramond 1990, Weinberg 1995, or Peskin & Schroeder 1995; a more accessible overview can be found in Auyang 1995.
 ^ Cf. Wald 1994 and Birrell & Davies 1984.
 ^ For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000.
 ^ Cf. chapter 3 in Wald 2001.
 ^ Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. section 2 in Carlip 2001.
 ^ E.g. p. 407ff. in Schutz 2003.
 ^ A timeline and overview can be found in Rovelli 2000.
 ^ See Donoghue 1995.
 ^ In particular, a technique known as renormalization, an integral part of deriving predictions which take into account higherenergy contributions, cf. chapters 17 and 18 of Weinberg 1996, fails in this case; cf. Goroff & Sagnotti 1985.
 ^ An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b.
 ^ At the energies reached in current experiments, these strings are indistinguishable from pointlike particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3 and 5.3.
 ^ E. g. Green, Schwarz & Witten 1987, sec. 4.2.
 ^ E.g. Weinberg 2000, ch. 31.
 ^ E.g. Townsend 1996, Duff 1996.
 ^ Cf. section 3 in Kuchař 1973.
 ^ These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987.
 ^ For a review, see Thiemann 2006; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003.
 ^ See e.g. the systematic expositions in Isham 1994 and Sorkin 1997.
 ^ See Loll 1998.
 ^ See Sorkin 2005.
 ^ See ch. 33 in Penrose 2004 and references therein.
 ^ Cf. Hawking 1987.
 ^ E.g. Ashtekar 2007, Schwarz 2007.
 ^ Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
 ^ Cf. the section Quantum gravity, above.
 ^ Cf. the section Cosmology, above.
 ^ See Nieto 2006.
 ^ See Friedrich 2005.
 ^ A review of the various problems and the techniques being developed to overcome them, see Lehner 2002.
 ^ See Bartusiak 2000 for an account up to that year; uptodate news can be found on the websites of major detector collaborations such as GEO 600 and LIGO.
 ^ For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2007; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006.
 ^ A good starting point for a snapshot of presentday research in relativity is the electronic review journal Living Reviews in Relativity.
References
 Alpher, R. A.; Herman, R. C. (1948), "Evolution of the universe", Nature 162: 774–775, doi:10.1038/162774b0
 Anderson, J. D.; Campbell, J. K.; Jurgens, R. F.; Lau, E. L. (1992), "Recent developments in solarsystem tests of general relativity", in Sato, H.; Nakamura, T., Proceedings of the Sixth Marcel Großmann Meeting on General Relativity, World Scientific, pp. 353–355, ISBN 9810209509
 Arnold, V. I. (1989), Mathematical Methods of Classical Mechanics, Springer, ISBN 3540968903
 Arnowitt, Richard; Deser, Stanley; Misner, Charles W. (1962), "The dynamics of general relativity", in Witten, Louis, Gravitation: An Introduction to Current Research, Wiley, pp. 227–265
 Arun, K.G.; Blanchet, L.; Iyer, B. R.; Qusailah, M. S. S. (2007), Inspiralling compact binaries in quasielliptical orbits: The complete 3PN energy flux, arΧiv:0711.0302
 Ashby, Neil (2002), "Relativity and the Global Positioning System" (PDF), Physics Today 55(5): 41–47, doi:10.1063/1.1485583, http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf
 Ashby, Neil (2003), "Relativity in the Global Positioning System", Living Reviews in Relativity 6, http://relativity.livingreviews.org/Articles/lrr20031/index.html, retrieved 20070706
 Ashtekar, Abhay (1986), "New variables for classical and quantum gravity", Phys. Rev. Lett. 57: 2244–2247, doi:10.1103/PhysRevLett.57.2244
 Ashtekar, Abhay (1987), "New Hamiltonian formulation of general relativity", Phys. Rev. D36: 1587–1602, doi:10.1103/PhysRevD.36.1587
 Ashtekar, Abhay (2007), Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions, arΧiv:0705.2222
 Ashtekar, Abhay; Krishnan, Badri (2004), "Isolated and Dynamical Horizons and Their Applications", Living Rev. Relativity 7, http://www.livingreviews.org/lrr200410, retrieved 20070828
 Ashtekar, Abhay; Lewandowski, Jerzy (2004), "Background Independent Quantum Gravity: A Status Report", Class. Quant. Grav. 21: R53–R152, doi:10.1088/02649381/21/15/R01, arΧiv:grqc/0404018
 Ashtekar, Abhay; MagnonAshtekar, Anne (1979), "On conserved quantities in general relativity", Journal of Mathematical Physics 20: 793–800, doi:10.1063/1.524151
 Auyang, Sunny Y. (1995), How is Quantum Field Theory Possible?, Oxford University Press, ISBN 0195093453
 Bania, T. M.; Rood, R. T.; Balser, D. S. (2002), "The cosmological density of baryons from observations of 3He+ in the Milky Way", Nature 415: 54–57, doi:10.1038/415054a
 Barack, Leor; Cutler, Curt (2004), "LISA Capture Sources: Approximate Waveforms, SignaltoNoise Ratios, and Parameter Estimation Accuracy", Phys. Rev. D69: 082005, doi:10.1103/PhysRevD.69.082005, arΧiv:grqc/031012
 Bardeen, J. M.; Carter, B.; Hawking, S. W. (1973), "The Four Laws of Black Hole Mechanics", Comm. Math. Phys. 31: 161–170, doi:10.1007/BF01645742, http://projecteuclid.org/euclid.cmp/1103858973
 Barish, Barry (2005), "Towards detection of gravitational waves", in Florides, P.; Nolan, B.; Ottewil, A., General Relativity and Gravitation. Proceedings of the 17th International Conference, World Scientific, pp. 24–34, ISBN 9812564241
 Barstow, M.; Bond, Howard E.; Holberg, J.B. (2005), "Hubble Space Telescope Spectroscopy of the Balmer lines in Sirius B", Mon. Not. Roy. Astron. Soc. 362: 1134–1142, doi:10.1111/j.13652966.2005.09359.x, arΧiv:astroph/0506600
 Bartusiak, Marcia (2000), Einstein's Unfinished Symphony: Listening to the Sounds of SpaceTime, Berkley, ISBN 9780425186206
 Begelman, Mitchell C.; Blandford, Roger D.; Rees, Martin J. (1984), "Theory of extragalactic radio sources", Rev. Mod. Phys. 56: 255–351, doi:10.1103/RevModPhys.56.255
 Beig, Robert; Chruściel, Piotr T. (2006), "Stationary black holes", in Francoise, J.P.; Naber, G.; Tsou, T.S., Encyclopedia of Mathematical Physics, Volume 2, Elsevier, arΧiv:grqc/0502041, ISBN 0125126603
 Bekenstein, Jacob D. (1973), "Black Holes and Entropy", Phys. Rev. D7: 2333–2346, doi:10.1103/PhysRevD.7.2333
 Bekenstein, Jacob D. (1974), "Generalized Second Law of Thermodynamics in BlackHole Physics", Phys. Rev. D9: 3292–3300, doi:10.1103/PhysRevD.9.3292
 Belinskii, V. A.; Khalatnikov, I. M.; Lifschitz, E. M. (1971), "Oscillatory approach to the singular point in relativistic cosmology", Advances in Physics 19: 525–573, doi:10.1080/00018737000101171 ; original paper in Russian: Belinsky, V. A.; Khalatnikov, I. M.; Lifshitz, E. M. (1970), "Колебательный Режим Приближения К Особой Точке В Релятивистской Космологии", Uspekhi Fizicheskikh Nauk (Успехи Физических Наук) 102(3) (11): 463–500
 Bennett, C. L.; Halpern, M.; Hinshaw, G.; Jarosik, N. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results", Astrophys. J. Suppl. 148: 1–27, doi:10.1086/377253, arΧiv:astroph/0302207
 Berger, Beverly K. (2002), "Numerical Approaches to Spacetime Singularities", Living Rev. Relativity' 5, http://www.livingreviews.org/lrr20021, retrieved 20070804
 Bergström, Lars; Goobar, Ariel (2003), Cosmology and Particle Astrophysics (2nd ed.), Wiley & Sons, ISBN 3540431284
 Bertotti, Bruno; Ciufolini, Ignazio; Bender, Peter L. (1987), "New test of general relativity: Measurement of de Sitter geodetic precession rate for lunar perigee", Physical Review Letters 58: 1062–1065, doi:10.1103/PhysRevLett.58.1062
 Bertotti, Bruno; Iess, L.; Tortora, P. (2003), "A test of general relativity using radio links with the Cassini spacecraft", Nature 425: 374–376, doi:10.1038/nature01997
 Bertschinger, Edmund (1998), "Simulations of structure formation in the universe", Annu. Rev. Astron. Astrophys. 36: 599–654, doi:10.1146/annurev.astro.36.1.599
 Birrell, N. D.; Davies, P. C. (1984), Quantum Fields in Curved Space, Cambridge University Press, ISBN 0521278589
 Blair, David; McNamara, Geoff (1997), Ripples on a Cosmic Sea. The Search for Gravitational Waves, Perseus, ISBN 0738201375
 Blanchet, L.; Faye, G.; Iyer, B. R.; Sinha, S. (2008), The third postNewtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasicircular orbits, arΧiv:0802.1249
 Blanchet, Luc (2006), "Gravitational Radiation from PostNewtonian Sources and Inspiralling Compact Binaries", Living Rev. Relativity 9, http://www.livingreviews.org/lrr20064, retrieved 20070807
 Blandford, R. D. (1987), "Astrophysical Black Holes", in Hawking, Stephen W.; Israel, Werner, 300 Years of Gravitation, Cambridge University Press, pp. 277–329, ISBN 0521379768
 Börner, Gerhard (1993), The Early Universe. Facts and Fiction, Springer, ISBN 0387567291
 Brandenberger, Robert H. (2007), Conceptual Problems of Inflationary Cosmology and a New Approach to Cosmological Structure Formation, arΧiv:hepth/0701111
 Brans, C. H.; Dicke, R. H. (1961), "Mach's Principle and a Relativistic Theory of Gravitation", Physical Review 124 (3): 925–935, doi:10.1103/PhysRev.124.925
 Bridle, Sarah L.; Lahav, Ofer; Ostriker, Jeremiah P.; Steinhardt, Paul J. (2003), "Precision Cosmology? Not Just Yet", Science 299: 1532–1533, doi:10.1126/science.1082158, arΧiv:astroph/0303180, PMID 12624255
 Bruhat, Yvonne (1962), "The Cauchy Problem", in Witten, Louis, Gravitation: An Introduction to Current Research, Wiley, pp. 130, ISBN 9781114291669
 Buchert, Thomas (2007), "Dark Energy from Structure—A Status Report", General Relativity and Gravitation 40: 467–527, doi:10.1007/s1071400705548, arΧiv:0707.2153
 Buras, R.; Rampp, M.; Janka, H.Th.; Kifonidis, K. (2003), "Improved Models of Stellar Core Collapse and Still no Explosions: What is Missing?", Phys. Rev. Lett. 90: 241101, doi:10.1103/PhysRevLett.90.241101, arΧiv:astroph/0303171
 Caldwell, Robert R. (2004), "Dark Energy", Physics World 17(5): 37–42
 Carlip, Steven (2001), "Quantum Gravity: a Progress Report", Rept. Prog. Phys. 64: 885–942, doi:10.1088/00344885/64/8/301, arΧiv:grqc/0108040
 Carroll, Bradley W.; Ostlie, Dale A. (1996), An Introduction to Modern Astrophysics, AddisonWesley, ISBN 0201547309
 Carroll, Sean M. (2001), "The Cosmological Constant", Living Rev. Relativity 4, http://www.livingreviews.org/lrr20011, retrieved 20070721
 Carter, Brandon (1979), "The general theory of the mechanical, electromagnetic and thermodynamic properties of black holes", in Hawking, S. W.; Israel, W., General Relativity, an Einstein Centenary Survey, Cambridge University Press, pp. 294–369 and 860–863, ISBN 0521299284
 Celotti, Annalisa; Miller, John C.; Sciama, Dennis W. (1999), "Astrophysical evidence for the existence of black holes", Class. Quant. Grav. 16: A3–A21, doi:10.1088/02649381/16/12A/301, arΧiv:astroph/9912186v1
 Chandrasekhar, Subrahmanyan (1983), The Mathematical Theory of Black Holes, Oxford University Press, ISBN 0198503709
 Charbonnel, C.; Primas, F. (2005), "The Lithium Content of the Galactic Halo Stars", Astronomy & Astrophysics 442: 961–992, doi:10.1051/00046361:20042491, arΧiv:astroph/0505247
 Ciufolini, Ignazio; Pavlis, Erricos C. (2004), "A confirmation of the general relativistic prediction of the LenseThirring effect", Nature 431: 958–960, doi:10.1038/nature03007
 Ciufolini, Ignazio; Pavlis, Erricos C.; Peron, R. (2006), "Determination of framedragging using Earth gravity models from CHAMP and GRACE", New Astron. 11: 527–550, doi:10.1016/j.newast.2006.02.001
 Coc, A.; VangioniFlam, E.; Descouvemont, P.; Adahchour, A.; Angulo, C. (2004), "Updated Big Bang Nucleosynthesis confronted to WMAP observations and to the Abundance of Light Elements", Astrophysical Journal 600: 544–552, doi:10.1086/380121, arΧiv:astroph/0309480
 Cutler, Curt; Thorne, Kip S. (2002), "An overview of gravitational wave sources", in Bishop, Nigel; Maharaj, Sunil D., Proceedings of 16th International Conference on General Relativity and Gravitation (GR16), World Scientific, arΧiv:grqc/0204090, ISBN 9812381716
 Dalal, Neal; Holz, Daniel E.; Hughes, Scott A.; Jain, Bhuvnesh (2006), "Short GRB and binary black hole standard sirens as a probe of dark energy", Phys.Rev. D74: 063006, doi:10.1103/PhysRevD.74.063006, arΧiv:astroph/0601275
 Danzmann, Karsten; Rüdiger, Albrecht (2003), "LISA Technology—Concepts, Status, Prospects" (PDF), Class. Quant. Grav. 20: S1–S9, doi:10.1088/02649381/20/10/301, http://www.srl.caltech.edu/lisa/documents/KarstenAlbrechtOverviewCQG202003.pdf
 Dirac, Paul (1996), General Theory of Relativity, Princeton University Press, ISBN 069101146X
 Donoghue, John F. (1995), "Introduction to the Effective Field Theory Description of Gravity", in Cornet, Fernando, Effective Theories: Proceedings of the Advanced School, Almunecar, Spain, 26 June–1 July 1995, Singapore: World Scientific, arΧiv:grqc/9512024, ISBN 9810229089
 Duff, Michael (1996), "MTheory (the Theory Formerly Known as Strings)", Int. J. Mod. Phys. A11: 5623–5641, doi:10.1142/S0217751X96002583, arΧiv:hepth/9608117
 Ehlers, Jürgen (1973), "Survey of general relativity theory", in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, ISBN 9027703698
 Ehlers, Jürgen; Falco, Emilio E.; Schneider, Peter (1992), Gravitational lenses, Springer, ISBN 3540665064
 Ehlers, Jürgen; Lämmerzahl, Claus, eds. (2006), Special Relativity—Will it Survive the Next 101 Years?, Springer, ISBN 3540345221
 Ehlers, Jürgen; Rindler, Wolfgang (1997), "Local and Global Light Bending in Einstein's and other Gravitational Theories", General Relativity and Gravitation 29: 519–529, doi:10.1023/A:1018843001842
 Einstein, Albert (1907), "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen" (PDF), Jahrbuch der Radioaktivitaet und Elektronik 4: 411, http://www.soso.ch/wissen/hist/SRT/E1907.pdf, retrieved 20080505
 Einstein, Albert (1915), "Die Feldgleichungen der Gravitation", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847, http://nausikaa2.mpiwgberlin.mpg.de/cgibin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb, retrieved 20060912
 Einstein, Albert (1916), "Die Grundlage der allgemeinen Relativitätstheorie" (PDF), Annalen der Physik 49, http://www.alberteinstein.info/gallery/gtext3.html, retrieved 20060903
 Einstein, Albert (1917), "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie", Sitzungsberichte der Preußischen Akademie der Wissenschaften: 142
 Ellis, George F R; van Elst, Henk (1999), "Cosmological models (Cargèse lectures 1998)", in LachièzeRey, Marc, Theoretical and Observational Cosmology, Kluwer, pp. 1–116, arΧiv:grqc/9812046
 Everitt, C. W. F.; Buchman, S.; DeBra, D. B.; Keiser, G. M. (2001), "Gravity Probe B: Countdown to launch", Gyros, Clocks, and Interferometers: Testing Relativistic Gravity in Space (Lecture Notes in Physics 562), Springer, pp. 52–82, ISBN 3540412360
 Everitt, C. W. F.; Parkinson, Bradford; Kahn, Bob (2007) (PDF), The Gravity Probe B experiment. Post Flight Analysis—Final Report (Preface and Executive Summary), Project Report: NASA, Stanford University and Lockheed Martin, http://einstein.stanford.edu/content/exec_summary/GPB_ExecSumscrn.pdf, retrieved 20070805
 Falcke, Heino; Melia, Fulvio; Agol, Eric (2000), "Viewing the Shadow of the Black Hole at the Galactic Center", Astrophysical Journal 528: L13–L16, doi:10.1086/312423, arΧiv:astroph/9912263
 Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J.Phys. 7: 204, doi:10.1088/13672630/7/1/204, arΧiv:grqc/0501041
 Font, José A. (2003), "Numerical Hydrodynamics in General Relativity", Living Rev. Relativity 6, http://www.livingreviews.org/lrr20034, retrieved 20070819
 FourèsBruhat, Yvonne (1952), "Théoréme d'existence pour certains systémes d'équations aux derivées partielles non linéaires", Acta Mathematica 88: 141–225, doi:10.1007/BF02392131
 Frauendiener, Jörg (2004), "Conformal Infinity", Living Rev. Relativity 7, http://www.livingreviews.org/lrr20041, retrieved 20070721
 Friedrich, Helmut (2005), "Is general relativity `essentially understood'?", Annalen Phys. 15: 84–108, doi:10.1002/andp.200510173, http://www.arxiv.org/abs/grqc/0508016
 Futamase, T.; Itoh, Y. (2006), "The PostNewtonian Approximation for Relativistic Compact Binaries", Living Rev. Relativity 10, http://www.livingreviews.org/lrr20072, retrieved 20080229
 Gamow, George (1970), My World Line, Viking Press, ISBN 0670503762
 Garfinkle, David (2007), "Of singularities and breadmaking", Einstein Online, http://www.einsteinonline.info/en/spotlights/singularities_bkl/index.html, retrieved 20070803
 Geroch, Robert (1996), Partial Differential Equations of Physics, arΧiv:grqc/9602055
 Giulini, Domenico (2005), Special Relativity: A First Encounter, Oxford University Press, ISBN 0198567464
 Giulini, Domenico (2006a), "Algebraic and Geometric Structures in Special Relativity", in Ehlers, Jürgen; Lämmerzahl, Claus, Special Relativity—Will it Survive the Next 101 Years?, Springer, pp. 45–111, arΧiv:mathph/0602018, ISBN 3540345221
 Giulini, Domenico (2006b), "Some remarks on the notions of general covariance and background independence", in Stamatescu, I. O., An assessment of current paradigms in the physics of fundamental interactions, Springer, arΧiv:grqc/0603087
 Gnedin, Nickolay Y. (2005), "Digitizing the Universe", Nature 435: 572–573, doi:10.1038/435572a
 Goenner, Hubert F. M. (2004), "On the History of Unified Field Theories", Living Rev. Relativity 7, http://www.livingreviews.org/lrr20042, retrieved 20080228
 Goroff, Marc H.; Sagnotti, Augusto (1985), "Quantum gravity at two loops", Phys. Lett. 160B: 81–86, doi:10.1016/03702693(85)914704
 Gourgoulhon, Eric (2007), 3+1 Formalism and Bases of Numerical Relativity, arΧiv:grqc/0703035
 Gowdy, Robert H. (1971), "Gravitational Waves in Closed Universes", Phys. Rev. Lett. 27: 826–829, doi:10.1103/PhysRevLett.27.826
 Gowdy, Robert H. (1974), "Vacuum spacetimes with twoparameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions", Ann. Phys. (N.Y.) 83: 203–241, doi:10.1016/00034916(74)903844
 Green, M. B.; Schwarz, J. H.; Witten, E. (1987), Superstring theory. Volume 1: Introduction, Cambridge University Press, ISBN 0521357527
 Greenstein, J. L.; Oke, J. B.; Shipman, H. L. (1971), "Effective Temperature, Radius, and Gravitational Redshift of Sirius B", Astrophysical Journal 169: 563, doi:10.1086/151174, http://esoads.eso.org/abs/1971ApJ...169..563G
 Hafele, J.; Keating, R. (1972a), "Around the world atomic clocks:predicted relativistic time gains", Science 177: 166–168, doi:10.1126/science.177.4044.166, PMID 17779917
 Hafele, J.; Keating, R. (1972b), "Around the world atomic clocks: observed relativistic time gains", Science 177: 168–170, doi:10.1126/science.177.4044.168, PMID 17779918
 Havas, P. (1964), "FourDimensional Formulation of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity", Rev. Mod. Phys. 36: 938–965, doi:10.1103/RevModPhys.36.938
 Hawking, Stephen W. (1966), "The occurrence of singularities in cosmology", Proceedings of the Royal Society of London A294 (1439): 511–521, http://links.jstor.org/sici?sici=00804630%2819661018%29294%3A1439%3C511%3ATOOSIC%3E2.0.CO%3B2Y
 Hawking, S. W. (1975), "Particle Creation by Black Holes", Communications in Mathematical Physics 43: 199–220, doi:10.1007/BF02345020
 Hawking, Stephen W. (1987), "Quantum cosmology", in Hawking, Stephen W.; Israel, Werner, 300 Years of Gravitation, Cambridge University Press, pp. 631–651, ISBN 0521379768
 Hawking, Stephen W.; Ellis, George F. R. (1973), The large scale structure of spacetime, Cambridge University Press, ISBN 0521099064
 Heckmann, O. H. L.; Schücking, E. (1959), "Newtonsche und Einsteinsche Kosmologie", in Flügge, S., Encyclopedia of Physics, 53, pp. 489
 Heusler, Markus (1998), "Stationary Black Holes: Uniqueness and Beyond", Living Rev. Relativity 1, http://www.livingreviews.org/lrr19986, retrieved 20070804
 Heusler, Markus (1996), Black Hole Uniqueness Theorems, Cambridge University Press, ISBN 0521567351
 Hey, Tony; Walters, Patrick (2003), The new quantum universe, Cambridge University Press, ISBN 0521564573
 Hough, Jim; Rowan, Sheila (2000), "Gravitational Wave Detection by Interferometry (Ground and Space)", Living Rev. Relativity 3, http://www.livingreviews.org/lrr20003, retrieved 20070721
 Hubble, Edwin (1929), "A Relation between Distance and Radial Velocity among ExtraGalactic Nebulae" (PDF), Proc. Nat. Acad. Sci. 15: 168–173, doi:10.1073/pnas.15.3.168, PMID 16577160, http://www.pnas.org/cgi/reprint/15/3/168.pdf
 Hulse, Russell A.; Taylor, Joseph H. (1975), "Discovery of a pulsar in a binary system", Astrophys. J. 195: L51–L55, doi:10.1086/181708, http://esoads.eso.org/abs/1975ApJ...195L..51H
 Ibanez, L. E. (2000), "The second string (phenomenology) revolution", Class. Quant. Grav. 17: 1117–1128, doi:10.1088/02649381/17/5/321, arΧiv:hepph/9911499
 Iorio, L. (2009), "An Assessment of the Systematic Uncertainty in Present and Future Tests of the LenseThirring Effect with Satellite Laser Ranging", Space Sci. Rev., doi:10.1007/s1121400894781
 Isham, Christopher J. (1994), "Prima facie questions in quantum gravity", in Ehlers, Jürgen; Friedrich, Helmut, Canonical Gravity: From Classical to Quantum, Springer, ISBN 3540583394
 Israel, Werner (1971), "Event Horizons and Gravitational Collapse", General Relativity and Gravitation 2: 53–59, doi:10.1007/BF02450518
 Israel, Werner (1987), "Dark stars: the evolution of an idea", in Hawking, Stephen W.; Israel, Werner, 300 Years of Gravitation, Cambridge University Press, pp. 199–276, ISBN 0521379768
 Janssen, Michel (2005), "Of pots and holes: Einstein’s bumpy road to general relativity" (PDF), Ann. Phys. (Leipzig) 14: 58–85, doi:10.1002/andp.200410130, http://www.tc.umn.edu/~janss011/pdf%20files/potsandholes.pdf
 Jaranowski, Piotr; Królak, Andrzej (2005), "GravitationalWave Data Analysis. Formalism and Sample Applications: The Gaussian Case", Living Rev. Relativity 8, http://www.livingreviews.org/lrr20053, retrieved 20070730
 Kahn, Bob (1996–2008), Gravity Probe B Website, Stanford University, http://einstein.stanford.edu/, retrieved 20080521
 Kahn, Bob (April 14, 2007) (PDF), Was Einstein right? Scientists provide first public peek at Gravity Probe B results (Stanford University Press Release), Stanford University News Service, http://einstein.stanford.edu/content/press_releases/SU/praps041807.pdf
 Kamionkowski, Marc; Kosowsky, Arthur; Stebbins, Albert (1997), "Statistics of Cosmic Microwave Background Polarization", Phys. Rev. D55: 7368–7388, doi:10.1103/PhysRevD.55.7368, arΧiv:astroph/9611125
 Kennefick, Daniel (2005), "Astronomers Test General Relativity: Lightbending and the Solar Redshift", in Renn, Jürgen, One hundred authors for Einstein, WileyVCH, pp. 178–181, ISBN 3527405747
 Kennefick, Daniel (2007), "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition", Proceedings of the 7th Conference on the History of General Relativity, Tenerife, 2005, arΧiv:0709.0685
 Kenyon, I. R. (1990), General Relativity, Oxford University Press, ISBN 0198519966
 Kochanek, C.S.; Falco, E.E.; Impey, C.; Lehar, J. (2007), CASTLES Survey Website, HarvardSmithsonian Center for Astrophysics, http://cfawww.harvard.edu/castles, retrieved 20070821
 Komar, Arthur (1959), "Covariant Conservation Laws in General Relativity", Phys. Rev. 113: 934–936, doi:10.1103/PhysRev.113.934
 Kramer, Michael (2004), "Millisecond Pulsars as Tools of Fundamental Physics", in Karshenboim, S. G., Astrophysics, Clocks and Fundamental Constants (Lecture Notes in Physics Vol. 648), Springer, pp. 33–54, arΧiv:astroph/0405178
 Kramer, M.; Stairs, I. H.; Manchester, R. N.; McLaughlin, M. A. (2006), "Tests of general relativity from timing the double pulsar", Science 314: 97–102, doi:10.1126/science.1132305, arΧiv:astroph/0609417, PMID 16973838
 Kraus, Ute (1998), "Light Deflection Near Neutron Stars", Relativistic Astrophysics, Vieweg, pp. 66–81, ISBN 3528069090
 Kuchař, Karel (1973), "Canonical Quantization of Gravity", in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 237–288, ISBN 9027703698
 Künzle, H. P. (1972), "Galilei and Lorentz Structures on spacetime: comparison of the corresponding geometry and physics", Ann. Inst. Henri Poincaré a 17: 337–362, http://www.numdam.org/item?id=AIHPA_1972__17_4_337_0
 Lahav, Ofer; Suto, Yasushi (2004), "Measuring our Universe from Galaxy Redshift Surveys", Living Rev. Relativity 7, http://www.livingreviews.org/lrr20048, retrieved 20070819
 Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. Quant. Grav. 22: S487–S492, doi:10.1088/02649381/22/10/048, arΧiv:grqc/0411071
 Lehner, Luis (2001), "Numerical Relativity: A review", Class. Quant. Grav. 18: R25–R86, doi:10.1088/02649381/18/17/202, arΧiv:grqc/0106072
 Lehner, Luis (2002), Numerical Relativity: Status and Prospects, arΧiv:grqc/0202055
 Linde, Andrei (1990), Particle Physics and Inflationary Cosmology, Harwood, arΧiv:hepth/0503203, ISBN 3718604892
 Linde, Andrei (2005), "Towards inflation in string theory", J. Phys. Conf. Ser. 24: 151–160, doi:10.1088/17426596/24/1/018, arΧiv:hepth/0503195
 Loll, Renate (1998), "Discrete Approaches to Quantum Gravity in Four Dimensions", Living Rev. Relativity 1, http://www.livingreviews.org/lrr199813, retrieved 20080309
 Lovelock, David (1972), "The FourDimensionality of Space and the Einstein Tensor", J. Math. Phys. 13: 874–876, doi:10.1063/1.1666069
 MacCallum, M. (2006), "Finding and using exact solutions of the Einstein equations", in Mornas, L.; Alonso, J. D., A Century of Relativity Physics (ERE05, the XXVIII Spanish Relativity Meeting), American Institute of Physics, arΧiv:grqc/0601102
 Maddox, John (1998), What Remains To Be Discovered, Macmillan, ISBN 068482292X
 Mannheim, Philip D. (2006), "Alternatives to Dark Matter and Dark Energy", Prog. Part. Nucl. Phys. 56: 340–445, doi:10.1016/j.ppnp.2005.08.001, arΧiv:astroph/0505266v2
 Mather, J. C.; Cheng, E. S.; Cottingham, D. A.; Eplee, R. E. (1994), "Measurement of the cosmic microwave spectrum by the COBE FIRAS instrument", Astrophysical Journal 420: 439–444, doi:10.1086/173574, http://adsabs.harvard.edu/abs/1994ApJ...420..439M
 Mermin, N. David (2005), It's About Time. Understanding Einstein's Relativity, Princeton University Press, ISBN 0691122016
 Messiah, Albert (1999), Quantum Mechanics, Dover Publications, ISBN 0486409244
 Miller, Cole (2002), Stellar Structure and Evolution (Lecture notes for Astronomy 606), University of Maryland, http://www.astro.umd.edu/~miller/teaching/astr606/, retrieved 20070725
 Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0716703440
 Narayan, Ramesh (2006), "Black holes in astrophysics", New Journal of Physics 7: 199, doi:10.1088/13672630/7/1/199, arΧiv:grqc/0506078
 Narayan, Ramesh; Bartelmann, Matthias (1997), Lectures on Gravitational Lensing, arΧiv:astroph/9606001
 Narlikar, Jayant V. (1993), Introduction to Cosmology, Cambridge University Press, ISBN 0521412501
 Nieto, Michael Martin (2006), "The quest to understand the Pioneer anomaly" (PDF), EurophysicsNews 37(6): 30–34, http://www.europhysicsnews.com/full/42/article4.pdf
 Nordström, Gunnar (1918), "On the Energy of the Gravitational Field in Einstein's Theory", Verhandl. Koninkl. Ned. Akad. Wetenschap., 26: 1238–1245, http://www.digitallibrary.nl/proceedings/search/detail.cfm?pubid=2146&view=image&startrow=1
 Nordtvedt, Kenneth (2003), Lunar Laser Ranging—a comprehensive probe of postNewtonian gravity, arΧiv:grqc/0301024
 Norton, John D. (1985), "What was Einstein's principle of equivalence?" (PDF), Studies in History and Philosophy of Science 16: 203–246, doi:10.1016/00393681(85)900020, http://www.pitt.edu/~jdnorton/papers/ProfE_reset.pdf, retrieved 20070611
 Ohanian, Hans C.; Ruffini, Remo (1994), Gravitation and Spacetime, W. W. Norton & Company, ISBN 0393965015
 Olive, K. A.; Skillman, E. A. (2004), "A Realistic Determination of the Error on the Primordial Helium Abundance", Astrophysical Journal 617: 29–49, doi:10.1086/425170, arΧiv:astroph/0405588
 O'Meara, John M.; Tytler, David; Kirkman, David; Suzuki, Nao (2001), "The Deuterium to Hydrogen Abundance Ratio Towards a Fourth QSO: HS0105+1619", Astrophysical Journal 552: 718–730, doi:10.1086/320579, arΧiv:astroph/0011179
 Oppenheimer, J. Robert; Snyder, H. (1939), "On continued gravitational contraction", Physical Review 56: 455–459, doi:10.1103/PhysRev.56.455
 Overbye, Dennis (1999), Lonely Hearts of the Cosmos: the story of the scientific quest for the secret of the Universe, Back Bay, ISBN 0316648965
 Pais, Abraham (1982), 'Subtle is the Lord...' The Science and life of Albert Einstein, Oxford University Press, ISBN 019853907X
 Peacock, John A. (1999), Cosmological Physics, Cambridge University Press, ISBN 052141072X
 Peebles, P. J. E. (1966), "Primordial Helium abundance and primordial fireball II", Astrophysical Journal 146: 542–552, doi:10.1086/148918, http://esoads.eso.org/abs/1966ApJ...146..542P
 Peebles, P. J. E. (1993), Principles of physical cosmology, Princeton University Press, ISBN 0691019339
 Peebles, P.J.E.; Schramm, D.N.; Turner, E.L.; Kron, R.G. (1991), "The case for the relativistic hot Big Bang cosmology", Nature 352: 769–776, doi:10.1038/352769a0
 Penrose, Roger (1965), "Gravitational collapse and spacetime singularities", Physical Review Letters 14: 57–59, doi:10.1103/PhysRevLett.14.57
 Penrose, Roger (1969), "Gravitational collapse: the role of general relativity", Rivista del Nuovo Cimento 1: 252–276
 Penrose, Roger (2004), The Road to Reality, A. A. Knopf, ISBN 0679454438
 Penzias, A. A.; Wilson, R. W. (1965), "A measurement of excess antenna temperature at 4080 Mc/s", Astrophysical Journal 142: 419–421, doi:10.1086/148307, http://esoads.eso.org/abs/1965ApJ...142..419P
 Peskin, Michael E.; Schroeder, Daniel V. (1995), An Introduction to Quantum Field Theory, AddisonWesley, ISBN 0201503972
 Peskin, Michael E. (2007), Dark Matter and Particle Physics, arΧiv:0707.1536
 Poisson, Eric (2004), "The Motion of Point Particles in Curved Spacetime", Living Rev. Relativity 7, http://www.livingreviews.org/lrr20046, retrieved 20070613
 Poisson, Eric (2004), A Relativist's Toolkit. The Mathematics of BlackHole Mechanics, Cambridge University Press, ISBN 0521830915
 Polchinski, Joseph (1998a), String Theory Vol. I: An Introduction to the Bosonic String, Cambridge University Press, ISBN 0521633036, http://en.wikipedia.org/wiki/Joseph_Polchinski
 Polchinski, Joseph (1998b), String Theory Vol. II: Superstring Theory and Beyond, Cambridge University Press, ISBN 0521633044
 Pound, R. V.; Rebka, G. A. (1959), "Gravitational RedShift in Nuclear Resonance", Physical Review Letters 3: 439–441, doi:10.1103/PhysRevLett.3.439
 Pound, R. V.; Rebka, G. A. (1960), "Apparent weight of photons", Phys. Rev. Lett. 4: 337–341, doi:10.1103/PhysRevLett.4.337
 Pound, R. V.; Snider, J. L. (1964), "Effect of Gravity on Nuclear Resonance", Phys. Rev. Lett. 13: 539–540, doi:10.1103/PhysRevLett.13.539
 Ramond, Pierre (1990), Field Theory: A Modern Primer, AddisonWesley, ISBN 0201546116
 Rees, Martin (1966), "Appearance of Relativistically Expanding Radio Sources", Nature 211: 468–470, doi:10.1038/211468a0
 Reissner, H. (1916), "Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie", Annalen der Physik 355: 106–120, doi:10.1002/andp.19163550905
 Remillard, Ronald A.; Lin, Dacheng; Cooper, Randall L.; Narayan, Ramesh (2006), "The Rates of Type I XRay Bursts from Transients Observed with RXTE: Evidence for Black Hole Event Horizons", Astrophysical Journal 646: 407–419, doi:10.1086/504862, arΧiv:astroph/0509758
 Renn, Jürgen, ed. (2007), The Genesis of General Relativity (4 Volumes), Dordrecht: Springer, ISBN 1402039999
 Renn, Jürgen, ed. (2005), Albert Einstein—Chief Engineer of the Universe: Einstein's Life and Work in Context, Berlin: WileyVCH, ISBN 3527405712
 Reula, Oscar A. (1998), "Hyperbolic Methods for Einstein's Equations", Living Rev. Relativity 1, http://www.livingreviews.org/lrr19983, retrieved 20070829
 Rindler, Wolfgang (2001), Relativity. Special, General and Cosmological, Oxford University Press, ISBN 0198508360
 Rindler, Wolfgang (1991), Introduction to Special Relativity, Clarendon Press, Oxford, ISBN 0198539525
 Robson, Ian (1996), Active galactic nuclei, John Wiley, ISBN 0471958530
 Roulet, E.; Mollerach, S. (1997), "Microlensing", Physics Reports 279: 67–118, doi:10.1016/S03701573(96)000208
 Rovelli, Carlo (2000), Notes for a brief history of quantum gravity, arΧiv:grqc/0006061
 Rovelli, Carlo (1998), "Loop Quantum Gravity", Living Rev. Relativity 1, http://www.livingreviews.org/lrr19981, retrieved 20080313
 Schäfer, Gerhard (2004), "Gravitomagnetic Effects", General Relativity and Gravitation 36: 2223–2235, doi:10.1023/B:GERG.0000046180.97877.32, arΧiv:grqc/0407116
 Schödel, R.; Ott, T.; Genzel, R.; Eckart, A. (2003), "Stellar Dynamics in the Central Arcsecond of Our Galaxy", Astrophysical Journal 596: 1015–1034, doi:10.1086/378122, arΧiv:astroph/0306214
 Schutz, Bernard F. (1985), A first course in general relativity, Cambridge University Press, ISBN 0521277035
 Schutz, Bernard F. (2001), "Gravitational radiation", in Murdin, Paul, Encyclopedia of Astronomy and Astrophysics, Grove's Dictionaries, ISBN 1561592684
 Schutz, Bernard F. (2003), Gravity from the ground up, Cambridge University Press, ISBN 0521455065
 Schwarz, John H. (2007), String Theory: Progress and Problems, arΧiv:hepth/0702219
 Schwarzschild, Karl (1916a), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.: 189–196
 Schwarzschild, Karl (1916b), "Über das Gravitationsfeld eines Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D. Wiss.: 424–434
 Seidel, Edward (1998), "Numerical Relativity: Towards Simulations of 3D Black Hole Coalescence", in Narlikar, J. V.; Dadhich, N., Gravitation and Relativity: At the turn of the millennium (Proceedings of the GR15 Conference, held at IUCAA, Pune, India, December 16–21, 1997), IUCAA, arΧiv:grqc/9806088, ISBN 8190037838
 Seljak, Uros̆; Zaldarriaga, Matias (1997), "Signature of Gravity Waves in the Polarization of the Microwave Background", Phys. Rev. Lett. 78: 2054–2057, doi:10.1103/PhysRevLett.78.2054, arΧiv:astroph/9609169
 Shapiro, S. S.; Davis, J. L.; Lebach, D. E.; Gregory, J. S. (2004), "Measurement of the solar gravitational deflection of radio waves using geodetic verylongbaseline interferometry data, 1979–1999", Phys. Rev. Lett. 92: 121101, doi:10.1103/PhysRevLett.92.121101
 Shapiro, Irwin I. (1964), "Fourth test of general relativity", Phys. Rev. Lett. 13: 789–791, doi:10.1103/PhysRevLett.13.789
 Shapiro, I. I.; Pettengill, Gordon H.; Ash, Michael E.; 1968 (1968), "Fourth test of general relativity: preliminary results", Phys. Rev. Lett. 20: 1265–1269, doi:10.1103/PhysRevLett.20.1265
 Singh, Simon (2004), Big Bang: The Origin of the Universe, Fourth Estate, ISBN 0007152515
 Sorkin, Rafael D. (2005), "Causal Sets: Discrete Gravity", in Gomberoff, Andres; Marolf, Donald, Lectures on Quantum Gravity, Springer, arΧiv:grqc/0309009, ISBN 0387239952
 Sorkin, Rafael D. (1997), "Forks in the Road, on the Way to Quantum Gravity", Int. J. Theor. Phys. 36: 2759–2781, doi:10.1007/BF02435709, arΧiv:grqc/9706002
 Spergel, D. N.; Verde, L.; Peiris, H. V.; Komatsu, E. (2003), "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters", Astrophys. J. Suppl. 148: 175–194, doi:10.1086/377226, arΧiv:astroph/0302209
 Spergel, D. N.; Bean, R.; Doré, O.; Nolta, M. R. (2007), "Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology", Astrophysical Journal Supplement 170: 377–408, doi:10.1086/513700, arΧiv:astroph/0603449
 Springel, Volker; White, Simon D. M.; Jenkins, Adrian; Frenk, Carlos S. (2005), "Simulations of the formation, evolution and clustering of galaxies and quasars", Nature 435: 629–636, doi:10.1038/nature03597
 Stairs, Ingrid H. (2003), "Testing General Relativity with Pulsar Timing", Living Rev. Relativity 6, http://www.livingreviews.org/lrr20035, retrieved 20070721
 Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003), Exact Solutions of Einstein's Field Equations (2 ed.), Cambridge University Press, ISBN 0521461367
 Synge, J. L. (1972), Relativity: The Special Theory, NorthHolland Publishing Company
 Szabados, László B. (2004), "QuasiLocal EnergyMomentum and Angular Momentum in GR", Living Rev. Relativity 7, http://www.livingreviews.org/lrr20044, retrieved 20070823
 Taylor, Joseph H. (1994), "Binary pulsars and relativistic gravity", Rev. Mod. Phys. 66: 711–719, doi:10.1103/RevModPhys.66.711
 Thiemann, Thomas (2006), Loop Quantum Gravity: An Inside View, arΧiv:hepth/0608210
 Thiemann, Thomas (2003), "Lectures on Loop Quantum Gravity", Lect. Notes Phys. 631: 41–135
 Thorne, Kip S. (1972), "Nonspherical Gravitational Collapse—A Short Review", in Klauder, J., Magic without Magic, W. H. Freeman, pp. 231–258
 Thorne, Kip S. (1994), Black Holes and Time Warps: Einstein's Outrageous Legacy, W W Norton & Company, ISBN 0393312763
 Thorne, Kip S. (1995), Gravitational radiation, arΧiv:grqc/9506086
 Townsend, Paul K. (1997), Black Holes (Lecture notes), arΧiv:grqc/9707012
 Townsend, Paul K. (1996), Four Lectures on MTheory, arΧiv:hepth/9612121
 Traschen, Jenny (2000), "An Introduction to Black Hole Evaporation", in Bytsenko, A.; Williams, F., Mathematical Methods of Physics (Proceedings of the 1999 Londrina Winter School), World Scientific, arΧiv:grqc/0010055
 Trautman, Andrzej (2006), "EinsteinCartan theory", in Francoise, J.P.; Naber, G. L.; Tsou, S. T., Encyclopedia of Mathematical Physics, Vol. 2, Elsevier, pp. 189–195, arΧiv:grqc/0606062
 Unruh, W. G. (1976), "Notes on Black Hole Evaporation", Phys. Rev. D 14: 870–892, doi:10.1103/PhysRevD.14.870
 Valtonen, M. J.; Lehto, H. J.; Nilsson, K.; Heidt, J.; Takalo, L. O. (2008), "A massive binary blackhole system in OJ 287 and a test of general relativity", Nature 452: 851–853, doi:10.1038/nature06896
 Wald, Robert M. (1975), "On Particle Creation by Black Holes", Commun. Math. Phys. 45: 9–34, doi:10.1007/BF02345020
 Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0226870332
 Wald, Robert M. (1994), Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press, ISBN 0226870278
 Wald, Robert M. (2001), "The Thermodynamics of Black Holes", Living Rev. Relativity 4, http://www.livingreviews.org/lrr20016, retrieved 20070808
 Walsh, D.; Carswell, R. F.; Weymann, R. J. (1979), "0957 + 561 A, B: twin quasistellar objects or gravitational lens?", Nature 279: 381, doi:10.1038/279381a0
 Wambsganss, Joachim (1998), "Gravitational Lensing in Astronomy", Living Rev. Relativity 1, http://www.livingreviews.org/lrr199812, retrieved 20070720
 Weinberg, Steven (1972), Gravitation and Cosmology, John Wiley, ISBN 0471925675
 Weinberg, Steven (1995), The Quantum Theory of Fields I: Foundations, Cambridge University Press, ISBN 0521550017
 Weinberg, Steven (1996), The Quantum Theory of Fields II: Modern Applications, Cambridge University Press, ISBN 0521550025
 Weinberg, Steven (2000), The Quantum Theory of Fields III: Supersymmetry, Cambridge University Press, ISBN 0521660009
 Weisberg, Joel M.; Taylor, Joseph H. (2003), "The Relativistic Binary Pulsar B1913+16"", in Bailes, M.; Nice, D. J.; Thorsett, S. E., Proceedings of "Radio Pulsars," Chania, Crete, August, 2002, ASP Conference Series
 Weiss, Achim (2006), "Elements of the past: Big Bang Nucleosynthesis and observation", Einstein Online (Max Planck Institute for Gravitational Physics), http://www.einsteinonline.info/en/spotlights/BBN_obs/index.html, retrieved 20070224
 Wheeler, John A. (1990), A Journey Into Gravity and Spacetime, Scientific American Library, San Francisco: W. H. Freeman, ISBN 0716760347
 Will, Clifford M. (1993), Theory and experiment in gravitational physics, Cambridge University Press, ISBN 0521439736
 Will, Clifford M. (2006), "The Confrontation between General Relativity and Experiment", Living Rev. Relativity, http://www.livingreviews.org/lrr20063, retrieved 20070612
 Zwiebach, Barton (2004), A First Course in String Theory, Cambridge University Press, ISBN 0521831431
Further reading
 Popular books
 Geroch, Robert (1981). General Relativity from A to B. Chicago: University of Chicago Press. ISBN 0226288641.
 Lieber, Lillian (2008). The Einstein Theory of Relativity: A Trip to the Fourth Dimension. Philadelphia: Paul Dry Books, Inc.. ISBN 9781589880443.
 Wald, Robert M. (1992). Space, Time, and Gravity: the Theory of the Big Bang and Black Holes. Chicago: University of Chicago Press. ISBN 0226870294.
 Beginning undergraduate textbooks
 Callahan, James J. (2000). The Geometry of Spacetime: an Introduction to Special and General Relativity. New York: Springer. ISBN 0387986413.
 Taylor, Edwin F.; Wheeler, John Archibald (2000). Exploring Black Holes: Introduction to General Relativity. Addison Wesley. ISBN 020138423X.
 Advanced undergraduate textbooks
 Cheng, TaPei (2005). Relativity, Gravitation and Cosmology: a Basic Introduction. Oxford and New York: Oxford University Press. ISBN 0198529570.
 Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 9780387691992
 Hartle, James B. (2003). Gravity: an Introduction to Einstein's General Relativity. San Francisco: AddisonWesley. ISBN 0805386629.
 Hughston, L. P. & Tod, K. P. (1991). Introduction to General Relativity. Cambridge: Cambridge University Press. ISBN 052133943X.
 d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0198596863.
 Graduate level textbooks
 Carroll, Sean M. (2004). Spacetime and Geometry: An Introduction to General Relativity. San Francisco: AddisonWesley. ISBN 0805387323. http://spacetimeandgeometry.net/.
 Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity. New York: Springer. ISBN 9780387691992.
 Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: ButterworthHeinemann, ISBN 0750627689
 Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0716703440
 Stephani, Hans (1990). General Relativity: An Introduction to the Theory of the Gravitational Field,. Cambridge: Cambridge University Press. ISBN 0521379415.
 Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0226870332
External links
 Courses/Lectures/Tutorials