1501:. Unlike the Cartesian coordinates of the plane, coordinate differences are not the same as distances on the surface, as shown in the diagram on the right: for someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,100 mi), while for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,200 mi). Coordinates therefore do not provide enough information to describe the geometry of a spherical surface, or indeed the geometry of any more complicated space or spacetime. That information is precisely what is encoded in the metric, which is a function defined at each point of the surface (or space, or spacetime) and relates coordinate differences to differences in distance. All other quantities that are of interest in geometry, such as the length of any given curve, or the angle at which two curves meet, can be computed from this metric function.
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1886:: clocks aboard satellites orbiting the Earth, and reference clocks stationed on the Earth's surface. General relativity predicts that these two sets of clocks should tick at slightly different rates, due to their different motions (an effect already predicted by special relativity) and their different positions within the Earth's gravitational field. In order to ensure the system's accuracy, either the satellite clocks are slowed down by a relativistic factor, or that same factor is made part of the evaluation algorithm. In turn, tests of the system's accuracy (especially the very thorough measurements that are part of the definition of
1863: â an astronomical object that emits a tight beam of radiowaves. These beams strike the Earth at very regular intervals, similarly to the way that the rotating beam of a lighthouse means that an observer sees the lighthouse blink, and can be observed as a highly regular series of pulses. General relativity predicts specific deviations from the regularity of these radio pulses. For instance, at times when the radio waves pass close to the other neutron star, they should be deflected by the star's gravitational field. The observed pulse patterns are impressively close to those predicted by general relativity.
2163:: over time, black holes retain only a minimal set of distinguishing features (these theorems have become known as "no-hair" theorems), regardless of the starting geometric shape. For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves.
1368:, the straightest possible lines in spacetime. But still there are crucial differences between them and the truly straight lines that can be traced out in the gravity-free spacetime of special relativity. In special relativity, parallel geodesics remain parallel. In a gravitational field with tidal effects, this will not, in general, be the case. If, for example, two bodies are initially at rest relative to each other, but are then dropped in the Earth's gravitational field, they will move towards each other as they fall towards the Earth's center.
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Einstein's master insight was that the constant, familiar pull of the Earth's gravitational field is fundamentally the same as these fictitious forces. The apparent magnitude of the fictitious forces always appears to be proportional to the mass of any object on which they act â for instance, the driver's seat exerts just enough force to accelerate the driver at the same rate as the car. By analogy, Einstein proposed that an object in a gravitational field should feel a gravitational force proportional to its mass, as embodied in
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1715:. Einstein's theory predicts a more complicated curve: the planet behaves as if it were travelling around an ellipse, but at the same time, the ellipse as a whole is rotating slowly around the star. In the diagram on the right, the ellipse predicted by Newtonian gravity is shown in red, and part of the orbit predicted by Einstein in blue. For a planet orbiting the Sun, this deviation from Newton's orbits is known as the
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958:; objects either float motionless or drift at constant speed. Since everything in the elevator is falling together, no gravitational effect can be observed. In this way, the experiences of an observer in free fall are indistinguishable from those of an observer in deep space, far from any significant source of gravity. Such observers are the privileged ("inertial") observers Einstein described in his theory of
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ages by that amount; each clock, in short, is in perfect agreement with all processes happening in its immediate vicinity. It is only when the clocks are compared between separate observers that one can notice that time runs more slowly for the lower observer than for the higher. This effect is minute, but it too has been confirmed experimentally in multiple experiments, as described
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1071:, that is, shifted towards higher frequencies. Einstein argued that such frequency shifts must also be observed in a gravitational field. This is illustrated in the figure at left, which shows a light wave that is gradually red-shifted as it works its way upwards against the gravitational acceleration. This effect has been confirmed experimentally, as described
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1976:, which does not give off light and can be observed only by its gravitational effects. One particularly interesting application are large-scale observations, where the lensing masses are spread out over a significant fraction of the observable universe, and can be used to obtain information about the large-scale properties and evolution of our cosmos.
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1406:, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and energy with momentum.
1423:, is the most famous consequence of special relativity. In relativity, mass and energy are two different ways of describing one physical quantity. If a physical system has energy, it also has the corresponding mass, and vice versa. In particular, all properties of a body that are associated with energy, such as its
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itself acts as a source of attractive (or, less commonly, repulsive) gravity. Einstein originally introduced this term in his pioneering 1917 paper on cosmology, with a very specific motivation: contemporary cosmological thought held the universe to be static, and the additional term was required for
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as the downwards pull of a gravitational force, but as the upwards push of external forces. These forces deflect all bodies resting on the Earth's surface from the geodesics they would otherwise follow. For objects massive enough that their own gravitational influence cannot be neglected, the laws of
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A more basic manifestation of the same effect involves two bodies that are falling side by side towards the Earth, with a similar position and velocity. In a reference frame that is in free fall alongside these bodies, they appear to hover weightlessly â but not exactly so. These bodies are not
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It is important to stress that, for each observer, there are no observable changes of the flow of time for events or processes that are at rest in his or her reference frame. Five-minute-eggs as timed by each observer's clock have the same consistency; as one year passes on each clock, each observer
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Several physicists, including
Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity. Only Einstein's theory proved to be consistent with experiments and observations. To understand the theory's basic ideas, it is instructive to follow Einstein's thinking
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There are several properties that make black holes the most promising sources of gravitational waves. One reason is that black holes are the most compact objects that can orbit each other as part of a binary system; as a result, the gravitational waves emitted by such a system are especially strong.
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and be deflected slightly so as to reach an observer on Earth, while light passing along the opposite side of that same galaxy is deflected as well, reaching the same observer from a slightly different direction. As a result, that particular observer will see one astronomical object in two different
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By cosmic standards, gravity throughout the solar system is weak. Since the differences between the predictions of
Einstein's and Newton's theories are most pronounced when gravity is strong, physicists have long been interested in testing various relativistic effects in a setting with comparatively
1458:. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In the theory's mathematical formulation, all these quantities are but aspects of a more general physical quantity called the
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Compared with planets and other astronomical bodies, the objects of everyday life (people, cars, houses, even mountains) have little mass. Where such objects are concerned, the laws governing the behavior of test particles are sufficient to describe what happens. Notably, in order to deflect a test
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The properties of geodesics differ from those of straight lines. For example, on a plane, parallel lines never meet, but this is not so for geodesics on the surface of the Earth: for example, lines of longitude are parallel at the equator, but intersect at the poles. Analogously, the world lines of
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to account for the acceleration experienced by the observer and objects around them. In the example of the driver being pressed into their seat, the force felt by the driver is one example; another is the force one can feel while pulling the arms up and out if attempting to spin around like a top.
974:. Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.
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had been accepted for more than two hundred years as a valid description of the gravitational force between masses. In Newton's model, gravity is the result of an attractive force between massive objects. Although even Newton was troubled by the unknown nature of that force, the basic framework was
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from its magnetic poles. As the pulsar rotates, its beam sweeps over the Earth, where it is seen as a regular series of radio pulses, just as a ship at sea observes regular flashes of light from the rotating light in a lighthouse. This regular pattern of radio pulses functions as a highly accurate
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as a whole. A key point is that, on large scales, our universe appears to be constructed along very simple lines: all current observations suggest that, on average, the structure of the cosmos should be approximately the same, regardless of an observer's location or direction of observation: the
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of light. Consider two observers aboard an accelerating rocket-ship. Aboard such a ship, there is a natural concept of "up" and "down": the direction in which the ship accelerates is "up", and free-floating objects accelerate in the opposite direction, falling "downward". Assume that one of the
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Einstein explained this in section XX of
Einstein 1961. He considered an object "suspended" by a rope from the ceiling of a room aboard an accelerating rocket: from inside the room it looks as if gravitation is pulling the object down with a force proportional to its mass, but from outside the
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with considerable accuracy. As the precision of experimental measurements gradually improved, some discrepancies with Newton's predictions were observed, and these were accounted for in the general theory of relativity. Similarly, the predictions of general relativity must also be checked with
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Beyond the challenges of quantum effects and cosmology, research on general relativity is rich with possibilities for further exploration: mathematical relativists explore the nature of singularities and the fundamental properties of
Einstein's equations, and ever more comprehensive computer
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Of these tests, only the perihelion advance of
Mercury was known prior to Einstein's final publication of general relativity in 1916. The subsequent experimental confirmation of his other predictions, especially the first measurements of the deflection of light by the sun in 1919, catapulted
1097:: in a gravitational field, light is deflected downward, to the center of the gravitational field. Quantitatively, his results were off by a factor of two; the correct derivation requires a more complete formulation of the theory of general relativity, not just the equivalence principle.
1519:, and the principle that "spacetime tells matter how to move, and matter tells spacetime how to curve" means that these quantities must be related to each other. Einstein formulated this relation by using the Riemann curvature tensor and the metric to define another geometrical quantity
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Although general relativity is not the only relativistic theory of gravity, it is the simplest one that is consistent with the experimental data. Nevertheless, a number of open questions remain, the most fundamental of which is how general relativity can be reconciled with the laws of
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The equivalence between inertia and gravity cannot explain tidal effects â it cannot explain variations in the gravitational field. For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.
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constructing static model universes within the framework of general relativity. When it became apparent that the universe is not static, but expanding, Einstein was quick to discard this additional term. Since the end of the 1990s, however, astronomical evidence indicating an
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General relativity is very successful in providing a framework for accurate models which describe an impressive array of physical phenomena. On the other hand, there are many interesting open questions, and in particular, the theory as a whole is almost certainly incomplete.
1180:. For gravitational fields, the absence or presence of tidal forces determines whether or not the influence of gravity can be eliminated by choosing a freely falling reference frame. Similarly, the absence or presence of curvature determines whether or not a surface is
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This effect can be derived directly within special relativity, either by looking at the equivalent situation of two observers in an accelerated rocket-ship or by looking at a falling elevator; in both situations, the frequency shift has an equivalent description as a
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In fact, when starting from the complete theory, Einstein's equation can be used to derive these more complicated laws of motion for matter as a consequence of geometry, but deriving from this the motion of idealized test particles is a highly non-trivial task, cf.
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and thus following a geodesic, which they would otherwise be doing without the chair there, or any other matter in between them and the center point of the Earth. In this way, general relativity explains the daily experience of gravity on the surface of the Earth
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In 1907, Einstein was still eight years away from completing the general theory of relativity. Nonetheless, he was able to make a number of novel, testable predictions that were based on his starting point for developing his new theory: the equivalence principle.
1317:: particles that are influenced by gravity, but are so small and light that we can neglect their own gravitational effect. In the absence of gravity and other external forces, a test particle moves along a straight line at a constant speed. In the language of
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in the 1850s. With the help of
Riemannian geometry, Einstein formulated a geometric description of gravity in which Minkowski's spacetime is replaced by distorted, curved spacetime, just as curved surfaces are a generalization of ordinary plane surfaces.
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Einstein hypothesized that the similar experiences of weightless observers and inertial observers in special relativity represented a fundamental property of gravity, and he made this the cornerstone of his theory of general relativity, formalized in his
1082:: Since the "higher" observer measures the same light wave to have a lower frequency than the "lower" observer, time must be passing faster for the higher observer. Thus, time runs more slowly for observers the lower they are in a gravitational field.
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The equivalence between gravitational and inertial effects does not constitute a complete theory of gravity. When it comes to explaining gravity near our own location on the Earth's surface, noting that our reference frame is not in free fall, so that
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are the centerpiece of general relativity. They provide a precise formulation of the relationship between spacetime geometry and the properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of
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Conversely, any effect observed in an accelerated reference frame should also be observed in a gravitational field of corresponding strength. This principle allowed
Einstein to predict several novel effects of gravity in 1907, as explained in the
2419:, chapters 2â4. It is impossible to say whether the problem of Newtonian gravity crossed Einstein's mind before 1907, but, by his own admission, his first serious attempts to reconcile that theory with special relativity date to that year, cf.
2497:, use the equivalence principle, the equivalence of gravity and inertial forces, and the results of special relativity for the propagation of light and for accelerated observers (the latter by considering, at each moment, the instantaneous
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are to be expected, provides a suitable explanation. But a freely falling reference frame on one side of the Earth cannot explain why the people on the opposite side of the Earth experience a gravitational pull in the opposite direction.
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1446:. In special relativity, just as space and time are different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call
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exist within the universe if the laws of general relativity were to hold without any quantum modifications. The best-known examples are the singularities associated with the model universes that describe black holes and the
1750:). In the framework of Newtonian gravity, a heuristic argument can be made that leads to light deflection by half that amount. The different predictions can be tested by observing stars that are close to the Sun during a
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each other and, as they do so, gradually lose energy by emitting gravitational waves. For ordinary stars like the Sun, this energy loss would be too small to be detectable, but this energy loss was observed in 1974 in a
2299:. These singularities are boundaries ("sharp edges") of spacetime at which geometry becomes ill-defined, with the consequence that general relativity itself loses its predictive power. Furthermore, there are so-called
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2295:, there is at present no consistent and complete theory. It has long been hoped that a theory of quantum gravity would also eliminate another problematic feature of general relativity: the presence of
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uses lensing effects as an important tool to infer properties of the lensing object. Even in cases where that object is not directly visible, the shape of a lensed image provides information about the
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Since light is deflected in a gravitational field, it is possible for the light of a distant object to reach an observer along two or more paths. For instance, light of a very distant object such as a
1127:. Consequently, there is a component of each body's motion towards the other (see the figure). In a small environment such as a freely falling lift, this relative acceleration is minuscule, while for
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on Earth, and is far from any source of gravity. The objects are being pulled towards the floor by the same "inertial force" that presses the driver of an accelerating car into the back of their seat.
2065:) which was launched in 2015. Gravitational wave observations can be used to obtain information about compact objects such as neutron stars and black holes, and also to probe the state of the early
1987:, a direct consequence of Einstein's theory, are distortions of geometry that propagate at the speed of light, and can be thought of as ripples in spacetime. They should not be confused with the
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simulations of specific spacetimes (such as those describing merging black holes) are run. More than one hundred years after the theory was first published, research is more active than ever.
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1450:. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal
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in 1993. Since then, several other binary pulsars have been found. The most useful are those in which both stars are pulsars, since they provide accurate tests of general relativity.
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While the equivalence principle is still part of modern expositions of general relativity, there are some differences between the modern version and
Einstein's original concept, cf.
2139:, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable. Notable examples of great interest to astronomers are
1297:, examine which properties of matter serve as a source for gravity, and, finally, introduce Einstein's equations, which relate these matter properties to the curvature of spacetime.
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1360:. These paths are certainly not straight, simply because they must follow the curvature of the Earth's surface. But they are as straight as is possible subject to this constraint.
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2327:. There have been several controversial proposals to remove the need for these enigmatic forms of matter and energy, by modifying the laws governing gravity and the dynamics of
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After he had realized the validity of this geometric analogy, it took
Einstein a further three years to find the missing cornerstone of his theory: the equations describing how
1159:. An example is the transition from an inertial reference frame (in which free particles coast along straight paths at constant speeds) to a rotating reference frame (in which
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Experiments and observations show that
Einstein's description of gravitation accounts for several effects that are unexplained by Newton's law, such as minute anomalies in the
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1770:, have confirmed Eddington's results with significantly better precision (the first such measurements date from 1967, the most recent comprehensive analysis from 2004).
1723:, dates back to 1859. The most accurate results for Mercury and for other planets to date are based on measurements which were undertaken between 1966 and 1990, using
1512:, the spacetime in the theory of relativity, is curved at each point. As has already been mentioned, the matter content of the spacetime defines another quantity, the
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2100: â a region of space with a gravitational effect so strong that not even light can escape. Certain types of black holes are thought to be the final state in the
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860:, regions of space where the gravitational effect is strong enough that even light cannot escape. Their strong gravity is thought to be responsible for the intense
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A spherical surface like that of the Earth provides a simple example. The location of any point on the surface can be described by two coordinates: the geographic
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observers is "higher up" than the other. When the lower observer sends a light signal to the higher observer, the acceleration of the ship causes the light to be
1344:, the science of measuring Earth's size and shape, a geodesic is the shortest route between two points on the Earth's surface. Approximately, such a route is a
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particle from its geodesic path, an external force must be applied. A chair someone is sitting on applies an external upwards force preventing the person from
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Einstein to international stardom. These three experiments justified adopting general relativity over Newton's theory and, incidentally, over a number of
1337:, which are "as straight as possible", that is, they follow the shortest path between starting and ending points, taking the curvature into consideration.
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rocket it looks as if the rope is simply transferring the acceleration of the rocket to the object, and must therefore exert just the "force" to do so.
2035:"clock". It can be used to time the double star's orbital period, and it reacts sensitively to distortions of spacetime in its immediate neighborhood.
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While Einstein was exploring the equivalence of gravity and acceleration as well as the role of tidal forces, he discovered several analogies with the
4580:"Ueber die Ablenkung eines Lichtstrals von seiner geradlinigen Bewegung, durch die Attraktion eines Weltkörpers, an welchem er nahe vorbei geht"
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For an overview of the history of black hole physics from its beginnings in the early 20th century to modern times, see the very readable account by
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Springel, Volker; White, Simon D. M.; Jenkins, Adrian; Frenk, Carlos S.; Yoshida, N; Gao, L; Navarro, J; Thacker, R; Croton, D; et al. (2005),
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B. P. Abbott; et al. (LIGO Scientific and Virgo Collaborations) (2016), "Observation of Gravitational Waves from a Binary Black Hole Merger",
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of the universe are obtained by combining these simple solutions to general relativity with theories describing the properties of the universe's
1482:. The metric encodes the information needed to compute the fundamental geometric notions of distance and angle in a curved space (or spacetime).
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between 1907 and 1915. The theory of general relativity says that the observed gravitational effect between masses results from their warping of
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Hartl, Gerhard (2005), "The Confirmation of the General Theory of Relativity by the British Eclipse Expedition of 1919", in Renn, JĂŒrgen (ed.),
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in the presence of massive bodies. In particular, starlight is deflected as it passes near the Sun, leading to apparent shifts of up to 1.75
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Objects are falling to the floor because the room is resting on the surface of the Earth and the objects are being pulled down by gravity.
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distribution responsible for the light deflection. In particular, gravitational lensing provides one way to measure the distribution of
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According to general relativity, light does not travel along straight lines when it propagates in a gravitational field. Instead, it is
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2026:. In such a system, one of the orbiting stars is a pulsar. This has two consequences: a pulsar is an extremely dense object known as a
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are each determined by several functions of the coordinates of spacetime, and the equations equate each of these component functions.
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satellite experiment launched in 2004, with results confirming relativity to within 0.5% and 15%, respectively, as of December 2008.
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Converging geodesics: two lines of longitude (green) that start out in parallel at the equator (red) but converge to meet at the pole
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4694:(1989), "The Rigidly Rotating Disk as the 'Missing Link in the History of General Relativity'", in Howard, D.; Stachel, J. (eds.),
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Kennefick, Daniel (2007), "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition",
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2462:, p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by
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motion are somewhat more complicated than for test particles, although it remains true that spacetime tells matter how to move.
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Kennefick, Daniel (2005), "Astronomers Test General Relativity: Light-bending and the Solar Redshift", in Renn, JĂŒrgen (ed.),
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For a more detailed definition of the metric, but one that is more informal than a textbook presentation, see chapter 14.4 of
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Indirectly, the effect of gravitational waves had been detected in observations of specific binary stars. Such pairs of stars
1398:, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their
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The gravitational redshift of a light wave as it moves upwards against a gravitational field (caused by the yellow star below)
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1727:. General relativity predicts the correct anomalous perihelion shift for all planets where this can be measured accurately (
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A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in
2319:. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely
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and time. Some then-accepted physical theories were inconsistent with that framework; a key example was Newton's theory of
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Objects are falling to the floor because the room is aboard a rocket in space, which is accelerating at 9.81 m/s, the
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1273:(or, more precisely, his field equations of gravity), he presented his new theory of gravity at several sessions of the
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Kramer, Michael (2004), "Millisecond Pulsars as Tools of Fundamental Physics", in Karshenboim, S. G.; Peik, E. (eds.),
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The metric function and its rate of change from point to point can be used to define a geometrical quantity called the
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Proceedings of the 16th International Conference on General Relativity and Gravitation, held 15â21 July 2001 in Durban
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expansion consistent with a cosmological constant â or, equivalently, with a particular and ubiquitous kind of
2220:. According to these models, our present universe emerged from an extremely dense high-temperature state â the
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4963:. Website featuring articles on a variety of aspects of relativistic physics for a general audience, hosted by the
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2200:. Such comparatively simple universes can be described by simple solutions of Einstein's equations. The current
1762:. Eddington's results were not very accurate; subsequent observations of the deflection of the light of distant
1758:, confirmed that Einstein's prediction was correct, and the Newtonian predictions wrong, via observation of the
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1188:. In the summer of 1912, inspired by these analogies, Einstein searched for a geometric formulation of gravity.
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on opposite sides of the Earth, the effect is large. Such differences in force are also responsible for the
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A good starting point for a snapshot of present-day research in relativity is the electronic review journal
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In order to map a body's gravitational influence, it is useful to think about what physicists call probe or
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2147:. Under the right conditions, falling matter accumulating around a black hole can lead to the formation of
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frame of reference. An observer in a closed room cannot tell which of the following two scenarios is true:
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2128:, and they play a key role in current models of how galaxies have formed over the past billions of years.
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3127:. A treatment that is more thorough, yet involves only comparatively little mathematics can be found in
1242:; the orbits of bodies moving at constant speed without changing direction correspond to straight lines.
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Ball falling to the floor in an accelerating rocket (left) and on Earth (right). The effect is identical.
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have to be introduced in order to explain particle motion): this is analogous to the transition from a
1067:
for the light than the first sent out. Conversely, light sent from the higher observer to the lower is
990:
Most effects of gravity vanish in free fall, but effects that seem the same as those of gravity can be
36:
space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are
3165:. Using undergraduate mathematics but avoiding the advanced mathematical tools of general relativity,
1293:. What this means is addressed in the following three sections, which explore the motion of so-called
5645:
5571:
5415:
5017:
4973:
4829:
4648:
4601:
4461:
4332:
4260:
4206:
4085:
4007:
3933:
3837:
3765:
3693:
3611:
3579:
3395:
3359:
3309:
3025:
2633:
2374:
2292:
1823:
space probe. One set of tests focuses on effects predicted by general relativity for the behavior of
1687:
experiment, and Einstein himself devised three tests now known as the classical tests of the theory:
1643:
1478:, in which the geometric properties of a space (or a spacetime) are described by a quantity called a
1246:
440:
4751:
3490:
2171:
1866:
One particular set of observations is related to eminently useful practical applications, namely to
19:
This article is a non-technical introduction to the subject. For the main encyclopedia article, see
6058:
5720:
5715:
5710:
5705:
5537:
5486:
5304:
5289:
5115:
5046:
4994:
4578:
2225:
2117:
1901:, not of general relativity itself. So far, general relativity has passed all observational tests.
1667:
1509:
1475:
1250:
1185:
635:
615:
605:
600:
585:
352:
5609:
1123:
falling in precisely the same direction, but towards a single point in space: namely, the Earth's
920:). Special relativity introduced a new framework for all of physics by proposing new concepts of
495:
6046:
6034:
5998:
5785:
5685:
5467:
5448:
5442:
5393:
5335:
5244:
5162:
5080:
5057:
5025:
4981:
4819:
4680:
4638:
4451:
4348:
4322:
4131:
4111:
4075:
4039:
4023:
3997:
3949:
3791:
3709:
3683:
3637:
3559:
3049:
3015:
2805:
2649:
2623:
2384:
2359:
2349:
2201:
2039:
1984:
1934:
1678:
No scientific theory is self-evidently true; each is a model that must be checked by experiment.
1659:
1200:
1060:
959:
933:
905:
877:
838:
834:
685:
570:
342:
247:
232:
193:
55:
20:
2588:
For elementary presentations of the concept of spacetime, see the first section in chapter 2 of
4292:
2941:
An accessible account of relativistic effects in the global positioning system can be found in
1690:
5795:
5665:
5627:
5619:
5220:
5177:
4857:
4794:
4773:
4720:
4699:
4672:
4609:
4564:
4541:
4523:
4505:
4489:
4400:
4377:
4222:
4180:
4155:
4101:
4057:
3979:
3904:
3882:
3845:
3814:
3783:
3743:
3725:
3476:
3453:
3448:(2005), "The Cassini Experiment: Investigating the Nature of Gravity", in Renn, JĂŒrgen (ed.),
3432:
3414:
3387:
3335:
3041:
2279:. The quest for a quantum version of general relativity addresses one of the most fundamental
2272:
2101:
1455:
1353:
1216:
1109:
Two bodies falling towards the center of the Earth accelerate towards each other as they fall.
1042:
795:
565:
515:
290:
6093:
6010:
5760:
5700:
5690:
5637:
5614:
5192:
4847:
4837:
4733:
4664:
4656:
4479:
4469:
4340:
4268:
4214:
4093:
4049:
4015:
3941:
3809:
The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory
3773:
3701:
3627:
3619:
3599:
3587:
3551:
3528:
3502:
3377:
3367:
3325:
3317:
3033:
2641:
2328:
2232:
2217:
2148:
1938:
1755:
1728:
1720:
1254:
1196:
1160:
1115:
1022:
1006:
826:
660:
640:
595:
575:
520:
2131:
Matter falling onto a compact object is one of the most efficient mechanisms for releasing
5775:
5750:
5675:
5670:
5553:
5514:
5476:
5420:
5294:
5230:
4742:
4502:
Albert Einstein â Chief Engineer of the Universe: Einstein's Life and Work in Context
4419:
3660:
3648:
3632:
3519:
3262:
A review of the various problems and the techniques being developed to overcome them, see
2284:
2213:
2160:
2093:
2062:
1851:
strong gravitational fields. This has become possible thanks to precision observations of
1844:
1828:
1820:
1800:
1767:
1724:
1524:
1220:
917:
901:
889:
803:
695:
670:
555:
550:
414:
295:
257:
32:
5558:
3602:(1997), "Local and Global Light Bending in Einstein's and other Gravitational Theories",
2988:
Introductions to gravitational lensing and its applications can be found on the webpages
2187:
One of the most important aspects of general relativity is that it can be applied to the
1694:
Newtonian (red) vs. Einsteinian orbit (blue) of a single planet orbiting a spherical star
465:
4833:
4652:
4605:
4465:
4336:
4264:
4245:
4210:
4089:
4011:
3937:
3841:
3769:
3697:
3615:
3583:
3363:
3313:
3029:
2744:
A simple explanation of massâenergy equivalence can be found in sections 3.8 and 3.9 of
2637:
6070:
5986:
5800:
5472:
5456:
5452:
5355:
5325:
5182:
5085:
4852:
4484:
4394:
4173:
3897:
3807:
3445:
3382:
3330:
2678:
Einstein's struggle to find the correct field equations is traced in chapters 13â15 of
2596:, p. 47â61. More complete treatments on a fairly elementary level can be found e.g. in
2209:
1992:
1926:
1840:
1816:
1747:
1616:
1432:
1428:
1345:
1219:, Einstein's former mathematics professor at the Swiss Federal Polytechnic, introduced
1124:
955:
738:
705:
700:
388:
252:
37:
4356:
4218:
3994:
Proceedings of the 7th Conference on the History of General Relativity, Tenerife, 2005
3591:
1897:; strictly speaking, all measurements of gravitational time dilation are tests of the
1489:
Distances, at different latitudes, corresponding to 30 degrees difference in longitude
6082:
5790:
5770:
5765:
5680:
5548:
5376:
5320:
5152:
5105:
4807:
4786:
4390:
4371:
4352:
4272:
4149:
4027:
3641:
3053:
2512:
2288:
2019:
1883:
1852:
1751:
1743:
1479:
1447:
1326:
1314:
1294:
665:
625:
580:
560:
485:
383:
262:
3953:
3713:
3506:
3102:. For an up-to-date account of the role of black holes in structure formation, see
2653:
6065:
6022:
5810:
5730:
5695:
5225:
5187:
4691:
4684:
4593:
4367:
4168:
4115:
3827:
3802:
3795:
3037:
2783:
The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of
2248:
2027:
1988:
1955:
1910:
1856:
1792:
1683:
1349:
1245:
The geometry of general curved surfaces was developed in the early 19th century by
1136:
995:
853:
610:
590:
2493:
More specifically, Einstein's calculations, which are described in chapter 11b of
1021:
An observer in an accelerated reference frame must introduce what physicists call
26:
4907:
4869:
3470:
3140:
An elementary introduction to the black hole uniqueness theorems can be found in
2023:
928:, which describes the mutual attraction experienced by bodies due to their mass.
5842:
5594:
5563:
5110:
4624:"Simulations of the formation, evolution and clustering of galaxies and quasars"
4145:
4019:
2324:
2320:
2252:
1973:
1954:
places in the night sky. This kind of focussing is well known when it comes to
1879:
1781:
1780:. It is also seen in astrophysical measurements, notably for light escaping the
1655:
1424:
1322:
925:
869:
799:
500:
470:
4097:
2791:. An introduction using only very simple mathematics is given in chapter 19 of
982:
5805:
5371:
5215:
5210:
4980:, featuring an elementary introduction to general relativity, black holes and
4712:
4554:
3623:
3216:
With a focus on string theory, the search for quantum gravity is described in
3157:
Detailed information can be found in Ned Wright's Cosmology Tutorial and FAQ,
2645:
2463:
2152:
2097:
2086:
2007:
1875:
1824:
1651:
1239:
857:
845:. Many of these predictions have been confirmed by experiment or observation,
710:
272:
198:
3674:
Friedrich, Helmut (2005), "Is general relativity 'essentially understood'?",
2315:
Other attempts to modify general relativity have been made in the context of
1913:; the success of these models is further testament to the theory's validity.
1682:
was accepted because it accounted for the motion of planets and moons in the
1249:. This geometry had in turn been generalized to higher-dimensional spaces in
936:
involving an observer in free fall to his fully geometric theory of gravity.
5202:
2316:
2197:
2180:
2151:, in which focused beams of matter are flung away into space at speeds near
2136:
1921:
1893:
A number of other tests have probed the validity of various versions of the
1639:
1498:
1330:
1321:, this is equivalent to saying that such test particles move along straight
1318:
1235:
1232:
1176:
1068:
1064:
951:
861:
807:
281:
41:
4861:
4676:
4493:
4442:
Poisson, Eric (2004), "The Motion of Point Particles in Curved Spacetime",
4226:
3945:
3787:
3705:
3651:(1917), "Kosmologische Betrachtungen zur allgemeinen RelativitÀtstheorie",
3391:
3339:
3045:
1795:
to altitudes of between tens and tens of thousands of kilometers (first by
4344:
2895:, Mercury on pp. 253â254, Einstein's rise to fame in sections 16b and 16c.
1815:
Further tests of general relativity include precision measurements of the
1746:
in the stars' positions in the sky (an arc second is equal to 1/3600 of a
1485:
1277:
in late 1915, culminating in his final presentation on November 25, 1915.
1269:
influences spacetime's curvature. Having formulated what are now known as
5745:
5095:
4668:
4643:
4044:
2309:
2221:
2188:
2070:
2066:
1784:
1494:
1451:
1443:
1436:
1373:
1334:
1204:
1192:
1181:
1174:
A deeper analogy relates tidal forces with a property of surfaces called
1152:
1128:
1056:
873:
650:
460:
300:
4951:
Additional resources, including more advanced material, can be found in
4842:
4660:
4474:
3664:
3372:
1754:. In this way, a British expedition to West Africa in 1919, directed by
4960:
4810:(2006), "The Confrontation between General Relativity and Experiment",
3563:
2854:; Soldner's original derivation in the framework of Newton's theory is
2614:
Marolf, Donald (1999), "Spacetime Embedding Diagrams for Black Holes",
2571:
Tides and their geometric interpretation are explained in chapter 5 of
2140:
2113:
2085:
Black hole-powered jet emanating from the central region of the galaxy
1712:
1646:
describes the geometry around a spherical, non-rotating mass such as a
1439:, contribute to that body's mass, and hence act as sources of gravity.
1357:
1341:
1291:
spacetime tells matter how to move; matter tells spacetime how to curve
130:{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}
4927:
was created from a revision of this article dated 9 May 2021
4824:
4456:
4327:
4080:
3688:
3321:
3276:
2628:
2283:
in physics. While there are promising candidates for such a theory of
1658:
describes a rotating black hole. Still other solutions can describe a
1305:
1105:
5425:
4373:'Subtle is the Lord ...' The Science and life of Albert Einstein
4038:, Lecture Notes in Physics, vol. 648, Springer, pp. 33â54,
3411:
Einstein's Unfinished Symphony: Listening to the Sounds of Space-Time
3220:; for an account from the point of view of loop quantum gravity, see
2515:
between certain inertial frames. For simple derivations of this, see
2243:
2205:
2132:
2125:
1950:
1946:
1930:
1860:
1763:
1704:
1410:
1266:
833:. General relativity also predicts novel effects of gravity, such as
830:
3778:
3555:
2238:
Einstein's equations can be generalized by adding a term called the
2183:
emitted no more than a few hundred thousand years after the Big Bang
872:). General relativity is also part of the framework of the standard
6017:
4070:
Lehner, Luis (2002), "Numerical Relativity: Status and Prospects",
4053:
3020:
2081:
4002:
2080:
2057:
are in operation, and a mission to launch a space-based detector,
2014:
1843:. The geodetic and frame-dragging effects were both tested by the
1732:
1700:
1689:
1484:
1304:
1228:
1208:
1041:
981:
963:
921:
852:
General relativity has developed into an essential tool in modern
822:
25:
3833:
The Fabric of the Cosmos: Space, Time, and the Texture of Reality
1890:) are testament to the validity of the relativistic predictions.
1585:{\displaystyle \mathbf {G} ={\frac {8\pi G}{c^{4}}}\mathbf {T} ,}
1262:
are used to illustrate curved spacetime in educational contexts.
5406:
3919:"Of pots and holes: Einstein's bumpy road to general relativity"
3182:; good descriptions of more modern developments can be found in
2822:
2228:
2176:
2105:
2096:
region of space, general relativity predicts the formation of a
1999:
1969:
1836:
1708:
1666:, an expanding universe. The simplest solution is the uncurved
1647:
1399:
1132:
5846:
4990:
2669:, chapters 8 and 9 for vivid illustrations of curved spacetime.
1819:
or gravitational time delay for light, measured in 2002 by the
1527:, which describes some aspects of the way spacetime is curved.
1289:, Einstein's geometric theory of gravity can be summarized as:
2808:; for a (technical) summary of our current understanding, see
2121:
1887:
856:. It provides the foundation for the current understanding of
45:
4986:
4772:, Scientific American Library, San Francisco: W. H. Freeman,
4698:, Einstein Studies, vol. 1, BirkhĂ€user, pp. 48â62,
2975:
The geometry of such situations is explored in chapter 23 of
2958:
An accessible introduction to tests of general relativity is
1958:, and hence the corresponding gravitational effect is called
1909:
Models based on general relativity play an important role in
1231:
but also time. The basic entity of this new geometry is four-
4905:
3653:
Sitzungsberichte der PreuĂischen Akademie der Wissenschaften
3431:(1989 reprinted ed.), Institute of Physics Publishing,
2061:, is currently under development, with a precursor mission (
3472:
Ripples on a Cosmic Sea. The Search for Gravitational Waves
3081:
The ongoing search for gravitational waves is described in
1839:). Another, which is related to rotating masses, is called
864:
emitted by certain types of astronomical objects (such as
3103:
4717:
Black Holes and Time Warps: Einstein's Outrageous Legacy
2575:. This part of the historical development is traced in
1766:
by the Sun, which utilize highly accurate techniques of
1719:. The first measurement of this effect, for the planet
1599:(which measures curvature) is equated with the quantity
1167:(in which the coordinate lines are straight lines) to a
4415:"The equivalence principle and the deflection of light"
3861:
A Non-mathematical Proof of Gravitational Time Dilation
3756:
Gnedin, Nickolay Y. (2005), "Digitizing the Universe",
1325:
in spacetime. In the presence of gravity, spacetime is
1207: â are traditionally defined in three-dimensional
3106:; a brief summary can be found in the related article
2825:
measurements of planetary positions; see chapter 5 of
1776:
was first measured in a laboratory setting in 1959 by
1442:
In special relativity, energy is closely connected to
5970:
2355:
Introduction to the mathematics of general relativity
1540:
73:
4175:
It's About Time. Understanding Einstein's Relativity
1622:
This equation is often referred to in the plural as
1078:
This gravitational frequency shift corresponds to a
888:
to produce a complete and self-consistent theory of
5952:
5904:
5878:
5658:
5523:
5495:
5364:
5313:
5275:
5254:
5243:
5201:
5145:
5129:
5071:
5035:
5024:
3542:Cowen, Ron (2001), "A Dark Force in the Universe",
3515:"How many different kinds of black hole are there?"
4172:
3896:
3806:
1584:
129:
3570:Disney, Michael (1998), "A New Look at Quasars",
2908:Gravity Probe B Science ResultsâNASA Final Report
2858:. For the most precise measurements to date, see
2804:The most important solutions are listed in every
1835:(high-precision measurements of the orbit of the
1670:, the spacetime described by special relativity.
30:High-precision test of general relativity by the
3899:The Little Book of the Big Bang. A Cosmic Primer
2415:. A precis of Newtonian gravity can be found in
1827:travelling through space. One of these effects,
966:travels along straight lines at constant speed.
4978:National Center for Supercomputing Applications
4598:Galaxies in the universe â An introduction
4246:"What was Einstein's principle of equivalence?"
3120:
2558:These and other tidal effects are described in
1171:(where coordinate lines need not be straight).
4965:Max Planck Institute for Gravitational Physics
4918:
4748:Max Planck Institute for Gravitational Physics
4738:"Gravitational redshift and White Dwarf stars"
4696:Einstein and the History of General Relativity
4425:Max Planck Institute for Gravitational Physics
4036:Astrophysics, Clocks and Fundamental Constants
3525:Max Planck Institute for Gravitational Physics
3295:"Relativity and the Global Positioning System"
2880:
2501:associated with such an accelerated observer).
2092:When mass is concentrated into a sufficiently
1595:i.e., up to a constant multiple, the quantity
5858:
5002:
4308:"The quest to understand the Pioneer anomaly"
3348:"Relativity in the Global Positioning System"
3086:
2542:
772:
8:
4253:Studies in History and Philosophy of Science
2432:This is described in detail in chapter 2 of
2267:In contrast to all other modern theories of
2159:Another reason follows from what are called
1859:orbit each other. At least one of them is a
1855:. In such a star system, two highly compact
1664:FriedmannâLemaĂźtreâRobertsonâWalker solution
2855:
2275:theory: it does not include the effects of
2124:are assumed to reside in the cores of most
1063:; the second observer will measure a lower
818:extremely successful at describing motion.
5865:
5851:
5843:
5251:
5032:
5009:
4995:
4987:
3666:Relativity. The special and general theory
2962:; a more technical, up-to-date account is
2380:Derivations of the Lorentz transformations
1015:
841:and an effect of gravity on time known as
779:
765:
331:
221:
51:
4874:, University of California at Los Angeles
4851:
4841:
4823:
4642:
4483:
4473:
4455:
4326:
4079:
4043:
4001:
3777:
3687:
3631:
3381:
3371:
3329:
3250:
3141:
3082:
3069:
3019:
2905:Everitt, C.W.F.; Parkinson, B.W. (2009),
2872:
2851:
2847:
2809:
2627:
2407:, p. 110ff., in chapters 9 through 15 of
1791:effect has been measured by transporting
1574:
1566:
1549:
1541:
1539:
1093:In a similar way, Einstein predicted the
118:
109:
97:
78:
72:
4935:, and does not reflect subsequent edits.
3238:
3187:
3179:
2859:
2545:; for a non-technical presentation, see
2516:
2255: â has steadily been accumulating.
2170:
1920:
1223:, a geometric formulation of Einstein's
1104:
5977:
3429:Principles of Cosmology and Gravitation
3234:
3204:
2993:
2989:
2784:
2771:
2758:
2745:
2732:
2718:
2692:
2666:
2601:
2572:
2559:
2471:
2459:
2433:
2412:
2396:
1603:(which measures matter content). Here,
932:between 1907 and 1915, from his simple
401:
375:
334:
280:
54:
3263:
3221:
3217:
3200:
3169:provides a more thorough presentation.
3158:
3145:
3128:
3124:
3107:
3099:
3065:
2976:
2929:
2792:
2788:
2705:
2597:
2593:
2589:
2546:
2529:
2470:and their gravitational analogue, cf.
2446:
2416:
2303:which predict that such singularities
2175:An image, created using data from the
2002:team announced that they had directly
1340:A simple analogy is the following: In
916:(the interaction between objects with
813:By the beginning of the 20th century,
3740:Special relativity. A first encounter
3183:
3166:
3162:
2946:
2942:
2879:. For the Sirius B measurements, see
2843:
2842:For the historical measurements, see
1949:can pass along one side of a massive
1227:where the geometry included not only
815:Newton's law of universal gravitation
7:
4770:A Journey Into Gravity and Spacetime
2963:
2959:
2892:
2876:
2830:
2826:
2679:
2576:
2494:
2420:
2408:
2404:
1799:in 1971; most accurately to date by
1707:traces around a perfectly spherical
1699:Newtonian gravity predicts that the
1135:in the Earth's oceans, so the term "
16:Theory of gravity by Albert Einstein
5581:TolmanâOppenheimerâVolkoff equation
5534:FriedmannâLemaĂźtreâRobertsonâWalker
2403:This development is traced e.g. in
2385:List of books on general relativity
1937:(the much closer foreground galaxy
1882:. Such systems rely on two sets of
1638:describes a particular geometry of
1402:. In Einstein's theory and related
4577:von Soldner, Johann Georg (1804),
3604:General Relativity and Gravitation
3161:; a very readable introduction is
2616:General Relativity and Gravitation
2053:Currently, a number of land-based
1929:: four images of the same distant
1810:alternatives to general relativity
1508:, which describes exactly how the
1238:. The orbits of moving bodies are
896:From special to general relativity
90:
14:
5351:HamiltonâJacobiâEinstein equation
4219:10.1038/scientificamerican0802-42
3592:10.1038/scientificamerican0698-52
1995:, which are a different concept.
1409:The equivalence between mass and
1095:gravitational deflection of light
847:most recently gravitational waves
6064:
6052:
6040:
6028:
6016:
6004:
5992:
5980:
5829:
5828:
4917:
4585:Berliner Astronomisches Jahrbuch
4195:"Does dark matter really exist?"
3976:One hundred authors for Einstein
3879:One hundred authors for Einstein
3450:One hundred authors for Einstein
2069:fractions of a second after the
1575:
1542:
1364:test particles in free fall are
1087:
1072:
746:
745:
732:
62:
3996:, vol. 0709, p. 685,
3978:, Wiley-VCH, pp. 178â181,
3881:, Wiley-VCH, pp. 182â187,
3724:, University of Chicago Press,
3452:, Wiley-VCH, pp. 402â405,
2038:The discoverers of PSR1913+16,
1998:In February 2016, the Advanced
1874:that are used for both precise
1396:Newton's description of gravity
1301:Probing the gravitational field
1139:" is used for this phenomenon.
5158:Massâenergy equivalence (E=mc)
4600:, Cambridge University Press,
4560:Three Roads to Quantum Gravity
4540:, Cambridge University Press,
4306:Nieto, Michael Martin (2006),
4288:Gravitational lensing webpages
4179:, Princeton University Press,
3722:General relativity from A to B
3633:11858/00-001M-0000-0013-5AB5-4
3207:, section 34.1 and chapter 30.
3038:10.1103/PhysRevLett.116.061102
2806:textbook on general relativity
2242:. When this term is present,
2161:black-hole uniqueness theorems
1899:weak version of that principle
1833:Lunar Laser Ranging Experiment
1413:, as expressed by the formula
1:
5873:Introductory science articles
4596:; Gallagher, John S. (2007),
4151:What Remains To Be Discovered
3178:Einstein's original paper is
2365:History of general relativity
1636:A solution of these equations
1147:From acceleration to geometry
1052:gravitational frequency shift
4953:General relativity resources
4719:, W W Norton & Company,
4273:10.1016/0039-3681(85)90002-0
4128:Imagine the Universe Website
3489:Caldwell, Robert R. (2004),
3352:Living Reviews in Relativity
3277:Living Reviews in Relativity
2179:satellite telescope, of the
2055:gravitational wave detectors
2004:observed gravitational waves
1868:satellite navigation systems
1275:Prussian Academy of Sciences
1225:special theory of relativity
1059:, as may be calculated from
1050:The first new effect is the
5173:Relativistic Doppler effect
4793:, Oxford University Press,
4536:Schutz, Bernard F. (2003),
4376:, Oxford University Press,
4130:, NASA GSFC, archived from
4020:10.1016/j.shpsa.2012.07.010
3858:Harrison, David M. (2002),
3742:, Oxford University Press,
3121:Sparke & Gallagher 2007
2499:inertial frame of reference
2370:Tests of general relativity
2333:modified Newtonian dynamics
1831:, has been tested with the
1789:gravitational time dilation
1165:Cartesian coordinate system
1080:gravitational time dilation
1028:Newton's law of gravitation
843:gravitational time dilation
243:Gravitational time dilation
6110:
5644:In computational physics:
5168:Relativity of simultaneity
4972:. Website produced by the
4871:Cosmology tutorial and FAQ
4736:; Barstow, Martin (2007),
4538:Gravity from the ground up
4500:Renn, JĂŒrgen, ed. (2005),
4193:Milgrom, Mordehai (2002),
4122:Lochner, Jim, ed. (2007),
4098:10.1142/9789812776556_0010
3738:Giulini, Domenico (2005),
3469:; McNamara, Geoff (1997),
3427:Berry, Michael V. (1989),
3409:Bartusiak, Marcia (2000),
2881:Trimble & Barstow 2007
2821:More precisely, these are
2271:, general relativity is a
2192:universe is approximately
1905:Astrophysical applications
1888:universal coordinated time
1717:anomalous perihelion shift
1611:of Newtonian gravity, and
1191:The elementary objects of
943:
363:MathissonâPapapetrouâDixon
204:Pseudo-Riemannian manifold
18:
5826:
5481:LenseâThirring precession
5063:Doubly special relativity
3513:Chrusciel, Piotr (2006),
3507:10.1088/2058-7058/17/5/36
3087:Blair & McNamara 1997
2543:Ehlers & Rindler 1997
2466:, Einstein also explored
2310:beginning of the universe
2032:electromagnetic radiation
1872:Global Positioning System
1619:from special relativity.
1465:
5341:Post-Newtonian formalism
5331:Einstein field equations
5267:Mathematical formulation
5091:Hyperbolic orthogonality
4244:Norton, John D. (1985),
3917:Janssen, Michel (2005),
3895:Hogan, Craig J. (1999),
3203:, pp. 52â59 and 98â122;
2269:fundamental interactions
2110:supermassive black holes
1812:that had been proposed.
1506:Riemann curvature tensor
1281:Geometry and gravitation
1169:curved coordinate system
978:Gravity and acceleration
904:published his theory of
368:HamiltonâJacobiâEinstein
348:Einstein field equations
171:Mathematical formulation
44:(blue lines) due to the
5052:Galilean transformation
5043:Principle of relativity
4970:NCSA Spacetime Wrinkles
4124:"Gravitational Lensing"
3720:Geroch, Robert (1978),
3624:10.1023/A:1018843001842
3008:Physical Review Letters
2945:; details are given in
2646:10.1023/A:1026646507201
2297:spacetime singularities
1966:Observational astronomy
1680:Newton's law of gravity
1662:or, in the case of the
1626:, since the quantities
1404:theories of gravitation
1378:the center of the Earth
910:Newton's laws of motion
5932:mathematical formalism
5137:Lorentz transformation
4913:
4893:Listen to this article
4520:Active galactic nuclei
4285:Newbury, Pete (1997),
3946:10.1002/andp.200410130
3706:10.1002/andp.200510173
2184:
2145:active galactic nuclei
2089:
2048:Nobel Prize in Physics
1942:
1774:Gravitational redshift
1695:
1644:Schwarzschild solution
1609:gravitational constant
1586:
1514:energyâmomentum tensor
1490:
1460:energyâmomentum tensor
1310:
1211:or on two-dimensional
1110:
1047:
987:
866:active galactic nuclei
238:Gravitational redshift
131:
49:
5605:WeylâLewisâPapapetrou
5346:Raychaudhuri equation
5285:Equivalence principle
4912:
4504:, Berlin: Wiley-VCH,
3233:For dark matter, see
2240:cosmological constant
2174:
2108:. On the other hand,
2084:
1960:gravitational lensing
1924:
1917:Gravitational lensing
1895:equivalence principle
1693:
1587:
1488:
1308:
1108:
1045:
1034:Physical consequences
985:
972:equivalence principle
962:: observers for whom
954:elevator experiences
946:Equivalence principle
940:Equivalence principle
839:gravitational lensing
526:WeylâLewisâPapapetrou
481:KerrâNewmanâde Sitter
301:EinsteinâRosen bridge
233:Gravitational lensing
189:Equivalence principle
132:
29:
5646:Numerical relativity
5487:pulsar timing arrays
4974:numerical relativity
4944:More spoken articles
4868:Wright, Ned (2007),
4812:Living Rev. Relativ.
4518:Robson, Ian (1996),
4444:Living Rev. Relativ.
3346:Ashby, Neil (2003),
3293:Ashby, Neil (2002),
3104:Springel et al. 2005
2375:Numerical relativity
2301:singularity theorems
2293:loop quantum gravity
1624:Einstein's equations
1538:
1471:Einstein's equations
1466:Einstein's equations
1271:Einstein's equations
1247:Carl Friedrich Gauss
456:EinsteinâRosen waves
182:Fundamental concepts
71:
5538:Friedmann equations
5432:HulseâTaylor binary
5394:Gravitational waves
5290:Riemannian geometry
5116:Proper acceleration
5101:Maxwell's equations
5047:Galilean relativity
4982:gravitational waves
4843:10.12942/lrr-2006-3
4834:2006LRR.....9....3W
4791:Was Einstein Right?
4661:10.1038/nature03597
4653:2005Natur.435..629S
4606:2007gitu.book.....S
4475:10.12942/lrr-2004-6
4466:2004LRR.....7....6P
4413:Pössel, M. (2007),
4396:The Road to Reality
4345:10.1051/epn:2006604
4337:2006ENews..37f..30N
4265:1985SHPSA..16..203N
4211:2002SciAm.287b..42M
4199:Scientific American
4090:2002grg..conf..210L
4012:2007arXiv0709.0685K
3938:2005AnP...517S..58J
3842:2004fcst.book.....G
3770:2005Natur.435..572G
3698:2006AnP...518...84F
3616:1997GReGr..29..519E
3584:1998SciAm.278f..52D
3572:Scientific American
3373:10.12942/lrr-2003-1
3364:2003LRR.....6....1A
3314:2002PhT....55e..41A
3237:; for dark energy,
3030:2016PhRvL.116f1102A
2829:and section 3.5 of
2638:1999GReGr..31..919M
2604:, chapters 8 and 9.
2224: â roughly 14
2202:cosmological models
2143:and other types of
2046:, were awarded the
1985:Gravitational waves
1980:Gravitational waves
1829:geodetic precession
1668:Minkowski spacetime
1642:; for example, the
1529:Einstein's equation
1510:Riemannian manifold
1476:Riemannian geometry
1431:of systems such as
1366:spacetime geodesics
1251:Riemannian geometry
1240:curves in spacetime
908:, which reconciles
900:In September 1905,
835:gravitational waves
410:KaluzaâKlein theory
296:Minkowski spacetime
248:Gravitational waves
6089:General relativity
5927:general relativity
5587:ReissnerâNordström
5505:BransâDicke theory
5336:Linearized gravity
5163:Length contraction
5081:Frame of reference
5058:Special relativity
4914:
3926:Annalen der Physik
3676:Annalen der Physik
3669:, Crown Publishers
2717:See chapter 10 of
2528:See chapter 12 of
2468:centrifugal forces
2360:Special relativity
2350:General relativity
2185:
2090:
1943:
1935:gravitational lens
1803:launched in 1976).
1797:Hafele and Keating
1696:
1660:gravitational wave
1582:
1491:
1390:Sources of gravity
1311:
1260:Embedding Diagrams
1111:
1061:special relativity
1048:
988:
960:special relativity
934:thought experiment
906:special relativity
792:General relativity
739:Physics portal
511:OppenheimerâSnyder
451:ReissnerâNordström
343:Linearized gravity
291:Spacetime diagrams
194:Special relativity
127:
56:General relativity
50:
40:by the warping of
21:General relativity
5968:
5967:
5960:systolic geometry
5944:quantum mechanics
5840:
5839:
5654:
5653:
5633:OzsvĂĄthâSchĂŒcking
5239:
5238:
5221:Minkowski diagram
5178:Thomas precession
5121:Relativistic mass
4910:
4808:Will, Clifford M.
4787:Will, Clifford M.
4734:Trimble, Virginia
4637:(7042): 629â636,
4615:978-0-521-85593-8
4107:978-981-238-171-2
4063:978-3-540-21967-5
3764:(7042): 572â573,
3600:Rindler, Wolfgang
3420:978-0-425-18620-6
3322:10.1063/1.1485583
3119:See chapter 8 of
2875:and chapter 3 of
2787:; cf. box 2.6 in
2757:See chapter 6 of
2562:, pp. 83â91.
2231:ago and has been
2112:with the mass of
1572:
1531:then states that
1523:, now called the
1354:line of longitude
1217:Hermann Minkowski
1161:fictitious forces
1125:center of gravity
1116:fictitious forces
1023:fictitious forces
789:
788:
422:
421:
308:
307:
6101:
6069:
6068:
6057:
6056:
6055:
6045:
6044:
6043:
6033:
6032:
6031:
6021:
6020:
6009:
6008:
6007:
5997:
5996:
5985:
5984:
5976:
5912:electromagnetism
5867:
5860:
5853:
5844:
5832:
5831:
5615:van Stockum dust
5387:Two-body problem
5305:Mach's principle
5252:
5193:Terrell rotation
5033:
5011:
5004:
4997:
4988:
4934:
4932:
4921:
4920:
4911:
4901:
4899:
4894:
4881:
4880:
4879:
4864:
4855:
4845:
4827:
4803:
4782:
4766:Wheeler, John A.
4761:
4760:
4759:
4750:, archived from
4729:
4708:
4687:
4646:
4644:astro-ph/0504097
4628:
4618:
4594:Sparke, Linda S.
4588:
4582:
4573:
4550:
4532:
4514:
4496:
4487:
4477:
4459:
4438:
4437:
4436:
4427:, archived from
4409:
4386:
4363:
4361:
4355:, archived from
4330:
4315:Europhysics News
4312:
4302:
4301:
4300:
4291:, archived from
4281:
4280:
4279:
4250:
4240:
4239:
4238:
4229:, archived from
4189:
4178:
4169:Mermin, N. David
4164:
4141:
4140:
4139:
4118:
4083:
4066:
4047:
4045:astro-ph/0405178
4030:
4005:
3988:
3970:
3969:
3968:
3962:
3956:, archived from
3923:
3913:
3902:
3891:
3873:
3872:
3871:
3866:
3854:
3823:
3812:
3798:
3781:
3752:
3734:
3716:
3691:
3670:
3661:Einstein, Albert
3656:
3649:Einstein, Albert
3644:
3635:
3598:Ehlers, JĂŒrgen;
3594:
3566:
3538:
3537:
3536:
3527:, archived from
3509:
3485:
3462:
3441:
3423:
3405:
3404:
3403:
3394:, archived from
3385:
3375:
3342:
3333:
3299:
3280:
3273:
3267:
3260:
3254:
3247:
3241:
3231:
3225:
3214:
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3191:
3176:
3170:
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3117:
3111:
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3090:
3079:
3073:
3063:
3057:
3056:
3023:
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2997:
2986:
2980:
2973:
2967:
2956:
2950:
2939:
2933:
2927:
2921:
2920:
2919:
2918:
2913:
2902:
2896:
2890:
2884:
2869:
2863:
2856:von Soldner 1804
2840:
2834:
2819:
2813:
2802:
2796:
2781:
2775:
2768:
2762:
2755:
2749:
2742:
2736:
2728:
2722:
2715:
2709:
2702:
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2683:
2676:
2670:
2663:
2657:
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2569:
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2491:
2485:
2481:
2475:
2456:
2450:
2443:
2437:
2430:
2424:
2401:
2329:cosmic expansion
2218:particle physics
2208:content, namely
1933:, produced by a
1760:May 1919 eclipse
1756:Arthur Eddington
1725:radio telescopes
1591:
1589:
1588:
1583:
1578:
1573:
1571:
1570:
1561:
1550:
1545:
1255:Bernhard Riemann
1007:standard gravity
781:
774:
767:
754:
749:
748:
741:
737:
736:
521:van Stockum dust
506:RobertsonâWalker
332:
222:
136:
134:
133:
128:
126:
125:
113:
105:
104:
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85:
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5971:
5969:
5964:
5948:
5900:
5874:
5871:
5841:
5836:
5822:
5650:
5554:BKL singularity
5544:LemaĂźtreâTolman
5519:
5515:Quantum gravity
5497:
5491:
5477:geodetic effect
5451:(together with
5421:LISA Pathfinder
5360:
5309:
5295:Penrose diagram
5277:
5271:
5246:
5235:
5231:Minkowski space
5197:
5141:
5125:
5073:
5067:
5027:
5020:
5015:
4961:Einstein Online
4948:
4947:
4936:
4930:
4928:
4925:This audio file
4922:
4915:
4906:
4903:
4897:
4896:
4892:
4889:
4884:
4877:
4875:
4867:
4806:
4801:
4785:
4780:
4764:
4757:
4755:
4743:Einstein Online
4732:
4727:
4711:
4706:
4690:
4626:
4621:
4616:
4592:
4576:
4571:
4553:
4548:
4535:
4530:
4517:
4512:
4499:
4441:
4434:
4432:
4420:Einstein Online
4412:
4407:
4399:, A. A. Knopf,
4389:
4384:
4366:
4359:
4310:
4305:
4298:
4296:
4284:
4277:
4275:
4248:
4243:
4236:
4234:
4192:
4187:
4167:
4162:
4144:
4137:
4135:
4121:
4108:
4074:, p. 210,
4069:
4064:
4033:
3991:
3986:
3973:
3966:
3964:
3960:
3921:
3916:
3911:
3894:
3889:
3876:
3869:
3867:
3864:
3857:
3852:
3836:, A. A. Knopf,
3826:
3821:
3801:
3779:10.1038/435572a
3755:
3750:
3737:
3732:
3719:
3682:(1â2): 84â108,
3673:
3659:
3647:
3597:
3569:
3556:10.2307/3981642
3550:(14): 218â220,
3541:
3534:
3532:
3520:Einstein Online
3512:
3488:
3483:
3465:
3460:
3446:Bertotti, Bruno
3444:
3439:
3426:
3421:
3408:
3401:
3399:
3345:
3297:
3292:
3288:
3283:
3274:
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3173:
3156:
3152:
3139:
3135:
3118:
3114:
3097:
3093:
3080:
3076:
3068:, pp. 317â321;
3064:
3060:
3005:
3004:
3000:
2987:
2983:
2974:
2970:
2957:
2953:
2940:
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2440:
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2427:
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2398:
2394:
2389:
2345:
2285:quantum gravity
2277:quantum physics
2261:
2259:Modern research
2169:
2135:in the form of
2079:
2063:LISA Pathfinder
1982:
1919:
1907:
1845:Gravity Probe B
1801:Gravity Probe A
1778:Pound and Rebka
1768:radio astronomy
1735:and the Earth).
1703:which a single
1676:
1562:
1551:
1536:
1535:
1525:Einstein tensor
1468:
1392:
1303:
1283:
1221:Minkowski space
1149:
1103:
1036:
980:
948:
942:
918:electric charge
914:electrodynamics
902:Albert Einstein
898:
890:quantum gravity
886:quantum physics
804:Albert Einstein
785:
744:
731:
730:
723:
722:
546:
545:
536:
535:
491:LemaĂźtreâTolman
436:
435:
424:
423:
415:Quantum gravity
402:Advanced theory
329:
328:
327:
310:
309:
258:Geodetic effect
219:
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5617:
5612:
5607:
5602:
5593:Axisymmetric:
5590:
5589:
5584:
5578:
5567:
5566:
5561:
5556:
5551:
5546:
5541:
5532:Cosmological:
5529:
5527:
5521:
5520:
5518:
5517:
5512:
5507:
5501:
5499:
5493:
5492:
5490:
5489:
5484:
5473:frame-dragging
5470:
5465:
5460:
5457:Einstein rings
5453:Einstein cross
5446:
5435:
5434:
5429:
5423:
5418:
5413:
5400:
5390:
5389:
5384:
5379:
5374:
5368:
5366:
5362:
5361:
5359:
5358:
5356:Ernst equation
5353:
5348:
5343:
5338:
5333:
5328:
5326:BSSN formalism
5323:
5317:
5315:
5311:
5310:
5308:
5307:
5302:
5297:
5292:
5287:
5281:
5279:
5273:
5272:
5270:
5269:
5264:
5258:
5256:
5249:
5241:
5240:
5237:
5236:
5234:
5233:
5228:
5223:
5218:
5213:
5207:
5205:
5199:
5198:
5196:
5195:
5190:
5185:
5183:Ladder paradox
5180:
5175:
5170:
5165:
5160:
5155:
5149:
5147:
5143:
5142:
5140:
5139:
5133:
5131:
5127:
5126:
5124:
5123:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5086:Speed of light
5083:
5077:
5075:
5069:
5068:
5066:
5065:
5060:
5055:
5049:
5039:
5037:
5030:
5022:
5021:
5016:
5014:
5013:
5006:
4999:
4991:
4985:
4984:
4967:
4937:
4923:
4916:
4904:
4891:
4890:
4888:
4887:External links
4885:
4883:
4882:
4865:
4804:
4799:
4783:
4778:
4762:
4730:
4725:
4709:
4704:
4688:
4619:
4614:
4590:
4574:
4569:
4551:
4546:
4533:
4528:
4522:, John Wiley,
4515:
4510:
4497:
4439:
4410:
4405:
4391:Penrose, Roger
4387:
4382:
4364:
4303:
4282:
4259:(3): 203â246,
4241:
4190:
4185:
4165:
4160:
4142:
4119:
4106:
4067:
4062:
4054:10.1007/b13178
4031:
3989:
3984:
3971:
3914:
3909:
3892:
3887:
3874:
3855:
3850:
3824:
3819:
3799:
3753:
3748:
3735:
3730:
3717:
3671:
3657:
3645:
3610:(4): 519â529,
3595:
3567:
3539:
3510:
3486:
3481:
3463:
3458:
3442:
3437:
3424:
3419:
3406:
3343:
3289:
3287:
3284:
3282:
3281:
3268:
3255:
3251:Friedrich 2005
3242:
3226:
3209:
3192:
3171:
3150:
3148:, pp. 272â286.
3142:Chrusciel 2006
3133:
3112:
3091:
3083:Bartusiak 2000
3074:
3070:Bartusiak 2000
3058:
2998:
2981:
2968:
2951:
2934:
2922:
2897:
2885:
2873:Kennefick 2005
2864:
2852:Kennefick 2007
2848:Kennefick 2005
2835:
2814:
2810:Friedrich 2005
2797:
2776:
2763:
2750:
2737:
2723:
2710:
2697:
2691:E.g. p. xi in
2684:
2671:
2658:
2622:(6): 919â944,
2606:
2581:
2579:, section 12b.
2564:
2551:
2534:
2521:
2503:
2486:
2476:
2451:
2438:
2425:
2395:
2393:
2390:
2388:
2387:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2346:
2344:
2341:
2331:, for example
2281:open questions
2260:
2257:
2210:thermodynamics
2168:
2165:
2078:
2075:
1993:fluid dynamics
1981:
1978:
1956:optical lenses
1927:Einstein Cross
1918:
1915:
1906:
1903:
1853:binary pulsars
1841:frame-dragging
1817:Shapiro effect
1805:
1804:
1787:. The related
1771:
1736:
1675:
1672:
1654:, whereas the
1617:speed of light
1593:
1592:
1581:
1577:
1569:
1565:
1560:
1557:
1554:
1548:
1544:
1467:
1464:
1429:binding energy
1391:
1388:
1374:falling freely
1315:test particles
1302:
1299:
1295:test particles
1282:
1279:
1253:introduced by
1193:geometry
1148:
1145:
1102:
1099:
1035:
1032:
1011:
1010:
1003:
979:
976:
956:weightlessness
950:A person in a
944:Main article:
941:
938:
897:
894:
787:
786:
784:
783:
776:
769:
761:
758:
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708:
703:
698:
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673:
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663:
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628:
623:
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613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
547:
543:
542:
541:
538:
537:
534:
533:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
437:
431:
430:
429:
426:
425:
420:
419:
418:
417:
412:
404:
403:
399:
398:
397:
396:
394:Post-Newtonian
391:
386:
378:
377:
373:
372:
371:
370:
365:
360:
355:
350:
345:
337:
336:
330:
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318:
317:
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312:
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306:
305:
304:
303:
298:
293:
285:
284:
278:
277:
276:
275:
270:
265:
260:
255:
253:Frame-dragging
250:
245:
240:
235:
230:
228:Kepler problem
220:
216:
215:
214:
211:
210:
207:
206:
201:
196:
191:
185:
181:
180:
179:
176:
175:
174:
173:
168:
166:
165:
160:
155:
149:
147:
139:
138:
124:
121:
117:
112:
108:
103:
100:
96:
92:
89:
84:
81:
77:
67:
59:
58:
15:
13:
10:
9:
6:
4:
3:
2:
6106:
6095:
6092:
6090:
6087:
6086:
6084:
6077:
6072:
6067:
6062:
6060:
6050:
6048:
6038:
6036:
6026:
6024:
6019:
6014:
6012:
6002:
6000:
5995:
5990:
5988:
5983:
5978:
5974:
5961:
5958:
5957:
5955:
5951:
5945:
5942:
5940:
5937:
5933:
5930:
5929:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5909:
5907:
5903:
5897:
5894:
5892:
5889:
5887:
5884:
5883:
5881:
5877:
5868:
5863:
5861:
5856:
5854:
5849:
5848:
5845:
5835:
5825:
5819:
5818:
5814:
5812:
5809:
5807:
5804:
5802:
5799:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5772:
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5726:Chandrasekhar
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5686:Schwarzschild
5684:
5682:
5679:
5677:
5674:
5672:
5669:
5667:
5664:
5663:
5661:
5657:
5647:
5643:
5642:
5639:
5636:
5634:
5631:
5629:
5625:
5624:
5621:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5600:
5596:
5592:
5591:
5588:
5585:
5582:
5579:
5577:
5573:
5572:Schwarzschild
5569:
5568:
5565:
5562:
5560:
5557:
5555:
5552:
5550:
5547:
5545:
5542:
5539:
5535:
5531:
5530:
5528:
5526:
5522:
5516:
5513:
5511:
5508:
5506:
5503:
5502:
5500:
5494:
5488:
5485:
5482:
5478:
5474:
5471:
5469:
5468:Shapiro delay
5466:
5464:
5461:
5458:
5454:
5450:
5447:
5444:
5440:
5437:
5436:
5433:
5430:
5427:
5424:
5422:
5419:
5417:
5414:
5412:
5411:collaboration
5408:
5404:
5401:
5399:
5395:
5392:
5391:
5388:
5385:
5383:
5380:
5378:
5377:Event horizon
5375:
5373:
5370:
5369:
5367:
5363:
5357:
5354:
5352:
5349:
5347:
5344:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5321:ADM formalism
5319:
5318:
5316:
5312:
5306:
5303:
5301:
5298:
5296:
5293:
5291:
5288:
5286:
5283:
5282:
5280:
5274:
5268:
5265:
5263:
5260:
5259:
5257:
5253:
5250:
5248:
5242:
5232:
5229:
5227:
5226:Biquaternions
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5208:
5206:
5204:
5200:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5159:
5156:
5154:
5153:Time dilation
5151:
5150:
5148:
5144:
5138:
5135:
5134:
5132:
5128:
5122:
5119:
5117:
5114:
5112:
5109:
5107:
5106:Proper length
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5084:
5082:
5079:
5078:
5076:
5070:
5064:
5061:
5059:
5056:
5053:
5050:
5048:
5044:
5041:
5040:
5038:
5034:
5031:
5029:
5023:
5019:
5012:
5007:
5005:
5000:
4998:
4993:
4992:
4989:
4983:
4979:
4976:group at the
4975:
4971:
4968:
4966:
4962:
4959:
4958:
4957:
4956:
4954:
4945:
4941:
4926:
4886:
4873:
4872:
4866:
4863:
4859:
4854:
4849:
4844:
4839:
4835:
4831:
4826:
4825:gr-qc/0510072
4821:
4817:
4813:
4809:
4805:
4802:
4800:0-19-286170-0
4796:
4792:
4788:
4784:
4781:
4779:0-7167-6034-7
4775:
4771:
4767:
4763:
4754:on 2011-08-28
4753:
4749:
4745:
4744:
4739:
4735:
4731:
4728:
4726:0-393-31276-3
4722:
4718:
4714:
4710:
4707:
4705:0-8176-3392-8
4701:
4697:
4693:
4692:Stachel, John
4689:
4686:
4682:
4678:
4674:
4670:
4669:2027.42/62586
4666:
4662:
4658:
4654:
4650:
4645:
4640:
4636:
4632:
4625:
4620:
4617:
4611:
4607:
4603:
4599:
4595:
4591:
4586:
4581:
4575:
4572:
4570:0-465-07835-4
4566:
4562:
4561:
4556:
4552:
4549:
4547:0-521-45506-5
4543:
4539:
4534:
4531:
4529:0-471-95853-0
4525:
4521:
4516:
4513:
4511:3-527-40571-2
4507:
4503:
4498:
4495:
4491:
4486:
4481:
4476:
4471:
4467:
4463:
4458:
4457:gr-qc/0306052
4453:
4449:
4445:
4440:
4431:on 2007-05-03
4430:
4426:
4422:
4421:
4416:
4411:
4408:
4406:0-679-45443-8
4402:
4398:
4397:
4392:
4388:
4385:
4383:0-19-853907-X
4379:
4375:
4374:
4369:
4368:Pais, Abraham
4365:
4362:on 2007-06-29
4358:
4354:
4350:
4346:
4342:
4338:
4334:
4329:
4328:gr-qc/0702017
4324:
4320:
4316:
4309:
4304:
4295:on 2012-12-06
4294:
4290:
4289:
4283:
4274:
4270:
4266:
4262:
4258:
4254:
4247:
4242:
4233:on 2011-06-10
4232:
4228:
4224:
4220:
4216:
4212:
4208:
4204:
4200:
4196:
4191:
4188:
4186:0-691-12201-6
4182:
4177:
4176:
4170:
4166:
4163:
4161:0-684-82292-X
4157:
4154:, Macmillan,
4153:
4152:
4147:
4143:
4134:on 2007-06-17
4133:
4129:
4125:
4120:
4117:
4113:
4109:
4103:
4099:
4095:
4091:
4087:
4082:
4081:gr-qc/0202055
4077:
4073:
4068:
4065:
4059:
4055:
4051:
4046:
4041:
4037:
4032:
4029:
4025:
4021:
4017:
4013:
4009:
4004:
3999:
3995:
3990:
3987:
3985:3-527-40574-7
3981:
3977:
3972:
3963:on 2017-07-13
3959:
3955:
3951:
3947:
3943:
3939:
3935:
3932:(S1): 58â85,
3931:
3927:
3920:
3915:
3912:
3910:0-387-98385-6
3906:
3901:
3900:
3893:
3890:
3888:3-527-40574-7
3884:
3880:
3875:
3863:
3862:
3856:
3853:
3851:0-375-41288-3
3847:
3843:
3839:
3835:
3834:
3829:
3828:Greene, Brian
3825:
3822:
3820:0-375-70811-1
3816:
3811:
3810:
3804:
3803:Greene, Brian
3800:
3797:
3793:
3789:
3785:
3780:
3775:
3771:
3767:
3763:
3759:
3754:
3751:
3749:0-19-856746-4
3745:
3741:
3736:
3733:
3731:0-226-28864-1
3727:
3723:
3718:
3715:
3711:
3707:
3703:
3699:
3695:
3690:
3689:gr-qc/0508016
3685:
3681:
3677:
3672:
3668:
3667:
3662:
3658:
3654:
3650:
3646:
3643:
3639:
3634:
3629:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3596:
3593:
3589:
3585:
3581:
3577:
3573:
3568:
3565:
3561:
3557:
3553:
3549:
3545:
3540:
3531:on 2011-04-14
3530:
3526:
3522:
3521:
3516:
3511:
3508:
3504:
3500:
3496:
3495:Physics World
3492:
3491:"Dark Energy"
3487:
3484:
3482:0-7382-0137-5
3478:
3474:
3473:
3468:
3464:
3461:
3459:3-527-40574-7
3455:
3451:
3447:
3443:
3440:
3438:0-85274-037-9
3434:
3430:
3425:
3422:
3416:
3412:
3407:
3398:on 2007-07-04
3397:
3393:
3389:
3384:
3379:
3374:
3369:
3365:
3361:
3357:
3353:
3349:
3344:
3341:
3337:
3332:
3327:
3323:
3319:
3315:
3311:
3307:
3303:
3302:Physics Today
3296:
3291:
3290:
3285:
3278:
3272:
3269:
3265:
3259:
3256:
3252:
3246:
3243:
3240:
3239:Caldwell 2004
3236:
3230:
3227:
3223:
3219:
3213:
3210:
3206:
3202:
3196:
3193:
3189:
3188:Caldwell 2004
3185:
3181:
3180:Einstein 1917
3175:
3172:
3168:
3164:
3160:
3154:
3151:
3147:
3143:
3137:
3134:
3130:
3126:
3122:
3116:
3113:
3109:
3105:
3101:
3095:
3092:
3088:
3084:
3078:
3075:
3071:
3067:
3062:
3059:
3055:
3051:
3047:
3043:
3039:
3035:
3031:
3027:
3022:
3017:
3014:(6): 061102,
3013:
3009:
3002:
2999:
2995:
2991:
2985:
2982:
2978:
2972:
2969:
2965:
2961:
2955:
2952:
2948:
2944:
2938:
2935:
2931:
2926:
2923:
2910:
2909:
2901:
2898:
2894:
2889:
2886:
2882:
2878:
2874:
2868:
2865:
2861:
2860:Bertotti 2005
2857:
2853:
2849:
2845:
2839:
2836:
2832:
2828:
2824:
2818:
2815:
2811:
2807:
2801:
2798:
2794:
2790:
2786:
2780:
2777:
2773:
2767:
2764:
2760:
2754:
2751:
2747:
2741:
2738:
2734:
2727:
2724:
2720:
2714:
2711:
2707:
2701:
2698:
2694:
2688:
2685:
2681:
2675:
2672:
2668:
2662:
2659:
2655:
2651:
2647:
2643:
2639:
2635:
2630:
2629:gr-qc/9806123
2625:
2621:
2617:
2610:
2607:
2603:
2599:
2595:
2591:
2585:
2582:
2578:
2574:
2568:
2565:
2561:
2555:
2552:
2548:
2544:
2538:
2535:
2531:
2525:
2522:
2518:
2517:Harrison 2002
2514:
2513:Doppler shift
2507:
2504:
2500:
2496:
2490:
2487:
2480:
2477:
2473:
2469:
2465:
2461:
2455:
2452:
2448:
2442:
2439:
2435:
2429:
2426:
2422:
2418:
2414:
2410:
2406:
2400:
2397:
2391:
2386:
2383:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2347:
2342:
2340:
2336:
2334:
2330:
2326:
2322:
2318:
2313:
2311:
2306:
2302:
2298:
2294:
2290:
2289:string theory
2286:
2282:
2278:
2274:
2270:
2265:
2258:
2256:
2254:
2250:
2245:
2241:
2236:
2234:
2230:
2227:
2223:
2219:
2215:
2211:
2207:
2203:
2199:
2195:
2190:
2182:
2178:
2173:
2166:
2164:
2162:
2156:
2154:
2153:that of light
2150:
2146:
2142:
2138:
2134:
2129:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2099:
2095:
2088:
2083:
2076:
2074:
2072:
2068:
2064:
2060:
2056:
2051:
2049:
2045:
2044:Joseph Taylor
2041:
2040:Russell Hulse
2036:
2033:
2029:
2025:
2021:
2020:binary pulsar
2016:
2011:
2009:
2005:
2001:
1996:
1994:
1990:
1989:gravity waves
1986:
1979:
1977:
1975:
1971:
1967:
1963:
1961:
1957:
1952:
1948:
1940:
1939:Huchra's lens
1936:
1932:
1928:
1923:
1916:
1914:
1912:
1904:
1902:
1900:
1896:
1891:
1889:
1885:
1884:atomic clocks
1881:
1877:
1873:
1869:
1864:
1862:
1858:
1857:neutron stars
1854:
1848:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1813:
1811:
1802:
1798:
1794:
1793:atomic clocks
1790:
1786:
1783:
1779:
1775:
1772:
1769:
1765:
1761:
1757:
1753:
1752:solar eclipse
1749:
1745:
1741:
1737:
1734:
1730:
1726:
1722:
1718:
1714:
1711:should be an
1710:
1706:
1702:
1698:
1697:
1692:
1688:
1685:
1681:
1673:
1671:
1669:
1665:
1661:
1657:
1656:Kerr solution
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1620:
1618:
1614:
1610:
1606:
1602:
1598:
1579:
1567:
1563:
1558:
1555:
1552:
1546:
1534:
1533:
1532:
1530:
1526:
1522:
1518:
1515:
1511:
1507:
1502:
1500:
1496:
1487:
1483:
1481:
1477:
1472:
1463:
1461:
1457:
1453:
1449:
1448:four-momentum
1445:
1440:
1438:
1434:
1430:
1426:
1422:
1421:
1418: =
1417:
1412:
1407:
1405:
1401:
1397:
1389:
1387:
1384:
1379:
1375:
1369:
1367:
1361:
1359:
1355:
1351:
1347:
1343:
1338:
1336:
1332:
1328:
1327:non-Euclidean
1324:
1320:
1316:
1307:
1300:
1298:
1296:
1292:
1288:
1285:Paraphrasing
1280:
1278:
1276:
1272:
1268:
1263:
1261:
1256:
1252:
1248:
1243:
1241:
1237:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1189:
1187:
1183:
1179:
1178:
1172:
1170:
1166:
1162:
1158:
1154:
1146:
1144:
1140:
1138:
1134:
1130:
1126:
1120:
1117:
1107:
1101:Tidal effects
1100:
1098:
1096:
1091:
1089:
1083:
1081:
1076:
1074:
1070:
1066:
1062:
1058:
1053:
1044:
1040:
1033:
1031:
1029:
1024:
1019:
1017:
1008:
1004:
1001:
1000:
999:
997:
993:
984:
977:
975:
973:
967:
965:
961:
957:
953:
947:
939:
937:
935:
929:
927:
923:
919:
915:
911:
907:
903:
895:
893:
891:
887:
881:
879:
875:
871:
867:
863:
859:
855:
850:
848:
844:
840:
836:
832:
828:
824:
819:
816:
811:
809:
805:
802:developed by
801:
797:
793:
782:
777:
775:
770:
768:
763:
762:
760:
759:
753:
743:
740:
735:
729:
728:
727:
726:
719:
718:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
656:Chandrasekhar
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
571:Schwarzschild
569:
567:
564:
562:
559:
557:
554:
552:
549:
548:
540:
539:
532:
531:HartleâThorne
529:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
446:
442:
441:Schwarzschild
439:
438:
434:
428:
427:
416:
413:
411:
408:
407:
406:
405:
400:
395:
392:
390:
387:
385:
382:
381:
380:
379:
374:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
344:
341:
340:
339:
338:
333:
323:
320:
319:
314:
313:
302:
299:
297:
294:
292:
289:
288:
287:
286:
283:
279:
274:
271:
269:
266:
264:
263:Event horizon
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
225:
224:
223:
213:
212:
205:
202:
200:
197:
195:
192:
190:
187:
186:
178:
177:
172:
169:
164:
161:
159:
156:
154:
151:
150:
148:
146:
143:
142:
141:
140:
122:
119:
115:
110:
106:
101:
98:
94:
87:
82:
79:
75:
65:
61:
60:
57:
53:
47:
43:
39:
35:
34:
28:
22:
6076:
6059:Solar System
5926:
5922:gauge theory
5816:
5510:KaluzaâKlein
5262:Introduction
5261:
5188:Twin paradox
4950:
4949:
4876:, retrieved
4870:
4815:
4811:
4790:
4769:
4756:, retrieved
4752:the original
4741:
4716:
4695:
4634:
4630:
4597:
4584:
4559:
4537:
4519:
4501:
4447:
4443:
4433:, retrieved
4429:the original
4418:
4395:
4372:
4357:the original
4321:(6): 30â34,
4318:
4314:
4297:, retrieved
4293:the original
4287:
4276:, retrieved
4256:
4252:
4235:, retrieved
4231:the original
4205:(2): 30â37,
4202:
4198:
4174:
4150:
4146:Maddox, John
4136:, retrieved
4132:the original
4127:
4071:
4035:
3993:
3975:
3965:, retrieved
3958:the original
3929:
3925:
3903:, Springer,
3898:
3878:
3868:, retrieved
3860:
3832:
3808:
3761:
3757:
3739:
3721:
3679:
3675:
3665:
3652:
3607:
3603:
3578:(6): 52â57,
3575:
3571:
3547:
3544:Science News
3543:
3533:, retrieved
3529:the original
3518:
3501:(5): 37â42,
3498:
3494:
3471:
3467:Blair, David
3449:
3428:
3410:
3400:, retrieved
3396:the original
3355:
3351:
3308:(5): 41â47,
3305:
3301:
3286:Bibliography
3271:
3258:
3245:
3235:Milgrom 2002
3229:
3212:
3205:Penrose 2004
3195:
3174:
3153:
3136:
3115:
3094:
3077:
3072:, pp. 70â86.
3061:
3011:
3007:
3001:
2994:Lochner 2007
2990:Newbury 1997
2984:
2971:
2954:
2937:
2925:
2915:, retrieved
2907:
2900:
2888:
2867:
2838:
2817:
2800:
2785:Wheeler 1990
2779:
2772:Penrose 2004
2766:
2759:Wheeler 1990
2753:
2746:Giulini 2005
2740:
2733:Poisson 2004
2726:
2719:Wheeler 1990
2713:
2700:
2693:Wheeler 1990
2687:
2674:
2667:Wheeler 1990
2661:
2619:
2615:
2609:
2602:Wheeler 1990
2584:
2573:Wheeler 1990
2567:
2560:Wheeler 1990
2554:
2537:
2524:
2506:
2489:
2479:
2472:Stachel 1989
2460:Janssen 2005
2454:
2441:
2434:Wheeler 1990
2428:
2413:Janssen 2005
2399:
2337:
2314:
2304:
2266:
2262:
2249:accelerating
2237:
2235:ever since.
2186:
2157:
2130:
2091:
2052:
2037:
2028:neutron star
2012:
1997:
1983:
1964:
1944:
1911:astrophysics
1908:
1892:
1870:such as the
1865:
1849:
1814:
1806:
1684:Solar System
1677:
1631:
1627:
1623:
1621:
1612:
1604:
1600:
1596:
1594:
1528:
1520:
1516:
1503:
1492:
1469:
1441:
1419:
1415:
1408:
1393:
1382:
1370:
1362:
1352:, such as a
1350:great circle
1339:
1312:
1290:
1287:John Wheeler
1284:
1264:
1259:
1244:
1190:
1175:
1173:
1150:
1141:
1137:tidal effect
1121:
1112:
1092:
1084:
1077:
1069:blue-shifted
1049:
1037:
1020:
1016:next section
1012:
991:
989:
968:
952:free-falling
949:
930:
899:
882:
870:microquasars
854:astrophysics
851:
820:
812:
791:
790:
716:
676:Raychaudhuri
145:Introduction
144:
31:
6047:Outer space
6035:Spaceflight
5999:Mathematics
5953:Mathematics
5599:KerrâNewman
5570:Spherical:
5439:Other tests
5382:Singularity
5314:Formulation
5276:Fundamental
5130:Formulation
5111:Proper time
5072:Fundamental
4713:Thorne, Kip
4555:Smolin, Lee
3813:, Vintage,
3475:, Perseus,
3413:, Berkley,
3264:Lehner 2002
3222:Smolin 2001
3218:Greene 1999
3201:Maddox 1998
3159:Wright 2007
3146:Thorne 1994
3129:Robson 1996
3125:Disney 1998
3108:Gnedin 2005
3100:Thorne 1994
3066:Schutz 2003
2977:Schutz 2003
2930:Kramer 2004
2793:Schutz 2003
2789:Thorne 1994
2706:Geroch 1978
2598:Mermin 2005
2594:Greene 2004
2590:Thorne 1994
2547:Pössel 2007
2530:Mermin 2005
2447:Norton 1985
2417:Schutz 2003
2325:dark matter
2321:dark energy
2253:dark energy
2244:empty space
2194:homogeneous
2104:of massive
2077:Black holes
1974:dark matter
1880:timekeeping
1876:positioning
1782:white dwarf
1744:arc seconds
1674:Experiments
1425:temperature
1323:world lines
1233:dimensional
1215:. In 1907,
1057:red-shifted
996:accelerated
858:black holes
800:gravitation
691:van Stockum
621:Oppenheimer
476:KerrâNewman
268:Singularity
6083:Categories
5751:Zel'dovich
5659:Scientists
5638:Alcubierre
5445:of Mercury
5443:precession
5372:Black hole
5255:Background
5247:relativity
5216:World line
5211:Light cone
5036:Background
5028:relativity
5018:Relativity
4940:Audio help
4931:2021-05-09
4878:2007-06-12
4758:2007-06-13
4435:2007-05-06
4299:2007-06-12
4278:2007-06-11
4237:2007-06-13
4138:2007-06-12
3967:2013-07-15
3870:2007-05-06
3535:2007-07-15
3402:2007-07-06
3184:Cowen 2001
3167:Berry 1989
3163:Hogan 1999
3021:1602.03837
2947:Ashby 2003
2943:Ashby 2002
2917:2009-05-02
2844:Hartl 2005
2464:Ernst Mach
2392:References
2287:, notably
2098:black hole
2024:PSR1913+16
2008:black hole
1825:gyroscopes
1652:black hole
1182:equivalent
829:and other
544:Scientists
376:Formalisms
324:Formalisms
273:Black hole
199:World line
6011:Astronomy
5886:evolution
5721:Robertson
5706:Friedmann
5701:Eddington
5691:de Sitter
5525:Solutions
5403:detectors
5398:astronomy
5365:Phenomena
5300:Geodesics
5203:Spacetime
5146:Phenomena
4587:: 161â172
4563:, Basic,
4353:118949889
4028:119203172
4003:0709.0685
3642:118162303
3054:124959784
2964:Will 2006
2960:Will 1993
2893:Pais 1982
2877:Will 1993
2831:Will 2006
2827:Will 1993
2680:Pais 1982
2577:Pais 1982
2495:Pais 1982
2423:, p. 178.
2421:Pais 1982
2411:, and in
2409:Pais 1982
2405:Renn 2005
2317:cosmology
2273:classical
2233:expanding
2198:isotropic
2181:radiation
2167:Cosmology
2137:radiation
2102:evolution
1740:deflected
1640:spacetime
1556:π
1499:longitude
1437:molecules
1335:geodesics
1319:spacetime
1236:spacetime
1205:triangles
1177:curvature
1129:skydivers
1065:frequency
878:cosmology
876:model of
862:radiation
808:spacetime
636:Robertson
601:Friedmann
596:Eddington
586:Nordström
576:de Sitter
433:Solutions
358:Geodesics
353:Friedmann
335:Equations
321:Equations
282:Spacetime
217:Phenomena
123:ν
120:μ
111:κ
102:ν
99:μ
91:Λ
83:ν
80:μ
42:spacetime
5939:M-theory
5891:genetics
5834:Category
5711:LemaĂźtre
5676:Einstein
5666:Poincaré
5626:Others:
5610:TaubâNUT
5576:interior
5498:theories
5496:Advanced
5463:redshift
5278:concepts
5096:Rapidity
5074:concepts
4942: ·
4862:28179873
4818:(1): 3,
4789:(1993),
4768:(1990),
4715:(1994),
4677:15931216
4557:(2001),
4494:28179866
4450:(1): 6,
4393:(2004),
4370:(1982),
4227:12140952
4171:(2005),
4148:(1998),
3954:10641693
3830:(2004),
3805:(1999),
3788:15931201
3714:37236624
3663:(1961),
3392:28163638
3358:(1): 1,
3340:28163638
3046:26918975
2654:12502462
2343:See also
2222:Big Bang
2214:nuclear-
2189:universe
2126:galaxies
2118:billions
2114:millions
2071:Big Bang
2067:universe
2010:merger.
1785:Sirius B
1495:latitude
1452:pressure
1444:momentum
1376:towards
1213:surfaces
1157:surfaces
1153:geometry
992:produced
874:Big Bang
752:Category
616:LemaĂźtre
581:Reissner
566:Poincaré
551:Einstein
496:TaubâNUT
461:Wormhole
445:interior
158:Timeline
48:'s mass.
6094:Gravity
6071:Science
5987:Physics
5973:Portals
5917:entropy
5905:Physics
5896:viruses
5879:Biology
5776:Hawking
5771:Penrose
5756:Novikov
5736:Wheeler
5681:Hilbert
5671:Lorentz
5628:pp-wave
5449:lensing
5245:General
5026:Special
4929: (
4900:minutes
4853:5256066
4830:Bibcode
4685:4383030
4649:Bibcode
4602:Bibcode
4485:5256043
4462:Bibcode
4333:Bibcode
4261:Bibcode
4207:Bibcode
4116:9145148
4086:Bibcode
4008:Bibcode
3934:Bibcode
3838:Bibcode
3796:3023436
3766:Bibcode
3694:Bibcode
3612:Bibcode
3580:Bibcode
3564:3981642
3383:5253894
3360:Bibcode
3331:5253894
3310:Bibcode
3144:and in
3085:and in
3026:Bibcode
2634:Bibcode
2600:and in
2226:billion
2141:quasars
2094:compact
2022:called
2006:from a
1821:Cassini
1764:quasars
1729:Mercury
1721:Mercury
1713:ellipse
1615:is the
1607:is the
1456:tension
1427:or the
1358:equator
1356:or the
1346:segment
1342:geodesy
926:gravity
831:planets
827:Mercury
671:Hawking
666:Penrose
641:Bardeen
631:Wheeler
561:Hilbert
556:Lorentz
516:pp-wave
153:History
38:delayed
33:Cassini
5817:others
5806:Thorne
5796:Misner
5781:Taylor
5766:Geroch
5761:Ehlers
5731:Zwicky
5549:Kasner
4860:
4850:
4797:
4776:
4723:
4702:
4683:
4675:
4631:Nature
4612:
4567:
4544:
4526:
4508:
4492:
4482:
4403:
4380:
4351:
4225:
4183:
4158:
4114:
4104:
4060:
4026:
3982:
3952:
3907:
3885:
3848:
3817:
3794:
3786:
3758:Nature
3746:
3728:
3712:
3640:
3562:
3479:
3456:
3435:
3417:
3390:
3380:
3338:
3328:
3052:
3044:
2850:, and
2652:
2592:, and
2458:E. g.
2206:matter
2133:energy
1951:galaxy
1947:quasar
1931:quasar
1861:pulsar
1748:degree
1705:planet
1480:metric
1433:nuclei
1411:energy
1331:curved
1267:matter
1197:points
994:by an
823:orbits
796:theory
750:
717:others
711:Thorne
701:Newman
681:Taylor
661:Ehlers
646:Walker
611:Zwicky
486:Kasner
6023:Stars
5811:Weiss
5791:Bondi
5786:Hulse
5716:Milne
5620:discs
5564:Milne
5559:Gödel
5416:Virgo
4820:arXiv
4681:S2CID
4639:arXiv
4627:(PDF)
4452:arXiv
4360:(PDF)
4349:S2CID
4323:arXiv
4311:(PDF)
4249:(PDF)
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