HOME WEB NEWS IMAGES CLASSIFIEDS YELLOW PAGESPOLLS - SURVEYS WIKI COUNTRIES PHOTOS US UK INDIA
Cities Stocks Weather Movies Coupons Travel Blogs Health Horoscope Maps ASIA EUROPE Classifieds Yellow Pages
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
 1
|
For a generally accessible and less technical introduction to the topic, see Introduction to general relativity.
A simulated Black Hole of ten solar masses as seen from a distance of 600 km with the Milky Way in the background (horizontal camera opening angle: 90°).
General relativity (GR) or General theory of relativity (GTR) is the geometric theory of gravitation published by Albert Einstein in 1915/16.Einstein 1915 and Einstein 1916. It unifies special relativity, Newton\'s law of universal gravitation, and the insight that gravitational acceleration can be described by the curvature of space and time, this latter being produced by the mass-energy and momentum content of the matter in spacetime.
General relativity is distinguished from other metric theories of gravitation by its use of the Einstein field equations to relate spacetime content and spacetime curvature.Einstein 1916Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8. The field equations are a system of partial differential equations whose solution gives the metric tensor of spacetime, describing its "shape". In the resulting geometry, an object moving inertially in a gravitational field is viewed as following a geodesic path that may be found using the Christoffel symbols of the metric. Solutions of the Einstein field equations model gravitating systems, especially important ones exhibiting spherical symmetry, notable examples being the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric.
General relativity is currently the most successful gravitational theory, being almost universally accepted and well-supported by observations. General relativity\'s first success was in explaining the anomalous perihelion precession of Mercury. In 1919, Sir Arthur Stanley Eddington announced that observations of stars near the eclipsed Sun confirmed general relativity\'s prediction that massive objects bend light. Other observations and experiments have since confirmed many of the predictions of general relativity, including gravitational time dilation, the gravitational redshift of light, signal delay, gravitational radiation and the expansion of the universe. Numerous observations are also interpreted as confirming one of general relativity\'s most mysterious and exotic predictions, the existence of black holes.
| General relativity | ||||||||||||
| Einstein field equations | ||||||||||||
| Introduction to... Mathematical formulation of...
| ||||||||||||
Contents |
The structure of general relativity, as well as the way the theory is formulated, are best understood by examining its similarities with, and departures from, classical physics.The following exposition re-traces that of Ehlers 1973, section 1.
At the basis of classical mechanics, there is the notion that in describing a body\'s motion, we can differentiate between a special type of motion commonly known as free (or inertial) motion and deviations from such free motion; such deviations are caused by external forces acting on a body in accordance with Newton\'s second law of motion: the force acting on a body is equal to that body\'s (inertial) mass times its acceleration.See, for instance, Arnold 1989, chapter 1. There is a direct connection between the preferred inertial motions and the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed; in modern parlance, their paths are geodesics, or straight world lines in space-time.See Ehlers 1973, pp. 5f..
Ball falling to the floor in an accelerated rocket (left), and on Earth (right)
Conversely, it would seem that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space and a time coordinate. However, there is an ambiguity once gravity comes into play. Following from Newton\'s law of gravity, and independently verified by experiments such as that of Eötvös and its successors, there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.See Will 1993, section 2.4 or Will 2006, section 2. A simplified version of this is embodied in Einstein\'s elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket.Cf. Wheeler 1992, chapter 2; similar accounts can be found in most other popular-science books on general relativity.
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time – in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry; however, as can be shown using simple thought experiments, the Newtonian connection is not integrable – space-time is curved. The result is a geometric formulation of Newtonian gravity in geometrical terms. The geometric concepts used are all covariant, in other words: this description can be formulated using any desired coordinate system.See Ehlers 1973, section 1.2, Havas 1964, and Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959.
In this geometric formulation, tidal effects – the relative acceleration of bodies in free fall – are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.See Ehlers 1973, pp. 10f..
Light cone
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics;Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006. in the language of symmetry: where gravity can be neglected, physics is Lorentz invariant, not Galilei invariant – and the differences between the two become significant when we are dealing with fast motions or high-energy phenomena.An in-depth comparison between the two symmetry groups can be found in Giulini 2006a.
Lorentz symmetry introduces an additional conformal structure, namely the set of lightcones (see the image on the left). For each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.For instance Rindler 1991, section 22; a thorough treatment can be found in Synge 1972, ch. 1 and 2.
Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. As gravity comes into play, assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles; translated into the language of space-time: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.Cf. Ehlers 1973, sec. 1.4. and Schutz 1985, sec. 5.1.
A priori, it is not clear whether the new local frames in free fall are indeed those in which the laws of special relativity hold – that theory, is based on the propagation of light, and thus on electromagnetism, and its preferred frames might not be the same as the local free-falling inertial frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift; the actual measurements show that free-falling frames in which light propagates as it does in special relativity.See Ehlers 1973, p. 17ff.; a derivation can be found e.g. in Mermin & 2005 ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below. The generalization of this statement, namely that the laws of special relativity hold, to good approximation, in freely falling (and non-rotating) reference frames, is known as the Einstein [equivalence principle], and is one of the guiding principles when it comes to generalizing special relativistic physics to include gravity.Cf. Rindler 2001, sec. 1.13; for an elementary account, see chapter 2 of Wheeler 1990; there are, however, some differences between the modern version and Einstein\'s original concept used in the historical derivation of general relativity, cf. Norton 1985.
However, again with reference to experimental data, it becomes clear that, for instance, proper time for clocks in a gravitational field, is not measured by the Minkowski metric of special relativity. As in the Newtonian case, this is suggestive of a more general geometry: where all reference frames in free fall are equivalent, and approximately Minkowskian, we are dealing with a curved generalization of Minkowski space: instead of Minkowskian, assume the metric tensor to be, more generally, semi-Riemannian Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, a Levi-Civita connection; assume this to be the connection implied by the universality of free fall.Ehlers 1973, sec. 1.4. for the experimental evidence, see once more section Gravitational time dilation and frequency shift.
While the preceding section shows the relativistic, geometric version of the effects of gravity, there is still the question of the source of gravity. In Newtonian gravity, the source is mass; in special relativity, mass is equivalent to energy, and in fact turns out to be part of a more inclusive quantity called energy-momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear).Cf. Ehlers 1973, p. 16; Kenyon 1990, sec. 7.2; Weinberg 1972, sec. 2.8. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this and the tensor that embodies tidal effects, the Ricci tensor. Adding a suitable geometric form for the conservation of energy-momentum, the simplest set of equations are what are called Einstein\'s (field) equations, which equate the energy-momentum tensor and a specific combination of the Ricci tensor and the metric known as the Einstein tensor:
where Gab is the Einstein tensor, Tab is the energy-momentum tensor (both written in abstract index notation).See Ehlers & 1973 pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. It can be shown that the Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the space-time of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. Matching the theory\'s prediction to observational results for planetary orbits, the proportionality constant has the value with the gravitational constant and the speed of light.E.g. Kenyon 1990, sec. 7.4. The tensors Gab and Tab are both rank-2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations; the fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations.E.g. Schutz 1985, sec. 8.3.
While the metric description of gravity follows rather straightforwardly from special relativity and the universality of free fall, it is worth mentioning that there are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Brans-Dicke theory, teleparallelism, and Einstein-Cartan theory.Cf. Brans & Dicke 1961 and section 3 in ch. 7 of Weinberg 1972, Goenner 2004, sec. 7.2, and Trautman 2006, respectively.
As a result of the derivation sketched in the previous section, we now have all the information needed to define and characterize general relativity.
General relativity is a metric theory of gravitation. Its core are Einstein\'s equations, which link the geometry of a four-dimensional, semi-Riemannian manifold representing space-time with the energy-momentum contained in that space-time.E.g. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other text-book on general relativity. Phenomena that, in classical mechanics, are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity.At least approximately, cf. Poisson 2004. The curvature is, in turn, caused by the energy-momentum of matter; paraphrasing the relativist John Archibald Wheeler, space-time tells matter how to move; matter tells space-time how to curve.E.g. p. xi in Wheeler 1990.
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases: for weak gravitational fields and slow speed relative to the speed of light, the theory\'s predictions converge on those of Newton\'s law of gravity.E.g. Wald 1984, sec. 4.4.
As it is constructed using tensors, general relativity exhibits general covariance, that is, its laws – and further laws formulated within the general relativistic framework – take on the same form in all coordinate systems.E.g. in Wald 1984, sec. 4.1. Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2006b. Locally, as expressed in the equivalence principle, space-time is Minkowskian, and the laws of physics have local Lorentz invariance.E.g. section 5 in ch. 12 of Weinberg 1972.
The core concept of general-relativistic model-building is that of a solution of Einstein\'s equations. Given both Einstein\'s equation and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates) on which are defined specific matter fields, in such a way that matter and geometry satisfy Einstein\'s equations, and that the matter satisfies whatever equations have been imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.Cf. the introductory chapters of Stephani et al. 2003.
Einstein\'s equations are non-linear partial differential equations and, as such, very difficult to solve.A review showing Einstein\'s equation in the broader context of other PDEs with physical significance is Geroch 1996. Nevertheless, a number of exact solutions are known, although only a few of them have direct physical applications.For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006. The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,E.g. chapters 3, 5, and 6 of Chandrasekhar 1983. and the Friedmann-Lemaître-Robertson-Walker and de Sitter universes, each describing an expanding cosmos.E.g. ch. 4 and sec. 3.3. in Narlikar 1993. Exact solutions of great theoretical interest include the Gödel universe, the Taub-NUT solution, and Anti-de Sitter space.Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973.
In addition, significant efforts are being made in the field of numerical relativity, where the goal is to find interesting numerical solutions describing, say, two black holes orbiting each other, with the help of powerful computers.See Lehner 2002 for an overview.
Also, there are different methods for finding approximate solutions in the context of perturbation theory. The best-known of these are linearized gravityFor instance Wald 1984, sec. 4.4. and its generalization, the Post-Newtonian expansion, which represents a systematic way of describing a space-time containing matter which is not particularly compact and moves but slowly compared with the speed of light; the description starts with Newtonian gravity and, in a systematic sequence, takes into account smaller and smaller effects arising from the difference between Newton\'s theory and general relativity.E.g. Will 1993, sec. 4.1 and 4.2. An extension of this expansion is the Parametrized Post-Newtonian (PPN) formalism, a framework of testing general relativity against alternative theories in a way that allows quantitative comparisons.Cf. section 3.2 of Will 2006 as well as Will 1993, ch. 4.
General relativity has a number of consequences, some following directly from the theory\'s axioms, others having become clear only in the course of the ninety years of research that followed Einstein\'s initial publication.
Schematic representation of the gravitational redshift of a light wave escaping from the surface of a massive body
In general relativity (and, in fact, in any theory in which the equivalence principle holdsCf. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. In fact, Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198.), gravity has an immediate influence on the passage of time. Imagine two observers Alice and Bob, both of which are at rest in a stationary gravitational field, with Alice closer to the source of gravity ("deeper in the gravity well") and Bob at a greater distance. Then for light sent from Alice to Bob or vice versa, Bob will measure a lower frequency than Alice: light sent down into a gravity well is blue-shifted, light climbing out of a gravity well is redshifted. Also, Alice\'s clocks tick more slowly than Bob\'s: whenever the two are compared (either by sending light signals back and forth, or by slowly transporting clocks from one location to the other), the result will be that Bob\'s clocks are running faster. This effect is not restricted to clocks, but applies to all processes (the rate at which Alice and Bob age, cook five-minute eggs, or play Chopin\'s Minute Waltz); it is known as gravitational time dilation.Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5..
The gravitational redshift was first measured in 1959 in a laboratory experiment by Pound and RebkaSee Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186. and later confirmed by astronomical observations.E.g. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow et al. 2005. There are numerous direct measurements of gravitational time dilation using atomic clocksStarting with the Hafele-Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186. while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS).GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistc effects, see Ashby 2002 and Ashby 2003. Tests in stronger gravitational fields are provided by the observation of binary pulsars.Reviews are given in Stairs 2003 and Kramer 2004. All results are in agreement with general relativity;General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, section 4.2. however, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.Cf. Ohanian & Ruffini 1994, pp. 164–172.
Deflection of light (sent out from the location shown in blue) near a compact body (shown in gray).
In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics – a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity.The fact that light follows null geodesics is not an independent axiom; it can be derived from Einstein\'s equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, section 5. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),A brief descriptions and pointers to the literature can be found in Blanchet 2006, section 1.3. several effects of gravity on light propagation emerge.
The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also be derived by extending the universality of free fall to light,See Rindler 2001, section 1.16; for the historical examples, Israel 1987, p. 202–204.; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997. the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity; from the standpoint of Einstein\'s theory they take into account the effect of gravity on time, but not its consequences for the warping of space.E.g. Rindler 2001, sec. 11.11. An important example of this is starlight being deflected as it passes the Sun; in consequence, the positions of stars observed in the Sun\'s vicinity during a solar eclipse appear shifted by up to 1.75 arc seconds. This effect was first measured by a British expedition directed by Arthur Eddington, and confirmed with significantly higher accuracy by subsequent measurements.Cf. Kennefick 2005; for an overview of more recent measurements, see Ohanian & Ruffini 1994, chapter 4.3. The most precise direct modern observations measure the deflection of the light of distant quasars by the Sun, cf. Shapiro et al. 2004.
Closely related to the bending of light is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. This effect was discovered through the observations of radar signals sent from Earth to planets such as Venus or Mercury and thence reflected back;Shapiro 1965; a pedagogical introduction can be found in Weinberg 1972, ch. 8, sec. 7. later, much more accurate measurements utilized signals sent to space probes and sent back using active transponders.The most recent measurements are Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200. In both the case of the planets and the probes, what was measured was the propagation of signals in the Sun\'s gravitational field. More recent measurements have detected the Shapiro effect in signals sent by a pulsar that is part of a binary system; in that case, the gravitational field causing the time delay is that of the other pulsar.Cf. Stairs 2003, section 4.4. In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called that reflects the influence of gravity on the geometry of space.Will 1993, sec. 7.1 and 7.2.
The angular deflection of a beam of light may be represented by the following equation: where A.D. is the angular deflection of the light beam, G is the gravitational constant, M is the mass of the object causing the deflection, c2 is the speed of light squared, and R is the radius of the object (such as a star) that is causing the deflection. This is equivalent to saying where C is 1.75 arc seconds (the amount of deflection caused by the sun), M is the mass of the star divided by the mass of the Sun, and R is the radius of the star divided by the radius of the Sun.
Ring of test particles floating in space
Ring of test particles influenced by gravitational wave
There are several analogies between weak-field gravity and electromagnetism. One is that, for electromagnetic waves, there are corresponding gravitational waves: ripples in spacetime that propagate at the speed of light.For an overview, see Misner, Thorne & Wheeler 1973, part VIII. Note, however that for gravitational waves, the dominant contribution is not the dipole, but the quadrupole cf. Schutz 2001.
The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right).Any textbook on general relativity will contain a description of these properties, e.g. Schutz 1981, ch. 9. Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.For example Jaranowski & Królak 2005.
It is, however, important to note that the linearized waves are only approximations. Generically, the non-linearity of the Einstein equations means that there is no linear superposition for gravitational waves. Describing such more general waves is not an easy task. There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty spaceRindler 2001, ch. 13. or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves,See Gowdy 1971, Gowdy 1974. while, when it comes to describing the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy.
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The most striking one concerns the relativistic apside shifts, orbital decay caused by the emission of gravitational waves, and effects that are due to the relativity of direction.
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star
In general relativity, the apsides of orbits (the points of an orbiting body closest approach to the system\'s center of mass) will precess – the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rosette-like shape (see image). Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle;Pais 1982, pp. 253–254 the result can also be obtained by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)See Rindler 2001, section241. or the much more general post-Newtonian formalism.See Will 1993, pp. 177–181. The effect is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body\'s gravity (in other words, the special kind of nonlinearity exhibited by Einstein\'s theory).In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms and , cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3.
An early success of general relativity was that the theory offered a straightforward explanation for an anomalous perihelion shift of the planet Mercury, which had been discovered by Urbain Le Verrier in 1859 but had remained mysterious.See Schutz 2003, pp. 48-49 and Pais 1982, pp. 253–254. This agreement between theory and experiment confirmed for Einstein that he had at last identified the correct form of the gravitational field equations. More recent observations have shown that the field equations predict the correct anomalous perihelion shift for all planets where this can be measured accurately (Mercury, Venus and the Earth).The most precise measurements are VLBI measurements of planetary positions; see Will 1993, chapter 5, Will 2006, section 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406-407. The effect has also been checked in binary pulsar systems where it is larger by five orders of magnitude.See Kramer, Stairs & Manchester 2006.
Orbital decay for PSR1913+16: time shift in seconds, tracked over three decades.
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period; since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation.See Stairs 2003 and Schutz 2003, pp. 317–321; an accessible account can be found in Bartusiak 2000, pp. 70–86.
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor using binary pulsar PSR1913+16 they had discovered in 1974; it amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993.An overview can be found in Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994. Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars.Cf. Kramer 2004.
Several relativistic effects are directly related to the relativity of direction.See e.g. Penrose 2004, §14.5, Misner, Thorne & Wheeler 1973, sec. §11.4. One is geodetic precession: for a gyroscope in free fall in curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars – even though its motion comes closest to keeping its axis direction constant ("parallel transport").See Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1982, sec. 7.8. For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging;See Bertotti, Ciufolini & Bender 1987 and, for a more recent review, Nordtvedt 2003. more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent.See Kahn 2007.
Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass gets "dragged around"; this is most extreme for rotating black holes where, for an object entering a zone known as the ergosphere, rotation is inevitable.E.g. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471. Such effects can again be tested through their influence on the orientation of a gyroscope in free fall:E.g. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004. somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction;E.g. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006; see the entry frame-dragging for an account of the debate. a precision measurement is the main aim of the Gravity Probe B mission, whose final results are expected in May 2008.A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt et al. 2007; further updates will be available on the mission website Kahn 1996–2007.
Einstein cross: four images of the same astronomical object, produced by a gravitational lens
The deflection of light by gravity can have an intriguing side effect: a massive object between the observer and a distant target object makes it possible for the observer to see multiple distorted images of the target. This and similar effects are known as gravitational lensingFor overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998. and, depending on the configuration, scale, and mass distribution, it can result in two images, a bright ring known as an Einstein ring, or partial rings called arcs.For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3. The earliest example was discovered in 1979;See Walsh, Carswell & Weymann 1979. since then, more than a hundred gravitational lenses have been observed.Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007. Images too close to be resolved can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object; a number of such "microlensing events" has been observed, as well.For an overview, see Roulet & Mollerach 1997.
Gravitational lensing has developed into a tool of observational astronomy. Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data are also used to understand the structural evolution of galaxies.See Narayan & Bartelmann 1997, sec. 3.7.
Artist\'s impression of the space-borne gravitational wave detector LISA
From observations of binary pulsars, there is strong indirect evidence for the existence of gravitational waves (see the section on Orbital decay, above). However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, this being one of the major goals of current relativity-related research.For an overview, Barish 2005; accessible accounts can be found in Bartusiak 2000 and Blair & McNamara 1997. To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO.An overview is given in Hough & Rowan 2000. A joint US-European mission to launch a space-based detector, LISA, is currently under development,See Danzmann & Rüdiger 2003. with a precursor mission (LISA Pathfinder) due for launch in late 2009.See Landgraf, Hechler & Kemble 2005.
Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation:Cf. Thorne 1995. Terrestrial detectors are expected to yield new information about inspiral phase and mergers of binary stellar mass black holes and binaries consisting of one such black hole and a neutron star (of interest as a candidate mechanism for gamma ray bursts); they could also detect signals from core-collapse supernovae and from periodic sources such as rotating neutron stars with small deformation. If there is truth to speculation about certain kinds of phase transitions or kink bursts from long cosmic strings in the very early universe (at cosmic times around seconds) these could also be detectable.See Cutler & Thorne 2001, sec. 2. Space-based detectors like LISA should detect objects such as binaries consisting of two White Dwarfs, and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star), and also observe the mergers of supermassive black holes and the inspiral of smaller objects (between one and a thousand solar masses) into such black holes. LISA should also be able to listen to the same kind of sources from the early universe as ground-based detectors, but at even lower frequencies and with greatly increased sensitivity.See Cutler & Thorne 2001, sec. 3.
Simulation based on the equations of general relativity: a star collapsing to form a black hole while emitting gravitational waves
Whenever an object becomes sufficiently compact, general relativity predicts the formation of a black hole: a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars with around 1.4 solar mass and so-called stellar black holes with a few to a few dozen solar masses are thought to be the final state for the evolution of massive stars.See Miller 2002, lectures 19 and 21. Supermassive black holes with between a few million and a few billion solar masses are now thought to be the rule rather than the exception in the centers of galaxies,E.g. Celotti, Miller & Sciama 1999, sec. 3. and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures.Cf. Springel & al. 2005 and the accompanying summary Gnedin 2005.
Astronomically, the most important property of compact objects is that they provide a superbly efficient mechanism for converting gravitational into radiation energy.Cf. Blandford 1987, section 8.2.4, Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as Microquasars.For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996. In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.For a review, see Begelman, Blandford & Rees 1984. General relativity plays a central role in modelling all these phenomena,For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. relativistic lensing effects being thought to play a role for the signals received from X-ray pulsars.Cf. Kraus 1998.
Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"),See Celotti, Miller & Sciama 1999. observations of stellar dynamics in the center of our own Milky Way galaxy,Cf. Schödel et al. 2003. and indications that at least some of the compact objects in question appear to have no solid surfaceExamination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 and, for an overview, Narayan 2006, sec. 5. provide strong indirect evidence for the existence of black holes. Direct evidence, such as observing the "shadow" of the Milky Way galaxy\'s central black hole horizon,Cf. Falcke, Melia & Agol 2000. is eagerly sought for.
Black holes are also sought-after targets in the search for gravitational waves (see the section Gravitational waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity;Cf. Seidel 1998. the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances;Cf. Dalal et al. 2006. the gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole\'s geometry.E.g. Barack & Cutler 2004.
Each solution of Einstein\'s equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. The current models are based on an extension of the original form of Einstein\'s equations which include the cosmological constant , an additional term that has an important influence on the large-scale dynamics of the cosmos,
where gab is the spacetime metric.Originally Einstein 1917; cf. the description in Pais 1982, pp. 285–288.
Image of radiation emitted no more than a few hundred thousand years after the big bang, captured with the satellite telescope WMAP
On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker solutions,See Carroll 2001, ch. 2. are built the models of modern cosmology in which the universe has evolved over the past 14 billion years from a hot, early Big bang phase.See Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991. Once a small number of parameters (for example the universe\'s mean matter density) have been fixed by astronomical observation,E.g. with WMAP data, see Spergel et al. 2003. further observational data can be used to put the models to the test: successful predictions include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,See Peebles 1966; for a recent account of predictions, see Coc et al. 2004; an accessible account can be found in Weiss 2006. which is in good agreement with astronomical observations;See Olive & Skillman 2004, Bania, Rood & Balser 2002, O\'Meara et al. 2001, and Charbonnel & Primas 2005. the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation,Cf. Alpher & Herman 1948 and, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965, andfor precision measurements by satellite observatories see Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). and the large-scale distribution of galaxies.A review can be found in Lahav & Suto 2004.
The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,See, e.g., fig. 2 in Bridle et al. 2003. and they have proven a sound basis to explaining the evolution of the universe\'s large-scale structure.For a review, see Bertschinger 1998; more recent results can be found in Springel et al. 2005. On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observationsThese additional observations involve the dynamics of galaxies and galaxy clusters cf. chapter 18 of Peebles 1993, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005.) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly); there is currently no generally accepted description of this new kind of matter within the framework of particle physicsSee Peacock 1999, ch. 12, and Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12. or otherwise.Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9. A similar open question is that of dark energy. Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, namely dark energy;See Carroll 2001; an accessible overview is given in Caldwell 2004. the nature of this new form of energy remains unclear.Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2007.
A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1. have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around seconds, known as an inflationary phase.A good introduction is Linde 1990; for a more recent review, see Linde 2005. While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,See Spergel et al. 2007, sec. 5 & 6. problems remain. There is a bewildering variety of possible inflationary scenarios not restricted by current observations.More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory. Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity; an authoritative answer would require a complete theory of quantum gravity, which does not exist at the momentSee Brandenberger 2007, sec. 2. (cf. the section Quantum gravity, below).
In general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. Hence, an exploration of all light worldlines (
