User:Madcoverboy/Sandbox/General relativity

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Illustration of the Cassini measurements of the Shapiro effect
Illustration of the Cassini measurements of the Shapiro effect

General relativity (GR) (also general theory of relativity) is the theory of gravitation published by Albert Einstein in 1916.[1][2] As one of the cornerstones of modern physics and cosmology, it unifies special relativity and Newton's law of universal gravitation through the insight that gravitational accelerations and forces can be described by the curvature of space and time.

The Einstein field equations that describe the geometry of space-time and its solutions predict various cosmological phenomena such as black holes, gravitational lensing, time dilation, and gravitational waves. So far, general relativity has been experimentally verified with space probes traveling around the Sun and planets and observations of light emitted by supernova or passing near distant neutron stars.[3] which However, there are irreconcilable differences between modern theories of relativity and quantum mechanics which suggests the need for alternative theoretical interpretations or a theory of quantum gravity.


General relativity
G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,
Einstein field equations
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[edit] History

Main articles: History of general relativity and Classical theories of gravitation

General relativity was developed by Einstein in a process that began in 1907 with the publication of an article on the influence of gravity and acceleration on the behavior of light in special relativity. Soon after, he began to think about how to incorporate gravity into his new relativistic framework. His considerations led him from a simple thought experiment involving an observer in free fall to a fully geometric theory of gravity.[4] Most of this work was done in the years 1911–1915, beginning with the publication of a second article on the effect of gravitation on light. By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. In December of 1915, these efforts culminated in Einstein's submission of a paper presenting the Einstein field equations, which are a set of differential equations.[2] Since 1915, the development of general relativity has focused on solving the field equations for various cases. The interpretation of the solutions and their possible experimental and observational testing also constitutes a large part of research in GR.

Starting in 1922, researchers found that cosmological solutions of the Einstein field equations call for an expanding universe. Einstein did not believe in an expanding universe, and so he added a cosmological constant to the field equations to permit the creation of static universe solutions. In 1929, Edwin Hubble found evidence that the universe is expanding. This resulted in Einstein dropping the cosmological constant, referring to it as "the biggest blunder in my career." Progress in solving the field equations and understanding the solutions has been ongoing. Notable solutions have included the Schwarzschild solution (1916), the Reissner-Nordström solution, the Friedmann-Robertson-Walker solution and the Kerr solution.

Observationally, general relativity has accounted for the discrepancy between the Newtonian prediction and observed perihelion precession of Mercury. In 1919, Eddington's announcement that his observations of stars near the eclipsed Sun confirmed Einstein's prediction for the deflection of light by the Sun helped to cement the status of general relativity as a likely true theory. Since then, many observations have confirmed the predictions of general relativity. These include observations of gravitational red shift, studies of binary pulsars, observations of radio signals passing the limb of the Sun, and the GPS system.

The Golden age of general relativity was a period between approximately 1960 and 1975 when study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics with concepts such as the big bang, black holes, quasars, and pulsars. While there were many contributors to general relativity, the "golden age" is generally regarded as having ending in 1980 when Steven Hawking proposed that black holes could radiate energy.

[edit] Overview

Further information: Equivalence principle
Ball falling to the floor in an accelerated rocket (left) and on Earth (right)
Ball falling to the floor in an accelerated rocket (left) and on Earth (right)

One of the defining features of general relativity is the idea that gravitational 'force' is replaced by geometry. In 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) are taken in general relativity to represent inertial motion within a curved geometry known as spacetime.

The justification for creating general relativity came from the equivalence principle, which dictates that free-falling observers are the ones in inertial motion. Roughly speaking, the principle states that the most obvious effect of gravity – things falling down – can be eliminated by making the transition to a reference frame that is in free fall, and that in such a reference frame, the laws of physics will be approximately the same as in special relativity.[5] A consequence of this insight is that inertial observers can accelerate with respect to each other. For example, a person in free fall in an elevator whose cable has been cut will experience weightlessness: objects will either float alongside him or her, or drift at constant speed. In this way, the experiences of an observer in free fall will be very similar to those of an observer in deep space, far away from any source of gravity, and in fact to those of the privileged ("inertial") observers in Einstein's theory of special relativity.[6] Einstein realized that the close connection between weightlessness and special relativity represented a fundamental property of gravity.

Einstein's key insight was that there is no fundamental difference between the constant pull of gravity we know from everyday experience and the fictitious forces felt by an accelerating observer (in the language of physics: an observer in a non-inertial reference frame).[7][8] So what people standing on the surface of the Earth perceive as the 'force of gravity' is a result of their undergoing a continuous physical acceleration which could just as easily be imitated by placing an observer within a rocket accelerating at the same rate as gravity (9.81 m/s²).

This redefinition is incompatible with Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity. To quote Einstein himself:

“ If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them." [9] ”

Thus the equivalence principle led Einstein to develop a gravitational theory which involves curved space-times. Paraphrasing John Wheeler, Einstein's geometric theory of gravity can be summarized thus: spacetime tells matter how to move; matter tells spacetime how to curve.[10]

[edit] Theoretical justification

General relativity is based upon a set of fundamental principles and assumptions which guided its development.[2][11]

[edit] General principle of relativity

Main article: General principle of relativity

The laws of physics must be the same for all observers (accelerated or not).

[edit] Principle of general covariance

Main article: Principle of general covariance

The laws of physics must take the same form in all coordinate systems. In addition, the principle of general covariance forces that mathematics to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.

[edit] Inertial and geodesic motion

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

Due to the expectation that spacetime is curved, Riemannian geometry (a type of non-Euclidean geometry) must be used. Because the motion of objects is influenced solely by the geometry of spacetime, inertial motion occurs along special paths known as the timelike and null geodesics. In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry (straight lines that start out as parallel will remain parallel, namely at a constant distance from each other), but instead objects that were initially traveling in parallel paths through spacetime come to travel in a non-parallel fashion. On a curved surface like that of the Earth, lines of longitude that are parallel at the equator will eventually intersect at the pole. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity.

On a curved two-dimensional surface, it is still possible to define lines that are as straight as possible, so called geodesics – for instance, on the spherical surface of the Earth, the lines of longitude, the equator, and other great circles. But the properties of these lines will differ from those of straight lines. Analogously, the world lines of test particles in free fall are spacetime geodesics – they are the straightest possible lines in spacetime, but there will be important differences between them and the truly straight lines in the gravity-free spacetime of special relativity: in special relativity, parallel geodesics remain parallel; in a gravitational field with tidal effects, such as the case of two bodies falling side by side towards the center of the Earth, initially parallel geodesics converge as the bodies move towards each other.[12]

[edit] Local Lorentz invariance

Main article: Local Lorentz invariance

The laws of special relativity apply locally for all inertial observers.

[edit] Geometry of spacetime

Main article: Einstein field equations

In 1907, Hermann Minkowski introduced a geometry that included not only the three (apparent) dimensions of space, but also a fourth dimension of time in order to present a geometric formulation of Einstein's special theory of relativity.[13] Building upon the mathematical work of Carl Friedrich Gauss and Bernhard Riemann, Einstein postulated that Minkowski's spacetime could be treated as a 4-dimensional manifold which is curved by the presence of mass, energy and momentum (which, taken together, is known as stress-energy). Because Einstein previously determined that mass and energy are equivalent (E = mc²), gravitation is not caused by mass alone (as predicted by Newtonian physics) but by the distortion of spacetime by a combination of mass, energy, and momentum. The Einstein field equations describe the relationship between stress-energy and the curvature of spacetime.[14]

The curvature of spacetime (caused by the presence of stress-energy of massive entities like the Sun or Earth) can be analogized by placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. The larger the mass, the bigger the amount of curvature. A relatively light object placed in the vicinity of the 'dent', such as a ping-pong ball, will accelerate towards the bowling ball in a manner governed by the 'dent'. Firing the ping-pong ball at some suitable combination of direction and speed towards the 'dent' will result in the ping-pong ball 'orbiting' the bowling ball. This is analogous to the Moon orbiting the Earth, for example.[15] Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to the ping-pong ball's response to the bowling ball's dent rather than the bowling ball itself), other massive objects respond to how the first massive object curves spacetime.

[edit] The mathematics of general relativity

Main articles: Mathematics of general relativity and Einstein field equations
Coordinates with the same difference in longitude at different latitudes are different absolute distances. Someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,051 mi); for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,181 mi)
Coordinates with the same difference in longitude at different latitudes are different absolute distances. Someone at the equator, moving 30 degrees of longitude westward (magenta line) corresponds to a distance of roughly 3,300 kilometers (2,051 mi); for someone at a latitude of 55 degrees, moving 30 degrees of longitude westward (blue line) covers a distance of merely 1,900 kilometers (1,181 mi)

A metric is a function used to describe the geometric properties of a space (or a spacetime), especially within Riemannian geometry. Just as a location on a flat plane can be expressed using coordinates, location on a curved sphere can be likewise defined using coordinates like latitude and longitude. Because coordinate differences on a curved surface do not correspond to equal absolute distances (see image on right), metrics are employed to calculate distances, angles, and other quantities between complex geometries.[16]

Using a combination of calculus and metrics called metric tensors, the Einstein field equations (EFE) describe how stress-energy causes curvature of spacetime and are usually written in abstract index notation (a mathematical shorthand) as:

G_{ab} = \kappa T_{ab} = \frac{8\pi G}{c^4}\ T_{ab}

where Gab is the Einstein tensor describing the curvature of spacetime, Tab is the stress-energy tensor describing the combination of mass, energy, and momentum within spacetime, and κ is a constant (combining the speed of light, gravitational constant, and pi). By setting these two objects equal to each other results in ten separate equations (expanded from the shorthand above) expressing in mathematically precise language just what is meant by "spacetime tells matter how to move, and matter tells spacetime how to curve."[17]

[edit] Solutions of EFEs

Main article: Solutions of the Einstein field equations

Any spacetime with an associated distribution of matter for which the curvature of spacetime and the energy and momentum of matter do satisfy these conditions is called a solution of Einstein's equations. These solutions are metrics of spacetime which describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. For any model universe in which matter and geometry are meant to obey the laws of general relativity, Einstein's equations define ten independent conditions that must be fulfilled simultaneously at each point in spacetime. Being non-linear differential equations, the EFE often defy exact solutions; however, many such solutions are known. The simplest solution is Minkowski spacetime, the spacetime of special relativity. Other important solutions describe the gravitational field around a spherically symmetric massive object (Schwarzschild solution, 1916) or gravity and geometry in an expanding universe (Friedmann-Lemaître-Robertson-Walker solution).[18]

The EFE are the identifying feature of general relativity. Other theories built out of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, teleparallelism, Rosen's bimetric theory, and Einstein-Cartan theory)

[edit] Modification of other theories

Main article: Physical theories modified by general relativity

Because general relativity enabled a paradigm shift away from Newtonian mechanics which had underpinned the physical sciences to that time, previous theories of inertia, gravitation, electromagnetism, and quantum mechanics had to adopt new geometrical framework and assumptions. While some of these have been validated, there is still a theoretical gap between the assumptions in quantum mechanics and general relativity.

[edit] Predictions of general relativity

[edit] Gravitational effects

[edit] Gravitational redshifting

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

The first of these effects is the gravitational redshifting of light. Under this effect, the frequency of light will decrease (shifting visible light towards the red end of the spectrum) as it moves to higher gravitational potentials (out of a gravity well).

Assume that there are two observers, both of them at rest relative to a massive body. When the observer closer to the massive object sends some light to a second observer that is at rest higher up, the light will be red-shifted; the second observer will measure a lower frequency for the light than the first. Conversely, light sent from the second observer to the first will be blue-shifted (shifted towards higher frequencies).[19] This is caused by an observer at a higher gravitational potential being accelerated (with respect to the local inertial frames of reference) away from the source of a beam of light as that light is moving towards that observer. Gravitational redshifting has been confirmed by the Pound-Rebka experiment.[20][21][22]

[edit] Gravitational time dilation

A related effect is gravitational time dilation, under which clocks will run slower at lower gravitational potentials (deeper within a gravity well). For the same light wave, the second observer measures a lower frequency than the first; evidently, the second observer's clocks are running faster than those of the first observer. The same effect can also be derived in other ways (notably by transporting clocks back and forth and reconstructing the effect of location on their tick rate). Generally, clocks that are further down in a gravitational field tick more slowly than those that are higher up.[23] This effect has been directly confirmed by the Hafele-Keating experiment[24][25] and GPS.

Gravitational time dilation has as a consequence another effect called the Shapiro effect (also known as gravitational time delay). Shapiro delay occurs when 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 signals from spacecraft and pulsars passing behind the Sun as seen from the Earth.[26][27]

[edit] Gravitational lensing

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

Gravitational lensing occurs when one distant object is in front of or close to being in front of another much more distant object. In that case, the bending of light by the nearer object can change how the more distant object is seen. The first example of gravitational lensing was the discovery of a case of two nearby images of the same pulsar. Since then many other examples of distant galaxies and quasars being affected by gravitational lensing have been found.

In a similar way, Einstein also derived another effect, the gravitational deflection of light where light rays are bent downward in a gravitational field. 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. Subsequent observations of the deflection of the light of distant quasars by the Sun, which utilize highly accurate techniques of radio astronomy, have confirmed Eddington's results with significantly higher accuracy.[28][29]

A special type of gravitational lensing occurs in Einstein rings and arcs. The Einstein ring is created when an object is directly behind another object with a uniform gravitational field. In that case, the light from the more distant object becomes a ring around the closer object. If the more distant object is slightly offset to one side and/or the gravitational field is not uniform, partial rings (called arcs) will appear instead.

Finally, in our own galaxy a star can appear to be brightened when compact massive foreground object is sufficiently aligned with it. In that case, the magnified and distorted images of the background star due to the gravitational bending of light cannot be resolved. This effect is called microlensing, and such events are now regularly observed.

Gravitational lensing has developed into a tool of observational astronomy, where it is used (among other things) to determine the masses of certain objects, detect the presence of dark matter, and provide an independent estimate of the Hubble constant.[30]

[edit] Orbital effects

Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star
Newtonian (red) vs. Einsteinian orbit (blue) of a lone planet orbiting a star
Main article: Kepler problem in general relativity

General relativity differs from classical mechanics in its predictions for orbiting bodies. The first difference is in the prediction that apsides of orbits will precess on their own. This is not called for by Newton's theory of gravity. Because of this, an early successful test of general relativity was its correctly predicting the anomalous perihelion precession of Mercury. More recently, perihelion precession has been confirmed in the large precessions observed in binary pulsar systems.

A related effect is geodetic precession. This is a precession of the poles of a spinning object due to the effects of parallel transport in a curved space-time. This effect is not expected in Newtonian gravity. The prediction of geodetic precession was tested and verified by the Gravity Probe B experiment to a precision of better than 1 percent[31].

Another effect is that of orbital decay due to the emission of gravitational radiation by a co-rotating system. It is observable in closely orbiting stars as an ongoing decrease in their orbital period. This effect has been observed in binary pulsar systems.

[edit] Frame dragging

Main article: Frame dragging

Frame dragging is where a rotating massive object "drags" space-time along with its rotation. In essence, an observer who is distant from a rotating massive object and at rest with respect to its center of mass will find that the fastest clocks at a given distance from the object are not those which are at rest (as is the case for a non-rotating massive object). Instead, the fastest clocks will be found to have component of motion around the rotating object in the direction of the rotation. Similarly, it will be found by the distant observer that light moves faster in the direction of the rotation of the object than against it. Frame dragging will also cause the orientation of a gyroscope to change over time. For a spacecraft in a polar orbit, the direction of this effect is perpendicular to the geodetic precession mentioned above. Gravity Probe B is using this feature to test both frame dragging and the geodetic precession predictions.

[edit] Black holes

Main article: Black hole
An illustration of a rotating black hole at the center of a galaxy
An illustration of a rotating black hole at the center of a galaxy

When mass is concentrated into a sufficiently compact region of space, general relativity predicts the formation of a black hole – a region of space with a gravitational attraction so strong that not even light can escape.

The disappearance of light and matter within a black hole may be thought of as their entering a region where all possible world lines point inwards. Stephen Hawking has predicted that black holes can "leak" mass,[32] a phenomenon called Hawking radiation, a quantum effect not in violation of general relativity. Certain types of black holes are thought to be the final state in the evolution of massive stars. Supermassive black holes are thought to be present in the cores of most galaxies, and are thought to play a key role in galactic evolution. Numerous black hole candidates are known. These include the supermassive object associated with Sagittarius A* at the center of our galaxy[33]

Matter falling onto a compact object is one of the most efficient mechanisms for releasing energy in the form of radiation, and matter falling onto black holes is thought to be responsible for some of the brightest astronomical phenomena imaginable, such as quasars and other types of active galactic nuclei.[34]

[edit] Cosmology

Main article: Physical cosmology

Although it was created as a theory of gravitation, it was soon realized that general relativity could be used to model the universe, and so gave birth to the field of physical cosmology. The central equations for physical cosmology are the Friedmann-Lemaître-Robertson-Walker metric, which are the cosmological solution of the Einstein field equations. This metric predicts that the universe must be dynamic: It must either be expanding, contracting, or switching between those states.

At the time of the discovery of the Friedmann-Lemaître-Robertson-Walker metric, Einstein could not abide by the idea of a dynamic universe. In an attempt to make general relativity accommodate a static universe, Einstein introduced an alternative form of the field equations to accommodate a static universe solution in his theory:[citation needed]

G_{ab} + \Lambda\ g_{ab} = \kappa\, T_{ab}

where Λ is the cosmological constant and gab is the spacetime metric.

However, the resultant static universe was unstable. Then in 1929 Edwin Hubble showed that the redshifting of light from distant galaxies indicates that they are receding from our own at a rate which is proportional to their distance from us.[35] [36]. This demonstrated that the universe is indeed expanding. Hubble's discovery ended Einstein's objections and his use of the cosmological constant.

The equations for an expanding universe become singular when one goes far enough back in time, and this primordial singularity marks the formation of the universe. That event has come to be called the Big Bang. In 1948, Ralph Asher Alpher and George Gamov published an article describing this event and predicting the existence of the cosmic microwave background radiation left over from the Big Bang. In 1965, Arno Penzias and Robert Wilson first observed the background radiation.[37], confirming the Big Bang theory.

Unsolved problems in physics: What is dark matter? Do the phenomena attributed to dark matter point not to some form of matter but actually to an extension of gravity?

Recently, observations of distant supernovae have indicated that the expansion of the universe is currently accelerating. This was unexpected since Friedmann-Lemaître-Robertson-Walker metric calls for a universe that only contains visible matter to have a decelerating expansion. In the modern cosmological models, most energy in the universe is in forms that have never been detected directly, namely dark energy and dark matter. However, for a universe that is 4% baryonic matter, 26% dark matter, and 70% dark energy, the Friedmann-Lemaître-Robertson-Walker metric takes on a form that is consistent with observation. There is also an irony in that the dark energy can be modeled using Einstein's cosmological constant, but with a value that enhances the dynamic nature of the universe instead of muting it.[citations needed] There have been several (controversial) proposals to obviate the need for these enigmatic forms of matter and energy by modifying the laws governing gravity and the dynamics of cosmic expansion (for example, modified Newtonian dynamics).[38]

[edit] Other predictions

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

General relativity predicts the equivalence of inertial mass and gravitational mass. A number of other tests have probed the validity of various versions of the equivalence principle (strictly speaking, all measurements of gravitational time dilation are tests of the weak version of that principle, not of general relativity itself). As embodied by the strong equivalence principle, even a self-gravitating object will respond to an external gravitational field in the same manner as a test particle would. (This is often violated by alternative theories.)

The theory also predicts exotic forms of Gravitational radiation. As gravity is substantially weaker than other physics forces, currently only extremely violent interactions (like merging neutron stars and/or black holes) are expected to emitting observable radiation. A number of land-based gravitational wave detectors are in operation, with the aim of detecting gravitational waves directly. Orbital decay, as described above, may be likened to gravitational radiation as well. Graviational radiation may also be observed in subatomic particles called gravitons or quadrupole and higher order multipole moments, but these have been experimentally observed or verified.

[edit] Validity

Because general relativity has passed every unambiguous observational and experimental test to which it has been subjected so far, it is almost universally accepted by the scientific community. However, while it is a highly successful model of gravitation and cosmology, there are substantial theoretical inconsistencies between general relativity, quantum mechanics, and the spacetime singularities associated with black holes.

[edit] Quantum mechanics

Main article: Quantum field theory in curved spacetime
Unsolved problems in physics: How can the theory of quantum mechanics be merged with the theory of general relativity to produce a so-called "theory of everything"?

While general relativity is very successful in that it provides an accurate description for an impressive array of physical phenomena, the theory is very likely incomplete.[39]

Notably, in contrast to all other modern theories of fundamental interactions, general relativity is a classical theory which does not include the effects of quantum physics. The question of what general relativity looks like at a quantum level is often called a theory of quantum gravity or Theory of everything. This subject remains one of the great open questions of physics. While there are promising candidates such as string theory and loop quantum gravity, there is at present no consistent and complete theory of quantum gravity which reconciles Einstein's geometric picture of gravity with the laws of the quantum world.

[edit] Spacetime signularities

Main article: Spacetime singularity

It is a longstanding hope that the theory of quantum gravity would also do away with spacetime singularities. Such singularities are boundaries ("ragged edges") of spacetime at which geometry becomes ill-defined; the best-known examples are the singularities inside black holes and at the Big Bang singularity at the beginning of the universe. If the laws of general relativity were to hold without any quantum modifications, then, by what are known as the singularity theorems, such singularities would indeed exist in our universe.[40]

[edit] Other anomolies

Unsolved problems in physics: What causes the apparent residual sunward acceleration of the Pioneer spacecraft?

The Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from what would be expected according to known effects (gravitational or otherwise). The possibility of new physics has not been ruled out, despite very thorough investigation in search of a more prosaic explanation.[41]

[edit] Alternative theories

Main article: Alternatives to general relativity

Well known classical theories of gravitation other than general relativity include:

  • Nordström's theory of gravitation (1913) was one of the earliest metric theories (an aspect brought out by Einstein and Fokker in 1914). Nordström soon abandoned his theory in favor of general relativity on theoretical grounds, but this theory, which is a scalar theory, and which features a notion of prior geometry, does not predict any light bending, so it is solidly incompatible with observation.
  • Alfred North Whitehead formulated an alternative theory of gravity that was regarded as a viable contender for several decades, until Clifford Will noticed in 1971 that it predicts grossly incorrect behavior for the ocean tides.
  • George David Birkhoff's (1943) yields the same predictions for the classical four solar system tests as general relativity, but unfortunately requires sound waves to travel at the speed of light. Thus, like Whitehead's theory, it was never a viable theory after all, despite making an initially good impression on many experts.
  • Like Nordström's theory, the gravitation theory of Wei-Tou Ni (1971) features a notion of prior geometry, but Will soon showed that it is not fully compatible with observation and experiment.
  • The Brans-Dicke theory and the Rosen bimetric theory are two alternatives to general relativity which have been around for a very long time and which have also withstood many tests. However, they are less elegant and more complicated than general relativity, in several senses.
  • There have been many attempts to formulate consistent theories which combine gravity and electromagnetism. The first of these, Weyl's gauge theory of gravitation, was immediately shot down (on a postcard) by Einstein himself,[citation needed] who pointed out to Hermann Weyl that in his theory, hydrogen atoms would have variable size, which they do not. Another early attempt, the original Kaluza-Klein theory, at first seemed to unify general relativity with classical electromagnetism, but is no longer regarded as successful for that purpose. Both these theories have turned out to be historically important for other reasons: Weyl's idea of gauge invariance survived and in fact is omnipresent in modern physics, while Kaluza's idea of compact extra dimensions has been resurrected in the modern notion of a braneworld.
  • The Fierz-Pauli spin-two theory was an optimistic attempt to quantize general relativity, but it turns out to be internally inconsistent. Pascual Jordan's work toward fixing these problems eventually motivated the Brans-Dicke theory, and also influenced Richard Feynman's unsuccessful attempts to quantize gravity.
  • Einstein-Cartan theory includes torsion terms, so it is not a metric theory in the strict sense.
  • Teleparallel gravity goes further and replaces connections with nonzero curvature (but vanishing torsion) by ones with nonzero torsion (but vanishing curvature).
  • The Nonsymmetric Gravitational Theory (NGT) of John W. Moffat is a dark horse in the race.

[edit] External links

Additional resources, including more advanced material, can be found in General relativity resources.

[edit] Quotes

Spacetime grips mass, telling it how to move, and mass grips spacetime, telling it how to curve — John Archibald Wheeler.
The theory appeared to me then, and still does, the greatest feat of human thinking about nature, the most amazing combination of philosophical penetration, physical intuition, and mathematical skill. But its connections with experience were slender. It appealed to me like a great work of art, to be enjoyed and admired from a distance. — Max Born.

[edit] See also

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[edit] References

For a more complete list of available publications on general relativity, please see general relativity resources.
  1. ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungun der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844-847. 
  2. ^ a b c Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. 
  3. ^ An accessible introduction to tests of general relativity is Will 1993; a more technical, up-to-date account is Will 2006.
  4. ^ This development is traced e.g. in Renn 2005, p. 110ff., in chapters 9 through 15 of Pais 1982, and in Janssen 2005.
  5. ^ 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. Norton 1985.
  6. ^ This is described in detail in chapter 2 of Wheeler 1990.
  7. ^ Einstein, Albert (1907). "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen". Jahrbuch der Radioaktivitaet und Elektronik 4. 
  8. ^ E. g. Janssen (2005), p. 64f. Einstein himself also explains this in section XX of his non-technical book Einstein 1961. Following earlier ideas by Ernst Mach, Einstein also explored centrifugal forces and their gravitational analogue, cf. Stachel 1989.
  9. ^ See http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/General_relativity.html
  10. ^ E.g. p. xi in Wheeler 1990.
  11. ^ Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8. 
  12. ^ A thorough, yet accessible account of basic differential geometry and its application in general relativity can be found in Geroch 1978.
  13. ^ For elementary presentations of the concept of spacetime, see the first section in chapter 2 of Thorne 1994, and Greene 2004, p. 47–61. More complete treatments on a fairly elementary level can be found e.g. in Mermin 2005 and in Wheeler 1990, chapters 8 and 9.
  14. ^ Einstein's struggle to find the correct field equations is traced in chapters 13–15 of Pais 1982.
  15. ^ See Wheeler 1990, chapters 8 and 9, for a give vivid illustrations of curved spacetime.
  16. ^ For a more rigorous definition of the metric, but one that is still below the level of presentation of textbooks, see chapter 14.4 of Penrose 2004.
  17. ^ The geometrical meaning of Einstein's equations is explored in chapters 7 and 8 of Wheeler 1990; cf. box 2.6 in Thorne 1994. An introduction using only very simple mathematics is given in chapter 19 of Schutz 2003.
  18. ^ The most important solutions are listed in every textbook on general relativity; for a (technical) summary of our current understanding, see Friedrich 2005.
  19. ^ 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 Doppler shift between certain inertial frames. For simple derivations of this, see Harrison 2002.
  20. ^ Pound, R. V.; G. A. Rebka Jr. (November 1, 1959). "Gravitational Red-Shift in Nuclear Resonance". Physical Review Letters 3 (9): 439-441. 
  21. ^ Pound, R. V.; G. A. Rebka Jr. (April 1, 1960). "Apparent weight of photons". Physical Review Letters 4 (7): 337-341. 
  22. ^ Pound, R. V.; J. L. Snider (November 2, 1964). "Effect of Gravity on Nuclear Resonance". Physical Review Letters 13 (18): 539-540. 
  23. ^ See chapter 12 of Mermin 2005.
  24. ^ Hafele, J.; Keating, R. (July 14, 1972). "Around the world atomic clocks:predicted relativistic time gains". Science 177 (4044): 166-168. doi:10.1126/science.177.4044.166. 
  25. ^ Hafele, J.; Keating, R. (July 14 1972). "Around the world atomic clocks:observed relativistic time gains". Science 177 (4044): 168-170. doi:10.1126/science.177.4044.168. 
  26. ^ Shapiro, I. I. (December 28, 1964). "Fourth test of general relativity". Physical Review Letters 13 (26): 789-791. 
  27. ^ Shapiro, I. I.; Gordon H. Pettengill, Michael E. Ash, Melvin L. Stone, William B. Smith, Richard P. Ingalls, and Richard A. Brockelman (May 27, 1968). "Fourth test of general relativity:preliminary results". Physical Review Letters 20 (22): 1265-1269. doi:10.1103/PhysRevLett.20.1265. 
  28. ^ See Kennefick 2005 and, for the most precise measurements to date, Bertotti 2005.
  29. ^ In contrast to the derivation of frequency shift and time dilation, this calculation leads to a slightly different result when repeated in the full theory of general relativity, cf. Ehlers & Rindler 1997; for a non-technical presentation, see Pössel 2007.
  30. ^ Introductions to gravitational lensing can be found on the webpages Newbury 1997 and Lochner 2007.
  31. ^ http://einstein.stanford.edu/content/press_releases/SU/pr-aps-041807.pdf
  32. ^ Hawking, Stephen (1975). "Particle creation by black holes". Communications in Mathematical Physics 43 (3): 199-220. 
  33. ^ See the Max Plank Institute page on stars orbiting the galaxy's central object.
  34. ^ For an overview of the history of black hole physics from its beginnings in the early twentieth century to modern times, see the very readable account by Thorne 1994.
  35. ^ Hubble, Edwin (1929-01-17). "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" (PDF). Proceedings of the National Academy of Sciences USA 15 (3): 168-173. 
  36. ^ Hubble, Edwin (1929-01-17). A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae. Retrieved on 2006-11-03.
  37. ^ Penzias, Arno; Wilson, R. W. (1965-01-07). "A Measurement of Excess Antenna Temperature at 4080 mc/s (Effective Zenith Noise Temperature of Horn-Reflector Antenna at 4080 mc Due to Cosmic Black Body Radiation, Atmospheric Aborption, etc)". Astrophysical Journal 142 (3): 419-421. 
  38. ^ For dark matter, see Milgrom 2002; for dark energy, Caldwell 2004.
  39. ^ Cf. Maddox 1998, pp. 52–59 and 98–122; Penrose 2004, section 34.1 and chapter 30.
  40. ^ With a focus on string theory, the search for quantum gravity is described in Greene 1999; for an account from the point of view of loop quantum gravity, see Smolin 2001.
  41. ^ See Nieto 2006.

[edit] Bibliography

  • Ohanian, Hans C.; Ruffini, Remo (1994). Gravitation and Spacetime. New York: W. W. Norton. ISBN 0-393-96501-5. 

[edit] External links

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