general relativity - Relationship to other physical theories, History, Status
A theory of gravity deriving almost entirely from Einstein (1916). It supersedes Newton's theory of gravitation, which is reproduced as a weak gravity, low velocity special case, and replaces the Newtonian notion of instantaneous action at a distance with the gravitational field as a distortion of spacetime due to the presence of mass. For example, as the Earth moves round the Sun there is distortion of spacetime by the Sun's greater mass. An analogy represents spacetime as a rubber sheet distorted by a heavy ball representing the Sun; a smaller ball rolling by, representing a planet, will tend to fall into this depression, apparently attracted. General relativity is supported by experiments which measure the bending of starlight due to the presence of the Sun's mass, and also the precession of Mercury's orbit. Other predictions include black holes and gravitational waves. Using general relativity Einstein predicted the impact of the Earth's spin on other rotating objects; this effect, called frame dragging, was first reported in 1997 by observing changes in satellite orbits.
General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. It unifies special relativity and Sir Isaac Newton's law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time, this curvature being produced by the mass-energy and momentum content of the 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.
Gravitational effects
Acceleration effects
These effects occur in any accelerated frame of reference, and are therefore independent of the curvature of spacetime.
Orbital effects
These are ways in which the celestial mechanics of general relativity differs from that of classical mechanics.
Non-Newtonian periapsis precession: The apsides of orbits precess more than expected under Newton's theory of gravity. Stephen Hawking has predicted that black holes can "leak" mass, a phenomenon called Hawking radiation, a quantum effect not in violation of general relativity. General relativity predicts that these will be spin-2 particles. Dipole gravitational radiation (prohibited by this prediction) is predicted by some alternative theories.Relationship to other physical theories
This section will use the Einstein summation convention.
Classical mechanics and special relativity
Classical mechanics and special relativity are lumped together here because special relativity is in many ways intermediate between general relativity and classical mechanics, and shares many attributes with classical mechanics.
In the following discussion, the mathematics of general relativity is used heavily. Also, under the principle of minimal coupling, the physical equations of special relativity can be turned into their general relativity counterparts by replacing the Minkowski metric (ηab) with the relevant metric of spacetime (gab) and by replacing any partial derivatives with covariant derivatives. Zero force means that inertial motion is just motion with zero acceleration:
The idea is the same in special relativity.
In both Newtonian mechanics and special relativity, space and then spacetime are assumed to be flat, and we can construct a global Cartesian coordinate system. In general relativity, these restrictions on the shape of spacetime and on the coordinate system to be used are lost. Since general relativity describes four-dimensional spacetime, this represents four equations, with each one describing the second derivative of a coordinate with respect to proper time.
Gravitation
For gravitation, the relationship between Newton's theory of gravity and general relativity is governed by the correspondence principle: General relativity must produce the same results as gravity does for the cases where Newtonian physics has been shown to be accurate.
Around a spherically symmetric object, the Newtonian theory of gravity predicts that objects will be physically accelerated towards the center on the object by the rule
where G is Newton's Gravitational constant, M is the mass of the gravitating object, r is the distance to the gravitation object, and is a unit vector identifying the direction to the massive object.
In the weak-field approximation of general relativity, an identical coordinate acceleration must exist.
Transition from Newtonian mechanics to general relativity
Some of the basic concepts of general relativity can be outlined outside the relativistic domain.
General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with Fermi-Walker transport along world lines in spacetime.
Transition from special relativity to general relativity
The basic structure of general relativity, including the geodesic equation and Einstein field equation, can be obtained from special relativity by examining the kinetics and dynamics of a particle in a circular orbit about the earth.
Conservation of energy-momentum
In classical mechanics, conservation laws for energy and momentum are handled separately in the two principles of conservation of energy and conservation of momentum.
Mathematically, the general relativity statement of the conservation of energy and momentum is:
where is the stress-energy tensor, the comma indicates a partial derivative and the semicolon indicates a covariant derivative.
Unlike classical mechanics and special relativity, it is not usually possible to unambiguously define the total energy and momentum in general relativity, so the tensorial conservation laws are local statements only (see ADM energy, though).
Electromagnetism
General relativity modifies the description of electromagnetic phenomena by employing a new version of Maxwell's equations.
The source equations of electrodynamics in curved spacetime are (in cgs units)
where F is a four-current representing the sources of the electromagnetic field.
The source-free equations are the same as their special relativity counterparts.
The effect of an electromagnetic field on a charged object is then modified to
,where q is the charge on the object, m is the rest mass of the object and P a is the four-momentum of the charged object. For Maxwell's equations in flat spacetime in curvilinear coodinates see or
Quantum mechanics
Quantum mechanics is viewed as the fundamental theory of physics along with general relativity, but combining quantum mechanics with general relativity has presented difficulties.
Quantum field theory in curved spacetime
Normally, quantum field theory models are considered in flat Minkowski space (or Euclidean space), which is an excellent approximation for weak gravitational fields like those on Earth. In the presence of strong gravitational fields, the principles of quantum field theory have to be modified. The spacetime is static so the theory is not fully relativistic in the sense of general relativity;
Einstein gravity is nonrenormalizable
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"?It is often said that general relativity is incompatible with quantum mechanics. This means that if one attempts to treat the gravitational field using the ordinary rules of quantum field theory, one finds that physical quantities are divergent. Such divergences are common in quantum field theories, and can be cured by adding parameters to the theory known as counterterms.
Many of the best understood quantum field theories, such as quantum electrodynamics, contain divergences which are canceled by counterterms that have been effectively measured. While nonrenormalizable theories are sometimes seen as problematic, the framework of effective field theories presents a way to get low-energy predictions out of non-renormalizable theories. The result is a theory that works correctly at low energies, though such a theory cannot be considered to be a theory of everything because it cannot be self-consistently extended to the high-energy realm.
Proposed quantum gravity theories
General relativity fits nicely into the effective field theory formalism and makes sensible predictions at low energies (Donoghue, 1995). However, high enough energies will "break" the theory.
It is generally held that one of the most important unsolved problems in modern physics is the problem of obtaining the true quantum theory of gravitation, that is, the theory chosen by nature, one that will work at all energies. Discarded attempts at obtaining such theories include supergravity, a field theory which unifies general relativity with supersymmetry. In this approach, one does not try to quantize the gravitational field as one quantizes other fields in quantum field theories.
Of these two proposals, M-theory is significantly more ambitious in that it also attempts to incorporate the other known fundamental forces of Nature, whereas loop quantum gravity "merely" attempts to provide a viable quantum theory of gravitation with a well-defined classical limit which agrees with general relativity.
Alternative theories
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 bi-metric 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, 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. The Nonsymmetric Gravitational Theory (NGT) of John W.Even for "weak field" observations confined to our Solar system, various alternative theories of gravity predict quantitatively distinct deviations from Newtonian gravity. In the weak-field, slow-motion limit, it is possible to define 10 experimentally measurable parameters which completely characterize predictions of any such theory. This system of these parameters, which can be roughly thought of as describing a kind of ten dimensional "superspace" made from a certain class of classical gravitation theories, is known as PPN formalism (Parametric Post-Newtonian formalism).
See in particular confrontation between Theory and Experiment in Gravitational Physics, a review paper by Clifford Will.
History
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. By 1912, Einstein was actively seeking a theory in which gravitation was explained as a geometric phenomenon. Since 1915, the development of general relativity has focused on solving the field equations for various cases.
The expansion of the universe created an interesting episode for general relativity.
Observationally, general relativity has a history too. Einstein's showing that general relativity could account for the discrepancy between the Newtonian prediction for the perihelion precession of Mercury and the observed was the first evidence that general relativity is correct. In 1919, Eddington's announcement that his observations of stars near the ecliped 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.
Status
The status of general relativity is decidedly mixed.
On the one hand, general relativity is a highly successful model of gravitation and cosmology.
On the other hand, general relativity is inconsistent with quantum mechanics, and the singularities of black holes also raise some disconcerting issues. So while it is accepted, there is also a sense that something beyond general relativity may yet be found.
Currently, better tests of general relativity are needed. Even the most recent binary pulsar discoveries only test general relativity to the first order of deviation from Newtonian projections in the post-Newtonian parameterizations. Some way of testing second and higher order terms is needed, and may shed light on how reality differs from general relativity (if it does). Introduction to the Effective Field Theory Description of Gravity. Lectures presented at the Advanced School on Effective Field Theories (Almunecar, Spain, June 1995), to be published in the proceedings.
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