Encyclopedia
General relativity is the
geometrical theory of
gravitation published by
Albert Einstein in 1915. It unifies
special relativity and
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.
Overview
Treatment of gravitation
In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of
mass,
energy and momentum within it. The relationship between stress-energy and the curvature of spacetime is described by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime occurs along special paths called timelike and null geodesics of spacetime.
Justification
The justification for creating general relativity came from the
equivalence principle, which dictates that
freefalling observers are the ones in inertial motion. A consequence of this insight is that inertial observers can accelerate with respect to each other. 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."
Thus the equivalence principle led Einstein to search for a gravitational theory which involves curved spacetimes.
Another motivating factor was the realization that relativity calls for gravitation to be expressed as a rank-two tensor, and not just a vector as was the case in Newtonian physics . Thus, Einstein sought a rank-two tensor means of describing curved spacetimes surrounding massive objects. This effort came to fruition with the discovery of the Einstein field equations in 1915.
- The general principle of relativity: The laws of physics must be the same for all observers .
- The principle of general covariance: The laws of physics must take the same form in all coordinate systems.
- The principle that inertial motion is geodesic motion: The world lines of particles unaffected by physical forces are timelike or null geodesics of spacetime.
- The principle of local Lorentz invariance: The laws of special relativity apply locally for all inertial observers.
- Spacetime is curved: This permits gravitational effects such as freefall to be described as a form of inertial motion.
- Spacetime curvature is created by stress-energy within the spacetime: This is described in general relativity by the Einstein field equations.
Spacetime as a curved Lorentzian manifold
In general relativity, the spacetime concept introduced by
Hermann Minkowski for special relativity is modified. More specifically, general relativity stipulates that spacetime is:
The term
non-Euclidean geometry describes hyperbolic [i], elliptic [i] ...
. In special relativity, spacetime is flat.
- Lorentzian: The metrics of spacetime must have a mixed metric signature. This is inherited from special relativity.
- four dimensional: to cover the three spatial dimensions and time. This is also inherited from special relativity.
The curvature of spacetime can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the top of this article. 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 just the right 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.
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 , other massive objects respond to how the first massive object curves spacetime.
The mathematics of general relativity
Due to the expectation that spacetime is curved, Riemannian geometry must be used. In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. 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. Similarly, two balls initially at rest with respect to and above the surface of the Earth come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to
their subsequent freefall.
The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian. In addition, the principle of general covariance forces that math 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 intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.
The Einstein field equations
The Einstein field equations describe how stress-energy causes curvature of spacetime and are usually written in tensor form as
where is the Einstein tensor, is the
stress-energy tensor and is a constant. The tensors and are both rank 2 symmetric tensors, that is, they can each be thought of as 4×4 matrices, each of which contains 10 independent terms.
An alternative form of the Einstein field equations includes a Cosmological Constant, .
where is the cosmological constant and is the spacetime metric. Einstein originally introduced the cosmological term to allow a flat spacetime solution to his field equations. However, later, after seeing
Edwin Hubble's evidence for an
expanding universe, he regretted adding the term, calling it the "biggest blunder" of his life. For many years the cosmological constant was almost universally considered to be 0. The cosmological term, however, is still interesting today as current cosmological studies indicate that the expansion of the universe may be accelerating.
The solutions of the EFE are metrics of spacetime. These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution. Being non-linear differential equations, the EFE often defy attempts to obtain an exact solution; however, many such solutions are known.
The EFE reduce to
Newton's law of gravity in the limiting cases of a weak gravitational field and slow speed relative to the speed of light. In fact, the value of in the EFE is determined to be by making these two approximations.
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 .
Coordinate vs. physical acceleration
One of the greatest sources of confusion about general relativity comes from the need to distinguish between coordinate and physical accelerations.
In
classical mechanics, space is preferentially mapped with a
Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a consistent coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in
special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronization procedure. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary.
In general relativity, the elegance of a flat spacetime and the ability to use a preferred coordinate system are lost . Consequently, coordinate and physical accelerations become sundered. For example: Try using a radial coordinate system in classical mechanics. In this system, an inertially moving object which passes by the origin point is found to first be moving mostly inwards, then to be moving tangentially with respect to the origin, and finally to be moving outwards, yet is moving in a straight line. This is an example of an inertially moving object undergoing a coordinate acceleration, and the way this coordinate acceleration changes as the object travels is given by the geodesic equations for the manifold and coordinate system in use.
Another more direct example is the case of someone standing on the Earth, where they are at rest with respect to the surface coordinates for the Earth but are undergoing a continuous physical acceleration because the mechanical resistance of the Earth's surface keeps them from free falling.
Predictions of general relativity
- .
Gravitational effects
Acceleration effects
These effects occur in any accelerated frame of reference, and are therefore independent of the curvature of spacetime.
- Gravitational redshifting of light: The frequency of light will decrease as it moves to higher gravitational potentials . Confirmed by the Pound-Rebka experiment.
- Gravitational time dilation: Clocks will run slower at lower gravitational potentials . Confirmed by the Hafele-Keating experiment and GPS.
- Shapiro effect : Signals will take longer than expected to move through a gravitational field. Confirmed through observations of signals from spacecraft and pulsars passing behind the Sun as seen from the Earth.
Bending of light
This bending also occurs in any accelerated frame of reference. However, the details of the bending and therefore the gravitational lensing effects are governed by spacetime curvature.
- The magnitude of this effect is twice the Newtonian prediction. It was confirmed by astronomical observations during eclipse
...
s of the Sun and observations of pulsars passing behind the Sun.
- Gravitational lensing: One distant object in front of or close to being in front of another much more distant object can change how the more distant object is seen. These effects include
- Multiple views of the same object: Observations of quasars whose light passes close to an intervening galaxy.
- Brightening of a star due to the focusing effects of a planet or another star passing in front of it: Such "microlensing
...
" events are now regularly observed.
- Einstein rings and arcs: One object directly behind another can make the more distant object's light appear as a ring. When almost directly behind, the result is an arc. Observed for distant galaxies.
Orbital effects
These are ways in which the celestial mechanics of general relativity differs from that of classical mechanics.
Rotational effects
These involve the behavior of spacetime around a rotating massive object.
- Frame dragging: A rotating object will drag the spacetime along with it. This will 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. This prediction is also being tested by Gravity Probe B.
Black holes
Black holes are objects which have gravitationally collapsed behind an
event horizon. In a "classical" black hole, nothing that enters can ever escape. The disappearance of light and matter within a black hole may be thought of as their entering a region where all null and timelike geodesic paths are warped so that they point inwards.
Stephen Hawking has predicted that black holes can "leak" mass, a phenomenon called Hawking radiation, a quantum effect not in violation of general relativity.
Cosmological effects
- Expansion of the universe: This is predicted by cosmological solutions of the Einstein Field Equations. Its existence was confirmed by Edwin Hubble in 1929.
- Big Bang: The arising of the universe from a primordial singularity.
...
in 1965.
- Dark energy: This is an energy field of unknown composition that may exist throughout the universe. Recent observations of distant supernovae indicate that the expansion of the universe is currently "accelerating" . The solutions of the Einstein field equations that call for this behavior for the current universe, which may require the reintroduction of the cosmological constant, are for a stress-energy which is at least 70% dark energy.
Other predictions
- The equivalence of inertial mass and gravitational mass: This follows naturally from freefall being inertial motion.
- 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.
- Gravitational radiation: Orbiting objects and merging neutron stars and/or black holes are expected to emit gravitational radiation.
- Orbital decay .
- Binary pulsar mergers: May create gravitational waves strong enough to be observed here on Earth. Several are in operation. However, there are no confirmed observations of gravitational radiation at this time.
- Gravitons: According to quantum mechanics, gravitational radiation must be composed of quanta called gravitons. General relativity predicts that these will be spin-2 particles. They have not been observed.
- Only quadrupole moments create gravitational radiation.
In physics, there are two kinds of
dipoles = double and polos = pivot)....
gravitational radiation is predicted by some alternative theories. It has not been observed.
Relationship to other physical theories
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.
Note that in the discussion which follows, the mathematics of general relativity is used heavily. Also note that under the principle of minimal coupling, the physical equations of special relativity can be turned into their general relativity equivalent by replacing the Minkowski metric with the relevant metric of spacetime and by replacing any regular derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied.
Inertia
Inertial motion is motion free of all forces. In Newtonian mechanics,
F=m a with the force
F equal to zero says that inertial motion is just motion with no acceleration. The idea is the same in special relativity. If we assume that the coordinates are
Cartesian,
inertial motion to be described mathematically as
where
- xa is the position coordinate,
- t is proper time.
In Newtonian mechanics,
t = t, the coordinate time.
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. Therefore a different definition of inertial motion is required. In relativity, inertial motion occurs along timelike or null geodesics as parameterized by proper time. This is expressed mathematically by the geodesic equation:
where is a Christoffel symbol. 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. In the case of flat space in Cartesian coordinates, we have , so this equation reduces to the simpler form .
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 the 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. For the Schwarzschild solution , the same acceleration as that which is created by gravity is obtained when a constant of integration is set equal to
2MG/c^2). For more information, see Deriving the Schwarzschild solution.
Transition from Newtonian mechanics to general relativity
Some of the basic concepts of general relativity can be outlined outside the
relativistic domain. In particular, the idea that mass/energy generates
curvature in space and that curvature affects the motion of masses can be illustrated in a
Newtonian setting.
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. The equations are also generalized to more complicated curvatures.
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. In terms of symmetry the transition involves replacing a global Lorentz covariance by a local Lorentz covariance.
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.
In special relativity, energy and momentum are joined in the four-momentum and the
stress-energy tensors. For any self-contained system or for any physical interaction, the total energy-momentum is conserved in the sense that:
, where
- is a partial derivative.
- is the stress-energy tensor.
For general relativity, this relationship is modified to account for curvature, becoming
, where
- is 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 . This often causes confusion in time-dependent spacetimes which apparently do not conserve energy, although the local law is always satisfied. Exact formulation of energy-momentum conservation on an arbitrary geometry requires use of a non-unique stress-energy-momentum pseudotensor.
Electromagnetism
Maxwell's equations, the equations of electrodynamics, in curved spacetime are
and
, where
- F ab is the electromagnetic field tensor, and
- J a is a four-current
- is the covariant derivative.
The effect of an electromagnetic field on a charged object is then
, where
- again, is the covariant derivative,
- q is the charge on the object,
- m is the mass of the object, and
- P a is the four-momentum of the charged object.
Maxwell's equations in flat spacetime are recovered by reverting the covariant derivatives to regular derivatives .
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 , 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; it is not
background independent nor generally covariant under the diffeomorphism group. The interpretation of excitations of quantum fields as particles becomes frame dependent. Hawking radiation is a prediction of this semiclassical approximation.
Einstein gravity is nonrenormalizable
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. Experimentalists must then measure the values of these counterterms in order to be able to use the quantum field theory in question to make predictions .
Many of the best understood quantum field theories, such as
quantum electrodynamics, contain divergences which are canceled by counterterms that have been effectively measured. One needs to say
effectively because the counterterms are formally infinite, however it suffices to measure observable quantities, such as physical particle masses and coupling constants, which depend on the counterterms in such a way that the various infinities cancel.
A problem arises, however, when the cancellation of all infinities requires the inclusion of an infinite number of counterterms. In this case the theory is said to be nonrenormalizable.
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 . 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 the second superstring revolution supergravity has come back into fashion, with its quantum completion rebranded with a new name:
M-theory.
A very different approach to that described above is employed by loop quantum gravity. In this approach, one does not try to quantize the gravitational field as one quantizes other fields in quantum field theories. Thus the theory is not plagued with divergences and one does not need counterterms. However it has not been demonstrated that the classical limit of loop quantum gravity does in fact contain flat space Einsteinian gravity. This being said, the universe has only one spacetime and it is not flat.
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 was one of the earliest metric theories . 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 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 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 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.
- Teleparallel gravity goes further and replaces connections with nonzero curvature by ones with nonzero torsion .
- The Nonsymmetric Gravitational Theory of John W. Moffat
...
is a dark horse in the race.
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 . Current bounds on the PPN parameters are compatible with GR.
See in particular , a review paper by Clifford Will.
History
- See also: Tests of general relativity
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. 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 . This paper was subsequently published in 1916. Since 1915, the development of general relativity has focused on solving the field equations for various cases. This generally means finding metrics which correspond to realistic physical scenarios. The interpretation of the solutions and their possible experimental and observational testing also constitutes a large part of research in GR.
The
expansion of the universe created an interesting episode for general relativity. 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 , the Reissner-Nordström solution and the Kerr solution.
Observationally, general relativity has a history too. The perihelion precession of Mercury was the first evidence that general relativity is correct. Eddington's 1919 expedition in which he 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 studies of binary pulsars, observations of radio signals passing the limb of the Sun, and even the
GPS system.
Status
The status of general relativity is decidedly mixed.
On the one hand, general relativity is a highly successful model of gravitation and cosmology. It has passed every unambiguous test that it has been subjected to so far, both observationally and experimentally. It is therefore almost universally accepted by the scientific community.
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 .
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
Notes
See also
- Basic introduction to the mathematics of curved spacetime
- Mathematics of general relativity
- Classical theories of gravitation
- David Hilbert
- Einstein-Hilbert action
- General relativity resources, an annotated reading list giving bibliographic information on some of the most cited resources.
- History of general relativity
- Golden age of general relativity
- Contributors to general relativity
References
- For a more complete list of available publications on general relativity, please see general relativity resources.
Lectures presented at the Advanced School on Effective Field Theories , to be published in the proceedings
