General relativity

General relativity (GR) 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 viewed as being due to a force (in the traditional sense) but rather a manifestation of curved space and time, this curvature being produced by the mass-energy content of the spacetime.

Overview

File:Spacetime curvature.png
Two-dimensional visualization of space-time distortion. The presence of matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.

In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy, and momentum (or stress-energy) within it. The relationship between stress-energy and the curvature of spacetime is governed by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime (inertial motion) occurs along special paths called timelike and null geodesics of spacetime.

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 freefall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion in a curved spacetime. 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 caused by the mechanical resistance of the surface that they are standing on.

Justification

The justification for creating general relativity comes 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. (Think of two balls falling on opposite sides of the Earth, for example.) This phenomenon violates Newton's first law of motion, and cannot be accounted for in the Euclidean geometry of special relativity either. To quote Einstein himself:

If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them. [1]

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 [2]. (An analogy is the electromagnetic field tensor of special relativity). 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.

Fundamental principles

General relativity is based on a set of fundamental principles which guided its development. These are:

(The equivalence principle, which was the starting point for the development of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.)

Spacetime as a curved Lorentzian manifold

In general relativity, the concept of spacetime (which was introduced by Hermann Minkowski for special relativity) is modified. In general relativity spacetime is

The curvature of spacetime (caused by the presence of stress-energy) 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 (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.

The mathematics of general relativity

Due to the expectation that spacetime is curved, Riemannian geometry (also known as non-Euclidean 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 (meaning that their velocities do not differ to first order in their separation) 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 (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent freefall. (Another way of looking at this is how a single ball moving in a purely timelike fashion parallel to the center of the Earth comes through geodesic motion to be moving towards the center of the Earth.)

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 (spacetime) 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 (EFE) describe how stress-energy causes curvature of spacetime and are usually written in tensor form (using abstract index notation) as

$G_{ab} = \kappa\, T_{ab} \$

where $G_{ab} \$ is the Einstein tensor, $T_{ab} \$ is the stress-energy tensor and $\kappa \$ is a constant. The tensors $G_{ab} \$ and $T_{ab} \$ are both rank 2 symmetric tensors, i.e. they can each be thought of as 4×4 matrices each of which contains 10 independent terms.

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 $\kappa \$ in the EFE is determined to be $\kappa = 8 \pi G / c^4 \$ by making these two approximations.

The EFE are the identifying feature of general relativity. Other theories built 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 bimetric theory, and Einstein-Cartan theory).

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 (due to stress-energy curving spacetime and the principle of general covariance). 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 (instead of through) 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 are 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 (latitude, longitude, and elevation) but are undergoing a continuous physical acceleration because the mechanical resistance of the Earth's surface keeps them from free falling.

Predictions of GR

(For more detailed information about tests and predictions of general relativity, see Tests 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. (Note however that spacetime curvature usually is the source the causative acceleration when these effects are being observed.)

• Gravitational redshifting of light: The frequency of light will decrease (shifting visible light towards the red end of the spectrum) as it moves to higher gravitational potentials. Confirmed by the Pound-Rebka experiment.
• Gravitational time dilation: Clocks will run slower at lower potentials than the observer in a gravitational field. Confirmed by the Haefele-Keating experiment and GPS.
• Shapiro effect (a.k.a. gravitational time delay): 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. Confirmed by astronomical observations during eclipses 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: Observed 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. However, Stephen Hawking has shown the black holes can "leak" energy, a phenomenon called Hawking radiation.

Other predictions

• The equivalence of inertial mass and gravitational mass: This follows naturally from freefall being inertial motion.
• Gravitational radiation: Orbiting objects and merging neutron stars and/or black holes are expected to emit gravitational radiation.
• Orbital decay (described above).
• Binary pulsar mergers: May create gravitational waves strong enough to be observed here on Earth. Several gravitational wave observatories are (or will soon be) 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.
• Dipole gravitational radiation (prohibited by this prediction) 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 (ηab) with the relevant metric of spacetime (gab) and by replacing any regular derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied.

Inertia

In both classical mechanics and special relativity, space and then spacetime were assumed to be flat. In the language of tensor calculus, this meant that Rabcd = 0, where Rabcd is the Riemann curvature tensor. In addition, the coordinate system itself was also assumed to be Cartesian. These restrictions permit inertial motion to be described mathematically as

$\ddot{x}^a = 0,$ where

• xa is a position vector,
• $\dot{} = \partial / \partial\tau$, and
• τ is proper time.

Note that in classical mechanics, xa is three-dimensional and τ ≡ t, where t is coordinate time.

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:

$\ddot{x}^a + {\Gamma^a}_{bc} \, \dot{x}^b \,\dot{x}^c = 0,$ where

Since x is a rank one tensor, these equations are four in number, with each one describing the second derivative of a coordinate with respect to proper time. (Note that under the Minkowski metric of special relativity, the values of the connections are all zeros. This is what turns the general relativity geodesic equations into $\ddot{x}^a = 0$ for special relativity.)

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 theory of gravity predicts that objects will be physically accelerated towards the center on the object by the rule $\mathbf{F} = M \mathbf{\hat{r}}/r^2$ where

• M is the mass of the gravitating object,
• r is the distance to the gravitation object, and
• $\mathbf{\hat{r}}$ 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 (which is the simplest possible spacetime surrounding a massive object), the same acceleration as that of the 'force of gravity' is obtained when constant of integration is set equal to 2m (where m=MG/c^2). For more information, see Deriving the Schwarzschild solution.

Conservation of energy-momentum

In classical mechanics, conservation of energy and momentum are handled separately.

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:

$\partial_b \, {T_a}^b = 0$, where

• $\partial$ is a partial derivative.
• ${T_a}^b$ is the stress-energy tensor.

For general relativity, this relationship is modified to account for curvature, becoming

$\nabla_b \, {T_a}^b = \partial_b \, {T_a}^b + {\Gamma^b}_{cb} \, {T_a}^c + {\Gamma^c}_{ab} \, {T_c}^b = 0$, where

Unlike classical mechanics and special relativity, it is not usually possible to unambiguously define the total energy and momentum in general relativity, so the conservation laws are local statements only (see ADM energy, though). This often causes confusion in time-dependent spacetimes which apparently do not conserve energy, although the local law is always satisfied.

Electromagnetism

Electromagnetism sounded the death knell for classical mechanics, since Maxwell's Equations are not Galilean invariant. This created a dilemma that was resolved by the advent of special relativity.

In tensor form, Maxwell's equations are

$\partial_a\,F^{\,ab} = (4\pi/c)\,J^{\,b}$ and
$\partial^{a}\,F^{\,bc} + \partial^{b} \, F^{\,ca} + \partial^{c} \, F^{\,ab} = 0$, where

The effect of an electromagnetic field on a charged object of mass m is then

$dP^a/d\tau = (q/m)\,P_b\,F^{\,ab}$, where

In general relativity, Maxwell's equations become

$\nabla_a\,F^{\,ab} = (4\pi/c)\,J^{\,b}$ and
$\nabla^a\,F^{\,bc} + \nabla^b \, F^{\,ca} + \nabla^c \, F^{\,ab} = 0$.

The equation for the effect of the electromagnetic field remains the same, although the change of metrics will modify its results.

Quantum mechanics

General relativity is incompatible with quantum mechanics; it is generally held that one of the most important unsolved problems in modern physics is the problem of obtaining a true quantum theory of gravitation. At present, leading contenders which may turn out to solve this problem include M-theory and loop quantum gravity. Of these two, M-theory is significantly more ambitious in that it attempts to unify gravitation with 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 Cliff 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 nowadays not 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 brane-world.
• 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 non-symmetric gravitation theory (NGT) of John W. Moffatt 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 (Parametric Post-Newtonian formalism). [3] Current bounds on the PPN parameters [4] are compatible with GR.

See in particular confrontation between Theory and Experiment in Gravitational Physics, a review paper by Clifford Will.

History

For more details on this topic, see History 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 19111915, 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 1915, these efforts culminated in the publication of the Einstein field equations, which are a set of differential equations.

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 (1916), 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. For more information, see the Tests of general relativity article.

Status

The status of general relativity is decidedly mixed. On the one hand, it is a highly successful model of gravitation and cosmology which 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 at the same time as it is accepted, there is also a sense that there may well be something beyond Einstein's theory still yet to 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 Einstein's theory (if it does).

Quotes

Spacetime grips mass, telling it how to move, and mass grips spacetime, telling it how to curveJohn 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

[1] In general relativity, the term "gravitation" (meaning the tendency of massive objects to accelerate towards each other) is used instead of "gravity" since gravity is by definition the force which causes gravitation.

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