General relativity

General relativity (GR) or general relativity theory (GRT) is a geometrical theory of gravitation and cosmology published by Albert Einstein in 1915. In this theory:

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.

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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.

Related Topics:
Mathematics of general relativity - Principle of minimal coupling

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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 permitted inertial motion to be described mathematically as

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ddot{x}^a = 0, where

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• x^a is a position vector,
• $dot\{\}\; =\; partial\; /\; partial\; au$, and
• τ is proper time.

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

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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. Mathematically, this is expressed using the following geodesic equation:

Related Topics:
Geodesic - Geodesic equation

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

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• $\{Gamma^a\}\_\{bc\}$ is a Christoffel symbol (otherwise known as a connection).

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.)

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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.

Related Topics:
Gravity - Correspondence principle

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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

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• 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.

Related Topics:
Weak-field approximation - Deriving the Schwarzschild solution

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Conservation of energy-momentum

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

Related Topics:
Conservation of energy - Momentum

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In special relativity, energy and momentum are joined in the four-momentum and the stress-energy tensors. So for any self-contained system or for a physical interaction,

Related Topics:
Four-momentum - Stress-energy

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partial_b , {T_a}^b = 0, where

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• $\{T\_a\}^b$ is the stress-energy tensor.

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

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

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• ∇ is the 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 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.

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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.

Related Topics:
Maxwell's Equations - Galilean invariant

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In tensor form, Maxwell's equations are

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partial_a,F^{,ab} = (4pi/c),J^{,b} and

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partial^{a},F^{,bc} + partial^{b} , F^{,ca} + partial^{c} , F^{,ab} = 0, where

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• F ^ab is the electromagnetic field tensor, and
• J ^a is a four-current.

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

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

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• P ^a is the four-momentum of the charged object.

In general relativity, Maxwell's equations become

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abla_a,F^{,ab} = (4pi/c),J^{,b} and

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abla^a,F^{,bc} + abla^b , F^{,ca} + abla^c , F^{,ab} = 0.

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

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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.

Related Topics:
Quantum mechanics - M-theory - Loop quantum gravity

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Alternative theories

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

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• 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 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). http://relativity.livingreviews.org/open?pubNo=lrr-2001-4&page=node8.html Current bounds on the PPN parameters http://wugrav.wustl.edu/people/CMW/expgravpage/ppnbounds.html are compatible with GR.

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

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