Einstein field equations

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The Einstein Field Equations (EFE) are a set of ten equations in Einstein's theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. The EFE are sometimes called Einstein's equation or Einstein's equations because of their appearance. Such usage is often discouraged as many people state E=mc² as being Einstein's equation.

The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor).

Being the most accurate field equation for gravitation, the EFE are often used to determine the curvature of spacetime resulting from the presence of mass and energy. That is, they determine the metric tensor of spacetime for a given arrangement of stress-energy in the spacetime. Because of the relationship between the metric tensor and the Einstein tensor, the EFE become a set of coupled, non-linear differential equations when used in this way.

1 Mathematical form of Einstein's field equation
2 Properties of Einstein's equation
3 The cosmological constant
4 Solutions of the field equations
5 Vacuum field equations
6 The linearised EFE
7 See also
8 References
9 External links

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Mathematical form of Einstein's field equation

The Einstein field equation (EFE) are usually written in the form

Here <math>R_{ab}</math> is the Ricci tensor, <math>R</math> is the Ricci scalar, <math>g_{ab}</math> is the metric tensor, <math>T_{ab}</math> is the stress-energy tensor, and the constants are <math>\pi</math> (pi), <math>G</math> (the gravitational constant) and <math>c</math> (the speed of light). The EFE is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here using the abstract index notation. Each tensor has 10 independent components. Given the freedom of choice of the four spacetime coordinates, the independent equations reduce to 6 in number.

The EFE are understood to be an equation for the metric tensor <math>g_{ab}</math> (given a specified distribution of matter and energy in the form of a stress-energy tensor). Despite the simple appearance of the equation it is, in fact, quite complicated. This is because both the Ricci tensor and Ricci scalar depend on the metric in a complicated nonlinear manner.

One can write the EFE in a more compact form by defining the Einstein tensor

which is a symmetric second-rank tensor that is a function of the metric. Working in geometrized units where G = c = 1, the EFE can then be written as

The expression on the left represents the curvature of spacetime as determined by the metric and the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how the curvature of spacetime is related to the matter/energy content of the universe.

These equations, together with the geodesic equation, form the core of the mathematical formulation of general relativity.

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Properties of Einstein's equation

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Conservation of energy and momentum

An important consequence of the EFE is the local conservation of energy and momentum; this result arises by using the differential Bianchi identity to obtain

<math>\nabla_b G^{ab}=G^{ab}{}_{;b}=0</math>

which, by using the EFE, results in

<math>\nabla_b T^{ab}= T^{ab}{}_{;b}=0</math>

which expresses the local conservation of stress-energy. This conservation law is a physical requirement. In designing the field equations, Einstein aimed at finding equations which automatically satisfied this conservation condition.

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Nonlinearity of the field equations

The EFE are a set of 10 coupled elliptic-hyperbolic nonlinear partial differential equations for the metric components. This nonlinear feature of the dynamical equations distinguishes general relativity from many other physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics which is linear in the wavefunction.

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The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations.

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The cosmological constant

One can modify the EFE by introducing a term proportional to the metric:

<math>R_{ab} - {1 \over 2}R g_{ab} + \Lambda g_{ab} = {8 \pi} T_{ab}</math>

The constant <math>\Lambda</math> is called the cosmological constant. Since <math>\Lambda</math> is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by Hubble a decade later confirmed that our universe is, in fact, not static but expanding. So <math>\Lambda</math> was abandoned, with Einstein calling it the "biggest blunder [he] ever made".

Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved astronomical techniques have found that a non-zero value of <math>\Lambda</math> is needed to explain some observations.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor:

<math>T_{ab}^{\mathrm{(vac)}} = -\frac{\Lambda}{8\pi}g_{ab}.</math>

The constant

<math>\rho_{\mathrm{vac}} = \frac{\Lambda}{8\pi}</math>

is called the vacuum energy. The existence of a cosmological constant is equivalent to the existence of a non-zero vacuum energy. The terms are now used interchangeably in general relativity.

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Solutions of the field equations

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The solutions of the Einstein field equations are metrics of spacetime. The solutions are hence often called 'metrics'. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post Newtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions.

The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.

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Vacuum field equations

If the energy-momentum tensor <math>T_{ab}</math> is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations, which can be written as:

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

The above vacuum equation assumes that the cosmological constant is zero. If it is taken to be nonzero then the vacuum equation becomes:

<math>R_{ab} = \Lambda g_{ab}\,</math>

Mathematicians usually refer to manifolds with a vanishing Ricci tensor as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

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The linearised EFE

Main articles: Linearised Einstein field equations, Linearized gravity

The nonlinearity of the EFE makes finding exact solutions quite difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric. This linearisation procedure can be used to discuss the phenomena of gravitational radiation.

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References

See General relativity resources.

Aczel, Amir D., 1999. God's Equation: Einstein, Relativity, and the Expanding Universe. Delta Science. A popular account.
Charles Misner, Kip Thorne, and John Wheeler, 1973. Gravitation. W H Freeman.
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers C. and Herlt, E., 2003. Exact Solutions of Einstein's Field Equations, 2nd ed. Cambridge Univ. Press. ISBN 0521461367

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