General Relativity/Einstein's equation

Main article: Einstein's field equation

The Einstein field equation or Einstein equation is a dynamical equation which describes how matter and energy change the geometry of spacetime, this curved geometry being interpreted as the gravitational field of the matter source. The motion of objects (with a mass much smaller than the matter source) in this gravitational field is described very accurately by the geodesic equation.

Mathematical form of Einstein's field equation

Einstein's field equation (EFE) is usually written in the form:

R_{\mu \nu }-{1 \over 2}{Rg_{\mu \nu }}+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }

Where

$R_{\mu \nu }$ is the Ricci curvature tensor
$R$ is the Ricci scalar (the tensor contraction of the Ricci tensor)
$g_{\mu \nu }$ is a (symmetric 4 x 4) metric tensor
$\Lambda$ is the Cosmological constant
$\pi$ is pi=3.1415..., the ratio between a circle's circumference and diameter
$G$ is the Gravitational constant
$c$ is the speed of light in free space
$T_{\mu \nu }$ is the energy-momentum stress tensor of matter

The EFE equation is a tensor equation relating a set of symmetric 4 x 4 tensors. It is written here in terms of components. 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 is understood to be an equation for the metric tensor $g_{\mu \nu }$ (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

G_{\mu \nu }=R_{\mu \nu }+(\Lambda -{1 \over 2}R)g_{\mu \nu }

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

G_{\mu \nu }=8\pi T_{\mu \nu }\,

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.

Properties of Einstein's equation

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

\nabla _{b}G^{\mu \nu }=G^{\mu \nu }{}_{;b}=0

which, by using the EFE, results in

\nabla _{b}T^{\mu \nu }=T^{\mu \nu }{}_{;b}=0

which expresses the local conservation law referred to above.

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 other physical theories.

For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields (i.e. the sum of two solutions is also a solution).

Another example is Schrodinger's equation of quantum mechanics where the equation is linear in the wavefunction.

The correspondence principle

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

The cosmological constant

The cosmological constant term $\Lambda g_{\mu \nu }$ 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 $\Lambda$ was abandoned (set to 0), with Einstein calling it the "biggest blunder he ever made".

Despite Einstein's misguided motivation for introducting the cosmological constant term, there is nothing wrong (i.e. inconsistent) with the presence of such a term in the equations. Indeed, quite recently, improved astronomical techniques have found that a non-zero value of $\Lambda$ 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:

T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda }{8\pi }}g_{\mu \nu }

The constant

\rho _{\mathrm {vac} }={\frac {\Lambda }{8\pi }}

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.

Solutions of the field equations

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.

Vacuum field equations

If the energy-momentum tensor $T_{\mu \nu }$ 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:

R_{ab}=0\,

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:

R_{\mu \nu }=\Lambda g_{\mu \nu }\,

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.

References

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972) ISBN 0471925675

Stephani, H., Kramer, D., MacCallum, M., Hoenselaers C. and Herlt, E. Exact Solutions of Einstein's Field Equations (2nd edn.) (2003) CUP ISBN 0521461367