Einstein field equations
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).
The EFE is 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 becomes a set of coupled, non-linear differential equations when used in this way.
Mathematical form of Einstein's field equation
The Einstein field equation (EFE) is usually written in the form
Here Rab is the Ricci tensor, R is the Ricci scalar, gab is the metric tensor, Tab is the stress-energy tensor, and the constants are π (pi), G (the gravitational constant) and c (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 is understood to be an equation for the metric tensor gab (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.
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
which, by using the EFE, results in
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.
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.
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.
The cosmological constant
One can modify the EFE by introducing a term proportional to the metric:
The constant Λ is called the cosmological constant. Since Λ 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 Λ 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 Λ 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:
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
- Main article: 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.
Vacuum field equations
If the energy-momentum tensor Tab 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:
The linearised EFE
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.
- Einstein-Hilbert action
- Exact solutions of Einstein's field equations
- History of general relativity
- Mathematics of general relativity
- 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
- Caltech Tutorial on Relativity — A simple introduction to Einstein's Field Equations.