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 are 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.
The Einstein field equations (EFE) may be written in the form:
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler. Some authors, including Einstein, have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
Notice that the sign of the (very small) cosmological term would change in both of these versions, if the author is using the +--- metric sign convention rather than the −+++ metric sign convention which we have adopted here (following MTW).
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.
Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations (if the dimension is clear).
Despite the simple appearance of the equations they are, in fact, quite complicated. Given a specified distribution of matter and energy in the form of a stress-energy tensor, the EFE are understood to be equations for the metric tensor , as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic partial differential equations.
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. The EFE can then be written as
where the cosmological term has been absorbed into the stress-energy tensor as dark energy.
Using geometrized units where G = c = 1, this can be re-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.
which may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace in the expression on the right with the Minkowski tensor without significant loss of accuracy).
which is necessary to be consistent with
which expresses the local conservation of stress-energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.
The nonlinearity of the EFE distinguishes general relativity from many other fundamental 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.
Newtonian gravitation can be written as the theory of a scalar field, , which is the gravitational potential in Joules per kilogram
where is the mass density. The orbit of a free-falling particle satisfies
In tensor notation, these become
In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
for some constant, K, and the geodesic equation
To see how the latter reduce to the former, we assume that the test particle's velocity is approximately zero
and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives
where two factors of have been divided out. This will reduce to its Newtonian counterpart, provided
Our assumptions force α=i and the time (0) derivatives to be zero. So this simplifies to
which is satisfied by letting
Turning to the Einstein equations, we only need the time-time component
the low speed and static field assumptions imply that
From the definition of the Ricci tensor
Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
Combining the above equations together
which reduces to the Newtonian field equation provided
which will occur if
The constant is called the cosmological constant. Since 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 was abandoned, with Einstein calling it the "biggest blunder [he] ever made". For many years the cosmological constant was almost universally considered to be 0.
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 positive value of 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.
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.
Taking the trace of this (contracting with ) and using the fact that , we get
Substituting back, we get an equivalent form of the vacuum field equations
In the case of nonzero cosmological constant, the equations are
yielding the equivalent form
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.
is used, then the Einstein field equations are called the Einstein-Maxwell equations:
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.