Einstein field equations

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.

The third sign above is related to the choice of convention for the Ricci tensor:

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:

Taking the trace with respect to the metric of both sides of the EFE one gets

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 and incorporated as part of the stress–energy tensor:

The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

General relativity is consistent with the local conservation of energy and momentum expressed as

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:

The definitions of the Ricci curvature tensor and the scalar curvature then show that

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.

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero

Turning to the Einstein equations, we only need the time-time component

Our simplifying assumptions make the squares of Γ disappear together with the time derivatives

A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).

Additionally, the covariant Maxwell equations are also applicable in free space:

Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written