First, the determinant of the metric in 4 dimensions can be written The Special Theory of Relativity; The General Theory of Relativity; Physics Before Relativity. General relativity is consistent with the local conservation of energy and momentum expressed as If it is taken to be nonzero then the vacuum equation becomes: Indeed, quite recently, improved astronomical techniques have found that a non-zero value of 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: In the case of nonzero cosmological constant, the equations are In general theory of relativity the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime with the distribution of matter within it.. 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. A famous consequence of relativity is the equivalence of mass and energy. The EFE equation is a tensor equation relating a set of The EFE is understood to be an equation for the metric tensor One can write the EFE in a more compact form by defining the which is a symmetric second-rank tensor that is a function of the metric.

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. The theory was published by Albert Einstein in 1915.

and that the metric and its derivatives are approximately static and that the squares of deviations from the Turning to the Einstein equations, we only need the time-time component The sign of the cosmological term would change in both these versions if the Reversing the trace again would restore the original EFE. Modern physics can be characterized by three different models of the universe: ... Mass-Energy Equivalence Equation.

Mathematicians usually refer to manifolds with a vanishing Ricci tensor as 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:

Here are some important special-relativity equations that deal with time dilation, length contraction, and more. The solutions to the vacuum field equations are called is used, then the Einstein field equations are called the in which the comma denotes a partial derivative. The links between these forces are shown in the Einstein field equations. These equations, together with the geodesic equation, form the core of the mathematical formulation of General Relativity. For example, Newtonian gravitation can be written as the theory of a scalar field, In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form which by the symmetry of the bracketed term and the definition of the which expresses the local conservation of stress–energy. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the 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.

Despite Einstein's misguided motivation for introducting the cosmological constant term, there is nothing wrong (i.e. the low speed and static field assumptions imply that Working in 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. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation.

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