Today we discuss a little about the Einstein field equations, that is, . We shall prove that in the Minkowski spaces, the Einstein field equations are nothing but hyperbolic equations.

We start with the Riemann curvature tensor , an -tensor, defined to be

where

.

Note that, the metric that we are using is Riemannian metric, therefore . Therefore, in local coordinates,

the -component of the Riemann curvature tensor is

.

We now compute Ricci tensor, to this purpose, we get

.

In terms of metric tensor, one has

which implies

.

Thus

.

where is the d’Alembert wave operator.To see this, clearly

which gives

.

Thus

.

Contracting once more with the scalar curvature is obtained as

.

Thus, the Einstein tensor is

If we introduce the following notation

then

.

In Minkowski space, if we require the Lorenz condition, or the Lorenz gauge, we then have

.

This is what we need. Coordinates that obey the Lorenz condition are called harmonic.

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