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.