Metric tensor

The metric tensor's elements are the coefficients read off of the line element

For special relativity rectilinear coordinate inertial frames are used which given

the metric tensor will be designated and the line element will be

and the Minkowski metric tensor elements given by

All other elements are 0.

Written as a matrix this is

The metric tensor acts a an index raising

and lowering

opperator. And as an inner product operator in 4d spacetime

There is an inverse relationship between the contravariant and covariant metric tensor elements

which can be expressed as the matrix

So solving for the contravariant metric tensor elements given the covariant ones and vica-versa can be done by simple matrix inversion.

The covariant derivative of the metric with respect to any coordinate is zero

where the covariant derivative is done with the use of Christoffel symbols. And so of course the covariant divergence of the metric is also zero