Fundamentals of Time and Relativity

Metric Tensor

Since the mathematics pounced on the relativity theory,
I no longer understand it myself.
Albert Einstein, 1908

Points of spacetime are specified uniquely by the set of four spacetime coordinates ${x^{μ}; μ = 0, 1, 2, 3}$ . With this notation, the invariant Minkowski distance eq. (4.1) between two nearby points of spacetime can be written in the compact form

4.3

{(Δ s)}^{2} = {(Δ x^{0})}^{2} - {(Δ x^{1})}^{2} - {(Δ x^{2})}^{2} - {(Δ x^{3})}^{2} = g_{μ ν} Δ x^{μ} Δ x^{ν}

for any choice of coordinates $Δ x^{μ}$ . This defines the Minkowski metric tensor (metric coefficients) as:

4.4

g_{μ ν} : = diag(1,-1,-1,-1) = g^{μ ν} μ = 0, 1, 2, 3

Note in equation (4.3) the Einstein summation convention that repeated indices are to be summed over.

There is never need to memorize the index positions in equations. One must only line up indices so that:

free indices at each side of an equation are in the same position;
summed (also called contracted or dummy) indices appear once up and once down.

The metric tensor (4.4) is used to raise or lower indices according to the rule: $x_{μ} : = g_{μ ν} x^{ν} x^{μ} : = g^{μ ν} x_{ν}$ . Upper/lower indices are often referred to as ‘contravariant’ and ‘covariant’, respectively.
There are two sign conventions in use in the relativity literature associated with the metric signatures $(+ 1, - 1, - 1, - 1)$ and $(- 1, + 1, + 1, + 1)$ .