ABSTRACT
In Special Relativity, the four-dimensional space-time continuum (Minkowski space) is in rectilinear coordinates, {xµ}, characterized by the metric tensor η = {ηµν}, which is the same in all space-time points, and given by the diagonal form
{ηµν} =
−1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
; (6.1)
see Eq. (3.4). The metric relates to the “infinitesimal squared distance,” ds2, associated with infinitesimal separated events in Minkowski space as follows:
ds2 = ηµνdx µdxν . (6.2)
The metric is called indefinite because ds2 can take on both positive and negative values, as is evident from Eq. (3.3). The lowering of indices by means of the covariant metric tensor, {ηµν}, is supplemented by a raising of indices with the help of the contravariant metric tensor, {ηµν}, as indicated for four-vectors in Eqs. (3.37) and (3.38). The two tensors are related by Eq. (3.50), and ηµν is numerically identical with ηµν : They are in matrix notation inverse matrices.
