Cartesian Tensors in Spaces with an Indefinite Metric with Applications to Relativity

Abstract

A new formalism for dealing with Cartesian tensor analysis in flat spaces with an indefinite metric with the same ease as in Euclidean spaces is introduced. It avoids the necessity of distinguishing covariant and contravariant indices and the consequent use of the metric tensor in raising and lowering indices; neither does it require the introduction of imaginary coordinates and components of tensors. It is based on the use of a modified Einstein summation convention combined with a modified differentiation with respect to tensor components in such a way that all formal manipulations are essentially identical with those employed in Cartesian tensor analysis in Euclidean spaces. The further introduction of a modified matrix multiplication and a modified definition of a determinant serves to round out the formal analogy with the Euclidean space. The convenience, simplicity, and typographical economy of the new formalism is illustrated by examples drawn from special relativity. The formalism can be trivially generalized to complex linear vector spaces with an indefinite metric.