Question

Is there some kind of completeness relation linking a set of four orthogonal (orthonormal) vectors in Minkowski space to the metric Diag(1, -1, -1, -1)? I tried to look this up online but found nothing.

Answer 1

A set of four orthogonal (or orthonormal) vectors in Minkowski space can be linked to the metric Diag(1, -1, -1, -1) through a completeness relation, which expresses the metric tensor as a sum of outer products of the orthogonal vectors.

Step-by-step explanation:

In Minkowski space, a set of four orthogonal (or orthonormal) vectors can be linked to the metric Diag(1, -1, -1, -1) through a completeness relation. The completeness relation expresses the metric tensor as a sum of outer products of the orthogonal vectors. In this case, the completeness relation is:

g = v1⊗v1 - v2⊗v2 - v3⊗v3 - v4⊗v4

Here, ⊗ denotes the outer product and g represents the metric tensor. The relation ensures that the metric tensor is properly represented by the given set of orthogonal vectors.