Question

Is the variation of a metric with respect to a metric with a different signature, zero?

Answer 1

I have a problem that involves calculating the variation of a metric ˉgαβ with +3 signature with respect to a metric gαβ with a signature of +1. Both metrics have the same spatial dimension of 3. The metrics are related by gαβ=ˉgαβ−2uαuβ where u is a unit vector in ˉg, i.e. (uαuα=1), and uαuα=−1in g. Since the signature of a metric is constant everywhere, it seems trivial that δˉgαβδgαβ=0. Am I missing something?

Answer 2

I have a problem that involves calculating the variation of a metric with +3 signature with respect to a metric with a signature of +1. Both metrics have the same spatial dimension of 3. The metrics are related by where u is a unit vector in , i.e. ( ), and
in g. Since the signature of a metric is constant everywhere, it seems trivial that . Am I missing something?