You have measured the intervals between a number of adjacent events in spacetime and thereby have deduced the metric g. Your friend claims that the metric is some other frame-independent tensor g̃ that differs from g. Suppose that your correct metric g and his wrong one g̃ agree on the forms of the light cones in spacetime (i.e., they agree as to which intervals are null, which are spacelike, and which are timelike), but they give different answers for the value of the interval in the spacelike and timelike cases:

Prove that g̃ and g differ solely by a scalar multiplicative factor, g̃ = ag for some scalar a. We say that g̃ and g are conformal to each other.

Transcribed Image Text:

g(Δ., Δx) #ğ(Δ., Δ.).