(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let (M,g) be a spacetime.

(a) Let A and A' be vector fields on M such that g(A,B)=g(A',B) for any future-pointing timelike vector field Y. Show that X=X'.

(b) Let w and w' be two two-forms on M. Suppose that i¬A w = i¬A w' for any future -pointing timelike vector field A on M, where i¬x denotes the contraction with A. Show that w=w'

2. Relevant equations

3. The attempt at a solution

I think that part (a) is based on the transformation law between two metrics. So if we are going from g(A,B) -> g(A',B') we use the tensorial transformation law:

g(A,B) = d(A',A)d(B',B) g(A',B')

where d(A',A) mean the partial derivative of A' with respect to A, etc.

But g(A',B')=g(A,B) (given)

so g(A,B) = d(A',A)d(B',B) g(A,B)

Thus d(A',A)d(B',B) = 1

But B'=B , so d(B',B) = Kroneckerdelta (B',B) = 1

So d(A',A)=1, thus A'=A.

I'm not sure if this is even the right way of tackling this question for part (a). As for Part (b), I'm completely lost :S

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# Vector fields, metrics and two forms on a spacetime.

Can you offer guidance or do you also need help?

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