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I'm now reading A Relativist's Toolkit by Eric Poisson.

In chapter 3, he say something about stationary and static spacetime.

For static spacetime, it's just like stationary (admits timelike Killing vector [itex] t^{\alpha} [/itex]) with addition that metric should be invariant under time reversal [itex] t \rightarrow -t [/itex]. Which implies in some specific coordinate [itex] g_{t\mu} = 0 [/itex].

He also say that, this also implies that [itex] t_{\alpha} = g_{tt}\partial_{\alpha}t [/itex] and he concludes that "a spacetime is static if the timelike Killing vector field is hypersurface orthogonal".

I'd like to ask about these two implications. I have no idea how these two become relevant with static spacetime. Because, as far as I know, hypersurface orthogonal is the congruence of geodesics which have no rotating part.

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# Question about static spacetime

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