- #36
latentcorpse
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bigubau said:Yes. Finally :)
Now for the final part, it is asking us to show that [itex]u_a[/itex] is parallel to [itex]k_a[/itex]. But we have a static spacetime and so [itex]k_a[/itex] will be hypersurface orthogonal. This means we are asked to show that [itex]u_a[/itex] is also hypersurface orthogonal.
So the question is asking us to show that [itex]u_{[a;b}u_{c]}=0[/itex], correct?
Well we know that this spacetime will have energy momentum tensor [itex]T_{ab}=(\rho+p)u_au_b+pg_{ab}[/itex]
So we have [itex]T_{ac;b}=(\rho+p) u_{a;b}u_c+u_au_{c;b}[/itex] since [itex]g_{ac;b}=0[/itex]
Then I tried writing [itex]u_{[a;b}u_{c]}[/itex] in terms of [itex]T[/itex]'s but it didn't get me anywhere. Can you give me a hint please?