Im reading Carroll's Spacetime and Geometry section 4.2 where he claims the following as the weak energy condition:(adsbygoogle = window.adsbygoogle || []).push({});

Given a energy momentum tensor T and a timelike vector t then T_{μν}t^{μ}t^{ν}≥ 0.

He claims that for a perfect fluid this is equivalent to the statement that ρ ≥ 0 and (ρ+P) ≥ 0. Where ρ is the density and P the pressure

He shows this by first showing that if the timelike vector t was the velocity of the fluid (i.e. we were in the rest frame of the fluid) then the weak energy condition reduces to ρ ≥ 0. So far so good. Then he claims that for a lightlike vector l, T_{μν}l^{μ}l^{ν}≥ 0 implies that (ρ+P) ≥ 0. I see that. He then says that I should convince myself that these conditions on ρ and P therefore hold for any timelike vectors as a consequnce, I dont see how, can anyone explain it to me?

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# I Equivalent statements of the weak energy condition

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