Sean Carroll talks about energy conditions in ch4 of his GR book. From what I understand we want to impose co-ordinate invariant restrictions so we need to form a scalar from the energy momentum tensor, which is done by just arbitrarily contracting with timelike/null vectors (why not spacelike?).

The WEC says that [tex] T_{\mu\nu}t^{\mu}t^{\nu}\geq 0 [/tex] for all [tex] t^{\mu} [/tex] timelike. If we consider a perfect fluid [tex] T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}[/tex], then Carroll says that because pressure is isotropic, then [tex] T_{\mu\nu}t^{\mu}t^{\nu} \geq 0[/tex] for timelike [tex] t^{\mu} [/tex] IF [tex] T_{\mu\nu}U^{\mu}U^{\nu}\geq 0[/tex] AND [tex] T_{\mu\nu}l^{\mu}l^{\nu}\geq 0[/tex] where l is null.

Despite him saying this is just adding vectors, I'm not sure how to see this...

thanks