Energy conditions in GR

Main Question or Discussion Point

Hi,

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

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

thanks