How do the energy conditions in GR relate to timelike and null vectors?

[LAHLH]

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

 

[PeterDonis]

why not spacelike?

Because such a contraction would not have a physical interpretation as "energy", since, while ordinary objects can travel on timelike worldlines and massless objects like light can travel on null worldlines, nothing can travel on spacelike worldlines.

I'm not sure how to see this

See previous discussion here:

https://www.physicsforums.com/threads/equivalent-statements-of-the-weak-energy-condition.862113/