- #1
LAHLH
- 405
- 2
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
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