LAHLH
- 405
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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 [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
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