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

In summary, Sean Carroll discusses energy conditions in ch4 of his GR book and explains that we want to impose coordinate invariant restrictions by forming a scalar from the energy momentum tensor. The WEC states that T_{\mu\nu}t^{\mu}t^{\nu}\geq 0 for all t^{\mu} timelike, while a perfect fluid has T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}. Carroll explains that, in order for the pressure to be isotropic, T_{\mu\nu}t^{\mu}t^{\nu} \geq 0 for timelike t^
  • #1
LAHLH
409
1
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
 
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  • #2

1. What are the energy conditions in General Relativity (GR)?

The energy conditions in GR are a set of mathematical inequalities that describe the properties of energy and stress in the theory of General Relativity. These conditions are used to prove the consistency and stability of the theory.

2. What is the Weak Energy Condition (WEC) in GR?

The Weak Energy Condition states that the energy density measured by any observer must be positive. In other words, energy cannot be negative or zero. This condition is satisfied by all known forms of matter and energy in the universe.

3. What is the Strong Energy Condition (SEC) in GR?

The Strong Energy Condition states that the sum of the energy density and the pressure measured by any observer must be positive. This condition is stricter than the WEC and is violated by exotic forms of matter with negative pressure, such as dark energy.

4. What is the Dominant Energy Condition (DEC) in GR?

The Dominant Energy Condition states that the energy flux measured by any observer must be positive or zero. In other words, energy cannot flow faster than the speed of light. This condition is satisfied by most forms of matter and energy, but can be violated in certain exotic scenarios.

5. What is the Null Energy Condition (NEC) in GR?

The Null Energy Condition states that the sum of the energy density and the pressure measured in any direction must be positive. This condition is the weakest among all the energy conditions and can be violated in some quantum field theories and vacuum fluctuations.

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