General Relativity: An Introduction for Physicists · Problem 8.7

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · strong-energy-condition

Problem

The strong energy condition (SEC) states that any energy–momentum tensor must satisfy

T_{\mu\nu}t^\mu t^\nu \geq \tfrac{1}{2}T^\rho{}_\rho t^\sigma t_\sigma

for all timelike vectors

t^\mu

. Show that for a perfect fluid the SEC implies that

\rho c^2 + p \geq 0 \qquad \text{and} \qquad \rho c^2 + 3p \geq 0.

Does the SEC imply the WEC in Exercise 8.7? Show further that, from the Einstein equations, the SEC implies that

R_{\mu\nu}t^\mu t^\nu \geq 0

, where

R_{\mu\nu}

is the Ricci tensor.