General Relativity: An Introduction for Physicists · Problem 8.6

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · weak-energy-condition

Problem

The weak energy condition (WEC) states that any energy–momentum tensor must satisfy

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

for all timelike vectors

t^\mu

. Show that for a perfect fluid the WEC implies that

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