5. Energy Condition
- Null Energy Condition : \(X^{\mu}X^{\nu}T_{\mu\nu}\ \ge0\ \) for null-like vector \(X^{\mu}\)
- Weak Energy Condition : \(u^{\mu}u^{\nu}T_{\mu\nu}\ge0\) for time-like vector \(u^{\mu}\)
- Strong Energy Condition : \(\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)u^{\mu}u^{\nu}\ \ge0\) for time-like vector.
If we define equation of state as \(\omega\left(t\right)\ \equiv\frac{P\left(t\right)}{\rho\left(t\right)}\) then
- NEC : \(\omega\ge-1\)
- SEC : \(\omega\ge-\frac{1}{3}\)
- WEC : \(\rho\ge0\)
6. Energy - Momentum Conservation