In curved spacetime, the notion of particles becomes observer-dependent because positive-frequency modes are defined relative to a chosen timelike vector field. This leads to phenomena such as the Unruh effect and particle creation in expanding universes.

Black holes are predicted to emit thermal radiation with temperature $T_H = \hbar c^3/(8\pi GMk_B)$. This links horizon geometry, quantum vacuum fluctuations, and thermodynamics.

A common approximation replaces $T_{\mu\nu}$ by its expectation value, yielding $$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\langle \hat T_{\mu\nu} \rangle.$$ This is powerful but incomplete, especially near singularities and in regimes of strong backreaction.