Renormalized stress-energy tensor near the horizon of a slowly evolving, rotating black hole

The renormalized expectation value of the stress-energy tensor 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ of a quantum field in an arbitrary quantum state near the future horizon of a rotating (Kerr) black hole is derived in two very different ways: One derivation (restricted for simplicity to a massless scalar field) makes use of traditional techniques of quantum field theory in curved spacetime, augmented by a variant of the ‘‘η formalism’’ for handling superradiant modes. The other derivation (valid for any quantum field) uses the equivalence principle to infer, from 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ in flat spacetime, what must be 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ren}}$ near the hole’s horizon. The two derivations give the same result—a result in accord with a previous conjecture by Zurek and Thorne: 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ren}}$, in any quantum state, is equal to that, 〈$T_{\mu \nu }$ ${〉}^{\mathrm{ZAMO}}$, which zero-angular-momentum observers (ZAMO’s) would compute from their own physical measurements near the horizon, plus a vacuum-polarization contribution $T_{\mu \nu }^{\mathrm{vac}\mathrm{pol}}$, which is the negative of the stress-energy of a rigidly rotating thermal reservoir with angular velocity equal to that of the horizon ${\Omega }_H$, and (red-shifted) temperature equal to that of the Hawking temperature $T_H$.