Finite curvature-coupling remainder of the renormalized stress-energy tensor near the horizon of a rotating black hole, and the regularity of the mirror-modified Hartle-Hawking state

Near the future horizon of a rotating (Kerr) black hole, Frolov and Thorne (Phys. Rev. D 39, 2125 (1989)) showed that the renormalized stress-energy tensor $\langle T_{\mu\nu}\rangle^{\rm ren}$ of a quantum field in any state equals the stress-energy measured by zero-angular-momentum observers plus a vacuum-polarization term equal to minus that of a rigidly rotating thermal reservoir at the horizon angular velocity $\Omega_H$ and the Hawking temperature $T_H$. The divergent, locally determined parts, and the horizon fluxes driving the hole's evolution, are thereby settled; in a Hartle-Hawking-like equilibrium state only a finite $O(\alpha^0)$ curvature-coupling remainder survives, where $\alpha$ is the lapse (the metric factor relating local proper time to coordinate time, which vanishes at the horizon). The source paper leaves that remainder, the transition region at distances of order $\kappa^{-1}$ ($\kappa$ the surface gravity) where it competes with the thermal terms, and the regularity of the modified equilibrium state, all unspecified. The original regularity dichotomy for unmodified Kerr was resolved negatively by Kay and Wald (Phys. Rep. 207, 49 (1991)): no isometry-invariant quasifree (Gaussian) state on Kerr is regular everywhere, the Hartle-Hawking state being singular at and beyond the speed-of-light surface. What remains open is the quantitative completion of the Frolov-Thorne picture: an explicit, general computation of the curvature-coupling corrections as functions of position, rotation parameter, and field type in the Hartle-Hawking-like state; the full $\langle T_{\mu\nu}\rangle^{\rm ren}$ through the $\kappa^{-1}$ transition region; and a rigorous existence and Hadamard-regularity proof for the mirror-modified Hartle-Hawking state, including the explicitly unanswered question (Duffy and Ottewill, Phys. Rev. D 77, 024007 (2008)) whether its deviation from exact thermality is regular on the horizon. No exact renormalized stress tensor is known for any four-dimensional rotating black hole: the existing results are approximations, other states (the Unruh state of an evaporating hole), other geometries (the 'subtracted' conformally modified Kerr-Newman metrics, and anti-de Sitter-type spacetimes), or numerics.

Background

In quantum field theory on curved spacetime, fields polarize the vacuum and the renormalized expectation value $\langle T_{\mu\nu}\rangle^{\rm ren}$ of their stress-energy tensor is the source term of semiclassical gravity, so its value near black-hole horizons controls backreaction. For a nonrotating (Schwarzschild) black hole the equilibrium state is the Hartle-Hawking state (Hartle and Hawking 1976): the quasifree state (a Gaussian state, fixed by its two-point function) whose Feynman propagator has positive-frequency behavior on the future horizon and negative-frequency behavior on the past horizon, describing a hole in equilibrium with a thermal bath at the Hawking temperature $T_H=\kappa/2\pi$, where $\kappa$ is the surface gravity (the red-shifted acceleration of the horizon's null generators). On Schwarzschild, $\langle T_{\mu\nu}\rangle^{\rm ren}$ in this state is known from mode-sum computations (building the expectation value as a divergent sum over angular modes and subtracting the analytically known divergent pieces), and it deviates from exact local thermality away from the horizon (Jensen, Mc Laughlin and Ottewill 1992).

For a rotating (Kerr) black hole the problem is harder. Superradiance, the amplification of modes with frequencies below $m\Omega_H$ (with $\Omega_H$ the horizon angular velocity and $m$ the azimuthal quantum number), makes the choice of positive-frequency convention and of stationary vacua ambiguous, and any equilibrium bath must rigidly co-rotate at $\Omega_H$. Frolov and Thorne (1989) derived the near-horizon structure of $\langle T_{\mu\nu}\rangle^{\rm ren}$ in an arbitrary state by two independent routes: a quantum-field-theory calculation for a massless scalar, using a variant of the $\eta$ formalism to handle the superradiant modes (a device for quantizing with an indefinite-metric auxiliary space), and an equivalence-principle argument valid for any field. Both give: $\langle T_{\mu\nu}\rangle^{\rm ren}$ equals the stress-energy $\langle T_{\mu\nu}\rangle_{\rm ZAMO}$ that zero-angular-momentum observers (ZAMOs, the local frames with zero conserved angular momentum) would measure, plus a vacuum-polarization term $T_{\mu\nu}^{\rm vac\,pol}$ equal to minus the stress-energy of a rigidly rotating thermal reservoir at angular velocity $\Omega_H$ and red-shifted temperature $T_H$. In the Hartle-Hawking-like state the divergent $O(\alpha^{-2})$ and $O(\alpha^{-1})$ pieces (where $\alpha$ is the lapse, $\alpha^2$ being of order the surface gravity squared times squared distance to the horizon, so $\alpha\to 0$ at the horizon) cancel, leaving $\langle T_{ll}\rangle^{\rm ren}=\langle T_{l\varphi}\rangle^{\rm ren}=0$ on the future horizon and only a finite $O(\alpha^0)$ curvature-coupling remainder. The equivalence principle is blind to this remainder: at distances of order $\kappa^{-1}$ above the horizon, curvature-coupling terms become comparable to the thermal contributions and the derivation fails without knowing them.

The rotating equilibrium state is itself pathological. The rigidly co-rotating thermal frame moves at or above the speed of light on and beyond the speed-of-light surface (the surface other than the horizon on which the Killing field $\partial_t+\Omega_H\partial_\varphi$ becomes null), and the Hartle-Hawking state is singular there, with divergent stress-energy and a malformed Hadamard two-point function (the standard short-distance singularity structure required of a physical state). Kay and Wald (1991) proved that unmodified Kerr admits no isometry-invariant quasifree state that is regular everywhere; Frolov and Thorne therefore proposed modifying the spacetime at radii bounded away from the horizon, for example by a perfectly reflecting mirror just inside the speed-of-light surface, producing a modified state $|H_M\rangle$ with the same near-horizon physics up to curvature-coupling corrections, and assumed its regularity without proof. Duffy and Ottewill (2008) later constructed such a mirror state, confirmed the near-horizon thermality numerically, and showed that the construction fails (via a Friedman-type instability) unless the mirror removes the entire region outside the speed-of-light surface. Numerical mode-sum regularization, the technique that solved the Schwarzschild problem, has only recently yielded the first stress-tensor values in Kerr, and only in the nonequilibrium Unruh state (the state describing an evaporating hole). The finite remainder, the transition region, and the rigorous status of the modified equilibrium state are the open questions recorded here.

Problem Statement

Near the future horizon of a rotating (Kerr) black hole, or of a slowly evolving rotating hole whose local geometry is Kerr to leading order, the divergent and flux-carrying parts of $\langle T_{\mu\nu}\rangle^{\rm ren}$ are settled by the Frolov-Thorne decomposition, and the non-existence of a regular isometry-invariant Hartle-Hawking state on unmodified Kerr is proven (Kay-Wald). Determine what remains: (1) the explicit, general form of the finite curvature-coupling corrections $(T_{\mu\nu}^{\rm vac\,pol})_{\rm curvature-coupling}$, the $O(\alpha^0)$ remainder in the ZAMO (zero-angular-momentum observer) frame, in the Hartle-Hawking-like equilibrium state, as functions of position, rotation parameter, and field type (at minimum a massless minimally coupled scalar, and ideally any field, as in the source paper's equivalence-principle result); (2) the full renormalized stress tensor through the transition region at distances of order $\kappa^{-1}$ ($\kappa$ the surface gravity) above the horizon, where curvature-coupling terms become comparable to the thermal contributions and the equivalence-principle derivation fails, including its matching to the global exterior solution; and (3) the rigorous mathematical status of the mirror-modified (or otherwise regularized) Hartle-Hawking state on which such computations rest, in particular a proof or disproof of its existence and Hadamard regularity (the standard short-distance singularity structure of the two-point function) on the horizons, resolving the explicitly open question of Duffy and Ottewill (2008) whether the deviation from exact thermality measured by rigidly rotating observers is regular on the horizon. The problem comprises the three parts (1)-(3) together.

References

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