Francois Englert and Philippe Spindel (JHEP 12 (2010) 065) proposed that the exact black hole S-matrix is unitary because post-selecting a generic exclusive final out-state drives the weak value of the stress tensor (the saddle-point source of the gravitational path integral) to Boulware-like, Planckian magnitudes within one Planck proper distance of the would-be event horizon, destroying the semiclassical geometry, while the inclusive (unpost-selected) geometry retains a regular horizon as a coarse-grained average over horizonless exclusive backgrounds, with Bekenstein-Hawking entropy SBH=ln⁡NS_{BH}=\ln N where NN counts the exclusive out-states. The paper's evidence rests on acknowledged approximations: no exact energy-conservation constraint, an emission-time regulator sent to infinity, s-wave truncation with only heuristic higher-partial-wave estimates, and an unperturbed background; its own footnote 6 concedes the singularities might be artifacts. The open problem is to assess this conjecture: (1) decide whether the post-selection-induced Planckian breakdown of the horizon is robust once these approximations are removed, or whether tuned or energy-conserving post-selections avoid it; (2) establish the inclusive coarse-graining and verify SBH=ln⁡NS_{BH}=\ln N by an operational counting in a controlled quantum-gravity setting; (3) extend the mechanism to cosmological and Rindler horizons. No resolution of the open problem is known in the literature to date. No work treats the energy-conservation caveat, computes the exclusive amplitudes, extends the analysis to other horizons, or realizes the counting; an independent direct computation by Kanai and Hosoya confirms the divergence in dynamical collapse but within the same class of approximation, and consonant arguments by Brustein and by Brustein and Medved use different reasoning.

Background

Black hole evaporation and unitarity. Semiclassical analysis shows that a black hole formed by gravitational collapse emits thermal radiation (Hawking radiation) and gradually loses mass (S. W. Hawking, Nature 248 (1974) 30; Commun. Math. Phys. 43 (1975) 199). If the radiation is exactly thermal and uncorrelated with the state that formed the hole, a system that begins in a pure quantum state ends as a mixed state, so the process cannot be described by a unitary S-matrix, i.e. the unitary map between asymptotic in- and out-states that governs scattering in ordinary quantum field theory. S. W. Hawking (Phys. Rev. D 14 (1976) 2460) argued that this breakdown of predictability is a genuine feature of quantum gravity; the resulting information paradox has shaped the field since. One long-standing response, initiated by G. 't Hooft (Nucl. Phys. B 256 (1985) 727), holds that the full black hole S-matrix is nevertheless unitary and that the thermal, horizon-endowed classical geometry is only an approximation to an underlying quantum description.

Weak values and post-selection. In the time-symmetric formulation of quantum measurement of Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz (Phys. Rev. 134 (1964) B1410), a system prepared in an initial state ∣i⟩|i\rangle (pre-selection) and later found in a final state ∣f⟩|f\rangle (post-selection) is assigned, for any operator A^\hat A, the weak value ⟨f∣A^∣i⟩/⟨f∣i⟩\langle f|\hat A|i\rangle/\langle f|i\rangle. Weak values can be complex, can lie far outside the operator's eigenvalue range, and diverge as the overlap ⟨f∣i⟩\langle f|i\rangle approaches zero. F. Englert (arXiv:gr-qc/9502039) proposed that operator weak values control the back-reaction of quantum matter on the geometry in black hole complementarity, and F. Englert, S. Massar, and R. Parentani (Class. Quantum Grav. 11 (1994) 2919) analyzed how vacuum fluctuations source the Hawking flux.

Exclusive and inclusive backgrounds; the Boulware divergence. Englert and Spindel evaluate the path integral for a matter-gravity amplitude ⟨f∣i⟩\langle f|i\rangle by saddle points: each stationary point defines an effective classical background geometry. Fixing both the in-state and a particular out-state on future null infinity J+\mathcal J^+ (the far-future null surface on which outgoing radiation is recorded) selects an 'exclusive' background sourced by the weak value ⟨f∣T^μν∣i⟩/⟨f∣i⟩\langle f|\hat T_{\mu\nu}|i\rangle/\langle f|i\rangle of the stress-energy tensor; leaving the out-state unspecified yields the 'inclusive' background sourced by the ordinary expectation value ⟨i∣T^μν∣i⟩\langle i|\hat T_{\mu\nu}|i\rangle, which reproduces the usual semiclassical Einstein equation. Among the standard quantum states on Schwarzschild spacetime, the Boulware vacuum, i.e. the state empty with respect to the static time coordinate, has a renormalized stress tensor that diverges at the horizon like an inverse power of the proper distance ρ\rho to it (P. Candelas, Phys. Rev. D 21 (1980) 2185), whereas the physical evaporation state is regular on the future horizon.

The 2010 result and conjecture. For a collapsing star forming a Schwarzschild black hole of mass MM, Englert and Spindel showed that post-selecting a 'generic' outgoing radiation state on J+\mathcal J^+ (one whose diagonal weak-value flux is of the same order as the thermal average, O(1/M2)O(1/M^2)) makes the weak-valued stress tensor carry the Boulware-type singularity. In two-dimensional s-wave truncation (keeping only the spherically symmetric, zero-angular-momentum partial wave of the field) the quadratic invariant behaves as ⟨T2⟩weak=O(1/(M4ρ4))\langle T^2\rangle_{\text{weak}}=O(1/(M^4\rho^4)), and a thermodynamic estimate including higher angular momenta (whose near-horizon thermal atmosphere contains roughly M2/ρ2M^2/\rho^2 modes) gives ⟨T2⟩weak=O(1/ρ8)\langle T^2\rangle_{\text{weak}}=O(1/\rho^8), reaching Planckian magnitude at a proper distance ρ=O(1)\rho=O(1), i.e. one Planck length, from the would-be horizon, where the linearized semiclassical back-reaction breaks down. Their conjecture: for generic exclusive amplitudes the event horizon does not survive; the black hole S-matrix is unitary and horizonless; and the inclusive geometry, which is regular and does possess a horizon, is a coarse-grained average of the exclusive ones, so that the classical event horizon and the Bekenstein-Hawking entropy SBH=4πM2S_{BH}=4\pi M^2 (one quarter of the horizon area in Planck units, the standard black hole entropy) have only thermodynamic, coarse-grained significance. This is a dynamical cousin of the fuzzball proposal (S. D. Mathur, Fortsch. Phys. 53 (2005) 793), in which black hole microstates are horizonless geometries; here the role of microstates is played by a complete set of NN exclusive out-states, with SBH=ln⁡NS_{BH}=\ln N.

The acknowledged approximations. The divergence is established for generic post-selections in the unperturbed Schwarzschild geometry, with the emission time regulated by a cutoff τ~\tilde\tau sent to infinity; imposing exact energy conservation would limit τ~\tilde\tau to the evaporation time O(M3)O(M^3), but the paper states this 'cannot be imposed in the unperturbed background'. The detailed computation is s-wave, with only order-of-magnitude estimates for higher partial waves. Footnote 6 of the paper concedes that the horizon singularities 'might be an artifact of the approximation', and post-selections whose weights satisfy a thermal-average cancellation condition (Eq. (3.72) of the paper) can cancel the Boulware singularity on the horizon. The path-integral measure, gauge-fixing and ghost issues, and the non-renormalizability of quantum gravity are bracketed by the assumption that only neighborhoods of saddle points are needed, an assumption whose status is unclear precisely where the weak-valued source becomes Planckian.

Problem Statement

Determine whether the Englert-Spindel hidden-horizon mechanism for black hole unitarity is correct and robust. The components below share one scientific objective, namely assessing the conjecture that the exact black hole S-matrix is unitary with horizonless exclusive saddle-point backgrounds while the classical event horizon emerges only as a coarse-grained inclusive average; component (1) is the load-bearing quantitative core on which components (2) and (3) depend. The hidden-horizon mechanism is defeated outright if the divergence is absent in a controlled treatment, or if the thermal-average cancellation class of Eq. (3.72) is generic rather than negligible.

(1) Robustness of the post-selection breakdown. Decide, in a controlled treatment that removes the approximations acknowledged in the source, whether the weak value of the stress tensor for a generic post-selected out-state on J+\mathcal J^+ (future null infinity, the far-future null surface on which outgoing radiation is recorded) still carries a Boulware-type divergence (the inverse-power divergence at the horizon characteristic of the Boulware vacuum, the quantum state that is empty with respect to the static time coordinate) and reaches Planckian magnitude within a proper distance of order one Planck length of the would-be event horizon of the collapsing black hole (of mass MM, in Planck units), so that the linearized semiclassical description of the geometry breaks down there. The approximations to be removed or controlled are: the absence of an exact global energy-conservation constraint (which would limit the emission-time regulator to the evaporation time O(M3)O(M^3)); infinite-time (rather than finite-time) boundary conditions; the s-wave truncation, with higher angular momenta so far estimated only through a thermodynamic-average argument; and the use of an unperturbed rather than self-consistently back-reacted background. In particular, determine whether, as footnote 6 of the source concedes, the horizon singularities are artifacts of this approximation, and whether post-selections whose weights satisfy the thermal-average cancellation condition (Eq. (3.72) of the source) form a negligible class or a generic one. Establish also whether the saddle-point metric description, including the bracketed questions of path-integral measure, gauge-fixing, and non-renormalizability, remains meaningful in the regime where the weak-valued source is Planckian.

(2) Coarse-graining and entropy. Establish or refute that the inclusive background, the saddle driven by the ordinary expectation value ⟨i∣T^μν∣i⟩\langle i|\hat T_{\mu\nu}|i\rangle which is regular and possesses an event horizon, is the coarse-grained average of the exclusive horizonless backgrounds; and verify the identification SBH=ln⁡NS_{BH}=\ln N, between the Bekenstein-Hawking entropy and the number NN of exclusive out-states of a complete basis, by an operational counting within a candidate theory of quantum gravity.

(3) Scope of the mechanism. Extend or fail to extend the weak-value post-selection analysis to cosmological horizons (for example the horizon of de Sitter space, the exponentially expanding solution with positive cosmological constant) and to Rindler horizons (the causal horizons perceived by uniformly accelerated observers in flat spacetime), and determine what it implies for the thermodynamic interpretation of those horizons, as the source defers this to a separate investigation that has not appeared.

The assessment is achievable without a full computation of the exclusive S-matrix amplitudes, which the source acknowledges would require an operational theory of quantum gravity.

References

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