Open problems in 't Hooft's antipodal-identification framework for black hole unitarity: the $\ell_{\max}(M)$ cutoff, the Euclidean instanton's field content, the Rindler limit, and second-quantized horizon variables

The 2016 proposal of black hole unitarity via antipodal identification of the horizon asserts that the unitarity of partial-wave evolution forces a topologically non-trivial horizon geometry, but leaves four load-bearing ingredients undelivered: (i) the microscopic mechanism and precise form — sharp or smooth — of the angular-momentum cutoff $\ell_{\max}(M)$ under which horizon state counting reproduces the Bekenstein–Hawking entropy; (ii) a field content and action within general relativity plus conventional matter that admits the proposed Euclidean instanton with $a(r)\approx\sqrt{r^{2}+\mu^{2}}$ and antipodal identification (pure gravity and minimally coupled scalars being excluded); (iii) the realization of the antipodal construction in the Rindler limit of an infinitely heavy black hole, including the role of the dimensional constant $R$; and (iv) a second-quantized, Standard-Model-compatible encoding of the horizon variables $u^{\pm},p^{\pm}$ that preserves unitarity and completes the CPT picture.

Background

Hawking's 1975 calculation finds the radiation emerging from a black hole to be thermal to an excellent approximation, which appears incompatible with unitary quantum evolution — the black hole information paradox. Gerard 't Hooft's long-standing S-matrix approach instead treats the horizon as a scattering device: in-going and out-going particles interact through gravitational backreaction so strongly near the horizon that their relation can be a unitary scattering matrix (S-matrix, the operator mapping asymptotic in-states to out-states). In modern form of this program one expands the horizon degrees of freedom — light-cone positions $u^{\pm}(\theta,\varphi)$ and momenta $p^{\pm}(\theta,\varphi)$ of particles grazing the horizon — in spherical partial waves $Y_{\ell m}(\theta,\varphi)$, eigenfunctions of angular momentum on the sphere, and each partial wave $(\ell,m)$ evolves independently as a one-dimensional quantum-mechanical system.

In "Black Hole Unitarity and Antipodal Entanglement" (Foundations of Physics 46, 1185, 2016), 't Hooft concludes from this partial-wave unitarity that spacetime, as seen by a distant observer, must be topologically non-trivial. In the maximally extended Schwarzschild geometry — depicted on a Penrose diagram with the exterior region I and the diametrically opposite exterior region II — region II must represent the antipodal points of region I, and on the horizon itself each point is identified with its antipode, $(\theta,\varphi)\sim(\pi-\theta,\varphi+\pi)$. This antipodal identification removes the black hole interior from the outside observer's description, makes Hawking particles from opposite hemispheres fully entangled (only locally thermal), and renders the global evolution unitary while agreeing locally with general relativity; the antipodal relation is expected to be a CPT relation (charge conjugation, parity, time reversal). The paper also proposes a candidate Euclidean gravitational instanton — a finite-action solution of the field equations after Wick rotation to imaginary time — for the formation and evaporation of "virtual" black holes:

$ds^2 = dr^2 + a(r)^2\left[d\eta^2 + \sin^2\eta\left(d\vartheta^2 + \sin^2\vartheta\, d\varphi^2\right)\right],\qquad -\infty < r < \infty,\quad a(r) \approx \sqrt{r^2 + \mu^2},$

together with the antipodal identification $(r,\eta,\vartheta,\varphi)\leftrightarrow(-r,\pi-\eta,\pi-\vartheta,\pi+\varphi)$: a sphere of radius $\mu$ is excised and geodesics crossing it continue at the antipodal point, so the $S^3$ never gets smaller than $\mu$.

Four ingredients of this construction are asserted rather than derived, and the paper itself flags each as open. First, counting horizon states so as to obtain the Bekenstein–Hawking entropy $S=A/4\ell_P^2$ (horizon area $A$ in Planck units) requires cutting off the partial-wave expansion at $\ell\le\ell_{\max}(M)$ for a black hole of mass $M$; the cutoff was introduced in the earlier diagonalization analysis (arXiv:1509.01695), but neither its microscopic mechanism nor its shape is settled — Mersini-Houghton argues for a smooth exponential suppression from a "quantum Zeno effect" (the freezing of a system's evolution under effectively continuous measurement), while 't Hooft considers a sharp cutoff more likely for state counting, and writes that how to perform the cutoff at maximal $\ell$ "is not precisely understood today". Second, pure Einstein gravity cannot support the instanton's topology (the ansatz forces $a(r)$ to vanish somewhere), and ordinary minimally coupled scalar fields are excluded by a convexity argument on the equation $2a''+a(V(\varphi)+2\varphi'^{2})=0$; whether a conformal coupling term $\lambda R\varphi^2$ ($R$ the Ricci scalar) or other matter fields could produce such instantons is left open. Third, the construction carries the dimensional constant $R=2GM$ and its infinite-black-hole limit is Rindler space — the spacetime of uniformly accelerated observers in flat spacetime, itself endowed with horizons — where the realization of the antipodal identification is unexplained. Fourth, the horizon variables $u^{\pm},p^{\pm}$ are real-number, single-particle-like coordinates; a direct second-quantized reading (a Fock space, the standard quantum-field-theoretic framework in which particles are field quanta) "emphatically, is not possible here (unitarity would get lost)", and the suggested workaround — treating the real numbers as binary-digit sequences whose shifts emulate fermionic second quantization — is explicitly inelegant and unconnected to known particle interactions.

Two families of related constructions fall outside this setting: conformal-dilaton and brane-world instanton solutions (arXiv:2212.07716; arXiv:2605.25532; arXiv:2608.03536) are formulated in modified gravitational theories, and the Einstein–Yang–Mills meron wormhole (arXiv:1711.03469), which realises a maximal antipodal correlation, does so on a different geometry.

Problem Statement

Complete the microscopic foundations of the antipodal-identification mechanism for unitary black hole evaporation, by resolving the four gaps that G. 't Hooft's "Black hole unitarity and antipodal entanglement" (Found. Phys. 46, 1185, 2016) itself flags as open. The four components share one objective — turning the asserted antipodal construction into a self-consistent physical theory — and are linked by the common horizon algebra of the partial-wave variables $u^{\pm}_{\ell m},p^{\pm}_{\ell m}$ and the antipodal/CPT structure; the four components are jointly posed, each at the source's own generality:

(i) Derive the physical mechanism and the precise form of the high-$\ell$ cutoff $\ell_{\max}(M)$ under which the horizon state counting reproduces the Bekenstein–Hawking entropy $S=A/4\ell_P^2$, including whether the cutoff is sharp or smooth, from the microscopic theory rather than imposing it by hand (the source derives only its necessity; Mersini-Houghton's smooth quantum-Zeno cutoff and 't Hooft's sharp-cutoff preference are competing unfixed alternatives).

(ii) Determine which field content and action admit a Euclidean instanton of the proposed type — a finite-action solution with the topology of the ansatz $a(r)\approx\sqrt{r^2+\mu^2}$, $-\infty<r<\infty$, together with the antipodal identification $(r,\eta,\vartheta,\varphi)\leftrightarrow(-r,\pi-\eta,\pi-\vartheta,\pi+\varphi)$, in a theory containing Einstein gravity plus conventional matter — or prove that no such theory exists; the source's exclusion covers only pure gravity and minimally coupled scalar fields, explicitly leaving open whether a conformal coupling $\lambda R\varphi^2$ or other matter fields suffice.

(iii) Clarify the antipodal construction in the Rindler (infinite-black-hole) limit: how the antipodal point is realized in Rindler space, whether Rindler space must be compactified analogously to the finite black hole horizon, and what plays the role of the C-number $R$ — a classical constant rather than an operator; here the horizon radius entering the commutation relations and Green functions of the finite-$R$ construction — in the limit $R\to\infty$.

(iv) Construct the mapping between the real-number horizon variables $u^{\pm},p^{\pm}$ and a second-quantized (quantum field theoretic) description of the particle degrees of freedom — ideally one making contact with the Standard Model, including electromagnetism and charge conjugation so that the antipodal relation becomes a complete CPT relation — in a way that does not destroy the unitarity of the horizon evolution (a naive second-quantized reading of $u^{\pm}$ is excluded by the paper's own unitarity argument; the binary-digit emulation of fermionic quantization is a placeholder, not a construction).

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