[Paper Review] What happens in a black hole when a particle meets its antipode

Gerard ’t Hooft|arXiv (Cornell University)|Apr 16, 2018

Black Holes and Theoretical Physics15 references3 citations

TL;DR

This paper proposes that antipodal identification on a black hole's horizon—where each point is identified with its opposite—enables unitary evolution without firewalls or information loss. By treating particles and their antipodes as entangled via a sign-reversed momentum relation, the model restores locality and unitarity in Schwarzschild black holes using only standard quantum field theory and gravity, avoiding reliance on AdS/CFT or chaotic models.

ABSTRACT

The notion of antipodal identification on the black hole horizon is further explained and elaborated. Contrasting with numerous attempts in the literature to make fuzzy, poorly motivated models for black holes, we explain how, with an absolute minimum of assumptions, known laws of local physics suffice to calculate the unitary evolution law for a Schwarzschild black hole. Earlier work by the author, which explains how firewalls can and must be avoided, while also the information paradox disappears, ran into its one remaining problem: how to explain it better to the community. Antipodal identification is a natural way to replace thermally mixed states by pure quantum states, without the need to hide our ignorance in "chaos". We do encounter a strange looking sign switch in the relation of particles to their antipodes near the horizon. This sign switch is necessary to recover complete unitarity without any loss of information anywhere, while restoring locality. Although there are some important remaining problems, we advertise our approach as a healthy alternative to the reliance on AdS/CFT conjectures, which, we claim, do not guarantee to provide reliable answers.

Motivation & Objective

• To resolve the black hole information paradox by restoring unitarity in a Schwarzschild black hole using only known local physics.
• To eliminate the need for firewalls or non-locality by introducing antipodal identification on the horizon.
• To explain how information from infalling particles is preserved and unitarily mapped to outgoing radiation.
• To challenge the reliance on AdS/CFT duality as a reliable framework for black hole dynamics.
• To address the apparent sign flip in momentum relations between particles and their antipodes as a necessary feature for unitarity.

Proposed method

• Uses light-cone coordinates on the future and past horizons to define Cauchy data via longitudinal momentum and position operators: $[u^{+}_{i}, p^{-}_{j}] = i\delta_{ij}$ and $[u^{-}_{i}, p^{+}_{j}] = \delta_{ij}$.
• Applies antipodal identification: $u^{\pm}(\Omega) = -u^{\pm}(-\Omega)$, ensuring that only one particle per projective point $\Omega$ exists, either at $\Omega$ or its antipode $-\Omega$.
• Imposes that all angular momentum modes $\ell$ are odd, which enforces a sign structure that enables unitarity and avoids double-counting.
• Derives momentum and position distributions in terms of spherical harmonics with odd $\ell$, ensuring the $\ell=0$ state is non-vanishing and physically meaningful.
• Maps Fock space states to horizon data using a grid in $\Omega$-space, with low-$\ell$ dominance due to grey-body factors.
• Draws analogy to string theory, where vertex operators on the worldsheet must be integrated over to recover amplitudes, suggesting a similar integration is needed to resolve particle state degeneracy.

Experimental results

Research questions

• RQ1 How can unitary evolution be preserved in black hole evaporation without violating locality or introducing firewalls?
• RQ2 What is the physical role of antipodal identification on the horizon in restoring information conservation?
• RQ3 Why does a sign flip in the momentum relation between particles and their antipodes emerge, and how does it enable complete unitarity?
• RQ4 How can Fock space states be consistently mapped to horizon data when multiple particles at the same $\Omega$ are indistinguishable?
• RQ5 To what extent can this approach constrain the Standard Model, particularly regarding $CP$ violation and lepton number conservation?

Key findings

• Antipodal identification allows the black hole to preserve unitarity by mapping incoming particles to outgoing states via a sign-reversed momentum relation, avoiding information loss.
• The sign flip in the $p^{-}_{i}$ to $p^{+}_{j}$ relation is not a flaw but a necessary feature to maintain unitarity and locality in the presence of horizon entanglement.
• Hawking radiation, dominated by $\ell=0,1,2$ modes, is consistently described because the odd-$\ell$ constraint ensures the $\ell=0$ contribution does not vanish.
• The model avoids the need for chaotic or non-local descriptions by using only standard quantum field theory on a curved background with perturbative gravity.
• The formalism suggests that $CP$ violation in the Standard Model must be spontaneous, not explicit, due to the $CP$-like symmetry of antipodal identification.
• The degeneracy of multi-particle states at a single $\Omega$-point is resolved only through a full integration over horizon locations, analogous to string theory amplitudes.