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Black to white hole transition as a change of the topology of the event horizon

Mattia Villani

TL;DR

The paper addresses a classical description of the black-to-white hole transition as a topology change of the event horizon, proposing a cobordism-based framework to construct the interpolating spacetime between a Kerr black hole and its time-reversed counterpart. It analyzes horizon topology via the Euler characteristic, showing that black holes have $\chi=2$ while white holes can exhibit a different $\chi$, signaling a topology change, and it provides a concrete non-compact cobordism construction for the first transition $S^2 \to T^2$, including a surgical resolution to obtain a differentiable Lorentzian interpolant. The full evolution is described as $S^2 \to T^2 \to \varnothing$, implying an $S^3$ boundary and $\pi_3=\mathbb{Z}$ for the interpolating manifold, with the second transition to $\varnothing$ modeled by a sequence of time-reversed Kerr spacetimes with shrinking mass to $M=0$ ending in Minkowski spacetime. The work offers a semiclassical, topological framework linking horizon topology changes to global spacetime structure and provides a basis for incorporating quantum-gravity effects in a controlled geometric setting.

Abstract

We prove that the black to white hole transition theorized in several papers can be described as a change in the topology of the event horizon. We also show, using the theory of cobordism due to Milnor and Wallace, how to obtain the full manifold containing the transition.

Black to white hole transition as a change of the topology of the event horizon

TL;DR

The paper addresses a classical description of the black-to-white hole transition as a topology change of the event horizon, proposing a cobordism-based framework to construct the interpolating spacetime between a Kerr black hole and its time-reversed counterpart. It analyzes horizon topology via the Euler characteristic, showing that black holes have while white holes can exhibit a different , signaling a topology change, and it provides a concrete non-compact cobordism construction for the first transition , including a surgical resolution to obtain a differentiable Lorentzian interpolant. The full evolution is described as , implying an boundary and for the interpolating manifold, with the second transition to modeled by a sequence of time-reversed Kerr spacetimes with shrinking mass to ending in Minkowski spacetime. The work offers a semiclassical, topological framework linking horizon topology changes to global spacetime structure and provides a basis for incorporating quantum-gravity effects in a controlled geometric setting.

Abstract

We prove that the black to white hole transition theorized in several papers can be described as a change in the topology of the event horizon. We also show, using the theory of cobordism due to Milnor and Wallace, how to obtain the full manifold containing the transition.
Paper Structure (5 sections, 10 equations, 3 figures)

This paper contains 5 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Why a topology change
  3. Determining the Euler characteristic
  4. The first transition
  5. Discussion and conclusion

Figures (3)

  • Figure 1: The idea behind non-compact manifold cobordism. The change in topology is confined in the cylinder $\mathcal{S}_1 \cup \mathcal{S}_2 \cup \mathcal{T}$.
  • Figure 2: The initial and final manifolds with the interpolating manifold $\mathcal{M}$. $\mathcal{S}_1$ lies on the initial Kerr spacetime (In. Kerr), while $\mathcal{S}_2$ lies in the final Kerr spacetime (Fin. Kerr). $P$ is the critical point.
  • Figure 3: The complete $S^2\rightarrow T^2\rightarrow\varnothing$ transition. The manifold $1$ is a Kerr spacetime; manifolds $2,3,\dots$ are time-reversed Kerr spacetimes, while $n$ is a Minkowski spacetime.

Theorems & Definitions (1)

  • Definition 1: Spherical modification