Black-to-White Hole Transition

• The process is defined as a quantum tunneling mechanism where a finite quantum region converts a classical black hole into a nonsingular white hole.
• Models utilize explicit spacetime metric constructions with precise gluing conditions and spin foam computations to detail the transition dynamics and timescales.
• The transition offers an information-preserving resolution to black hole paradoxes, predicting unique astrophysical signatures and revised evaporation models.

The black-to-white hole transition refers to a nonperturbative quantum gravitational process in which a spacetime geometry that would classically give rise to a black hole is instead completed by a finite “quantum region” where the geometry tunnels, via quantum effects, to that of a white hole. The resulting spacetime is singularity-free, does not possess a classical event horizon, and generically leads to observable differences from standard black hole evaporation. Such transitions have been rigorously formulated via explicit geometries, effective toy models, and candidate computations in both canonical and covariant (spin foam) loop quantum gravity, as well as in semiclassical and thermodynamical frameworks.

1. Construction of the Global Metric and Geometric Regions

The central component of these models is the spacetime metric, divided into “classical” and “quantum” regions with precise coordinate patching and gluing conditions. In the Haggard–Rovelli “fireworks” construction (Haggard et al., 2014), the spacetime is composed of:

• Region I (interior): Flat Minkowski, described in null coordinates $(u_I, v_I,\theta, \phi)$ as:

$$ds^2 = -du_I dv_I + r_I^2 (d\theta^2 + \sin^2\theta d\phi^2), \qquad r_I(u_I, v_I) = \frac{v_I - u_I}{2}$$

• Region II (exterior): A Schwarzschild (or Kruskal–Szekeres) patch,

$$ds^2 = -F(u,v) du dv + r^2(u,v)[d\theta^2 + \sin^2\theta d\phi^2],$$

with

$$F(u, v) = \frac{32 m^3}{r} e^{-r/2m}, \qquad (1 - r/2m) e^{r/2m} = uv$$

• Region III (“quantum” region): An ansatz with a metric of the same general form, with $F(u_q, v_q) = 32 m^3 / r_q \, e^{r_q/2m}$ and $r_q = (v_q - u_q)/2$, the explicit details of which are fixed by the requirement of smooth matching to the classical regions.

The key feature is that the Einstein equations are solved exactly everywhere except within a compact spacetime region—typically a “diamond” in the Penrose diagram—where quantum gravitational effects become non-negligible and provide the dynamical “tunneling” between the black-hole and white-hole sectors.

Boundary Conditions and Gluing: The metric is glued across null surfaces and a time-symmetric surface, ensuring continuity of the induced metric and, where possible, of the extrinsic curvature. Recent work has improved on earlier constructions by prescribing explicit coordinate transformations (including use of Kruskal–Penrose coordinates and matching functions with desired smoothness properties) that eliminate defects such as conical singularities at the gluing interface (Rovelli et al., 2018).

2. Dynamics of the Quantum Tunneling Transition

The black-to-white hole transition is fundamentally a quantum tunneling process where quantum gravity effects, though locally small, accumulate over macroscopic timescales outside the Planckian region. At Planck-scale curvatures, a repulsive quantum gravitational force causes the collapsing matter to “bounce.” The quantum sector is constrained to affect the geometry not only within but also slightly outside the classical horizon, a requirement justified by the observation that cumulative quantum corrections integrated over large times can become order unity even if the instantaneous effect is minuscule (Haggard et al., 2014).

The cumulative parameter,

$q \sim l_P \, \mathcal{R} \, \tau,$

(devoted to the accumulation of quantum curvature $\mathcal{R}$ over proper time $\tau$) reaches unity on timescales

$\tau_{\text{outside}} \sim \frac{m^2}{l_P},$

which is far shorter than the evaporation time ($\sim m^3$) but vastly longer than the bounce time measured by an infalling observer. The process, for all practical purposes, is instantaneous at the quantum region itself due to extreme redshift effects.

Some models predict symmetry between the black and white hole phases with total duration $\sim m^2$; others (following the “improved fireworks” scenario (Lorenzo et al., 2015)) require a highly asymmetric evolution, with a long black phase (accumulation time $\sim m^2$) and a very short explosion time for the white hole ($\sim m \log m$), motivated by both quantum stress-energy divergences and classical stability arguments.

3. Causality, Stability, and Validity of Approximations

A significant controversy is whether quantum corrections can physically “leak outside” the horizon without violating causality or the semiclassical approximation. The cumulative argument demonstrates that even in the region $r > 2m$ (where curvatures are low), corrections scale as $l_P^2 \mathcal{R}^2$ per unit time; when integrated over a time of order $m^2/l_P$, these become order unity. Hence, globally, a quantum transition can occur without violating local energy conditions or causality, and the already “piled-up” corrections can trigger the tunneling outside the region causally connected to the quantum core.

Stability is a critical issue for long-lived white hole phases. Rigorous analyses (Barceló et al., 2015) show that classical and quantum instabilities (e.g., exponential blueshifting leading to the divergence of the renormalized stress-energy tensor near the white hole horizon), as well as the “Eardley” instability induced by arbitrarily small accretion, suffocate long-delayed transitions. Only transitions that occur on timescales comparable to the light-crossing time or as short as $m \log m$ are found to be stable to generic perturbations, as required for astrophysical viability (Lorenzo et al., 2015).

4. Thermodynamic and Information-Theoretic Implications

The black-to-white hole transition offers a radical departure from the traditional semiclassical picture of black hole evaporation as a slow, information-losing process. In tunneling models, the entire content of the initial trapped region (including information) can eventually be released in the white hole phase. The process, realized via quantum gravity, is unitary and does not require an information-destroying singularity or a permanent event horizon.

Entropy and entropy bounds play a central role. In some scenarios, white hole remnants are shown to maintain a huge internal volume and thus the capacity to store the “missing” information (with entropy not dictated by the shrinking horizon area but by the cumulative interior volume), which bounces and is slowly released (Bianchi et al., 2018). In the context of thermodynamic fluctuation arguments and coordinate-invariant tunneling calculations, the entropy of the final white hole is assigned either a negative value (e.g., $S_\text{WH} = -S_\text{BH}$ (Volovik, 2021, Phat et al., 2022)) or is interpreted operationally in terms of the information retrieval process.

5. Spin Foam Models and Transition Amplitude Calculations

First-principles calculations of the black-to-white hole tunneling amplitude have been carried out using covariant spin foam models of loop quantum gravity (Soltani, 2021, Han et al., 3 Apr 2024). The problem is formulated as a path integral over a 2-complex dual to a triangulation of the “quantum region” separating black and white phases. The boundary data are encoded with Thiemann’s complexifier coherent states, and, critically, superpositions over opposite orientations are included—motivated by the non-invariance of the quantum theory under reversal of the tetrad orientation, while the classical intrinsic geometry remains unchanged.

Dominant contributions to the amplitude are found to come from histories that include a change of orientation, i.e., “tunneling” not only between macroscopic black and white geometries but also between distinct quantum sectors (distinguished by the sign of $e^a_i$). This “orientation change” is interpreted as a quantum gravitational barrier penetration (Han et al., 3 Apr 2024).

Advanced numerical techniques, including identification of complex critical points of the effective spin foam action, are employed for practical computation of amplitudes and for extraction of quantitative scales such as the crossing (transition) time. The predicted crossing times in the deep quantum regime are independent of extrinsic boundary geometry, confirming the universality of the quantum tunneling process (Frisoni, 2023).

6. Model Variants and Effective Descriptions

Effective descriptions range from explicit metric constructions—both non-singular, mass-independent, and with minimal violation of energy conditions (Feng et al., 2023)—to equations of state frameworks that connect black and white holes via thermodynamic control parameters (e.g., the electric charge in charged AdS black holes) (Phat et al., 2022). These models often exhibit rapid transitions, bounded curvature invariants even at the core, and only local violations of classical energy conditions.

Some approaches numerically solve the semiclassical Einstein equations including the renormalized stress-energy tensor for collapsing matter with charge. These demonstrate that the trapped region (defined by inner and outer horizons) can shrink and merge due to quantum backreaction, with the subsequent emergence of an anti-trapped region interpreted as a long-lived white hole interior (Boyanov et al., 5 Jun 2025), even in absence of a full quantum gravity treatment.

Geodesic analyses confirm the smooth passage of matter (and information) through the regularized core from black to white region. Carter–Penrose diagrams constructed with careful conformal matching at thin shell interfaces verify the possibility of globally regular spacetimes realizing the bounce (Lin et al., 2023).

7. Phenomenological and Observational Signatures

Models incorporating a rapid black-to-white transition predict possible observable astrophysical phenomena, such as explosive outbursts or distinct signatures in gravitational waveforms—although such signals, delayed by enormous external timescales, may remain challenging to detect.

The possibility of a zero-action transition (“gray” horizons as quantum superpositions of causal structures) is proposed as a dynamically allowed process potentially unsuppressed in the Feynman path integral (Gaur et al., 2023). Thermodynamical treatments further expand the scope by considering negative temperature and negative entropy white holes, especially in the AdS/CFT context (Phat et al., 2022).

Fundamentally, the theoretical achievement is the articulation and explicit construction—classically, semiclassically, and in candidate quantum gravity theories—of a non-singular, information-preserving, time-asymmetric spacetime evolution describing a black hole’s quantum transition to its white hole remnant, unifying dynamical, global causal, and information-theoretic perspectives.

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Table: Key Geometric and Physical Characteristics

Feature	Classical Model (e.g. Schwarzschild)	Black-to-White Hole Transition
Singularity	Inevitable at $r=0$	Resolved via quantum region, no singularity
Event Horizon	Permanent, traps information	No permanent horizon; global Cauchy surface connects past and future
Time Scale	Infinite evaporation (Hawking, $\sim m^3$)	Bounce time $\sim m^2/l_P$ (depends on model); white phase often rapid ($\sim m \log m$)
Information Fate	Lost or released over long times, paradox	Preserved, ultimately released in white hole phase
Global Geometry	Schwarzschild/Kruskal	Piecewise classical + quantum region; smooth manifold, extendible geodesics
Energy Conditions	Satisfied except near $r=0$	Locally violated near quantum region only, elsewhere regular
Stability	Stable (BH), white holes forbidden	Only short-lived white holes stable under perturbations
Calculation Methods	Classical GR, semiclassical QFT	Quantum tunneling, effective models, spin foam amplitudes