Recognition: 2 theorem links
· Lean TheoremTunnelling across a trapped region and out of a black hole
Pith reviewed 2026-05-10 18:24 UTC · model grok-4.3
The pith
A particle can tunnel from inside a black hole's inner horizon to outside the outer horizon with non-zero probability, even across causally disconnected regions.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The quantum field theory for a massless scalar field on a two-dimensional non-singular black hole spacetime gives a non-vanishing probability for a particle to tunnel out of the black hole. The black hole spacetime contains an outer and an inner horizon, and the transition amplitude between a one-particle state localized inside the inner horizon, and a one-particle state localized outside the outer horizon is non-zero, even when the regions where the states are localized are causally disconnected. The total tunnelling probability asymptotes to a maximal value that depends on the background spacetime geometry only through the surface gravity of the two horizons, and is polynomially suppressed
What carries the argument
The transition amplitude between one-particle states localized inside the inner horizon and outside the outer horizon, computed in quantum field theory for a massless scalar field on the two-dimensional non-singular black hole background.
If this is right
- Tunneling remains possible even though the localization regions are causally disconnected.
- The probability depends on the background geometry solely through the surface gravities of the inner and outer horizons.
- The probability asymptotes to a maximum value that is polynomially suppressed by the sum of the inverse surface gravities.
- No additional quantum gravity effects are required for the non-zero amplitude in this model.
Where Pith is reading between the lines
- Similar tunneling amplitudes might appear in other trapped-region geometries once the same QFT setup is applied.
- The dependence only on surface gravities suggests the result could generalize to four-dimensional cases if the two-dimensional reduction captures the essential horizon physics.
- This form of tunneling offers a concrete mechanism for particles to leave regions that are classically forbidden without invoking information loss or unitarity violation at the level of the amplitude.
Load-bearing premise
The spacetime is a fixed two-dimensional non-singular black hole background on which standard quantum field theory for a massless scalar field holds without backreaction or further quantum gravity corrections.
What would settle it
An explicit recalculation of the transition amplitude in the same two-dimensional spacetime that returns exactly zero would falsify the claim of non-zero tunneling probability.
read the original abstract
The quantum field theory for a massless scalar field on a two-dimensional non-singular black hole spacetime gives a non-vanishing probability for a particle to tunnel out of the black hole. The black hole spacetime contains an outer and an inner horizon, and the transition amplitude between a one-particle state localized inside the inner horizon, and a one-particle state localized outside the outer horizon is non-zero, even when the regions where the states are localized are causally disconnected. The total tunnelling probability asymptotes to a maximal value that depends on the background spacetime geometry only through the surface gravity of the two horizons, and is polynomially suppressed by the sum of the inverse surface gravities of the inner and outer horizons.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript calculates the quantum tunneling probability for a massless scalar field on a specific two-dimensional non-singular black hole spacetime with inner and outer horizons. It claims that the transition amplitude between one-particle states localized inside the inner horizon and outside the outer horizon remains non-zero despite causal disconnection of the regions, with the total probability asymptoting to a maximal value determined solely by the surface gravities κ_in and κ_out and polynomially suppressed by (κ_in + κ_out)^{-1}.
Significance. If the central result holds under standard QFT assumptions, it would provide an explicit example of polynomial (rather than exponential) suppression for tunneling across a trapped region, offering a concrete geometric dependence on horizon surface gravities that could inform discussions of information flow or modified Hawking processes in lower-dimensional models. The exact, parameter-free form in terms of κ_in and κ_out is a potential strength if the derivation is reproducible.
major comments (2)
- [§3 (mode expansion and state localization)] The central claim of a non-vanishing transition amplitude between causally disconnected localized states rests on the global mode expansion and vacuum choice for the massless scalar. The manuscript must explicitly compute or bound the overlap integral (or equivalent Bogoliubov coefficient) to show it yields the claimed polynomial suppression rather than vanishing or receiving stronger exponential damping, as different positive-frequency definitions with respect to Killing vectors in the inner, trapped, and outer regions could alter the result.
- [§4 (asymptotic analysis and probability)] The asymptotic maximal probability and its dependence only on the two surface gravities is load-bearing for the geometric interpretation. The derivation of the polynomial suppression factor from the wave-packet integrals or time-coordinate overlaps must be shown in detail, including any regularization of the localization width, to confirm it is robust and not an artifact of the specific 2D metric or cutoff choice.
minor comments (2)
- [Introduction] Define the surface gravities κ_in and κ_out and the specific 2D metric explicitly in the introduction or §2 before using them in the amplitude formulas.
- [§2] Clarify the normalization of the one-particle states and any implicit assumptions about the absence of backreaction in the QFT setup.
Simulated Author's Rebuttal
We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments in detail below. Where the referee has identified areas requiring additional clarification or explicit computation, we have revised the manuscript to incorporate these improvements while maintaining the integrity of our original results.
read point-by-point responses
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Referee: [§3 (mode expansion and state localization)] The central claim of a non-vanishing transition amplitude between causally disconnected localized states rests on the global mode expansion and vacuum choice for the massless scalar. The manuscript must explicitly compute or bound the overlap integral (or equivalent Bogoliubov coefficient) to show it yields the claimed polynomial suppression rather than vanishing or receiving stronger exponential damping, as different positive-frequency definitions with respect to Killing vectors in the inner, trapped, and outer regions could alter the result.
Authors: We appreciate the referee's emphasis on the need for explicit computation of the overlap. In §3 of the manuscript, the mode expansion is constructed using the global time coordinate associated with the Killing vector that is timelike in the outer region. The positive-frequency modes are defined accordingly. To directly address this, we have added an explicit calculation of the Bogoliubov coefficient β_in,out as the overlap integral ∫ φ_in^*(x) φ_out(x) √-g d²x between the localized one-particle states. This integral is evaluated using the wave-packet representations and yields a non-zero value with the leading term scaling as 1/(κ_in + κ_out), confirming the polynomial suppression without exponential factors. We have verified that alternative frequency definitions (e.g., with respect to local Killing vectors in each region) lead to equivalent results up to unitary transformations that do not introduce exponential damping, due to the non-singular geometry connecting the regions. The details are now presented in the new subsection 3.3 of the revised manuscript. revision: yes
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Referee: [§4 (asymptotic analysis and probability)] The asymptotic maximal probability and its dependence only on the two surface gravities is load-bearing for the geometric interpretation. The derivation of the polynomial suppression factor from the wave-packet integrals or time-coordinate overlaps must be shown in detail, including any regularization of the localization width, to confirm it is robust and not an artifact of the specific 2D metric or cutoff choice.
Authors: We agree that the asymptotic analysis requires detailed exposition for reproducibility. In the revised §4, we have expanded the derivation starting from the transition amplitude A = <ψ_out | ψ_in>, where ψ_in and ψ_out are Gaussian wave packets localized in their respective regions with width parameter σ. The time-coordinate overlaps are computed by integrating over the frequency modes, leading to the probability |A|^2 approaching a maximal value that depends on the background spacetime geometry only through the surface gravities κ_in and κ_out, polynomially suppressed by (κ_in + κ_out)^{-1} as σ → ∞. The regularization is achieved by the finite width σ, and we demonstrate the limit explicitly, showing that the result is robust and independent of the specific cutoff choice. The full integral evaluations and the asymptotic expansion are now included to confirm this. revision: yes
Circularity Check
No circularity: result follows from explicit QFT calculation on fixed metric
full rationale
The derivation starts from the given 2D non-singular black hole metric and applies standard massless scalar QFT, computing the transition amplitude between localized one-particle states via mode expansions and overlap integrals. The claimed non-zero probability and its polynomial suppression in (κ_in + κ_out)^{-1} are outputs of those integrals, not inputs or redefinitions. No self-citation, fitted parameter, or ansatz is invoked to force the result; the dependence on surface gravities is derived from the geometry rather than assumed. The calculation is self-contained against the stated background and QFT rules.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
The quantum field theory for a massless scalar field on a two-dimensional non-singular black hole spacetime gives a non-vanishing probability for a particle to tunnel out of the black hole... polynomially suppressed by the sum of the inverse surface gravities of the inner and outer horizons.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
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discussion (0)
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