Tunnelling across a trapped region and out of a black hole

This paper demonstrates that a massless scalar field on a two-dimensional non-singular black hole spacetime exhibits a non-vanishing probability for particles to tunnel from the causally disconnected interior of the inner horizon to the exterior of the outer horizon, with the total probability asymptoting to a maximal value determined solely by the surface gravities of the two horizons.

The Gist

Imagine a black hole not as a bottomless pit from which nothing can ever escape, but more like a one-way street with a hidden, secret exit door that is extremely difficult to find, but not impossible to use if you know the rules of the quantum world.

This paper by Edward Wilson-Ewing explores a fascinating possibility: Can a particle trapped deep inside a black hole "tunnel" its way back out?

Here is the story broken down into simple concepts, using everyday analogies.

1. The Setup: The Black Hole with Two Doors

In classical physics (the rules of big things like planets and stars), a black hole is a prison. Once you cross the "event horizon" (the point of no return), you are stuck. You can only move inward until you hit the center.

However, the author suggests that if we fix the math to account for Quantum Gravity (the rules of the very tiny), the center of the black hole isn't a crushing singularity. Instead, the black hole has two horizons:

• The Outer Horizon: The famous "point of no return."
• The Inner Horizon: A second boundary deep inside.

Between these two doors is a "trapped region." Classically, once you are in this middle zone, you are doomed. You can't go back out, and you can't go forward to the center. It's a dead end.

2. The Magic Trick: Quantum Tunneling

In the quantum world, particles are weird. They don't act like solid billiard balls; they act like fuzzy clouds of probability.

The Analogy: Imagine you are in a room with a very thick, solid wall. In the real world, you can't walk through it. But in the quantum world, there is a tiny, non-zero chance that you could simply appear on the other side of the wall without breaking it. This is called tunneling.

Usually, we think of tunneling as particles jumping over small barriers. But this paper asks: Can a particle tunnel across the entire "trapped region" of a black hole, jumping from the inner horizon all the way to the outside world?

3. The Calculation: The "Ghost" Connection

The author does the math using a simplified 2D model (like looking at a flat map instead of a 3D globe). He treats the black hole like a stage where a massless particle (like a photon of light) is playing a game.

He calculates the probability of a particle starting deep inside (just past the inner door) and ending up outside (just past the outer door).

The Result:

• Yes, it is possible. The probability is not zero.
• It is very small. It's like winning the lottery, but the odds are even worse than that.
• It depends on the "temperature" of the horizons. In physics, horizons have a property called "surface gravity" (think of it as how "steep" the gravity well is). The chance of escaping depends on how steep the inner and outer slopes are.

4. The "Fuzzy" Wave Packet

To make the math work, the author imagines the particle isn't a pinpoint dot, but a "wave packet"—a little fuzzy blob of energy.

• If you make the blob wider (like spreading a drop of ink in water), the chance of it tunneling out increases.
• If you have many particles (a crowd) trying to escape, the odds that at least one of them makes it out get much better.

5. Why Does This Matter? (The Big Picture)

This isn't just a math puzzle; it might solve one of the biggest mysteries in physics: The Information Loss Paradox.

The Problem:
According to Stephen Hawking, black holes slowly evaporate and disappear. But if a black hole swallows a book (information) and then vanishes, where did the information go? Quantum physics says information cannot be destroyed. This is a contradiction.

The Solution Proposed Here:
If particles can tunnel out from the inside of the black hole, they can carry information with them.

• Imagine the black hole is a shredder. Hawking radiation is like the paper coming out the back, but it's shredded (random).
• This new idea suggests that some of the original paper (the information) might "tunnel" through the shredder's blades and come out the other side intact.

The Catch:
The math shows this tunneling happens at a rate that is surprisingly similar to the rate at which black holes evaporate. This suggests that tunneling might be the "secret mechanism" that allows black holes to release their secrets without breaking the laws of physics.

Summary

• Classical View: Black holes are one-way prisons. Nothing gets out.
• Quantum View: Because of quantum weirdness, there is a tiny, non-zero chance for a particle to "teleport" through the walls of a black hole and escape.
• The Implication: This "quantum escape hatch" might be how black holes preserve information, solving a decades-old mystery about the universe.

While we can't test this on a real black hole (they are too far away and too big), the math suggests that if you look closely enough at the quantum rules of gravity, the universe is far less "final" than it appears. The walls of the black hole are not brick; they are more like a very thick, very stubborn fog that a determined quantum particle can eventually walk through.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in quantum field theory (QFT) on curved spacetime: Can a particle localized deep inside a black hole (specifically in the region bounded by an inner horizon) tunnel out of the black hole, traversing the classically forbidden "trapped region" to reach the exterior?

• Classical Context: In classical General Relativity, the trapped region (between the outer and inner horizons) is causally disconnected from the exterior. Particles inside the inner horizon cannot escape to the outer region.
• Quantum Context: While QFT allows for tunneling across potential barriers and particles to propagate outside their lightcones (with polynomial suppression), it is unclear if this mechanism applies to the global geometry of a black hole, particularly one that is non-singular and possesses an inner horizon.
• Motivation: Standard black hole solutions (like Schwarzschild) contain a singularity. However, quantum gravity models (e.g., Loop Quantum Gravity) suggest that singularities are resolved, resulting in non-singular black holes with both an outer and an inner horizon. The author investigates whether QFT on such a background permits information or particles to escape from the deep interior.

2. Methodology

The author employs a rigorous QFT approach on a simplified, non-singular background to calculate transition amplitudes.

• Spacetime Geometry:

  • The study uses a 2-dimensional, non-singular black hole spacetime modeled using Painlevé-Gullstrand coordinates.
  • The metric is given by $ds^2 = -dt^2 + (dx - v dt)^2$, where the flow velocity $v(x)$ creates a trapped region ($v < -1$) bounded by an outer horizon ($x_o$) and an inner horizon ($x_i$).
  • The spacetime is assumed to be approximately stationary, containing three distinct regions: outer ($x > x_o$), trapped ($x_i < x < x_o$), and inner ($x < x_i$).
  • To handle the coordinate singularities at the horizons, the author constructs a global null coordinate system ($U, V$) that covers all three regions smoothly, utilizing surface gravities $\alpha_o$ and $\alpha_i$.

• Quantum Field Setup:

  • A massless scalar field $\phi$ is quantized on this background.
  • Due to conformal invariance in 2D, the Klein-Gordon equation reduces to the flat-space form $\partial_U \partial_V \phi = 0$.
  • The field is expanded in plane-wave modes, and the Fock vacuum $|0\rangle$ is defined relative to the time coordinate $T$ (where $T = (U+V)/2$).

• Calculation of Tunneling:

  • The author calculates the propagator (two-point correlation function) between a point just inside the inner horizon and a point just outside the outer horizon.
  • Transition Amplitudes: The probability of tunneling is derived by calculating the transition amplitude between localized single-particle states (using delta-function and flat wave-packet approximations) in the inner region and states in the outer region.
  • The probability $P_{out}$ is computed by integrating the squared modulus of the transition amplitude over the exterior region.

3. Key Contributions

• Proof of Concept for Global Tunneling: The paper demonstrates that in a non-singular black hole spacetime, the transition amplitude between a state localized inside the inner horizon and a state outside the outer horizon is non-zero, even though the regions are causally disconnected.
• Analytical Derivation of Probability: The author derives an explicit analytical expression for the asymptotic tunneling probability, showing it is not exponentially suppressed (as in standard barrier tunneling) but polynomially suppressed.
• Dependence on Surface Gravity: The result establishes that the tunneling probability depends on the background geometry only through the surface gravities of the two horizons ($\alpha_o$ and $\alpha_i$).
• Scaling Laws: The paper provides scaling laws for the tunneling probability in 2D and extrapolates them to 4D, suggesting a cubic suppression in higher dimensions.

4. Results

• Non-Vanishing Probability: The propagator between causally disconnected regions (inner vs. outer) is non-zero, confirming that tunneling across the trapped region is possible in QFT.
• Asymptotic Probability Formula:
  For a wave packet of width $W$, the total tunneling probability $P_{out}$ asymptotes to a maximal value as the time difference $\Delta T \to \infty$:
  $$P_{out} \approx \frac{W}{4\pi^2} \left( \frac{1}{\alpha_i} + \frac{1}{\alpha_o} \right)$$
  • The probability is polynomially suppressed by the sum of the inverse surface gravities.
  • It increases with the width of the wave packet ($W$) and is amplified if the surface gravity of either horizon is small.
• Multi-Particle Enhancement: If $N$ indistinguishable particles are localized in the inner region, the probability for at least one to tunnel out scales linearly with $N$ (amplified by a factor of $N$).
• 4D Extrapolation: While the calculation is in 2D, the author argues that in 4D, the probability should scale as $P_{4D} \sim (W / (\alpha_i^{-1} + \alpha_o^{-1}))^3$.
• Evaporation Rate Estimate: Applying these results to a spherically symmetric vacuum black hole (assuming Loop Quantum Gravity parameters where the inner horizon is near the Planck scale), the author estimates the mass loss rate due to tunneling:
  $$\frac{dM}{dt} \sim -M^{-2}$$
  This rate is comparable to the standard Hawking evaporation rate.

5. Significance and Implications

• Information Loss Problem: The existence of a unitary tunneling mechanism from the interior to the exterior suggests a potential pathway to resolve the black hole information paradox. If particles (and thus information) can tunnel out, the information is not lost to a singularity.
• Purification of Hawking Radiation: The author hypothesizes that tunneling could help "purify" Hawking radiation. Specifically, negative-energy partners of Hawking pairs that fall into the black hole might tunnel back out, potentially entangling with the outgoing radiation and restoring unitarity.
• Non-Singular Black Holes: The results reinforce the physical relevance of non-singular black hole models (with inner horizons). If singularities are resolved by quantum gravity, the inner horizon survives, enabling this tunneling process.
• Analogue Gravity: The paper suggests that these effects could be tested in analogue gravity systems (e.g., fluid dynamics or Bose-Einstein condensates) that simulate trapped regions with inner and outer horizons.

Conclusion:
Wilson-Ewing's work provides a rigorous theoretical demonstration that quantum tunneling allows particles to escape from the deep interior of a non-singular black hole. While the probability is small and polynomially suppressed, the mechanism is non-vanishing and scales in a way that could make it a significant contributor to black hole evaporation and the preservation of quantum information.