Evaporating Black Hole Interior and Complexity Evolution

This paper demonstrates that in an evaporating Jackiw-Teitelboim black hole model, the subsystem complexity—probed via geodesic length—grows linearly to a peak at the Page time before decaying exponentially, a behavior driven by non-perturbative gravitational effects that cause a loss of self-averaging at late times.

The Gist

The Big Picture: A Black Hole That Shrinks

Imagine a black hole not as a static, eternal monster, but as a hot cup of coffee left on a table. Over time, it loses heat (energy) to the room. In physics, this is called "evaporation." As the black hole shrinks, it spits out particles (radiation) into the universe.

The big question this paper asks is: What happens inside the black hole as it shrinks? specifically, does the "interior" get bigger, smaller, or stay the same?

To answer this, the authors use a simplified model of gravity (called JT gravity) and a clever trick involving a "magic wall" (an End-of-the-World brane) behind the black hole's event horizon. They treat the black hole's interior like a complex puzzle that gets more complicated as time goes on, until it suddenly starts simplifying again.

The Main Characters and Tools

1. The Black Hole and the Radiation:
   Think of the black hole as a backpack and the radiation it emits as items being taken out of the backpack.

   • Early on: The backpack is full, and the items (radiation) are few. The backpack is the "big" part of the system.
   • Late on: The backpack is nearly empty, and the pile of items on the floor (radiation) is huge. The pile of items is now the "big" part.

2. The "Interior Length" (Complexity):
   The authors measure the size of the black hole's interior not by volume in cubic meters, but by Complexity.

   • Analogy: Imagine the interior is a tangled ball of yarn. "Complexity" is a measure of how knotted and messy the yarn is.
   • In standard physics, we expect a black hole to get more tangled (more complex) over time, eventually reaching a maximum knotiness and staying there forever.

3. The "Page Time":
   This is the moment when the backpack has lost half its contents. Before this, the backpack is bigger than the pile of items. After this, the pile of items is bigger than the backpack. This is a famous turning point in black hole physics.

What They Found: A Surprising Twist

The authors calculated how the "tangled yarn" (complexity) changes as the black hole evaporates. Their results are very different from what happens to a black hole that doesn't evaporate.

1. The Early Days (Before Page Time):

• What happens: The black hole is still the dominant system. The interior complexity grows steadily, just like a knot getting tighter and tighter.
• The Analogy: You are actively tying knots in the yarn. The mess is increasing linearly.

2. The Turning Point (At Page Time):

• What happens: The complexity hits a peak. It reaches its maximum knotiness right around the time the black hole has lost half its mass.
• The Surprise: Instead of staying at this maximum knotiness, the complexity immediately starts to decrease.

3. The Late Days (After Page Time):

• What happens: The complexity drops rapidly, exponentially. The tangled yarn suddenly starts to untangle itself.
• The Analogy: Imagine the backpack is now so empty that it's almost just a simple, flat piece of fabric. The "mess" inside is gone because the black hole has become a "maximally mixed" state—a state of pure randomness with no specific information left inside. It's no longer a complex knot; it's just a smooth, simple sheet.

The Result:

• Non-evaporating black hole: Complexity grows $\rightarrow$ Plateaus (stays high).
• Evaporating black hole: Complexity grows $\rightarrow$ Peaks $\rightarrow$ Crashes down to near zero.

The "Fluctuation" Surprise: When the Average Lies

The paper also looked at how reliable this average picture is. They asked: "If we look at one specific black hole, does it behave like the average?"

• Before Page Time: Yes. The average is a good description of what's happening. The "knot" is growing steadily in almost every case.
• After Page Time: No. The average says the complexity is low, but this is a trick.
  • The Analogy: Imagine a room full of people. Most people have a very simple, smooth piece of paper (low complexity). But, hidden in the room, there is one person holding a massive, incredibly complex knot of yarn.
  • If you take the average complexity of the room, it looks low because most people have simple paper.
  • However, the "average" is being dragged down by the fact that most people are simple, while the "rare" complex cases are the only ones that matter for the physics of the black hole.
  • The Conclusion: After the Page time, the "average" complexity is no longer a good description of a typical black hole. The system has lost its "self-averaging" property. The behavior is dominated by rare, unusual configurations rather than the typical ones.

Summary of the Story

1. Setup: They modeled an evaporating black hole as a system entangled with a growing pile of radiation.
2. Measurement: They measured the "complexity" (interior messiness) of the black hole.
3. Discovery: Unlike a permanent black hole that stays messy forever, an evaporating black hole gets messy, hits a peak, and then gets clean again.
4. Why? As the black hole shrinks, it loses its information to the radiation. Once it's small enough, it becomes a simple, random state, and the "knot" unties.
5. Caveat: After the peak, the "average" calculation becomes misleading because it is dominated by rare, weird scenarios rather than what a typical black hole looks like.

In short: Black holes that evaporate don't just stay complex; they eventually simplify and "clean up" their interiors as they disappear.

Technical Summary

Technical Summary: Evaporating Black Hole Interior and Complexity Evolution

Problem Statement
The paper investigates the evolution of the interior volume of an evaporating black hole, specifically probing whether geometric observables behind the horizon exhibit significant departures from classical spacetime geometry, similar to the non-trivial evolution seen in entanglement measures (e.g., the Page curve). While semiclassical gravity and replica wormholes have successfully explained the unitary evolution of radiation entropy, it remains unclear how these non-perturbative effects manifest in observables unrelated to entanglement, such as the interior volume or quantum complexity. The authors address this by studying a simple toy model of an evaporating black hole within two-dimensional Jackiw-Teitelboim (JT) gravity coupled to an end-of-the-world (EoW) brane.

Methodology
The authors employ a model where the black hole interior is represented by an EoW brane behind the horizon. Evaporation is modeled by entangling the internal states of the brane with an auxiliary radiation system $R$. The state of the combined system is given by a maximally entangled state between the black hole/brane states $|\psi_i\rangle_B$ and the radiation states $|i\rangle_R$, where the dimension of the radiation Hilbert space, $k$, grows with time ($k(t) = e^{S_{\text{rad}}(t)}$).

To probe the interior, the authors define a geodesic length $L$ connecting the asymptotic AdS boundary to the EoW brane. This length is extracted from the renormalized derivative of a boundary-to-brane two-point function of a probe field with conformal dimension $\Delta$ as $\Delta \to 0$. This length is interpreted as a measure of the subsystem complexity of the black hole.

The computation relies on a quenched disorder average over the random couplings specifying the brane states and the random Hamiltonian of the dual matrix model. This necessitates the use of the replica trick to compute the ensemble-averaged free energy. The calculation includes two distinct non-perturbative gravitational contributions:

1. Spacetime wormholes: Arising from the genus expansion of the pure JT gravity path integral.
2. Replica wormholes: Arising from the ensemble average over the brane data, which encode the correlations responsible for the Page transition.

The authors perform the calculation in both the microcanonical and canonical ensembles, utilizing the spectral density of JT gravity and the properties of complete Bell polynomials to sum over the topologies of the replica geometries.

Key Contributions and Results

1. Complexity Evolution in the Evaporating Case:
   Unlike non-evaporating black holes, where complexity grows linearly and then plateaus at a time scale $\sim O(e^{S_{BH}})$, the ensemble-averaged complexity of the evaporating black hole exhibits a qualitatively different behavior:

   • Early Times ($t \ll t_{\text{Page}}$): The complexity grows linearly.
   • Page Time ($t \sim t_{\text{Page}}$): The complexity reaches a maximum.
   • Late Times ($t > t_{\text{Page}}$): The complexity decays exponentially.

   The authors derive an explicit formula for the renormalized geodesic length (complexity) in the canonical ensemble:
   $$C(t) = \frac{2\pi}{\beta} t \frac{e^{2S_{BH}(\beta)}}{(k(t) + e^{S_{BH}(\beta)})^2}$$
   This result shows that the growth rate is suppressed as the radiation Hilbert space dimension $k(t)$ increases. The turnover occurs parametrically close to the Page time, defined by the condition where the black hole entropy equals the radiation entropy ($S_{BH} \approx S_{\text{rad}}$).

2. Role of Replica Wormholes and the Kernel $I(n)$:
   A central technical result is the identification of a new kernel, $I(n)$, in the energy integral determining the geodesic length. This kernel arises from summing over all weighted partitions of replica geometries into connected and disconnected components. In the microcanonical ensemble, $I'(0)$ is shown to be the expectation value of the function $f(X) = X/(X+1)^2$ for a Poisson-distributed random variable $X$ with mean $a = e^{S_{BH}}/k$. The non-monotonic behavior of this function, peaking when $S_{BH} \approx S_{\text{rad}}$, drives the turnover in complexity.

3. Fluctuations and Loss of Self-Averaging:
   The authors compute the variance of the interior length. They find that:

   • Before Page Time: Relative fluctuations are parametrically small ($\sigma/\langle L \rangle \sim \langle L \rangle^{-1/2}$), meaning the ensemble average faithfully represents typical realizations.
   • After Page Time: As the mean complexity decays exponentially, the relative fluctuations grow to order one and eventually become large. This signals a loss of self-averaging. The late-time ensemble average is dominated by rare configurations (specifically, exponentially rare values of the Poisson variable $X$) rather than typical realizations.

Significance
The paper claims that the same non-perturbative gravitational effects (replica wormholes) that restore unitarity and produce the Page curve for entanglement entropy also leave a sharp, observable imprint on geometric quantities probing the black hole interior. Specifically, the interior volume (interpreted as complexity) does not continue to grow indefinitely but instead turns over and decays as the black hole subsystem becomes increasingly mixed.

The results suggest that the "smooth" post-Page time decrease in complexity is a signature of the fact that the entanglement wedge of the black hole does not change abruptly. Furthermore, the loss of self-averaging at late times indicates that the ensemble-averaged description becomes dominated by rare configurations, a feature that may have implications for understanding the typicality of the black hole interior in the context of the information paradox. The authors note that these findings are consistent with recent conjectures regarding subsystem complexity in bipartite systems and provide a concrete gravitational realization of these ideas.