Complexity Growth in Black Holes: A Comparison of the Volume and Action Proposals

This paper investigates the late-time growth of holographic complexity in various black hole spacetimes using both volume and action prescriptions, revealing that while the action proposal yields a universal thermodynamic scaling, the complexity growth rate exhibits non-trivial, process-dependent variations under physical perturbations like the Penrose process and particle accretion that highlight the limitations of equilibrium-based treatments.

The Gist

Imagine a black hole not just as a cosmic vacuum cleaner, but as a giant, invisible computer. In the world of physics, there's a concept called "complexity," which is basically a measure of how many steps it takes to build a specific state from scratch. For a long time, scientists wondered: What happens to this "computational complexity" inside a black hole as time goes on?

This paper by Suraj Maurya and colleagues acts like a comparative study of two different ways to measure how fast this black hole computer is "thinking" or growing in complexity. They looked at four different types of black holes (some spinning, some charged, some in different types of space) to see if there's a universal rule.

Here is the breakdown of their findings using simple analogies:

The Two Rulers: Volume vs. Action

The researchers used two different "rulers" to measure the black hole's growth. Think of these as two different ways to estimate how busy a factory is:

1. The "Volume" Ruler (CV): This measures the size of the factory floor. In black hole terms, it looks at the volume of the space inside the event horizon.
   • The Finding: This ruler is a bit picky. It gives different results depending on the shape of the black hole. If the black hole is spinning or has a charge, the "volume" calculation changes its scale. It's like measuring a room with a tape measure that stretches differently depending on the color of the walls.
2. The "Action" Ruler (CA): This measures the work or "effort" the universe is putting into the black hole's existence over time.
   • The Finding: This ruler is much more consistent. No matter what kind of black hole they looked at (spinning, charged, or stationary), this method gave a result that was directly proportional to the black hole's Temperature × Entropy. It's like a universal speedometer that reads the same way for a Ferrari, a truck, and a bicycle.

The Universal Rule: Heat and Chaos

The most exciting discovery is that for both rulers, the rate at which complexity grows is tied to the black hole's Temperature and Entropy (a measure of disorder or the number of ways the black hole can be arranged).

• The Analogy: Imagine a hot, chaotic kitchen. The hotter it is and the more ingredients (disorder) it has, the faster the chefs (the black hole's interior) are working.
• The paper confirms that the "speed" of complexity growth is basically Temperature × Entropy. This holds true even for black holes in our own universe (flat space), not just in the theoretical "Anti-de Sitter" space often used in these theories.

What Happens When You Mess with the Black Hole?

The researchers didn't just look at calm, quiet black holes. They simulated what happens when you poke the black hole with various physical processes, like throwing things in or spinning it faster.

1. The Penrose Process & Superradiance (Stealing Energy):

   • Scenario: Imagine spinning a top and somehow stealing energy from its spin without stopping it.
   • Result: In these cases, the complexity growth rate increases. The black hole gets "busier" as it loses energy and angular momentum in these specific ways.

2. Particle Accretion (Dropping Things In):

   • Scenario: Dropping a particle into the black hole.
   • Result: This is tricky. If the particle is spinning the opposite way of the black hole, complexity goes up. But if the particle is spinning the same way and has a lot of angular momentum, the complexity growth rate can actually slow down or even appear to go negative (in their calculations).
   • The Catch: The authors warn that a "negative" result here is a red flag. It suggests that the black hole is in a state of chaos (out of equilibrium) and our simple "steady-state" math isn't capturing the full picture. It's like trying to measure the speed of a car while it's crashing; the math breaks down because the situation is too messy.

3. Hawking Radiation (Evaporation):

   • Scenario: The black hole slowly leaking energy and shrinking.
   • Result: The math gets complicated here. The growth rate depends on a delicate balance between losing mass and losing spin. The paper admits they need more work to fully understand this specific scenario.

The Big Takeaway

The paper concludes that while we don't yet have a perfect "microscopic" definition of what a black hole is thinking (since we don't have a full theory of quantum gravity for flat space), these geometric measurements are powerful tools.

• The "Action" method (CA) seems to be the more reliable, universal tool that respects the laws of thermodynamics.
• The "Volume" method (CV) is useful but depends heavily on the specific geometry of the black hole.

In short: Black holes are constantly "computing" or evolving. The speed of this evolution is governed by how hot and chaotic they are. While the exact math changes depending on how you measure it, the underlying rule—that heat and disorder drive the growth of complexity—seems to be a fundamental law of the universe, even for black holes right here in our neighborhood.

Technical Summary

Technical Summary: Complexity Growth in Black Holes

Problem Statement
The paper addresses the challenge of characterizing the late-time growth of holographic complexity in black hole interiors, specifically extending analyses beyond the standard asymptotically Anti-de Sitter (AdS) spacetimes to include asymptotically flat geometries. While the AdS/CFT correspondence provides a rigorous microscopic definition for boundary conformal field theory (CFT) states, no such precise dual is known for asymptotically flat black holes (e.g., Schwarzschild, Kerr). Consequently, the authors adopt a conservative viewpoint: rather than asserting a microscopic definition of computational complexity in non-AdS settings, they treat the Complexity–Volume (CV) and Complexity–Action (CA) prescriptions as "bulk geometric diagnostics." The central problem is to determine how these geometric measures of complexity growth behave across diverse spacetime geometries (BTZ, Schwarzschild, Reissner–Nordström, and Kerr) and how they respond to non-equilibrium physical processes such as energy extraction and accretion.

Methodology
The authors employ a comparative analysis of two primary holographic complexity proposals:

1. Complexity–Volume (CV) Duality: Complexity is related to the maximal volume ($V$) of a spacelike hypersurface anchored at the boundary. The growth rate is calculated as $dC/dt \sim \frac{1}{\hbar G \ell} \frac{dV}{dt}$, where $\ell$ is the AdS radius for AdS black holes and the horizon radius $r_+$ for asymptotically flat cases.
2. Complexity–Action (CA) Duality: Complexity is related to the on-shell gravitational action ($A$) evaluated on the Wheeler–DeWitt (WDW) patch. The growth rate is $dC/dt = \frac{1}{\pi \hbar} \frac{dA}{dt}$.

The study systematically evaluates these rates for:

• BTZ Black Holes: (2+1)-dimensional rotating solutions with negative cosmological constant.
• Schwarzschild Black Holes: Four-dimensional static, uncharged, asymptotically flat solutions.
• Reissner–Nordström Black Holes: Four-dimensional static, charged, asymptotically flat solutions.
• Kerr Black Holes: Four-dimensional rotating, asymptotically flat solutions.

For the Kerr geometry, the authors utilize a more general, angle-dependent Reinhart radius to determine the maximal interior volume, moving beyond the constant-$r$ approximations used in prior literature. Furthermore, the paper analyzes the variation in the complexity growth rate ($\delta \dot{C}$) under specific physical processes: the Penrose process, superradiance, particle accretion, and Hawking radiation. These variations are computed under a quasi-equilibrium approximation, assuming the black hole evolves slowly.

Key Contributions and Results

• Thermodynamic Scaling ($T_H S_H$):

  • CV Prescription: The complexity growth rate scales with the product of horizon temperature and entropy ($T_H S_H$) for all black holes studied. However, the proportionality constant is geometry-dependent. For example, the coefficient varies between BTZ, Schwarzschild, and Kerr cases, reflecting the foliation-dependent nature of the interior volume and the specific structure of the maximal hypersurface.
  • CA Prescription: In contrast, the CA prescription yields a universal proportionality constant (up to simple numerical factors) relating $dC/dt$ to $T_H S_H$ across all geometries, including non-AdS cases. This suggests that action growth is a more robust, geometry-independent measure of interior dynamics than volume growth.

• Response to Physical Processes (Kerr Black Hole):
  The paper investigates how $\delta \dot{C}$ responds to dynamical processes:

  • Penrose Process & Superradiance: In both cases, where energy and angular momentum are extracted, the variation $\delta \dot{C}$ is strictly positive. The complexity growth rate increases.
  • Particle Accretion: The behavior of $\delta \dot{C}$ depends on the angular momentum ($\delta J$) of the infalling particle. If $\delta J < 0$ (counter-rotating), $\delta \dot{C} > 0$. If $\delta J > 0$ (co-rotating), $\delta \dot{C}$ can be positive, zero, or negative depending on whether $\delta J$ exceeds a specific threshold relative to the mass increase ($\delta M$).
  • Hawking Radiation: The sign of $\delta \dot{C}$ is inconclusive at the leading order of approximation due to competing terms in the mass and angular momentum loss.

• Non-Equilibrium Limitations:
  The analysis reveals that under particle accretion, the quasi-equilibrium approximation can yield negative values for $\delta \dot{C}$, which contradicts the expectation that complexity should monotonically increase in classical processes. The authors interpret this negative result not as a physical violation but as a signal of the breakdown of the equilibrium approximation. They argue that in highly dynamical regimes, transient "hair" and horizon stresses become significant, rendering the growth of complexity path-dependent and necessitating a fully dynamical treatment beyond static thermodynamic data.

Significance and Claims
The paper claims that its results provide a controlled framework for assessing the scope and limitations of holographic complexity proposals outside the original AdS/CFT context.

• Universality vs. Geometry Dependence: The study highlights a fundamental distinction between the two proposals: while CV duality probes the global spatial evolution of the interior (and is thus sensitive to geometric slicing), CA duality captures a covariant, horizon-controlled dynamic that exhibits universal thermodynamic scaling.
• Diagnostic Utility: Even without a known microscopic dual for flat space, the authors assert that these geometric quantities serve as effective diagnostics for black hole interior dynamics. The persistence of the $T_H S_H$ scaling suggests that horizon thermodynamics governs complexity growth more fundamentally than the details of the bulk geometry.
• Future Directions: The paper modestly concludes that the appearance of negative $\delta \dot{C}$ in certain accretion scenarios motivates a shift from equilibrium-based treatments to fully dynamical analyses incorporating horizon stresses and transient hair. It suggests that a membrane or fluid-dynamical description of the horizon may be required to fully understand complexity growth in non-equilibrium regimes.

The authors do not claim to have derived a new microscopic definition of complexity for flat space, nor do they propose experimental tests. Instead, they frame their work as a systematic geometric investigation that clarifies which features of holographic complexity are universal and which are artifacts of specific spacetime geometries or equilibrium assumptions.