[Paper Review] Holographic complexity growth for a charged AdS-dilaton black holes with fixed and dynamical boundary respectively

Ai-chen Li|arXiv (Cornell University)|Jul 18, 2020

Black Holes and Theoretical Physics55 references4 citations

TL;DR

This paper investigates holographic complexity growth in charged AdS-dilaton black holes using the Complexity-Volume (CV) conjecture, comparing fixed and dynamical boundaries. For a fixed AdS boundary, the late-time complexity growth rate is bounded and increases with the dilaton coupling constant α, approaching a finite upper limit of $\frac{8\pi M}{\sqrt{3}}$ in the large-α limit. For a dynamical boundary via a self-gravitating brane, the growth rate monotonically decreases after peaking, with late-time behavior dominated by brane velocity, suggesting model-independence.

ABSTRACT

The holographic complexity conjectures are considered in a Einstein-Maxwell-Dilaton gravity, by using the "Complexity-Volume" proposal. Specifically, we calculate the growth rate of complexity for an eternal charged AdS-dilaton black holes with fixed and dynamical boundaries respectively. The dynamical boundary is achieved by introducing a moving self-graviting brane on which the induced metric has an exact FLRW form. In case of fixed AdS boundary, there exists a bound for evolution of growth rate on late time, while this bound will become larger as the dilaton coupling constant $α$ increases. In large $α$ limit, we analytically prove that this bound is a finite value which is proportional to the black hole mass. In case of dynamical boundary, namely the brane-bulk system, the growth rate decreases monotonously on late time, after reaching a maximum value at a certain time. We find that the evolution of growth rate for brane-bulk system on late time is dominated by the velocity of the moving brane. We guess this result is model-independent.

Motivation & Objective

• To investigate the growth rate of holographic complexity in Einstein-Maxwell-Dilaton gravity using the CV conjecture.
• To compare complexity dynamics between static bulk spacetime with fixed AdS boundary and a dynamical boundary via a moving self-gravitating brane.
• To examine the influence of the dilaton coupling constant α on complexity growth in both fixed and dynamical boundary scenarios.
• To assess whether the late-time behavior of complexity growth in the brane-bulk system is dominated by brane velocity, suggesting model-independence.

Proposed method

• Adopt the Complexity-Volume (CV) conjecture to relate boundary quantum complexity to the maximal volume of the Einstein-Rosen bridge in the bulk.
• Use exact solutions of five-dimensional Einstein-Maxwell-Dilaton gravity to model eternal charged AdS-dilaton black holes.
• Implement a moving, self-gravitating Randall-Sundrum brane with FLRW-induced metric to model a dynamical boundary.
• Apply the Israel junction conditions and projection method (Shiromizu-Maeda) to derive the effective Einstein equations on the brane and determine its motion.
• Calculate the time evolution of the complexity growth rate using the CV formula in both fixed and dynamical boundary setups.
• Analyze the late-time behavior of the growth rate numerically and analytically, especially in the large-α limit.

Experimental results

Research questions

• RQ1 What is the behavior of holographic complexity growth rate on late times for a charged AdS-dilaton black hole with a fixed AdS boundary, and how does it depend on the dilaton coupling constant α?
• RQ2 Does the complexity growth rate remain bounded in the large-α limit, and if so, what is the analytical form of the upper bound?
• RQ3 How does introducing a dynamical, self-gravitating brane as the boundary affect the time evolution of the complexity growth rate?
• RQ4 Is the late-time decay of complexity growth in the brane-bulk system dominated by the velocity of the moving brane, and is this behavior model-independent?
• RQ5 How do self-gravitating effects of the brane alter the dynamics compared to a non-self-gravitating brane?

Key findings

• For a fixed AdS boundary, the late-time complexity growth rate is bounded, and this bound increases with the dilaton coupling constant α.
• In the large-α limit, the upper bound on the complexity growth rate is analytically proven to be $\frac{8\pi M}{\sqrt{3}}$, a finite value proportional to the black hole mass M.
• With a dynamical boundary via a self-gravitating brane, the complexity growth rate increases initially, reaches a maximum, and then decreases monotonically on late times.
• The late-time decay of the complexity growth rate in the brane-bulk system is dominated by the velocity of the moving brane, suggesting a generic, model-independent behavior.
• The self-gravitating effects of the brane render its motion insensitive to changes in α, unlike in non-self-gravitating models where α strongly affects brane dynamics.
• The initial position of the brane must satisfy $a(0) > r_h$ to avoid crossing the past horizon, ensuring physical consistency in the Penrose diagram.