Black Hole Entropy and Complexity Growth in Horndeski Gravity within the AdS/BCFT Framework

TL;DR

<3-5 sentence high-level summary> This work extends the Complexity=Action (CA) holographic duality to Horndeski gravity within the AdS/BCFT framework, incorporating scalar-tensor couplings and BCFT boundary data. It demonstrates that the linear growth of complexity, with rate proportional to the product of black hole temperature and entropy, remains valid across planar, rotating BTZ, and charged AdS black holes, including boundary contributions, and it analyzes how entanglement entropy and phase transitions reflect this growth. The entropy is computed via Wald formulas and holographic renormalization, ensuring compatibility with the first law in Horndeski+BCFT settings, while shock-wave tests confirm the robustness of CA and the switchback effect under modified causal structure. The results reveal a form of universality for CA in modified gravity and illuminate how BCFT boundary data and Horndeski couplings affect holographic complexity, entanglement, and phase structure.

Abstract

This work investigates the connection between quantum complexity and gravitational dynamics within the framework of Horndeski gravity, extending the AdS/BCFT correspondence to include scalar-tensor interactions. By refining the ``\(complexity = action\)'' conjecture we investigate how Horndeski gravity modifies the Wheeler-DeWitt patch and the causal structure of the black hole. Our analysis reveals that the linear growth of complexity, proportional to the product of black hole entropy and temperature, remains valid across various black hole configurations, including those of rotating and charged black holes. Moreover we study the impact of shock waves on the growth of complexity, which shows the appearance of the ``switchback effect''. These results show the universality of the complexity = action conjecture and its validity in modified gravitational theories.

Figures (12)

• Figure 1: Left side, AdS bulk is represented as a hyperbolic space, with the black hole located at its center. The boundary of the AdS space hosts the boundary conformal field theory (BCFT), which encodes the dual description of the bulk gravitational dynamics. Right side, the Penrose diagram illustrates the causal structure of the black hole, including the event horizon, singularity, and AdS boundary.
• Figure 2: In the right side, eternal two-sided Anti-de Sitter (AdS) black hole and its one-sided counterpart formed through shockwave collapse represent distinct holographic frameworks with profound implications for Boundary Conformal Field Theories (BCFTs). The two-sided configuration corresponds to an entangled thermofield double state across dual CFTs at opposing boundaries. In the left side, the one-sided variant maps to a single BCFT system.
• Figure 3: AdS/CFT correspondence in the presence of boundary hypersurface.
• Figure 4: Organized scheme of CFT space.
• Figure 5: Planar black hole in $d=3$: (a) Entanglement entropy plotted as a function of the half-width of the boundary interval. (b) Examples of representative minimal surfaces. Both plots are generated using $g = \mu = \beta = G = 1$, with the following values of $\gamma$: $\gamma = 0$ (blue), $\gamma = -0.5$ (dark green), and $\gamma =0.5$ (dark purple).