Strong energy condition and complexity growth bound in holography

TL;DR

This work addresses whether the strong energy condition suffices to guarantee the holographic action-growth bound $\mathrm{d}\mathcal{A}/\mathrm{d}t \le 2M$ for eternal AdS black holes with matter outside the horizon. It develops a geometry-based analysis of the Wheeler-DeWitt patch in static maximal-extension spacetimes, decomposing late-time action growth into bulk, boundary, and joint contributions, and treats case (b2) in detail to obtain $\mathrm{d}\mathcal{A}/\mathrm{d}t = 2m_H$, which combined with $M \ge m_H$ yields the desired bound $\mathrm{d}\mathcal{A}/\mathrm{d}t \le 2M$; Schwarzschild-AdS saturates this bound when $(T_{\mu\nu}-\tfrac{1}{2}Tg_{\mu u})\xi^\mu\xi^\nu=0$. The paper then extends the reasoning to other horizon configurations (cases (a), (b1), (c)) and to higher dimensions, arguing that the bound remains valid under the stated energy and horizon-structure conditions, thereby providing robust classical GR support for the CA conjecture and insights into when complexity growth saturates. This contributes a concrete, largely geometry-driven proof strategy for holographic complexity constraints and clarifies the role of energy conditions and horizon topology in complexity dynamics.

Abstract

This paper proves that if eternal neutral black holes satisfy some general conditions and matter fields only appear in the outside of the Killing horizon, the strong energy condition is a sufficient condition to insure that the vacuum Schwarzschild black hole has the fastest action growth of the same total energy. This result is consistent with the bound of computational complexity growth rate and gives a strong evidence for the holographic complexity-action conjecture.

Figures (5)

• Figure 1: The Penrose diagram for Schwarzschild AdS black hole and the WDW patch. For given particular time slices at the two boundaries $t_L$ and $t_R$, the WDW patch is the yellow region with its boundary, which is just the domain of development of any spacelike slice which connects $t_L$ and $t_R$.
• Figure 2: The 2+1 decomposition of $\Sigma_t$ near the singularity. The red circles stands for the surfaces $\Psi=r$ and the red dashed circle stand for the singulary $\Psi=0$.
• Figure 3: The joint terms in the WDW patch. $H_+$ and $H_-$ stand for future ONKH and past ONKH, respectively. $k^\mu$ is affinely parameterized normal vector field of null surfaces characterized by $v=$costant. $\bar{k}^\mu$ is affinely parameterized normal vector field of null surfaces characterized by $u=$constant. $B(t)$ is the joint at the time $t_L=t$ and the flow $B(0)\mapsto B(t)$ is generated by vector field $\zeta^\mu$. The surface $u=$costant is fixed but the surface $v=$constant evolves with time $t$. At the late time limit, the suface $u=$costant coincidents with $H_-$. Hypersurface $\Sigma'$ stands for the timelike 3-surface in inner region of ONKH, which is orthogonal to $\xi^\mu$ and intersects with $H_\pm$ at $S_{H_+}$.
• Figure 4: An example of WDW patch in a black hole with multiple nondegenerated Killing horizons. For given particular time-slices $t_L$ and $t_R$ at the two boundaries, the WDW patch is the yellow region with its boundary. $H_1$ is the ONKH and $H_2$ is the sub-ONKH. The codimension-2 surfaces $P$ and $F$ are the joints of two past and future null sheets coming from the boundary slices $t_L$ and $t_R$, respectively. At the late time limit $t_R\rightarrow\infty$, the null sheet from $t_R$ to P will coincide with one branch of the ONKH and the null sheet from the $t_R$ to F will coincide with one branch of the sub-ONKH.
• Figure 5: An example of WDW patch in a black hole with only one bifurcate Killing horizon and null singularity. For given particular time-slices $t_L$ and $t_R$ at the two boundaries, the WDW patch is the yellow region with its boundary. $H$ is the ONKH. The codimension-2 surfaces $P$ and $F$ are the joints of two past and future null sheets coming from the boundary slices $t_L$ and $t_R$, respectively. At the late time limit $t_R\rightarrow\infty$, the null sheet from $t_R$ to P will coincide with one branch of the ONKH and the null sheet from the $t_R$ to F will coincide with the null singularity.