Upper bound about cross-sections inside black holes and complexity growth rate

TL;DR

This paper proposes a universal bound on the size of black hole interior cross-sections in AdS spacetimes, supporting a conjecture that relates interior geometry to energy and complexity growth, with implications for black hole physics and quantum information.

Contribution

It introduces a conjectured universal inequality for black hole interior cross-sections and provides proofs for specific symmetric cases, connecting interior geometry with complexity growth bounds.

Findings

Bound on cross-section size in Schwarzschild-AdS black holes

Complexity growth rate upper bound consistent with quantum information theory

Estimate of maximum interior volume for large evaporating black holes

Abstract

This paper studies cross-sections inside black holes and conjectures a universal inequality: in a static $(d+1)$-dimensional asymptotically planar/spherical Schwarzschild-AdS spacetime of given energy $E$ and AdS radius $\ell_{\text{AdS}}$, the ``size of cross-section'' inside black holes is bounded by $8\pi E\ell_{\text{AdS}}/(d-1)$. To support this conjecture, it gives the proofs for cases with spherical/planar symmetries and some special cases without planar/spherical symmetries. As one corollary, it shows that the complexity growth rate in complexity-volume conjecture satisfies the upper bound argued by quantum information theory. This makes a first step towards proving the conjecture that the vacuum black hole has fastest complexity growth in the systems of same energy. It also finds a similar bound for asymptotically flat black holes, which gives us an estimation on the largest interior volume of a large evaporating black hole.