Phys. Rev. D 102, 106001 (2020)

Upper bound on cross sections inside black holes and complexity growth rate

Run-Qiu Yang

Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Yaguan Road 135, Jinnan District, 300350 Tianjin, People’s Republic of China

ABSTRACT

This paper studies cross sections inside black holes and conjectures a universal inequality: in a static ( )-dimensional asymptotically planar/spherical Schwarzschild-AdS spacetime of given energy   and AdS radius  , the “size of cross section” inside black holes is bounded by  . 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 toward 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.