# Holographic Cavalieri Principle as a Universal relation between Holographic Complexity and Holographic Entanglement Entropy

## TL;DR

This paper proposes a universal relation between holographic complexity and entanglement entropy, supported by various AdS solutions, suggesting that different complexity proposals describe the same physics and that black holes are the fastest computers.

## Contribution

It introduces a universal relation between holographic complexity and entanglement entropy based on the AdS Cavalieri principle, applicable to various solutions including Janus and circular disk.

## Key findings

- The relation holds for all metrics asymptotic to AdS3.
- It is valid for Janus solutions in type IIB string theory.
- It supports the equivalence of different holographic complexity proposals.

## Abstract

In this paper, we will propose a universal relation between the holographic complexity (dual to a volume in AdS) and the holographic entanglement entropy (dual to an area in AdS). We will explicitly demonstrate that our conjuncture hold for all a metric asymptotic to AdS$_3$, and then argue that such a relation should hold in general due to the AdS version of the Cavalieri principle. We will demonstrate that it holds for Janus solution, which have been recently been obtained in type IIB string theory. We will also show that this conjecture holds for a circular disk. This conjecture will be used to show that the proposal that the complexity equals action, and the proposal that the complexity equal volume can represent the same physics. Thus, using this conjecture, we will show that the black holes are fastest computers, using the proposal that complexity equals volume.