# Phases of Holographic Hawking Radiation on spatially compact spacetimes

## TL;DR

This paper investigates the different equilibrium phases of Hawking radiation in holographic CFTs on compact spaces with two black holes, revealing complex phase transitions and the need for new phases.

## Contribution

It provides a detailed analysis of black funnel and droplet phases, including numerical construction and phase diagrams, highlighting novel phenomena in holographic Hawking radiation.

## Key findings

- Fat non-uniform funnel dominates at small R
- Uniform funnel dominates at large R
- Discontinuity in free energy at R_turn

## Abstract

We study phases of equilibrium Hawking radiation in $d$-dimensional holographic CFTs on spatially compact spacetimes with two black holes. In the particular phases chosen the dual $(d+1)$-dimensional bulk solutions describe a variety of black funnels and droplets. In the former the CFT readily conducts heat between the two black holes, but it in the latter such conduction is highly suppressed. While the generic case can be understood in certain extreme limits of parameters on general grounds, we focus on CFTs on specific geometries conformally equivalent to a pair of $d \ge 4$ AdS${}_d$-Schwarzschild black holes of radius $R$. Such cases allow perturbative analyses of non-uniform funnels associated with Gregory-Laflamme zero-modes. For $d=4$ we construct a phase diagram for pure funnels and droplets by constructing the desired bulk solutions numerically. The fat non-uniform funnel is a particular interesting phase that dominates at small $R$ (due to having lowest free energy) despite being sub-dominant in the perturbative regime. The uniform funnel dominates at large $R$, and droplets and thin funnels dominate at certain intermediate values. The thin funnel phase provides a mystery as it dominates over our other phases all that way to a critical $R_{\mathrm{turn}}$ beyond which it fails to exist. The free energy of the system thus appears to be discontinuous at $R_{\mathrm{turn}}$, but such discontinuities are forbidden by the 2nd law. A new more-dominant phase is thus required near $R_{\mathrm{turn}}$ but the nature of this phase remains unclear.