Influence Through Mixing: Hotspots as Benchmarks for Basic Black-Hole Behaviour

TL;DR

This paper introduces a simple, exactly solvable model of a scalar field interacting with a hotspot of many degrees of freedom, providing benchmarks for understanding black-hole effective theories.

Contribution

It presents a Gaussian model with explicit solutions for mode and correlation functions, enabling comparison with perturbative and renormalization-group methods.

Findings

Exact solutions for mode functions and correlations

Validation of perturbative expansion domains

Demonstration of resummation via EFT methods

Abstract

Effective theories are being developed for fields outside black holes, often with an unusual open-system feel due to the influence of large number of degrees of freedom that lie out of reach beyond the horizon. What is often difficult when interpreting such theories is the absence of comparisons to simpler systems that share these features. We propose here such a simple model, involving a single external scalar field that mixes in a limited region of space with a `hotspot' containing a large number of hot internal degrees of freedom. Since the model is at heart gaussian it can be solved explicitly, and we do so for the mode functions and correlation functions for the external field once the hotspot fields are traced out. We compare with calculations that work perturbatively in the mixing parameter, and by doing so can precisely identify its domain of validity. We also show how renormalization-group EFT methods can allow some perturbative contributions to be resummed beyond leading order, verifying the result using the exact expression.