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.
Highlights
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
We show how renormalization-group Effective field theories (EFTs) methods can allow some perturbative contributions to be resummed beyond leading order, verifying the result using the exact expression
At long last the detection of gravitational waves [1] has made near-horizon black-hole physics an experimental science, and this is very likely to deepen our understanding of General Relativity (GR) and/or end its hundred-year reign as the paradigm of choice when describing gravity
Summary
At long last the detection of gravitational waves [1] has made near-horizon black-hole physics an experimental science, and this is very likely to deepen our understanding of General Relativity (GR) and/or end its hundred-year reign as the paradigm of choice when describing gravity. EFT methods underline that the microscopic length is a regulator scale and so drops out of all physical predictions (as regulators always do), and this makes the EFT framework useful for understanding the physical significance of the length-scales involved in these types of models This type of EFT can be studied within the hotspot framework by allowing the radius ξ of the interaction sphere Sξ remain larger than the cutoff scale: do not pursue this variant further in this paper. Using the Heisenberg picture allows us to work in position space where we can follow the passage of the initial transient wave (generated by the turn-on of the couplings) as well as watch how the φ field settles down at later times in the on-going presence of the hotspot coupling Computing both exact and perturbative results allows us to identify precisely which small dimensionless parameter controls the perturbative expansion. Many of the calculational details are given in a collection of appendices
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