[Paper Review] Asymptotics in the time-dependent Hawking and Unruh effects

Benito A. Juárez-Aubry|arXiv (Cornell University)|Aug 30, 2017

Advanced Differential Geometry Research4 citations

TL;DR

This thesis investigates the asymptotic behavior of the Hawking and Unruh effects in time-dependent spacetimes using a novel Unruh-DeWitt detector model coupled to the proper time derivative of a scalar field. It establishes that thermalization occurs in polynomial time for high-energy detectors in Rindler motion, but switching function details critically affect thermalization rates, with non-uniform limits between infinite time and large energy gaps.

ABSTRACT

In this thesis, we study the Hawking and Unruh effects in time-dependent situations, as registered by localised spacetimes observers in several asymptotic situations. (Full abstract inside document.)

Motivation & Objective

• To analyze the onset and asymptotic behavior of the Hawking and Unruh effects in time-dependent spacetimes, particularly focusing on detector response and thermalization.
• To resolve the non-uniformity between infinite-time and large-energy-gap limits in the thermalization of detectors under the KMS condition.
• To characterize the divergence strength of transition rates and renormalized energy density near spacetime singularities and Cauchy horizons.
• To develop a detector model insensitive to infrared ambiguities and with correct massive-to-massless field limits in 1+1 dimensions.
• To assess the limitations of 1+1D models for extrapolating to full 3+1D quantum field theory in curved spacetime.

Proposed method

• Formulated a new Unruh-DeWitt detector model coupled to the proper time derivative of a scalar field, avoiding infrared ambiguities and ensuring correct massive-to-massless limits.
• Applied the detector model in three 1+1D scenarios: exponentially receding mirror spacetime, infalling detector in Schwarzschild spacetime, and trajectory near the Cauchy horizon of a generalized Reissner-Nordström spacetime.
• Used smooth switching functions of compact support to model detector interactions and analyzed the transition rate in the adiabatic limit.
• Proved that the Kubo-Martin-Schwinger (KMS) condition and detailed balance are equivalent only in the infinite-time limit, not uniformly in the energy gap.
• Employed asymptotic analysis and bounds on Fourier transforms of switching functions to estimate decay rates of response functions.
• Analyzed the interplay between switching tail behavior and thermalization time-scales, showing that polynomial thermalization is impossible under fixed switching tails.

Experimental results

Research questions

• RQ1 How does the transition rate of a detector in a time-dependent spacetime evolve asymptotically, particularly near singularities or Cauchy horizons?
• RQ2 What is the thermalization time-scale for a detector in Rindler motion when the energy gap is large, and how does it depend on the switching function?
• RQ3 Why is the equivalence between the KMS condition and detailed balance not uniform in the large-energy-gap limit?
• RQ4 Can a detector coupled to a massless scalar field via a smooth switching function achieve thermalization in polynomial time in the large-energy regime?
• RQ5 How do the details of the switching function affect the thermalization time-scale, especially when the constant interaction time is polynomially large?

Key findings

• The transition rate of a detector falling into a Schwarzschild black hole diverges as $ r^{-3/2} $ near the singularity.
• Near the Cauchy horizon of a generalized Reissner-Nordström spacetime, the transition rate and renormalized local energy density diverge with the same $ r^{-3/2} $ behavior as in the Schwarzschild case.
• For a detector in Rindler motion with a smooth switching function, thermalization occurs in a time scale that is polynomially large in the energy gap, under technical conditions on the switching function.
• The infinite-time and large-energy-gap limits do not commute: the KMS condition and detailed balance are not uniformly equivalent.
• If the constant interaction time is polynomially large in the energy gap with fixed switching tails, polynomially fast thermalization cannot occur.
• The overall response function decays faster than any polynomial in energy due to exponential suppression in the integrand, particularly in high-energy regimes.