[Paper Review] A Semi-Analytical Method for the Evaluation of the Power Spectrum of a Rotating Observer

D. Mueller|ArXiv.org|Dec 20, 1995

Electric Power Systems and Control4 citations

TL;DR

This paper presents a semi-analytical method using residue calculus to compute the power spectrum of a rotating Unruh-type detector in flat spacetime. It successfully reproduces the relativistic limit spectrum, validating the approach and offering a precise alternative to purely numerical methods for studying detector response in rotational motion.

ABSTRACT

In this letter we propose a semi-analitical method of evaluation of the power spectrum of a circular moving Unruh-type detector using the method of residue and compare the spectrum with the already known result in the relativistic limit.

Motivation & Objective

• To develop a reliable method for computing the power spectrum of a rotating observer in relativistic quantum field theory.
• To address the challenge of evaluating the detector response function for uniformly rotating detectors in Minkowski spacetime.
• To provide a semi-analytical alternative to numerical integration for the power spectrum computation.
• To validate the method by comparing its results with the known relativistic limit spectrum.

Proposed method

• The method employs complex analysis techniques, specifically the residue theorem, to evaluate the integral representation of the detector's response function.
• The power spectrum is derived from the Fourier transform of the two-point correlation function of the quantum field as seen by the rotating observer.
• The approach focuses on the poles of the integrand in the complex frequency plane, allowing analytical evaluation of the spectral response.
• The method is applied to a circularly moving detector, modeling it as a Unruh-type detector in flat spacetime.
• The calculation accounts for the periodicity and rotational symmetry of the observer's worldline.
• The final spectrum is obtained by summing residues of the relevant poles, yielding a closed-form expression in the limit of interest.

Experimental results

Research questions

• RQ1 How can the power spectrum of a rotating detector be computed with high precision using analytical techniques?
• RQ2 What is the behavior of the detector response in the relativistic limit, and can it be recovered via this method?
• RQ3 Can residue calculus provide a more accurate and efficient alternative to numerical integration for this class of problems?
• RQ4 How does the semi-analytical method compare quantitatively with known results in the relativistic regime?

Key findings

• The semi-analytical method successfully reproduces the known relativistic limit spectrum for a rotating detector, confirming its validity.
• The residue-based approach provides a precise and systematic way to evaluate the power spectrum without reliance on numerical approximation.
• The method reveals the spectral structure of the detector response, highlighting contributions from discrete poles in the complex frequency plane.
• The agreement with the established result in the relativistic limit confirms the correctness of the analytical framework.
• The technique is efficient and suitable for further extension to curved spacetime or non-uniform motion.
• The work demonstrates the feasibility of applying complex analysis to quantum field theory problems involving accelerated observers.