Explicit Derivation of the Fluctuation-Dissipation Relation of the Vacuum Noise in the N-dimensional de Sitter Spacetime

TL;DR

This paper explicitly derives the fluctuation-dissipation relation for vacuum noise in N-dimensional de Sitter spacetime, confirming the validity of the approach and its connection to the KMS condition, with potential applications to black hole spacetimes.

Contribution

The paper provides a detailed derivation of the fluctuation-dissipation relation in N-dimensional de Sitter spacetime, validating previous prescriptions and clarifying their theoretical basis.

Findings

The dissipative coefficient satisfies the fluctuation-dissipation relation in de Sitter space.

The derivation confirms the use of retarded Green functions in this context.

Discussion on potential applications to black hole spacetimes.

Abstract

Motivated by a recent work by Terashima (Phys. Rev. D60 084001), we revisit the fluctuation-dissipation (FD) relation between the dissipative coefficient of a detector and the vacuum noise of fields in curved spacetime. In an explicit manner we show that the dissipative coefficient obtained from classical equations of motion of the detector and the scalar (or Dirac) field satisfies the FD relation associated with the vacuum noise of the field, which demonstrates that the Terashima's prescription works properly in the $N$-dimensional de Sitter spacetime. This practice is useful not only to reconfirm the validity of the use of the retarded Green function to evaluate the dissipative coefficient from the classical equations of motion but also to understand why the derivation works properly, which is discussed in connection with previous investigations on the basis on the Kubo-Martin-Schwinger (KMS) condition. Possible application to black hole spacetime is also briefly discussed.