Horizons and Correlation Functions in 2D Schwarzschild-de Sitter Spacetime

TL;DR

This paper analyzes quantum field correlations in 2D Schwarzschild-de Sitter spacetime, revealing linear growth in correlations related to horizon surface gravities and the transition to the Bunch-Davies vacuum.

Contribution

It provides an exact computation of the scalar two-point function in this spacetime and explores the influence of horizons on quantum correlations.

Findings

Correlation functions grow linearly over time with a rate equal to the sum of surface gravities.

The state approaches the Bunch-Davies vacuum in the far-field limit.

A peak in the velocity two-point function occurs between the horizons and outside the cosmological horizon.

Abstract

Two-dimensional Schwarzschild-de Sitter is a convenient spacetime in which to study the effects of horizons on quantum fields since the spacetime contains two horizons, and the wave equation for a massless minimally coupled scalar field can be solved exactly. The two-point correlation function of a massless scalar is computed in the Unruh state. It is found that the field correlations grow linearly in terms of a particular time coordinate that is good in the future development of the past horizons, and that the rate of growth is equal to the sum of the black hole plus cosmological surface gravities. This time dependence results from additive contributions of each horizon component of the past Cauchy surface that is used to define the state. The state becomes the Bunch-Davies vacuum in the cosmological far field limit. The two point function for the field velocities is also analyzed and a peak is found when one point is between the black hole and cosmological horizons and one point is outside the future cosmological horizon.