Multiplicative Noise in Euclidean Schwarzschild Manifold

TL;DR

This paper investigates how multiplicative noise, modeling Hawking temperature fluctuations, affects a scalar field in Euclidean Schwarzschild space, leading to phase transitions and modified correlation functions.

Contribution

It introduces a novel analysis of multiplicative noise effects on scalar fields in Schwarzschild spacetime, revealing phase transitions and new correlation structures.

Findings

Fluctuations can induce first-order phase transitions.

Effective coupling constants vary with noise strength.

Correlation functions are modified by noise, resembling Rindler space results.

Abstract

We discuss a $\lambda\varphi^{4}+\rho\varphi^{6}$ scalar field model defined in the Euclidean section of the Schwarzschild solution of the Einstein equations in the presence of multiplicative noise. The multiplicative random noise is a model for fluctuations of the Hawking temperature. We adopt the standard procedure of averaging the noise dependent generating functional of connected correlation functions of the model. The dominant contribution to this quantity is represented by a series of the moments of the generating functional of correlation functions of the system. Positive and negative effective coupling constants appear in these integer moments. Fluctuations in the Hawking temperature are able to generate first-order phase transitions. Using the Gaussian approximation, we compute $\langle\varphi^{2}\rangle$ for arbitrary values of the strength of the noise. Due to the presence of the multiplicative noise, we show that $\langle\varphi^{2}\rangle$ near the horizon must be written as a series of the the renormalized two-point correlation functions associated to a free scalar field in Euclidean Rindler manifold.