Vacuum Fluctuations and Boundary Conditions in a Global Monopole

TL;DR

This paper investigates how boundary conditions affect vacuum fluctuations of a scalar field in a global monopole spacetime, revealing that only Dirichlet conditions yield expected Minkowski limits, while others produce nonzero effects due to the singularity.

Contribution

It demonstrates the impact of boundary conditions on vacuum fluctuations in monopole spacetimes and identifies Dirichlet conditions as physically consistent with Minkowski limits.

Findings

Dirichlet boundary condition yields zero vacuum fluctuations in Minkowski limit.

Other boundary conditions produce nonzero contributions due to the monopole's singularity.

Vacuum fluctuations are proportional to the monopole parameter ta^2.

Abstract

We study the vacuum fluctuations of a massless scalar field $\hat{\Psi}$ on the background of a global monopole. Due to the nontrivial topology of the global monopole spacetime, characterized by a solid deficit angle parametrized by $\eta^2$, we expect that $\left<\hat{\Psi}^2\right>_{\text{ren}}$ and $\left<\hat{T}_{\mu\nu}\right>_{\text{ren}}$ are nonzero and proportional to $\eta^2$, so that they annul in the Minkowski limit $\eta\to0$. However, due to the naked singularity at the monopole core, the evolution of the scalar field is not unique. In fact, they are in one to one correspondence with the boundary conditions which turn into self-adjoint the spatial part of the wave operator. We show that only Dirichlet boundary condition corresponds to our expectations and gives zero contribution to the vacuum fluctuations in Minkowski limit. All other boundary conditions give nonzero contributions in this limit due to the nontrivial interaction between the field and the singularity.