Vacuum Fluctuations and the Renormalized Stress-Energy Tensor on a Cone with Arbitrary Boundary Conditions

TL;DR

This work analyzes vacuum fluctuations and the renormalized stress-energy tensor of a massive scalar field on a 1+2 dimensional conical spacetime with a non-Dirichlet apex boundary condition encoded by a self-adjoint-extension scale $q$. A discrete bound mode exists for $M>q$, enabling a covariant model of an extended Unruh–DeWitt detector, whose presence modifies the local vacuum structure. The authors decompose the two-point function and the stress-energy tensor into Dirichlet, bound, and boundary-condition sectors, performing Hadamard renormalization for the Dirichlet part and using numerical methods to handle the bc sector; they show the boundary-condition contribution can dominate near the apex and verify covariant conservation. The results provide a quantitative framework to study detector backreaction on spacetime and suggest pathways to couple the system to linearized gravity and to explore backreaction via Einstein equations in future work.

Abstract

We analyze the vacuum fluctuations and the stress-energy tensor of a scalar field of mass $M$ in a conical spacetime, where the topological singularity at the apex requires boundary conditions for the field equation. The necessity of boundary conditions was established by Kay and Studer in the early 1990s, but the consequences of their arbitrariness, represented here by a parameter $q$, for renormalized observables have not been examined. While for $M=0$ stability is achieved only under Dirichlet boundary conditions, for $M>q$ the field is stable and a localized mode emerges. This mode admits a natural interpretation as a covariant model of an extended particle detector, which allows us to investigate how such detectors modify the local vacuum structure. In this framework, the renormalized stress-energy tensor offers a natural way to quantify the influence of the detector on the surrounding spacetime.

Figures (4)

• Figure 1: $\langle \Psi^2\rangle_{\textrm{Dirichlet}}$ (dotted line) as a function of $r$ for $M=2$ and $\langle \Psi^2\rangle$ (solid line) for $\alpha=0.9$, $q=1$ and $M=2$. As $r\to\infty$, we recover the Minkowski value $-M/(4\pi)$ (dashed line); as $r\to 0$, the boundary-condition contribution dominates the Dirichlet contribution.
• Figure 2: Renormalized stress–energy tensor components under Dirichlet boundary conditions for $\alpha=0.9$ and $M=2$. From top to bottom: $\langle T_{tt}\rangle_{\text{Dirichlet}}$, $\langle T_{rr}\rangle_{\text{Dirichlet}}$, and $\langle T_{\theta\theta}\rangle_{\text{Dirichlet}}$ as functions of the radial coordinate $r$.
• Figure 3: Integrand used in the calculation of $\langle T_{00}\rangle_{\textrm{bc}}$. The explicit expression for $f(\lambda)$ is provided in the Appendix. Note that $f(\lambda)$ tends to zero as $\lambda \to \infty$, albeit with very slow convergence.
• Figure 4: From top to bottom: $\langle T_{tt}\rangle_{\text{Dirichlet}}$ (dotted) and $\langle T_{tt}\rangle$ (solid), $\langle T_{rr}\rangle_{\text{Dirichlet}}$ (dotted) and $\langle T_{rr}\rangle$ (solid), $\langle T_{\theta\theta}\rangle_{\text{Dirichlet}}$ (dotted) and $\langle T_{\theta\theta}\rangle$ (solid), as functions of the radial coordinate $r$. The boundary-condition contribution clearly dominates over the Dirichlet one.