Mode-sum regularization of $\left\langle \phi^{2} \right\rangle$ in the angular-splitting method

TL;DR

This paper introduces an angular-splitting mode-sum regularization method for computing the renormalized scalar field expectation value in spherically symmetric spacetimes, demonstrating its effectiveness in Schwarzschild background and aiming to handle dynamical black hole scenarios.

Contribution

The paper develops and validates an angular-splitting approach for mode-sum regularization, extending previous t-splitting methods to more general dynamical spherically symmetric spacetimes.

Findings

Excellent agreement with t-splitting results and other methods.

Effective in Schwarzschild background for various quantum states.

Suitable for dynamical black hole spacetimes.

Abstract

The computation of the renormalized stress-energy tensor or $\left\langle\phi^{2}\right\rangle_{ren}$ in curved spacetime is a challenging task, at both the conceptual and technical levels. Recently we developed a new approach to compute such renormalized quantities in asymptotically-flat curved spacetimes, based on the point-splitting procedure. Our approach requires the spacetime to admit some symmetry. We already implemented this approach to compute $\left\langle \phi^{2}\right\rangle _{ren}$ in a stationary spacetime using t-splitting, namely splitting in the time-translation direction. Here we present the angular-splitting version of this approach, aimed for computing renormalized quantities in a general (possibly dynamical) spherically-symmetric spacetime. To illustrate how the angular-splitting method works, we use it here to compute $\left\langle \phi^{2}\right\rangle _{ren}$ for a quantum massless scalar field in Schwarzschild background, in various quantum states (Boulware, Unruh, and Hartle-Hawking states). We find excellent agreement with the results obtained from the t-splitting variant, and also with other methods. Our main goal in pursuing this new mode-sum approach was to enable the computation of the renormalized stress-energy tensor in a dynamical spherically symmetric background, e.g. an evaporating black hole. The angular-splitting variant presented here is most suitable to this purpose.