Computation of $\langle \Phi^2\rangle$ and quantum fluxes at the polar interior of a spinning black hole

TL;DR

This paper develops and applies a tailored point-splitting regularization method to compute renormalized quantum field quantities inside a rotating black hole, addressing unique interior challenges and validating results with an alternative approach.

Contribution

It extends the pragmatic mode-sum regularization method for the black hole interior, specifically handling divergence issues and providing detailed calculations of quantum fluxes and field squares.

Findings

Calculated renormalized fluxes near the inner horizon

Demonstrated convergence of fluxes to previous inner horizon results

Validated the method with an alternative divergence subtraction approach

Abstract

Renormalization of physical quantities for quantum field theories in curved spacetimes can be achieved via the consistent subtraction of counterterms within a regularization scheme such as a point-splitting method. Pragmatic mode-sum regularization (PMR) is a point-splitting method which is particularly suitable for rotating black hole spacetimes. We extend and tailor the t-splitting variant of PMR specifically for the interior of a Kerr black hole on the axis of rotation, focusing on a minimally-coupled massless scalar field in the physically-motivated Unruh state. The method addresses unique challenges in the black hole interior that do not occur outside. In particular, while the infinite sum over multipolar number l converges in the black hole exterior, it diverges in the interior, necessitating the subtraction of a so-called intermediate divergence which includes introducing an additional "small" split in the direction of the polar angle. This procedure is outlined and justified, along with the standard PMR method's counterterms subtraction. We apply this method to calculate the renormalized energy-momentum fluxes $\langle T_{uu}\rangle^U_\text{ren}$, $\langle T_{vv}\rangle^U_\text{ren}$ (where u and v are the standard Eddington coordinates) and the renormalized field square $\langle \Phi^2\rangle^U_\text{ren}$ throughout the black hole interior, spanning from (just off) the event horizon to (just off) the inner horizon. Special emphasis is placed on the inner horizon vicinity, where our t-splitting results for the fluxes asymptote to those obtained directly at the inner horizon using a different method in a previous work. In an Appendix, we develop an alternative t-splitting PMR variant which does not include the intermediate divergence subtraction. We utilize it for independent computations that are used to verify the standard t-splitting variant presented in the main text.