Renormalization of physical quantities for quantum field theories in curved spacetimes can be achieved via the subtraction of counterterms in a consistent manner 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 within 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 $\theta$. This procedure is outlined and justified, along with the standard PMR method’s subtraction of counterterms mode-by-mode. We apply this method to calculate the renormalized energy-momentum fluxes ${⟨T_{uu}⟩}_{\mathrm{ren}}^U$, ${⟨T_{vv}⟩}_{\mathrm{ren}}^U$ (where $u$ and $v$ are the standard Eddington coordinates) and the renormalized field square ${⟨{\mathrm{\Phi }}^2⟩}_{\mathrm{ren}}^U$ throughout the Kerr black hole interior, spanning from (just off) the event horizon to (just off) the inner horizon. Special emphasis is placed on the vicinity of the inner horizon, where the $t$-splitting results for ${⟨T_{uu}⟩}_{\mathrm{ren}}^U$ and ${⟨T_{vv}⟩}_{\mathrm{ren}}^U$ asymptote to those obtained directly at the inner horizon using a different method in a previous work. In an Appendix, we develop an alternative variant of the $t$-splitting PMR method, dubbed the analytic extension variant, which does not include the intermediate divergence subtraction. We utilize it to perform independent computations that are used to verify the standard $t$-splitting variant presented in the main text.

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• General relativity
• Quantum fields in curved spacetime
• Quantum aspects of black holes