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

Technical Summary: Computation of $\langle \Phi^2 \rangle$ and Quantum Fluxes at the Polar Interior of a Spinning Black Hole

Problem Statement
The paper addresses the challenge of computing renormalized physical quantities for a quantum field theory in the interior of a rotating (Kerr) black hole. Specifically, the authors focus on a minimally-coupled massless scalar field in the Unruh state (modeling an evaporating astrophysical black hole) along the axis of rotation ($\theta = 0$). The primary goal is to calculate the renormalized field square (vacuum polarization), $\langle \Phi^2 \rangle^U_{\text{ren}}$, and the renormalized energy-momentum flux components, $\langle T_{uu} \rangle^U_{\text{ren}}$ and $\langle T_{vv} \rangle^U_{\text{ren}}$, spanning the region from just outside the event horizon (EH) to just outside the inner horizon (IH).

A significant technical obstacle arises in the black hole interior that is absent in the exterior: the mode-sum over the multipolar number $l$ diverges. In the exterior, the effective potential acts as a barrier, causing mode contributions to decay exponentially with $l$. In the interior, the potential acts as a well, leading to a non-convergent sum. This necessitates a novel regularization approach to handle this "intermediate divergence" (ID) before standard renormalization can proceed.

Methodology
The authors extend and tailor the Pragmatic Mode-Sum Regularization (PMR) method, specifically the $t$-splitting variant, for the Kerr interior. The methodology involves several critical steps:

1. Mode Decomposition: The unrenormalized bi-tensor is expressed as a sum over modes ($\omega, l, m$). Due to the symmetry at the pole, only $m=0$ modes contribute.
2. Intermediate Divergence (ID) Subtraction:
   • The authors identify that for large $l$, the bare mode contribution $E_{\omega l}$ behaves as $c_0 + c_1\omega^2 + c_2 l(l+1)$. This divergent piece, termed the ID, prevents the convergence of the $l$-sum.
   • To address this, they introduce an additional "small" split in the polar angle $\theta$ (taken to zero before the $t$-split limit). This allows for the analytic identification and subtraction of the ID term ($E^{\text{div}}_{\omega l}$ or $T^{\text{div}}_{yy}(\omega l)$) from the bare mode sum.
   • The coefficients for the ID are derived analytically for the leading orders ($c_0$ for the field square; $c_2$ for the fluxes) and verified numerically for subleading terms.
3. $t$-Splitting Regularization:
   • After ID subtraction, the remaining "basic integrand" is still divergent in the frequency $\omega$ domain due to the point-splitting counterterms.
   • The authors subtract the frequency-dependent PMR counterterms ($E_{\text{sing}}$ or $T^{\text{sing}}_{yy}$) and a finite counterterm ($e$ or $e_{yy}$), derived from the DeWitt-Schwinger expansion.
   • The final renormalized quantity is obtained by integrating the regularized difference over $\omega$.
4. Numerical Implementation:
   • Due to the slow convergence of the $l$-sum (scaling as $1/l$) even after ID subtraction, the authors compute modes up to $l_{\text{max}} \approx 300$ with extremely high precision ($\sim 250$ decimal digits).
   • They fit the tail of the sequence to inverse powers of $l$ to extract the convergent sum.
   • An alternative "analytic extension variant" of the method is developed in Appendix D to independently verify the results at specific points, bypassing the ID subtraction by analytically extending exterior modes to the interior.

Key Contributions

• Extension of PMR to the Kerr Interior: The paper successfully adapts the $t$-splitting PMR method to the black hole interior, a regime where standard techniques fail due to the non-convergent nature of the mode sums.
• Resolution of the Intermediate Divergence: The authors introduce and justify a specific procedure to subtract the intermediate divergence (ID) arising from the $l$-sum in the interior. They provide both analytic derivations and numerical evidence for the form of this divergence.
• First Numerical Results for the Interior: The paper presents the first computational results for $\langle \Phi^2 \rangle^U_{\text{ren}}$ and the energy fluxes $\langle T_{uu} \rangle^U_{\text{ren}}$ and $\langle T_{vv} \rangle^U_{\text{ren}}$ throughout the Kerr interior (from near the EH to near the IH) for two spin parameters ($a/M = 0.8$ and $0.9$).
• Validation of State-Subtraction: The results obtained via $t$-splitting near the inner horizon are extrapolated to the horizon and compared with independent calculations performed directly at the IH using the state-subtraction method (from Ref. [3]). The agreement validates the state-subtraction approach used in previous work.

Results

• Fluxes ($\langle T_{uu} \rangle^U_{\text{ren}}$ and $\langle T_{vv} \rangle^U_{\text{ren}}$):
  • Event Horizon (EH): The results confirm the expected regularity of the Unruh state at the EH. $\langle T_{uu} \rangle^U_{\text{ren}}$ vanishes as $(r-r_+)^2$, consistent with the regularity of the Kruskal coordinate $U$. $\langle T_{vv} \rangle^U_{\text{ren}}$ approaches finite negative values.
  • Inner Horizon (IH): The fluxes exhibit non-trivial behavior, including oscillations and sign changes as they approach the IH. The extrapolated values at the IH agree with state-subtraction results to four significant digits. For $a/M=0.8$, the fluxes are positive at the IH; for $a/M=0.9$, they are negative.
  • Conserved Quantity: The difference $\langle T_{uu} \rangle^U_{\text{ren}} - \langle T_{vv} \rangle^U_{\text{ren}}$ is shown to be consistent with the conserved quantity $F_0$ (Hawking outflux per unit solid angle).
• Field Square ($\langle \Phi^2 \rangle^U_{\text{ren}}$):
  • EH: The field square approaches finite values at the EH, showing regularity. The sign depends on the spin: positive for $a/M=0.8$ and negative for $a/M=0.9$.
  • IH: The behavior near the IH is complex, showing a rapid decrease and non-monotonic oscillations. While the authors could not numerically resolve the asymptotic limit directly via $t$-splitting due to computational limitations near the horizon, they note that state-subtraction analysis (from Ref. [3]) suggests a finite asymptotic value approached with an $r_*^{-3}$ tail.
• Verification: The "analytic extension variant" developed in the Appendix yields results consistent with the standard $t$-splitting method at specific radii, providing an independent check of the numerical implementation.

Significance
The paper provides a crucial quantitative picture of semiclassical effects inside a rotating black hole. By successfully computing the renormalized stress-energy tensor components and field square along the axis of rotation, the work offers essential data for understanding the backreaction of quantum fields on the spacetime geometry, particularly near the Cauchy horizon (inner horizon).

The authors emphasize that their results serve as a rigorous test for the state-subtraction method previously used to analyze the inner horizon, confirming its validity. Furthermore, the demonstration of regularity at the event horizon provides numerical support for the Unruh state's physical viability in the Kerr geometry, a property that has not yet been rigorously proven analytically. The development of the ID subtraction technique represents a necessary methodological advancement for performing quantum field theory calculations in the non-static, non-spherically symmetric interiors of rotating black holes.