Determine the exact near-extremal Tomimatsu–Koyama limit

Determine whether the Tomimatsu–Koyama near-extremal Reissner–Nordström horizon-polarization limit is ultimately attained as the surface gravity approaches zero, including whether the peak amplitude converges to the predicted value despite the observed finite values below that prediction.

Background

The paper investigates the near-extremal behavior of the renormalized scalar vacuum polarization and compares its numerical results with the Tomimatsu–Koyama prediction for massive fields near an extreme Reissner–Nordström horizon. The predicted resonance occurs near mr_+=0.38 and has asymptotic amplitude Y=0.0424.

Numerical sequences approaching extremality show that the resonance location tends toward the predicted value, but the measured amplitude remains at least approximately 25% below the leading-order prediction and appears to approach a lower finite value. The accessible calculations stop at κr_+ approximately 10-3 because the inter-horizon separation becomes too small for the near-horizon mode-sum window, leaving the limiting behavior unresolved.

References

Whether the TK limit is ultimately attained cannot be decided with the present method below $\kappa r_+\sim10{-3}$, where the inter-horizon separation falls beneath the near-horizon window accessible to the mode sum.

— Vacuum Polarization in Quantum-Corrected and Effective Black Hole Geometries: a High Performance Approach  (2608.17670 - Flachi, 18 Aug 2026) in Section 5, paragraph “Comparison with the Tomimatsu--Koyama horizon law”