Extend the calculation to the renormalized stress tensor

Develop the extended point-splitting implementation for computing the renormalized stress tensor $\langle T_{\mu\nu}\rangle$ in quantum-corrected and regular black-hole geometries, in order to characterize the quantity governing quantum back-reaction.

Background

The paper computes the renormalized scalar vacuum polarization for five quantum-corrected, effective, and regular black-hole geometries using a metric-agnostic high-performance implementation of extended point splitting. The renormalized stress tensor is the next higher-rank local quantum expectation value needed to study the back-reaction of quantum fields on the geometry.

The authors explicitly identify extending the implementation to the stress tensor as an open direction. Such an extension would leverage the existing general dependence on the metric functions f and g and would enable analysis of quantum back-reaction in the same family of black-hole backgrounds.

References

Attaining said solutions is highly non-trivial, as it means solving the open problem of semiclassical backreaction: how quantum matter influences the classical gravitational field and vice versa.

— Looking inside a quantum black hole  (2609.20788 - Coviello et al., 17 Sep 2026) in Introduction

We leave explicit verification of Wald's axioms for future work, but we give some arguments below.

— Entropy, area, and the choice of regulator during gravitational collapse  (2609.20663 - Guenther et al., 17 Sep 2026) in Appendix, Section “The log-polynomial renormalization scheme”

Extending this approach to the computation of the renormalized stress tensor $\langle T_{\mu\nu}\rangle$, which controls back-reaction is the other open direction; the generality of the present implementation in the metric functions $f$ and $g$ makes its extension to further quantum-corrected and regular black holes -- and the assembly of a uniform $$ data set across geometries -- feasible.

— Vacuum Polarization in Quantum-Corrected and Effective Black Hole Geometries: a High Performance Approach  (2608.17670 - Flachi, 18 Aug 2026) in Section 6, Conclusions

Therefore, it is possible that a full analysis of the backreaction, encompassing both sets of equations, may rectify this issue.

— Quantum effects of charged massive scalar fields on charged black hole space-times  (2608.19359 - Breen et al., 19 Aug 2026) in Section 7, Conclusions