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  • Open Access

Cyclic Kruskal universe: A quantum-corrected Schwarzschild black hole in unitary unimodular gravity

Steffen Gielen* and Sofie Ried†

  • School of Mathematical and Physical Sciences, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom

  • *Contact author: s.c.gielen@sheffield.ac.uk
  • †Contact author: sried1@sheffield.ac.uk
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Phys. Rev. D 113, 026030 – Published 30 January, 2026

DOI: https://doi.org/10.1103/kpf8-52c9

Abstract

We analyze the physical properties of an analytical, nonsingular quantum-corrected black hole solution recently derived in a minisuperspace model for unimodular gravity under the assumption of unitarity in unimodular time. We show that the metric corrections compared to the classical Schwarzschild solutions only depend on a single new parameter, corresponding to a minimal radius where a black hole-white hole transition occurs. While these corrections substantially alter the structure of the spacetime near this minimal radius, they fall off rapidly toward infinity, and we show in various examples how physical properties of the exterior spacetime are very close to those of the Schwarzschild solution. We derive the maximal analytic extension of the initial solution, which corresponds to an infinite sequence of Kruskal spacetimes connected via black-to-white hole transitions, and compare with some other proposals for nonsingular black hole metrics. The metric violates the achronal averaged null energy condition, which indicates that we are capturing physics beyond the semiclassical approximation. Finally, we include some thoughts on how to go beyond the simple eternal black hole-white hole model presented here.

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Phys. Rev. D 113, 026030– Published 30 January, 2026
  • Received 3 October 2025
  • Accepted 24 December 2025
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