Cyclic Kruskal Universe: a quantum-corrected Schwarzschild black hole in unitary unimodular gravity

TL;DR

The work shows that enforcing unitarity in unimodular time within a minisuperspace quantisation of a spherically symmetric spacetime yields a quantum-corrected Schwarzschild metric governed by a single new scale $r_{ m min}$, which fixes a black-hole to white-hole transition. By constructing a Kruskal-like extension, the authors demonstrate a Cyclic Kruskal Universe consisting of an infinite sequence of connected black-hole and white-hole regions, while exterior regions remain effectively Schwarzschild. The exterior spacetime closely matches classical predictions, but curvature invariants remain finite at the minimal surface, confirming singularity resolution as a boundary-condition effect of unitarity. The spacetime violates the achronal averaged null energy condition, highlighting the presence of genuine quantum-geometry effects beyond the semiclassical regime and suggesting directions for incorporating dynamical collapse and evaporation scenarios.

Abstract

We analyse 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 towards 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 non-singular 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.

Figures (3)

• Figure 1: The tortoise coordinate $r^*$ as a function of $r$. The blue solid curve gives the numerical result for the parameters $\beta = 2$, $k_c=1$, $\sigma_{\Lambda} = 5$. The yellow dashed line shows the analytical approximation for $r\approx r_{\rm min}$ and the green dashed line shows the classical solution for $r^*$.
• Figure 2: $g_{TT}$ for $r_{\rm min}<r<r_H$. The solid lines show the numerical result for the parameters $2\leq\beta\leq3$, $k_c=1$, $\sigma_{\Lambda} = 5$. The dashed lines show the classical Schwarzschild solution.
• Figure 3: Conformal diagram to illustrate the causal structure of our Cyclic Kruskal Universe. The purple region is "cut out" and the interior region of the black hole is glued to the interior region of the white hole along the hypersurface with coordinate $r_{\rm min}$. The exterior regions are disconnected. This chain of black and white holes can continue infinitely far above and below.