Nonsingular Schwarzschild-de Sitter black holes in finite conformal quantum gravity

TL;DR

The paper addresses the problem of singularities in GR by embedding Schwarzschild-de Sitter spacetimes into a finite, conformal quantum gravity framework with weakly non-local form factors. It shows SdS is an exact solution without an explicit cosmological constant, with $\Lambda$ emerging from higher-derivative couplings, and uses conformal rescalings to construct a regular, geodesically complete spacetime for a real parameter $N>\tfrac{3}{4}$. The authors analyze geodesics for both conformally coupled and uncoupled probes and map the global causal structure with a Penrose diagram, finding an SdS-like diagram with an extended but transitively inaccessible region at $r=0$. These results connect UV-complete quantum gravity to observable cosmological and astrophysical data, suggesting ways to constrain the theory’s couplings via measurements of expansion and black-hole phenomenology.

Abstract

In this work, we prove that the classical Schwarzschild-de Sitter spacetime is an exact solution of a class of weakly non-local, UV finite conformal quantum gravity theories, without the necessity of including a cosmological constant term in the action, thus associating the effective cosmological constant $Λ$ appearing in the metric with the coupling constants of the quantum gravity theory. Furthermore, exploiting the inherent conformal symmetry of the theory, we take advantage of the natural enlargement of the exact solutions to motivate the construction of a regular spacetime via conformal rescaling of the Schwarzschild-de Sitter spacetime. Moreover, we ensure the spacetime completeness by investigating the regularity of the curvature invariants and the geodesic completeness of conformally/non-conformally coupled massive and massless particles. We also study the global causal structure by explicitly constructing the Penrose diagram of the regular spacetime. Furthermore, as a result of the spacetime completeness analysis, we generalize the range of conformal factors that generate regular spacetimes, by considering the $N$ parameter of the conformal factor as a real parameter with a lower bound, and not only a positive integer, as constrained in previous studies on regular Schwarzschild/Kerr black holes. Thus, the present analysis broadens the range of solutions of the finite conformal quantum theory and opens the window to more precise observational tests of the theory using astrophysical data, by considering the accelerated expansion of the universe.

Figures (4)

• Figure 1: Ricci scalar $R$ as a function of a) $r$ and $l$ with $\Lambda=0.01$, $M=1$ and $N=1$, b) $r$ and $N$ with $\Lambda=0.01$, $M=1$ and $l=1$, and c) $r$ and $\Lambda$ with with $l=M=N=1$. In the bottom plots we have the Kretschmann scalar $\mathcal{K}$ as a function of d) $r$ and $l$ with $\Lambda=0.01$, $M=1$ and $N=1$, e) $r$ and $N$ with $\Lambda=0.01$, $M=1$ and $l=1$, and f) $r$ and $\Lambda$ with with $l=M=N=1$. We observe that both curvature invariants are regular at $r=0$ in all the plots. Nevertheless, the Kretschmann and Ricci scalars present an odd behavior for $N$ close to $0$, since its value goes to infinity as we approach small values of $N$.
• Figure 2: Proper time (or affine parameter for null particles) as a function of $r$ for massive (left panel), massless (middle panel) and conformally coupled massive particle (right panel), with an effective mass $\tilde{m} = \rho\kappa_{4}$. We see that the geodesics of the spacetime generated by the conformal transformation (blue curve) requires an infinite amount of proper time to reach the former physical singularity at $r=0$, which is reached in a finite proper time in the classical SdS spacetime (red curve).
• Figure 3: Affine parameter behavior for small $N$ with $l=1$, $\tilde{E}=1$. We have that the particle now approaches $r=0$, but the functions corresponding to $N<1/4$ have a domain $\mathbb{R}\backslash\{0\}$, therefore, the geodesics for $N<1/4$ are not defined for all the values of their affine parameter $\lambda$.
• Figure 4: Penrose-Carter diagram for the singularity-free Schwarzschild spacetime. The conformal factor $S(r)$ does not change the general form of the diagram. Nevertheless, the spacetime is regular at $r=0$. The dots indicates that the displayed pattern repeats itself infinitely in those directions. The red regions are prohibited for all particles, in classical SdS due to the singularity at $r=0$, and in nonsingular SdS due to the fact that the required proper time to reach $r=0$ is infinite.