Regular black hole formation in four-dimensional non-polynomial gravities

TL;DR

The paper presents four-dimensional non-polynomial gravity theories (NPQTs) whose spherical sector reduces to two-dimensional Horndeski dynamics, yielding second-order equations and a Birkhoff theorem. By summing an infinite tower of higher-curvature corrections, the authors show generic resolution of Schwarzschild and cosmological singularities, obtaining explicit regular black hole and bouncing cosmological solutions; they further demonstrate RBH formation via dynamical collapse in both Oppenheimer–Snyder and thin-shell setups, using modified junction conditions. This provides a concrete, covariant framework to study RBH dynamics in 4D and suggests a viable pathway for singularity resolution within gravitational effective field theories. The results connect higher-curvature resummations with dynamical processes, offering insights with potential astrophysical and theoretical implications for the nature of black holes and early-universe cosmology.

Abstract

We construct four-dimensional gravity theories that resolve the Schwarzschild singularity and enable dynamical studies of nonsingular gravitational collapse. The construction employs a class of nonpolynomial curvature invariants that produce actions with (i) second-order equations of motion in spherical symmetry and (ii) a Birkhoff theorem, ensuring uniqueness of the spherically symmetric solution. Upon spherical reduction to two dimensions, these theories map to a particular subclass of Horndeski scalar-tensor models, which we use to explicitly verify the formation of regular black holes as the byproduct of the collapse of pressureless stars and thin-shells. We also show that linear perturbations on top of maximally symmetric backgrounds are governed by second-order equations.

Figures (3)

• Figure 1: We plot $f(r)$ for the unique spherically symmetric vacuum solution of a NPQT theory with infinitely many higher-derivative terms with couplings given by Eq. (\ref{['coup']}) and $\alpha=1/8$. The lower blue curve corresponds to a regular black hole with two horizons ($M=2M_{\rm cr}$). The red curve is the usual Schwarzschild black hole metric function with the same mass. The intermediate dotted blue curve represents an extremal black hole with a single horizon ($M=M_{\rm cr}$). The upper blue curve corresponds to a horizonless regular spacetime ($M=\frac{5}{8}M_{\rm cr}$).
• Figure 2: We plot the coordinate radius (normalized by its initial value $R_0$) of a pressureless dust star as a function of its proper time as it undergoes gravitational collapse for Einstein gravity (red) and for a NPQT theory with gravitational couplings given by (\ref{['coup']}) (blue). In GR, the star leaves behind a Schwarzschild black hole and reaches zero size (highlighted by a red star) after a finite proper time. On the other hand, for NPQT, the star reaches a minimum size inside the inner horizon of the regular black hole it creates, undergoing a bounce. It starts growing again and crosses the inner horizon and outer horizons of a white hole in a new universe, where it reaches its original size. The process is then restarted and repeated indefinitely. In the plot we have set $\rho_0=9/(32\pi G_{\rm N})$, $\alpha=1/8$.
• Figure 3: We plot the coordinate radius (normalized by its initial value, $R_0$) of a spherical thin shell as a function of its proper time as it undergoes gravitational collapse for Einstein gravity (red) and for a NPQT theory with gravitational couplings given by (\ref{['coup']}) (blue). The behavior is completely analogous to the one explained in the caption of Fig. \ref{['fig:modiH0']} for the collapse of a pressureless dust star. In the plot we have set $G_{\rm N}M=1$, $G_{\rm N}m=6/5$, $\alpha=1/8$.