Regular Geometries from Singular Matter in Quasi-Topological Gravity

Abstract

Vacuum quasi-topological gravity with infinitely many terms in the action satisfies Markov's limiting curvature hypothesis: the spherically symmetric solutions are regular and all curvature invariants are bounded by solution-independent scales. We study how this picture changes when the theory is coupled to matter. We find that minimally coupled matter spoils the scaling properties of the vacuum equations that lead to the validity of Markov's hypothesis, but the corresponding geometries often remain regular. We make this precise by developing a set of sufficient conditions on general static, spherically symmetric stress-tensors such that the corresponding solutions have bounded curvature. These conditions cover regular matter sectors but also singular matter profiles that are sufficiently singular in a sense we quantify. Our conclusions hold independently of the matter field equations and include configurations in which matter exhibits divergent energy density and pressure at finite radius or at Killing horizons, results that may have implications for mass inflation in these models. We then explore non-minimal couplings, focusing on theories with infinite towers of higher-curvature and electromagnetic terms in the action. In this class, Markov's hypothesis can be restored: we present theories admitting a universal upper bound on curvature, independent of the mass and charge. Overall, our results highlight subtleties in coupling quasi-topological gravity to matter and suggest Markov's hypothesis as a potential selection criterion for resummed gravity-matter effective theories.

Figures (3)

• Figure 1: Logarithm of the Kretschmann scalar associated to various solutions of a five-dimensional EMQT with characteristic function $h_y^{\rm I}(\psi)$, as defined in \ref{['eq:EMQTlch']}. We explicitly observe that the Kretschmann scalar attains larger and larger values as $P \rightarrow 0$. Also, we see that the maximum appears for smaller values of $r$ as the charge diminishes. For the sake of concreteness, we have used $\kappa=\alpha=2\mathsf{M}$.
• Figure 2: Kretschmann scalar associated to various solutions of a five-dimensional EMQT with characteristic function $h_y^{\rm II}(\psi)$, as defined in \ref{['eq:ejemlcheq']}. In the left plot, the maximum value of the Kretschmann scalar, $K_{\rm max}$, corresponds to $\sim 62/\alpha^2$, and we have used $\kappa=\mathsf{M}=\alpha$. In the right plot, $K_{\rm max} \sim 200/\alpha^2$ and we have used $2\kappa=\mathsf{M}/3=\alpha$. We explicitly observe that the limiting curvature hypothesis is satisfied: curvature invariants are bounded by a universal quantity, independent of the mass and the charge.
• Figure 3: Plots of the Kretschmann scalar $\alpha^2R_{abcd}R^{abcd}$ for various $(D,\mathrm{N})$-Hayward-like QT theories. In each graph, the curvature invariant is depicted in blue for the vacuum spacetime and in green for the FLRW spacetime, as a function of $\alpha\psi$ or $\alpha\Phi$, respectively, over their domain $(0,1)$. In all the cases presented, the maximum curvature is reached by the vacuum solution and it becomes more pronounced as $D$ and/or $\mathrm{N}$ increase.