Buchdahl limits in theories with regular black holes

TL;DR

This work generalizes Buchdahl’s compactness bounds to D-dimensional gravity with infinite higher-curvature corrections (Quasi-topological gravities) whose vacuum solutions are regular black holes. By analyzing static perfect-fluid stars, the authors derive analytic results for constant-density configurations and show that maximum compactness is controlled by three regimes: divergent central pressure, vanishing central pressure, and interior radii coinciding with a black hole inner horizon; for generic decreasing density profiles, the maximum occurs when the interior becomes singular. Under certain conditions, a novel Buchdahl limit is obtained, attained by a specific constant-density profile, indicating that stars in QT theories can be more compact than in GR. However, while vacuum curvature invariants are universally bounded in these theories, ordinary matter can reach arbitrarily high curvatures unless additional constraints such as the dominant energy condition are imposed, highlighting the nontrivial role of matter couplings in extending curvature bounds beyond vacuum.

Abstract

We study generalizations of Buchdahl's compactness limits for perfect-fluid star solutions of $D$-dimensional Einstein gravity coupled to higher-curvature corrections. We focus on Quasi-topological theories involving infinite towers of terms for which the unique vacuum spherically symmetric solutions correspond to regular black holes. We solve analytically the problem of constant-density stars and find that the space of solutions is bounded by: configurations with divergent central-pressure, corresponding to the most compact stars; configurations which possess zero central-pressure; and configurations for which the sizes of the stars coincide with the inner-horizon radii of the would-be regular black holes. In the more general case of perfect-fluid stars for which the mean density decreases with increasing radius, we show that, for each density profile, maximum compactness is reached when the metric becomes singular at the center. Under certain additional conditions, we find a novel Buchdahl limit for the maximum compactness of stars, attained by a specific constant-density profile. We show, in particular, that stars in these theories may be more compact than in Einstein gravity. While the vacuum solutions of these theories are such that all curvature invariants take mass-independent maximum finite values, we argue that there exist ordinary matter stars with finite central pressures for which such bounds can be violated -- namely, arbitrarily high curvatures can be reached -- unless additional constraints, such as the dominant energy condition, are imposed on the fluid.

Figures (9)

• Figure 1: Maximum compactness limit for constant-density stars in Einstein gravity as a function of the spacetime dimension. Purple dots indicate the value of the minimal radius (infinite-central-pressure), black dots for black holes, arrowheads for the limit as the dimension approaches to infinity, and we shade the regions accordingly: white for regular stars, and purple for stars that are physically unacceptable because the pressure $\mathsf{p}(r)$ diverges at some interior point $r$. (Left) Minimal radius divided by the Schwarzschild radius. We observe that in $D=6$, this quantity reaches its maximum value, and the ratio saturates as $D\to\infty$. (Right) Metric function at the limiting star's surface $f(R_\mathrm{min})$. This value increases monotonically with $D$, so it reaches its minimum at $D=4$, meaning that the gravitational effects are strongest in four dimensions. Likewise, for large $D$, the metric function approaches 1.
• Figure 2: We plot a family of pressure profiles for different star parameters in $D=6$ Einstein gravity (the plot is qualitatively analogous $\forall\,D\neq 6$). The yellow arrow indicates the direction in which the star radius $R$ decreases, starting from a pressureless star with a very large radius (X-axis), all the way to an unphysical ultracompact star with a Schwarzschild radius (black vertical line). Orange lines correspond to physical stars and purple dashed lines to unphysical ones. The limiting case of a star saturating the compactness limit is represented by a solid purple line. This has a point-like singularity at $r=0$ (indicated with a dot).The purple dashed lines represent would-be finite-volume singularities.
• Figure 3: We plot the Einstein-Gauss-Bonnet black hole metric function $f(r)$ in $D=5$ for a fixed value of $\alpha$ and different values of the mass. There exists a specific value, $\mathsf{M}_{\ast}=\alpha/2$, (third plot) beyond which smaller values of $\mathsf{M}$ correspond to naked singularities (fourth plot). For greater values of $\mathsf{M}$, on the other hand, the solution describes a black hole with a horizon radius smaller than $r_{\rm S}$ and a curvature singularity milder than Schwarzschild's at $r=0$ (second plot). The red curve in the first plot is the Einstein gravity result.
• Figure 4: Maximum compactness limit for constant-density stars in single Lovelock gravities as as a function of the spacetime dimension $D$. We follow the color convention of Fig \ref{['fig_GR_Buchdahl']}. In both plots, we include again the general relativity limit, which corresponds to the case ${n_\mathrm{max}}=1$. Furthermore, we have depicted the limiting cases for the three first orders of single Lovelock gravities, corresponding to ${n_\mathrm{max}}=2,3,4$, where the second order is Gauss-Bonnet. In both plots, the arrowheads indicate this limit for all the cases as the dimension approaches infinity. Although Lovelock theories can be built with ${n_\mathrm{max}}\leq\lfloor\frac{D-1}{2}\rfloor$, it has been shown that there are no static stars in odd dimensions when $D=2{n_\mathrm{max}}+1$Dadhich:2015rea. Therefore, the minimal dimension that allows stable stars in a single Lovelock theory of order ${n_\mathrm{max}}$ is $D=3{n_\mathrm{max}}+1$.
• Figure 5: We plot the metric function $f(r)$ of the five-dimensional Hayward spacetime for different values of the mass. In the first plot (red curve), the coupling constant is set to zero, representing the GR solution of the singular black hole (Schwarzschild-Tangherlini solution). In the subsequent three plots (blue curves), we show the regular solution for different values of the total mass: a) when $\mathsf{M}>\mathsf{M}_\mathrm{cr}$, the solution has both an outer and an inner horizon. Note that the gravitational radius of this black hole is smaller than $r_\mathrm{S}$; b) when $\mathsf{M}=\mathsf{M}_\mathrm{cr}$, both horizons degenerate, which corresponds to the extremal Hayward black hole, with mass $\mathsf{M}_\mathrm{cr}$ and gravitational radius $r_\mathrm{cr}$; c) when $\mathsf{M}<\mathsf{M}_\mathrm{cr}$, the solution is horizonless.