Regular Black Hole Cores via Gravitational Evanescence of Collapsing Matter

Abstract

Modified Einstein's equations implementing both an energy-dependent Newtonian and cosmological constant can be obtained via a modified action characterised by a non-minimal gravity-matter coupling. We show how different dynamics for the high energy regime - that nevertheless have in common the vanishing of the Newton constant - result in models of gravitational collapse which source the formation of non-singular geometries with different types of asymptotic core: de Sitter, Minkowski, and with "steep pressure". As we categorise their defining properties, one feature, which we prove to be actually independent of both the specific form of the non-minimal coupling and the equation of state of the collapsing fluid, stands out: the formation of a geometry with Minkowski core occurs only if the Newton constant, before finally vanishing, assumes negative values.

Figures (6)

• Figure 1: Potential of the OSD collapse, for $m_{0}=1$, plotted against the potential of a non-singular dynamics; the intersections of the latter with the line $V(a)=-K$ occur at $a_{\text{min}}$ and $a_{\text{max}}$; for illustrative purpose we choose the dynamics \ref{['dynChip']} with $p=2$, and $m_{0}=1$, $\xi=0.01$, however the qualitative behaviour holds true for all the non-singular cases (left panel). Scale factor for the OSD collapse, for $K=0$ and $m_{0}=1$, plotted against the bound, and marginally bound, non-singular collapse, for $p=3$, $m_{0}=1$ and $\xi=0.1$ (right panel). In all plots in the paper we set $8\pi G_{\text{N}}=1$.
• Figure 2: Curve defined by \ref{['ahcurve']} for the OSD model, with $m_0=1$, plotted against the one for the non-singular dynamics with $m_0=1$, $\xi=0.01$ and $p=3/2$ (left panel). The three different qualitative possibilities depending on the value of the parameter $r_{b}$. They correspond to different evolutions of the causal structure for the interior geometry (right panel).
• Figure 3: Phase diagram of the parameters space of the model with dust equation of state and $K=0$, for the non-singular dynamics corresponding to $p=3/2$, however the qualitative structure holds true for all the non-singular cases (left panel). The three qualitatively different causal structures that are possible for the exterior geometry, for $m_0=1$ and $\xi=0.01$ ($r_{{\text{ah}}_{\text{cr}}}=0.553$, $M_{0_{\text{cr}}}=0.085$). The blue curve represents an extremal black hole (right panel). The points in parameters space and the possible causal structures of the global spacetime resulting from the collapse are related by a bijective map.
• Figure 4: Penrose diagram for the OSD collapse. The grey and white regions represents, respectively, the interior and exterior spacetime (left panel). Possible Penrose diagram for the non-singular marginally bound (central panel) and bound (right panel) collapse, for $M_{0}>M_{0_{\text{cr}}}$. Unlike the trajectory $r_{b}$ of the collapsing boundary, the trajectory $r_{\text{ah}}$ is not a world-line, but rather a mathematical curve indicating that a certain spherical shell layering the matter ball becomes a trapped (an untrapped) surface. The formation of the inner horizon is a direct product of the non-singular dynamics, and affects the global structure of spacetime. The construction of the diagram for the bounded case involves the use of Kruskal-Szekeres coordinates, and potentially gives rise to nested universes, traced by the periodic occurrence of the spacetime region dubbed with Roman number “I”.
• Figure 5: Dependence of Newton coupling (left panel) and cosmological coupling (right panel) on matter energy-density, for the three dynamics studied, for $\xi=0.1$. In the dynamics generating the non-singular spacetime with Minkowski core, $G(\epsilon)$ assumes negative values before vanishing and has a negative minimum when $\Lambda(\epsilon)$ has a positive maximum. Here, for illustrative purpose, $\epsilon$ follows a linear scale, instead of solving \ref{['dust_energy_density']} with an actual solution $a(t)$.