Gravitational Landscapes: black holes with linear equations of state in asymptotically safe gravity

TL;DR

This work analyzes black hole formation in asymptotically safe gravity by incorporating a linear equation of state and a running gravitational coupling $G(\epsilon)$ and running cosmological constant $\Lambda(\epsilon)$. Using an interior FLRW patch matched to an exterior static spacetime, the authors derive a density-dependent coupling $G_w(\epsilon)$ and a Misner–Sharp mass $m_w(R)$ that depend on the EOS parameter $w$, revealing a rich set of exterior BH solutions, including regular (non-singular) cores for certain $w$. The Friedmann equations acquire hypergeometric dependencies on density, and a Hamiltonian formalism shows that the dynamical stability of the collapse depends sensitively on $w$, with stable configurations for $w=0,1/2,1$ and instability for some negative values. Overall, AS gravity can qualitatively alter collapse and horizon formation, offering potential routes to singularity resolution and new BH phenomenology with possible observational implications.

Abstract

We study black holes with linear equation of state within the framework of asymptotically safe gravity. This study extends previous work on gravitational collapse in asymptotically safe gravity (that has been done for a dust fluid) by considering into account the pressure of stellar matter. We derive modified field equations containing the running gravitational coupling and the cosmological constant as functions of energy density. The interior space-time of collapsing star is modeled by the Friedmann-Lemaître-Robertson-Walker metric, while the exterior is described by a static spherically symmetric space-time. Different equations of state from ordinary matter to exotic phantom energy are considered to investigate their impact on black hole structure and horizon formation. Our results illustrate that asymptotically safe gravity can introduce non-singular black hole solutions under specific conditions. These results provide new insights into black hole physics and the avoidance of singularities within the asymptotically safe gravity framework.

Figures (20)

• Figure 1: The behaviour of $G(k)$ as a function of the momentum scale $k$ for different values of $g_{*}$, with $8\pi G_{0} = 1$. At higher energy scales, the running gravitational coupling decreases, while at sufficiently low energy scales, it asymptotically approaches $1/8\pi$.
• Figure 2: The behaviour of $G(R)$ as a function of the radial distance $R$ for different values of $\varrho$, with $8\pi G_{0} = 1$. The running gravitational coupling decreases at small distances, while it approaches $1/8\pi$ at sufficiently large distances.
• Figure 3: Contour plot of $G_{w}(\epsilon)$ as a function of the parameters $\epsilon$ and $w$. Lighter regions correspond to lower values of $G_{w}(\epsilon)$, whereas darker regions indicate higher values. In general $G_{w}(\epsilon)$ decreases with increasing $\epsilon$. The rate of this decrease is significantly affected by $w$. The contour lines denote regions of constant $G_{w}(\epsilon)$, providing a detailed representation of the function's behavior in parameter space. The parameters are fixed to $8\pi G_{0}=1$ and $\alpha=10$.
• Figure 4: Contours of $\chi_{w}(\epsilon)$ as a function of $\epsilon$ and $w$. The color gradient represents the magnitude of $\chi_{w}$, with darker shades corresponding to higher values and lighter shades to lower values. Contour lines represent loci of constant $\chi_{w}(\epsilon)$, revealing structural patterns and transitions in the parameter space. The system parameters are set to $G_{0}=\alpha=\Lambda_{0}=1$
• Figure 5: Contours of $\Lambda_{w}(\epsilon)$ as a function of the energy density $\epsilon$ and the parameter $w$. Contour lines indicate regions of constant $\Lambda_{w}(\epsilon)$, while the color gradient reflects its magnitude, with darker shades corresponding to higher values. The parameters are fixed as $\Lambda_0=G_0 =\alpha = 1$.