Shadow of black holes with consistent thermodynamics

TL;DR

This work studies a Kerr black hole with a scale-dependent Newton coupling $G(z)$ chosen to satisfy the thermodynamic first-law integrability, yielding the constraint $G(r_h,M,a) = G(r_h^2 + a^2, M r_h)$. By parameterizing $G(z)$ with $z = \left(\frac{G_0}{r^2 + a^2}\right)^l \frac{1}{(Mr)^p}$ and analyzing four representative $(l,p)$ choices, the authors compute shadow critical curves and two observables, $\delta$ (size) and $\Delta C$ (shape), to constrain the parameter space using Sgr A* data and to identify horizon-based signatures of the underlying quantum corrections. They find that the circularity deviation $\Delta C$ closely tracks an extremality parameter $\epsilon$ defined on the horizon, suggesting that horizon geometry leaves imprints on shadow morphology; importantly, a universal lower bound $\delta_{\min} \gtrsim -0.21$ emerges for $|a_*| \le 1$, beyond which the D-shaped shadow may no longer be preserved. The paper also discusses scenarios where the Kerr bound can be locally violated with cuspy shadows due to stable photon orbits, highlighting rich phenomenology possible in thermodynamically consistent, scale-dependent black holes and offering observational avenues for probing quantum gravity effects near horizons.

Abstract

Quantum effects in general induce scale dependence in the coupling constants. We explore this possibility in gravity, with a scale-dependent Newton coupling. When applied to Kerr black holes with such a running coupling, the consistency of black hole thermodynamics requires that the Newton coupling have a specific dependence on the black hole parameters. In this work, we consider such a class of Newton couplings and look for the possible observational implications on the highly lensed images of the black holes. In addition to placing constraints on the parameter space of the model through the latest Sgr A* images, we find that the variations in the shape of shadows in a large portion of the parameter space can be qualitatively captured by a quantity solely defined by the event horizon. Most importantly, the consistency of thermodynamics suggests a lower bound on the shadow size, beyond which either horizon disappears, or the shadow cannot keep the standard D-shaped structure. The possibility that the black holes in this model could spin faster than the Kerr bound, and the physical implications of the resulting cuspy shadows, are also discussed.

Figures (15)

• Figure 1: The shadow critical curves (Left) for $b_1=a_2=0$ and different values of $b_2$, and (Right) for $b_1=b_2=0$ and different values of $a_2$.
• Figure 2: (Left) The critical curves for fixed $a_2=0$; the blue and the red curves with $(b_1,b_2)=(-1,0.1)$ and $(b_1,b_2)=(-1,-1)$, respectively. (Right) The critical curves for fixed $b_2=0$; the blue and the red curves with $(b_1,a_2)=(-1,-0.1)$ and $(b_1,a_2)=(-1,1)$, respectively. In each panel, the shaded regions represent the results of contours scanned when taking $b_1\ge-1$. For $b_1\rightarrow\infty$, the critical curves converge to the Kerr results (black curves).
• Figure 3: The fractional diameter deviation $\delta$ is shown by color in the parameter space of $(a_*,b_2)$ (left) and $(b_1,b_2)$ (right). The black and green curves represent the upper bounds obtained from the Keck and VLTI priors on the mass-to-distance ratio of the Sgr A*, i.e., Eq. \ref{['deltabound']}. The parameter space toward the blue region is more observationally preferred.
• Figure 4: The logarithm of the circularity deviation $\Delta C$ is shown by color in the parameter space of $(a_*,b_2)$ (left) and $(b_1,b_2)$ (right). The black contours represent the values of the deviation parameter $\epsilon$ from extremality defined in Eq. \ref{['defepsilon']}. On the left panel, the region to the left of the red curve represents the black hole spacetimes which only have one event horizon.
• Figure 5: The critical curves with $d_1=0$ and different choices of $(c_1/G_0M^2,c_2/G_0^2M^4,d_2/G_0^2M^4)$. The shaded regions on the left and right panels correspond to the results of contours scanned for varied $c_2/G_0^2M^4$ and $d_2/G_0^2M^4$, respectively. The red-shaded region on the left (right) panel represents the results with $c_2/G_0^2M^4\in(-0.1,0.1)$ ($d_2/G_0^2M^4\in(-0.1,0.1)$). On the other hand, the blue-shaded region on the left (right) panel represents the results with $c_2/G_0^2M^4\in(-1,1)$ ($d_2/G_0^2M^4\in(-1,1)$).