On vacuum and charged asymptotically (A)dS black holes in quadratic gravity

TL;DR

This work analyzes static, spherically symmetric black holes in quadratic gravity with a cosmological constant, uncovering that for large $|\Lambda|$ the spacetimes generically exhibit (A)dS asymptotics and constitute a three-parameter family $(\bar{r}_h, \Lambda, b)$. For smaller $|\Lambda|$ asymptotics generally require fine-tuning of the Bach parameter $b$, yielding two two-parameter families: (A)dS-Schwarzschild and FT(A)dS-Schwarzschild-Bach; these distinctions persist in both the vacuum and electrovacuum cases, though electric charge $q$ adds a fourth free parameter in the generic regime. The authors extend the framework to charged black holes, obtaining similar asymptotic structures with $q$ entering the horizon-expansion coefficients and producing a four-parameter charged family in generic regions. Their analysis uses the conformal-to-Kundt ansatz, a constant-$R$ reduction, and horizon-centered power-series expansions, with detailed recurrence relations that classify solution-classes and map parameter regions supporting generic versus tuned (A)dS behavior. Overall, the results reveal qualitative departures from Reissner–Nordström behavior and provide a systematic approach to distinguishing physical, generic asymptotics from fine-tuned cases in both vacuum and charged quadratic gravity.

Abstract

In this paper, we study asymptotic properties of static spherically symmetric black holes in quadratic gravity with a cosmological constant $Λ$. We find that for sufficiently large values of $|Λ|$ these black holes are generically asymptotically (A)dS and form a three-parameter family of black holes, with free parameters being the horizon radius, $Λ$, and Bach parameter $b$. For smaller values of $|Λ|$, fine-tuning of the Bach parameter is necessary to recover asymptotically (A)dS behaviour, resulting in two distinct two-parameter families of spherically symmetric black holes with (A)dS asymptotics. One family is the (A)dS-Schwarzschild solution, the other is a fine-tuned asymtotically (A)dS-Schwarzschild-Bach solution. We also generalise the above solutions to electrically charged black holes, obtaining qualitatively similar asymptotic behaviour. This adds the electric charge as an additional free parameter of these black holes. Both fine-tuned three-parameter families are distinct from Reissner-Nordström black holes.

Figures (10)

• Figure 1: Ratios of consecutive coefficients, $a_i/a_{i-1}$ (blue) and $c_i/c_{i-1}$ (yellow), for $[0,1]$ solutions with $r_h=-1$, $k=1/2$, and $\kappa'=1$. The remaining parameters are $q=0.3$, $b=-0.2$, $\Lambda=2$, $a_0=1$, $c_0=1$ (left) and $q=0.8$, $b=-0.4$, $\Lambda=-1$, $a_0=1$, $c_0=-2$ (right).
• Figure 2: The metric functions $\Omega(r)$ and $\mathcal{H}(r)$ for the $[0,1]$ solution with $r_h=-1$, $k=1/2$, $\kappa'=1$, $q=0.3$, $b=-0.2$, $\Lambda=2$, $a_0=1$, and $c_0=1$. Expansions with 20 (red), 30 (orange), 40 (green), and 800 (blue) terms are shown with the black dashed lines indicating the interval of convergence.
• Figure 3: The metric functions $\Omega(r)$ and $\mathcal{H}(r)$ for the $[0,1]$ solution with $r_h=-1$, $k=1/2$, $\kappa'=1$, $q=0.8$, $b=-0.4$, $\Lambda=-1$, $a_0=1$, $c_0=-2$. Expansions with 20 (red), 30 (orange), 40 (green), and 800 (blue) terms are shown with the black dashed lines indicating the interval of convergence.
• Figure 4: Plot of $f(\bar{r})/\bar{r}^2$ illustrating asymptotic behaviour of $[0,1]$ solutions with $r_h=-1$, $k=1/2$, and $\kappa'=1$. Expansions with 100 (red), 200 (orange), 300 (green), and 800 (blue) terms are shown for $q=0.3$, $b=-0.2$, $\Lambda=2$, $a_0=1$, $c_0=1$ (left) and $q=0.8$, $b=-0.4$, $\Lambda=-1$, $a_0=1$, $c_0=-2$ (right).
• Figure 5: Plot of $\Lambda$ vs $b$ for $[0,1]$ solutions with common parameters $r_h=-1$, $k=1/2$, $\kappa'=1$, $q=0$, $a_0=1$, and $c_0=-1$. The blue and red regions contain asymptotically de Sitter and anti-de Sitter solutions respectively, with $f(\bar{r})/\bar{r}^2$ approaching $-\Lambda/3$, in each case without the need for fine-tuning. These are G(A)dS Schwarzschild--Bach black holes (or for large positive $\Lambda$, these could be naked singularities surrounded by a cosmological horizon, c.f. Sec. \ref{['SchHor']}). Between these regions, there are two families of fine-tuned solutions with AdS (red), dS (blue), and flat (green) asymptotic behaviour. In addition, the plot contains several curves indicating where various early terms in the expansions vanish: $a_1=0$ on the blue line ($\Lambda=b+1$); $c_1=0$ on the orange line ($\Lambda=3b+2$); $c_2=0$ on the yellow line ($\Lambda=3b+2\pm\sqrt{3b+1}$). Note that below the blue line, solutions are expanded around the black-hole horizon, while above it, they are expanded around the cosmological horizon. Intersections of these lines are discussed in the main text.