Exact Bachian singularity in quadratic gravity

TL;DR

The work addresses the existence of a fully explicit global static, spherically symmetric solution in four-dimensional quadratic gravity with constant scalar curvature, revealing a central Bachian singularity and a horizon structure controlled by the sign of the cosmological parameter $\Lambda$. The authors employ the conformal-to-Kundt metric form $ds^2=\Omega^2(r)[d\theta^2+\sin^2\theta d\phi^2-2du dr+\mathcal{H}(r)du^2]$ and derive an exact solution under a fine-tuned coupling with $\Lambda=-\frac{9}{4k}$, giving $\Omega(r)=wr+v$ and $\mathcal{H}(r)=\frac{1}{w^2}\left(\frac{\Lambda}{9}\Omega^4-\frac{1}{7}\Omega^2-\frac{24}{49\Lambda}\right)$. The paper analyzes curvature invariants, horizon properties, thermodynamics via Wald entropy, geodesics and geodesic deviation, and demonstrates the solution’s relation to known Frobenius-type expansions and Schwarzschild–(A)dS with Bach corrections, thereby providing a rigorous global geometry and a testing ground for methods in quadratic gravity. This explicit example illuminates how Bach terms modify horizon and asymptotic behavior and offers a benchmark beyond local or numerical approaches for validating analytical techniques. Overall, the result connects previous approximation schemes with an exact global configuration, clarifies gauge issues, and serves as a robust reference for future studies of spherically symmetric spacetimes in quadratic gravity.

Abstract

For specifically coupled values of the quadratic gravity parameters, we present a fully explicit static spherically symmetric solution. It contains the central singularity surrounded by the black-hole or the cosmological horizon for the negative or positive cosmological parameter, respectively. This spacetime, thus, belongs to the already analyzed classes of solutions expressed in terms of the Frobenius expansions, continuous fractions, or numerical simulations; however, it has not been explicitly identified before. The purpose of the presented highly-constrained somehow unphysical model is to reveal a global geometric picture that may occur in spherical spacetimes within quadratic gravity, which is typically hidden in more general approximative solutions. At the same time, it can serve as a solid benchmark for evaluating the accuracy of the methods employed to obtain such solutions.

Figures (7)

• Figure 1: Dependence (\ref{['H_sol_gauge']}) of the function $\mathcal{H}$ scaled by the parameter $w^2$ on $\Omega$ for different values of $\Lambda$ related to the theory coupling constants by (\ref{['cont_relation']}). The signum of $\Lambda$ determines the staticity of the spacetime regions separated by the horizon $\mathcal{H}=0$ via condition (\ref{['eq:KillingVec']}).
• Figure 2: For $\Lambda=1$, we plot the circular photon orbit (blue -- partially covered by nearby non-circular red and green trajectories) at ${r_{\mathrm{po}}=3/\sqrt{14}}\approx0.8$ and a pair of typical nearby light rays starting with ${\dot{r}(0)=0}$ at ${r_0=r_{\mathrm{po}}\pm10^{-9}}$. The inner one (red) spirals into the central singularity, while the outer one (green) radially escapes due to $\Lambda>0$. The motion in the equatorial plane $\theta=\pi/2$ is considered. The cosmological horizon $r_h$ is plotted as a grey dashed line.
• Figure 3: We visualize typical timelike ($\sigma=-1$) trajectories for different values of the parameters $E$ and $L_z=L$ (so that the particles remain in equatorial plane which is allowed due to the spherical symmetry) in the case with $\Lambda=-1$. The two quadruplets of test particles start at two different radii (indicated by dotted semicircles), and all trajectories end up in the central singularity. Particular values of the initial data are plotted in the figure. The maximal value of the radial coordinate is restricted by (\ref{['eq:E_L_constraints']}). The black-hole horizon $r_h$ is plotted as a dashed line.
• Figure 4: For $\Lambda=1$ we plot typical timelike ($\sigma=-1$) trajectories for different values of the parameters $E$ and $L_z=L$. The cosmological horizon at $r=r_h$ is plotted as a dashed line. The two quadruplets of test particles outside the horizon start at two different radii (indicated by dotted arcs), and all trajectories escape the centre. Particular values of the initial data are plotted in the figure. Below the horizon pair of dotted circles represents radii (\ref{['circular_boundaries']}) forming the boundary of a region admitting circular time orbits, where the data are given by (\ref{['eq:co_general_E']}) and (\ref{['eq:co_general_L']}). One such orbit is plotted in green. The remaining trajectories slightly violate the data constraint for circular orbits. In particular, the inner curve ending in the singularity has $L$ less than the circular value (\ref{['eq:co_general_L']}), while three trajectories leaving the central region have $E$ greater than (\ref{['eq:co_general_E']}).
• Figure 5: Visualisation of the metric function $h(\bar{r})$ scaled by the factor $w^2$ corresponding to the time-scaling freedom of the standard spherical coordinates (\ref{['metric:SchwCoord']}).