Some sixty years ago Buchdahl pioneered a program in search of static spherically symmetric metrics for pure ${\mathcal{R}}^2$ gravity in vacuo [H. A. Buchdahl, Nuovo Cimento 23, 141 (1962).]. Surpassing several obstacles, his work culminated in a nonlinear second-order ordinary differential equation (ODE) which required being solved. However, Buchdahl deemed the ODE intractable and abandoned his pursuit for an analytical solution. We have finally managed to overcome this remaining hurdle and bring his program to fruition. Reformulating Buchdahl’s ODE, we obtain a novel class of metrics (which we shall call the Buchdahl-inspired metrics hereafter) in a compact and transparent expression: ${ds}^2=e^{k\int \frac{dr}{rq(r)}}\{p(r)[-\frac{q(r)}{r}d{t}^2+\frac{r}{q(r)}d{r}^2]+r^{2}d{\mathrm{\Omega }}^2\},$ in which the pair $\{p,q\}$ are two functions of the radial coordinate $r$ obeying the evolution rules $\frac{dp}{dr}=\frac{3k^2}{4r}\frac{p}{q^2},\frac{dq}{dr}=(1-\mathrm{\Lambda }{r}^2)p$, and the Ricci scalar is $\mathcal{R}(r)=4\mathrm{\Lambda }{e}^{-k\int \frac{dr}{rq(r)}}$. We are able to verify ex post, via direct inspection, that the metric given above satisfies the ${\mathcal{R}}^2$ vacuo field equation $\mathcal{R}({\mathcal{R}}_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }\mathcal{R})+(g_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu })\mathcal{R}=0,$ hence establishing its validity. The compact form above casts the Buchdahl-inspired metric in a parallel resemblance with the classic Schwarzschild–de Sitter (SdS) metric, with the case $k=0$ corresponding to the SdS metric. We show why the Buchdahl-inspired metric, which exhibits nonconstant scalar curvature when $k\ne 0$, defeats a “no-go” theorem proved in Kehagias et al. [J. High Energy Phys. 05 (2015) 143.], which posits that pure ${\mathcal{R}}^2$ gravity vacua are restricted to the Einstein spaces, ${\mathcal{R}}_{\mu \nu }=\mathrm{\Lambda }{g}_{\mu \nu }$, and the vanishing Ricci scalar spaces, $\mathcal{R}=0$. The aforementioned “no-go” theorem assumes a rapid asymptotic falloff for the metric as $r\to \infty$. However, we find that the Buchdahl-inspired metric evades that central assumption, which is overly restrictive. A product of a fourth-derivative gravity, a Buchdahl-inspired metric is specified by four parameters: $\mathrm{\Lambda }$ measuring the scalar curvature at largest distances, $k$ effecting the variation of the curvature on the manifold, and $\{p_0,q_0\}$ initiating the “evolution” of $\{p(r),q(r)\}$ along the radial direction, forming a two-dimensional phase space. The class of Buchdahl-inspired metrics is exhaustive as it covers all “nontrivial” static spherically symmetric metrics admissible for pure ${\mathcal{R}}^2$ gravity in vacuo, with the SdS metric being a special case, $k=0$. Transparently, the quartet $\{\mathrm{\Lambda },k,p_0,q_0\}$ spans a topological space with all members in the class of Buchdahl-inspired metrics being smoothly connected to the SdS metrics when $k$ is continuously tuned to 0. In this respect, the Buchdahl-inspired metrics constitute a natural enlargement suitably regarded as a framework “beyond Schwarzschild–de Sitter.” Our novel solution thereby completes Buchdahl’s six-decades-old program. We also explore the mathematical properties of the Buchdahl-inspired metric in the limit of small $k$ and in the region around the coordinate origin.

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