Non-triviality of asymptotically flat Buchdahl-inspired metrics in pure $R^2$ gravity

In Phys. Rev. D $\mathbf{\text{107}}$, 104008 (2023) we reported a novel exact closed-form solution which describes asymptotically flat spacetimes in pure $R^2$ gravity. The solution is Ricci scalar flat, viz. $R\equiv 0$ everywhere. Whereas any metric with a null Ricci scalar would $\mathit{\text{trivially}}$ satisfy the $R^2$ vacuo field equation, $R(R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R)+g_{\mu \nu }\mathrm{\square }R-{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }R=0$, in this article, we shall show that our solution satisfies a “stronger” version of the $R^2$ vacuo field equation, viz. $R_{\mu \nu }-\frac{1}{4}{g}_{\mu \nu }R+R^{-1}(g_{\mu \nu }\mathrm{\square }R-{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }R)=0$, despite the term $R^{-1}$ being $\mathit{\text{singular}}$. Even though $R$ identically vanishes, for our solution, the combinations $R^{-1}{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }R$ and $R^{-1}\mathrm{\square }R$ are $\mathit{\text{free}}\mathit{\text{of}}\mathit{\text{singularity}}$. This exceptional property sets our solution apart from the set of null-Ricci-scalar metrics and makes it a genuinely $\mathit{\text{non-trivial}}$ solution. We further demonstrate that, as a member of a larger class of asymptotically de Sitter metrics, our solution is resilient against perturbations in the scalar curvature at largest distances, making it relevant for physical situations where the background deviates from asymptotic flatness.