标题： 超越施瓦茨希尔德-德西特时空：I. 一种由布赫达尔启发的纯$R^2$引力的新全面度规类，以紧凑形式表示 显示英文标题

标题： Beyond Schwarzschild-de Sitter spacetimes: I. A new exhaustive class of metrics inspired by Buchdahl for pure $R^2$ gravity in a compact form

摘要： 大约六十年前，Buchdahl 开创了一个寻找纯$R^{2}$引力静态球对称真空的程序（Nuovo Cimento，第 23 卷，第 1 期，第 141-157 页（1962 年）；[https://link.springer.com/article/10.1007/BF02733549]）。 克服了多个障碍后，他的工作最终得出一个非线性的二阶常微分方程（ODE），需要被求解。 然而 Buchdahl 认为这个 ODE 难以处理，并过早地放弃了寻找解析解的尝试。 我们终于设法打破了这个持续六十年的僵局并完成了他的目标。 重新表述 Buchdahl 的 ODE，我们得到了一种新颖的全面度规类（我们将在此之后称之为 Buchdahl 启发的度规）以紧凑的形式。 我们能够通过直接检查事后验证，此处得到的度规满足$R^{2}$真空场方程，从而确立其有效性。 我们说明了为什么 Buchdahl 启发的度规，其具有非恒定标量曲率，能够击败之前针对二次引力证明的 Lichnerowicz 类无结果定理，通过避开在空间无限远处度规渐近衰减的过于严格的限制。 我们的新解因此完成了 Buchdahl 持续六十年的程序。 我们还探讨了 Buchdahl 启发的度规在小 k 极限和坐标原点附近区域的数学性质。 显示英文摘要

摘要： Some sixty years ago Buchdahl pioneered a program in search of static spherically symmetric vacua for pure $R^{2}$ gravity (Nuovo Cimento, Vol 23, No 1, pp 141-157 (1962); [https://link.springer.com/article/10.1007/BF02733549]). Surpassing several obstacles, his work culminated in a non-linear second-order ordinary differential equation (ODE) which required being solved. However Buchdahl deemed the ODE intractable and prematurely abandoned his pursuit for an analytical solution. We have finally managed to break this six-decades-old impasse and accomplish his goal. Reformulating the Buchdahl ODE, we obtain a novel exhaustive class of metrics (which we shall call the Buchdahl-inspired metrics hereafter) in a compact form. We are able to verify ex post, via direct inspection, that the metric obtained herein satisfies the $R^{2}$ vacuo field equation, hence establishing its validity. We show why the Buchdahl-inspired metrics, which exhibit non-constant scalar curvature, defeat a Lichnerowicz-type no-go theorem, previously proved for quadratic gravity, by evading the overly strong restriction on the asymptotic falloff of the metric at spatial infinity. Our novel solution thereby completes Buchdahl's six-decades-old program. We also explore the mathematical properties of the Buchdahl-inspired metrics in the limit of small k and in the region around the coordinate origin.