论文标题
超越Schwarzschild-de保姆空间：I。由Buchdahl启发的新的详尽指标，以紧凑的形式以纯$ r^2 $重力
Beyond Schwarzschild-de Sitter spacetimes: I. A new exhaustive class of metrics inspired by Buchdahl for pure $R^2$ gravity in a compact form
论文作者
论文摘要
大约六十年前，布赫达尔（Buchdahl）开创了一个计划，以寻找静态的球形对称真空吸尘器，以供纯$ r^{2} $重力（Nuovo Cimento，第23卷，第1卷，第1页，第141-157页（1962）; [htttps：//link.springer.com/article.com/article/10.1007/1007/bff543549]。他的工作超过了几个障碍，最终以非线性二阶的普通微分方程（ODE）达到顶峰，该方程（ODE）需要解决。然而，布赫达尔认为颂歌是顽强的，过早地放弃了他对分析解决方案的追求。我们终于设法打破了这个六个历史的僵局并实现了他的目标。重新设计了布赫达尔颂歌，我们获得了一种新颖的详尽指标（以下称为布赫达尔风格的指标），以紧凑的形式获得。我们能够通过直接检查来验证EX POST，以确保本文获得的指标满足$ r^{2} $ vacuo字段方程，从而确定其有效性。我们展示了为什么表现出非稳定标量曲率的Buchdahl启发的指标击败了Lichnerowicz-type no-Go定理，以前证明是通过对二次重力的证明，通过避免对空间无限度量的标准差异的过度限制。我们的小说解决方案从而完成了布赫达尔的六年历史计划。我们还探索了布赫达尔启发的指标的数学特性，在小k的极限以及坐标起源周围的区域中。
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.
