The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.

中文翻译：

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关于de Sitter-Schwarzschild度量的对称性和守恒定律分类以及相应的波和Klein-Gordon方程

这项工作的主要目的是集中讨论德西特-史瓦西时空度量所承认的李对称性，以及在德西特-史瓦西几何中构造的相应波或克莱因-戈登方程。对获得的对称性进行分类，并通过自然拉格朗日量获得与这些对称性相关的变分（Noether）守恒定律。在度量的情况下，与导致额外守恒定律的 Killing 向量相比，我们获得了额外的变分，对于波和 Klein-Gordon 方程，就不变性研究而言，变分对称涉及较少繁琐的计算。