关于互反解的Buchdahl定理的推广
Extension of Buchdahl's Theorem on Reciprocal Solutions
AI总结：
本研究将Buchdahl的互反解方法推广至含宇宙学常数的情形及更广泛的标量-张量引力理论，通过共形变换得到带二次势的Brans-Dicke理论的多种同步解，拓展了该引力框架下的真空解族。
摘要：
自Brans-Dicke引力理论发展以来，众所周知，度规的共形变换可以重新表述该理论，将标量场的耦合从Ricci标量转移到物质 sector。具体而言，在这个被称为Einstein frame（爱因斯坦框架）的新框架中，Brans-Dicke引力被重新表述为带有额外标量场的广义相对论。1959年，Hans Adolf Buchdahl利用一种巧妙的技术，在该引力框架下推导了真空场方程的一组解。本文中，我们将Buchdahl的方法推广到包含宇宙学常数的情况，以及超出Brans-Dicke原型理论的标量-张量情形，进而通过度规的共形变换，得到了包含二次势的Brans-Dicke理论版本的解。更具体地说，我们在以下情形中得到了同步解：具有无质量标量场的标量-张量引力、带二次势的Brans-Dicke理论——在此我们得到了对应Schwarzschild-de Sitter度规、Nariai解和双曲叶状解的特定同步度规。
英文摘要：
Since the development of Brans-Dicke gravity, it has become well-known that a conformal transformation of the metric can reformulate this theory, transferring the coupling of the scalar field from the Ricci scalar to the matter sector. Specifically, in this new frame, known as the Einstein frame, Brans-Dicke gravity is reformulated as General Relativity supplemented by an additional scalar field. In 1959, Hans Adolf Buchdahl utilized an elegant technique to derive a set of solutions for the vacuum field equations within this gravitational framework. In this paper, we extend Buchdahl's method to incorporate the cosmological constant and to the scalar-tensor cases beyond the Brans-Dicke archetypal theory, thereby, with a conformal transformation of the metric, obtaining solutions for a version of Brans-Dicke theory that includes a quadratic potential. More specifically, we obtain synchronous solutions in the following contexts: in scalar-tensor gravity with massless scalar fields, Brans-Dicke theory with a quadratic potential, where we obtain specific synchronous metrics to the Schwarzschild-de Sitter metric, the Nariai solution, and a hyperbolically foliated solution.
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