Brans-Dicke引力理论的弱场近似

Weak-field approximation of Brans-Dicke’s theory of gravitation

Brans-Dicke引力理论是重要的修正引力理论之一,其对于研究星体运动、宇宙演化以及解释宇宙现象等起着重要的作用.寻找Brans-Dicke引力理论场方程的解对于Brans-Dicke引力理论的研究和发展具有重要意义.但由于Brans-Dicke引力理论场方程本身的高度非线性性,使得一般情况下精确求解非常困难,特殊情况下也只能求得部分精确解.幸运的是大多数情况下引力场比较弱,且在低速的条件下求场方程的近似解相对容易.本文基于弱场低速条件,详细地求解了Brans-Dicke引力理论的弱场低速近似解.首先基于弱场条件,将标量场和度规写为一阶微扰展开的形式;然后将标量场和度规代入相应的场方程得到相应的线性场方程,通过选取特定的规范条件进一步简化线性场方程;最后求解出简化的线性场方程的低速近似解.本文的求解方法可以为Brans-Dicke引力理论的教学和研究提供有益的参考.

As one of the most important modified theories of gravity, Brans-Dicke’s theory of gravitation plays an important role in studying the motions of stars and cosmic evolution, as well as in explaining cosmic phenomena. There is no doubt that solutions to Brans-Dicke’s field equations are of great significance in this theory. However, it is very difficult to obtain exact solutions to the Brans-Dicke’s field equations in general cases because of the non-linearity of these equations, although some special solutions can be obtained in some cases. Fortunately, in these cases that the gravitational field is weak and the source of the gravitational field moves slowly, one can easily find approximate solutions to these field equations. In this paper, we aim at the weak-field approximate solutions to Brans-Dicke’s field equations. Firstly, based on the weak gravitational field, we expand the Brans-Dicke’s scalar field and spacetime metric in the corresponding background fields. Then, we substitute the scalar field and space-time metric into Brans-Dicke’s field equations to get these corresponding linearized field equations, and we further simplify one of them using a specific gauge condition. Finally, we solve the linearized field equations under the slow-motion condition, and obtain the weak-field, slow-motion approximate solutions to Brans-Dicke’s field equations. These approximation solutions obtained in this paper can provide useful reference for the teaching and research in Brans-Dicke’s theory of gravitation.