Abstract:

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 spacetime

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.

Key words: Brans-Dicke's theory of gravitation, The weak-field approximation, The linearized field equations