linear independence of field equations in the brans-dicke theory

In solving the brans-dicke (bd) equations in the bd theory of gravity, their linear independence is important. this is due to fact that in solving these equations in cosmology, if the number of unknown quantities is equal to the number of independent equations, then the unknowns can be uniquely determined. in the bd theory, the tensor field gµν and the bd scalar field ϕ are not two separate fields, but they are coupled together. the reason behind this is a corollary that proposed by v. b. johri and d. kalyani in cosmology, which states that the cosmic scale factor of the universe, a, and the bd scalar field ϕ are related by a power law. therefore, when the principle of least action is used to derive the bd equations, the variations δgµν and δϕ should not be considered as two independent dynamical variables. so, there is a constraint on δgµν and δϕ that causes the number of independent bd equations to decrease by one unit, in such a way that in the equations that have been known as bd equations, one of them is redundant. in this paper, we prove this issue, that is, we show that one of these equations, which we choose as the modified klein-gordon equation, is not an independent equation, but a result establishing other bd equations, the law of conservation of energy-momentum of matter and bianchi's identity. therefore, we should not look at the modified klein-gordon equation as an independent field equation in the bd theory, but rather it is included in the other bd equations and should not be mentioned separately as one of the bd equations once again.