Black holes and a scalar field in an expanding universe

We consider a model of an inhomogeneous universe with the presence of a mas sless scalar field, where the inhomogeneity is assumed to consist of many b lack holes. This model can be constructed by following Lindquist and Wheele r, which has already been investigated without the presence of a scalar fie ld to show that an averaged scale factor coincides with that of the Friedma nn model in Einstein gravity. In this paper we construct the inhomogeneous universe with a massless scalar field, where it is assumed that the average d scale factor and scalar field are given by those of the Friedmann model i ncluding the scalar field. All of our calculations are carried out within t he framework of Brans-Dicke gravity. In constructing the model of an inhomo geneous universe, we define the mass of a black hole in the Brans-Dicke exp anding universe which is equivalent to the ADM mass in the epoch of the adi abatic time evolution of the mass, and obtain an equation relating our mass with the averaged scalar field and scale factor. We find that the mass has an adiabatic time dependence in a sufficiently late stage of the expansion of the universe; that is our mass is equivalent to the ADM mass. The other result is that its time dependence is qualitatively different according to the sign of the curvature of the universe: the mass increases (deceleratin g) in the closed universe case, is constant in the flat case and decreases (decelerating) in the open case. It is also noted that the mass in the Eins tein frame depends on time. Our results that the mass has a time dependence should be retained even in the general scalar-tensor gravities with a scal ar field potential. Furthermore, we discuss the relation of our model of th e inhomogeneous universe to the uniqueness theorem of black hole spacetime and the gravitational memory effect of black holes in scalar-tensor graviti es.