Black holes and a scalar field in an expanding

We consider a model of inhomogeneous universe with the presence of a massless scalar field, where the inhomogeneity is assumed to consist of many black holes. This model can be constructed by following Lindquist and Wheeler, which has already been investigated without the presence of scalar field to show that an averaged scale factor coincides with that of the Friedmann model in Einstein gravity. In this paper we construct the inhomogeneous universe with a massless scalar field, where it is assumed that the averaged scale factor and scalar field are given by those of the Friedmann model including the scalar field. All of our calculations are carried out within the framework of Brans-Dicke gravity. In constructing the model of an inhomogeneous universe, we define the mass of a black hole in the Brans-Dicke expanding universe which is equivalent to the ADM mass in the epoch of the adiabatic time evolution of the mass, and obtain an equation relating our mass with the averaged scalar field and scale factor. As the results 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 ADM mass. The other result is that its time dependence is qualitatively diffenrent according to the sign of the curvature of the universe: the mass increases in a decelerating fashion in the closed universe case, is constant in the flat case, and decreases in a decelerating fashion in the open case. It is also noted that the mass in the Einstein 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 scalar field potential. Furthermore, we discuss the relation of our model of the inhomogeneous universe with the uniqueness theorem of black hole spacetime and the gravitational memory effect of black holes in scalar-tensor gravities. PACS: 04.70.-s, 04.70.Bw, 98.80.Hw E-mail: [email protected] E-mail: [email protected]