Abstract
We consider a model of an inhomogeneous universe with the presenceof a massless scalar field, where the inhomogeneity is assumed toconsist of many black holes. This model can be constructed byfollowing Lindquist and Wheeler, which has already beeninvestigated without the presence of a scalar field to show that anaveraged scale factor coincides with that of the Friedmann model inEinstein gravity. In this paper we construct the inhomogeneousuniverse with a massless scalar field, where it is assumed that theaveraged scale factor and scalar field are given by those of theFriedmann model including the scalar field. All of our calculationsare carried out within the framework of Brans-Dicke gravity. Inconstructing the model of an inhomogeneous universe, we define the massof a black hole in the Brans-Dicke expanding universe which isequivalent to the ADM mass in the epoch of the adiabatic time evolutionof the mass, and obtain an equation relating our mass with theaveraged scalar field and scale factor. We find thatthe mass has an adiabatic time dependence in a sufficiently late stageof the expansion of the universe; that is our mass is equivalent tothe ADM mass. The other result is that its time dependence isqualitatively different according to the sign of the curvature ofthe universe: the mass increases (decelerating) in the closed universecase, is constant in the flat case and decreases (decelerating) in the opencase. It is also noted that the mass in the Einstein framedepends on time. Our results that the mass has a time dependenceshould be retained even in the general scalar-tensor gravities witha scalar field potential. Furthermore, we discuss the relation ofour model of the inhomogeneous universe to the uniqueness theorem ofblack hole spacetime and the gravitational memory effect of blackholes in scalar-tensor gravities.
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