Black hole area in Brans-Dicke theory

We show that the dynamics of the scalar field $\phi \mathrm{}\left(x\right)=``G^{-1\mathrm{}}\mathrm{}\left(x\right)\text{'}\text{'}$ in Brans-Dicke theories of gravity makes the surface area of the black hole horizon oscillatory during its dynamical evolution. It explicitly explains why the area theorem does not hold in Brans-Dicke theory. However, we show that there exists a certain nondecreasing quantity defined on the event horizon which is proportional to the black hole entropy for the case of stationary solutions in Brans-Dicke theory. Some numerical simulations are demonstrated for Oppenheimer-Snyder collapse in Brans-Dicke theory.