Black Hole Area in Brans-Dicke Theory

We have shown that the dynamics of the scalar field $\varphi (x)=\mathrm{‘}\mathrm{‘}{G}^{-1}(x)\mathrm{"}$ in Brans-Dicke theories of gravity makes the surface area of the black hole horizon {\it 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 non-decreasing 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 have been demonstrated for Oppenheimer-Snyder collapse in Brans-Dicke theory.