Responses of the Brans–Dicke field due to gravitational collapses

Abstract

We study responses of the Brans–Dicke field due to gravitational collapses of scalar field pulses using numerical simulations. Double-null formalism is employed to implement the numerical simulations. If we supply a scalar field pulse, it will asymptotically form a black hole via dynamical interactions of the Brans–Dicke field. Hence, we can observe the responses of the Brans–Dicke field by two different regions. First, we observe the late time behaviors after the gravitational collapse, which include formations of a singularity and an apparent horizon. Second, we observe the fully dynamical behaviors during the gravitational collapse and view the energy–momentum tensor components. For the late time behaviors, if the Brans–Dicke coupling is greater (or smaller) than −1.5, the Brans–Dicke field decreases (or increases) during the gravitational collapse. Since the Brans–Dicke field should be relaxed to the asymptotic value with the elapse of time, the final apparent horizon becomes time-like (or space-like). For the dynamical behaviors, we observed the energy–momentum tensors around ω ∼ −1.5. If the Brans–Dicke coupling is greater than −1.5, the Tuu component can be negative at the outside of the black hole. This can allow an instantaneous inflating region during the gravitational collapse. If the Brans–Dicke coupling is less than −1.5, the oscillation of the Tvv component allows the apparent horizon to shrink. This allows a combination that violates weak cosmic censorship. Finally, we discuss the implications of the violation of the null energy condition and weak cosmic censorship.