Tolman-Bondi collapse in scalar-tensor theories as a probe of gravitational memory

In cosmological models with a varying gravitational constant, it is not clear whether primordial black holes preserve the value of $G$ at their formation epoch. We investigate this question by using the Tolman-Bondi model to study the evolution of a background scalar field when a black hole forms from the collapse of dust in a flat Friedmann universe. Providing the back reaction of the scalar field on the metric can be neglected, we find that the value of the scalar field at the event horizon very quickly assumes the background cosmological value. This suggests that there is very little gravitational memory.