Nonlinear scalar field dynamics in the Schwarzschild geometry

The dynamics of the nonlinear massless scalar field described by $\square \Phi =4\mathrm{}\lambda {\Phi }^3$ is numerically studied in the Schwarzschild geometry. It is found that such a scalar field is entirely radiated away during the gravitational collapse of a scalar charged star. Sawyer has argued that a quantum fluctuation can give rise to a quasistatic component of the scalar field which will trap subsequent scalar radiation and grow to a singular field. A phenomenological model used to study this situation leads to no such singular growth. All results, in fact, suggest that the dynamics of weak nonlinear fields in a black-hole background is qualitatively similar to that of linear fields.