Possibility of a static scalar field in the Schwarzschild geometry

Some effects of a nonlinear coupling, ${\mathfrak{L}}_1 = -\lambda {\varphi }^4$, on a massless scalar field in a Schwarzschild geometry are studied. There exists a classical time-dependent solution (probably unique) which blows up as $4^{-1\mathrm{}}{\left(2\mathrm{}MG\right)}^{\frac{-1\mathrm{}}{2}}{\lambda }^{\frac{-1\mathrm{}}{2}}{\mathrm{}(r-2\mathrm{MG})\mathrm{}}^{\frac{-1\mathrm{}}{2}}$ near the event horizon and goes to zero at infinity faster than $r^{-1\mathrm{}}$. It is argued that (a) such behavior is admissible up to some very short distance from the horizon, at which point small couplings to other fields should come into play, and (b) a field which at some initial time has an arbitrarily small singular amplitude will develop in time into this classical solution. The effective potential for the propagation of scalar waves is drastically modified in the presence of the classical background field. This should lead to significant changes in the rate of emission of scalar quanta from a black hole.