On the non-blow up of energy critical nonlinear massless scalar fields in ‘$3+1$’ dimensional globally hyperbolic spacetimes: light cone estimates

Abstract

Here we prove a global existence theorem for the solutions of the semi-linear wave equation with critical non-linearity admitting a positive definite Hamiltonian. Formulating a parametrix for the wave equation in a globally hyperbolic curved spacetime, we derive an apriori pointwise bound for the solution of the nonlinear wave equation in terms of the initial energy, from which the global existence follows in a straightforward way. This is accomplished in two steps. First, based on Moncrief's light cone formulation we derive an expression for the scalar field in terms of integrals over the past light cone from an arbitrary spacetime point to an `initial', Cauchy hypersurface and additional integrals over the intersection of this cone with the initial hypersurface. Secondly, we obtain apriori estimates for the energy associated with three quasi-local approximate time-like conformal Killing and one approximate Killing vector fields. Utilizing these naturally defined energies associated with the physical stress-energy tensor together with the integral equation, we show that the spacetime $L^{\infty}$ norm of the scalar field remains bounded in terms of the initial data and continues to be so as long as the spacetime remains singularity/Cauchy-horizon free.