THE SCALAR WAVE EQUATION IN A NON-COMMUTATIVE SPHERICALLY SYMMETRIC SPACE-TIME

Abstract

Recent work in the literature has studied a version of non-commutative Schwarzschild black holes where the effects of non-commutativity are described by a mass function depending on both the radial variable r and a non-commutativity parameter θ. The present paper studies the asymptotic behavior of solutions of the zero-rest-mass scalar wave equation in such a modified Schwarzschild space-time in a neighborhood of spatial infinity. The analysis is eventually reduced to finding solutions of an inhomogeneous Euler–Poisson–Darboux equation, where the parameter θ affects explicitly the functional form of the source term. Interestingly, for finite values of θ, there is full qualitative agreement with general relativity: the conformal singularity at spacelike infinity reduces in a considerable way the differentiability class of scalar fields at future null infinity. In the physical space-time, this means that the scalar field has an asymptotic behavior with a fall-off going on rather more slowly than in flat space-time.