The scalar wave equation in a non-commutative spherically symmetric space-time. (English) Zbl 1192.83041
Summary: 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 \(\theta\). 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 \(\theta\) affects explicitly the functional form of the source term. Interestingly, for finite values of \(\theta\), there is full qualitative agreement with general relativity: the conformal singularity at space-like 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.
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