Hardy inequalities and uncertainty principles in the presence of a black hole

In this paper, we establish Hardy and Heisenberg uncertainty-type inequalities for the exterior of a Schwarzschild black hole. The weights that appear in both inequalities are tailored to fit the geometry, and can both be compared to the related Riemannian distance from the event horizon to yield inequalities for that distance. Moreover, in both cases the classic Euclidean inequalities with a point singularity can be recovered in the limit where one stands “far enough” from the black hole, as expected from the asymptotic flatness of the metric.