Two-particle quantum mechanics in gravity using non-commuting coordinates

We find that the momentum conjugate to the relative distance between two gravitating particles in their centre-of-mass frame is a hyperbolic angle. This fact suggests that momentum space can be defined consistently on a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non-commuting Hermitian operators. We also find that there is a smallest distance between the two particles of one quarter of the Planck length.