From parabolic to loxodromic BMS transformations

Abstract

Half of the Bondi–Metzner–Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Möbius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated $$2 \times 2$$ matrix in the projective version of the $$SL(2,\mathbb {C})$$ group. The resulting particular forms of $$SL(2,\mathbb {C})$$ matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches $$\infty $$ , correspond to the fulfillment of limit-point condition for singular Sturm–Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics.