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 2×2 matrix in the projective version of the SL(2,C) group. The resulting particular forms of SL(2,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∞, 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.
From parabolic to loxodromic BMS transformations / Esposito, G; Alessio, F. - In: GENERAL RELATIVITY AND GRAVITATION. - ISSN 0001-7701. - 50:11(2018), pp. 141-1-141-13. [10.1007/s10714-018-2465-2]
From parabolic to loxodromic BMS transformations
ESPOSITO GPrimo
;
2018
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 2×2 matrix in the projective version of the SL(2,C) group. The resulting particular forms of SL(2,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∞, 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.File | Dimensione | Formato | |
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