From parabolic to loxodromic BMS transformations

  • Giampiero EspositoEmail author
  • Francesco Alessio
Research Article


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.


Fractional linear transformations Loxodromic Bondi–Metzner–Sachsgroup Limit-point limit-circle criterion 



The authors are grateful to the Dipartimento di Fisica “Ettore Pancini” of Federico II University for hospitality and support. G. Esposito is grateful to V. F. Bellino for conversations [39].


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Authors and Affiliations

  1. 1.INFN Sezione di Napoli, Complesso Universitario di Monte S. AngeloNaplesItaly
  2. 2.Dipartimento di Fisica “Ettore Pancini”Federico II University, Complesso Universitario di Monte S. AngeloNaplesItaly

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