Infinitesimal relative position vector fields for observers in a reference frame and applications to conformally stationary spacetimes

Abstract

We introduce and analyze the concept of infinitesimal relative position vector field between “infinitesimally nearby” observers, showing the equivalence between different definitions. Through the Fermi–Walker derivative of infinitesimal relative position vector fields along an observer in a reference frame, we characterize spacetimes admitting an umbilic foliation. Sufficient and necessary conditions for those spacetimes to be a conformally stationary spacetime are given. Finally, the important class of cosmological models known as generalized Robertson–Walker spacetimes is characterized.

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Correspondence to Magdalena Caballero.

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The authors are partially supported by Spanish MINECO and FEDER Project MTM2016-78807-C2-1-P.

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Caballero, M., de la Fuente, D. & Rubio, R.M. Infinitesimal relative position vector fields for observers in a reference frame and applications to conformally stationary spacetimes. Anal.Math.Phys. 9, 1977–1990 (2019). https://doi.org/10.1007/s13324-019-00293-y

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Keywords

  • Irrotational vector fields
  • Conformal Killing vector fields
  • Spatially conformal Killing vector fields
  • Observers
  • Fields of observers

Mathematics Subject Classification

  • 53C50
  • 53B50
  • 83C99