Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 1

Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes

Pages: 25 – 45



I. P. Costa e Silva (Department of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil)

J. L. Flores (Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Spain)

J. Herrera (Departamento of Matemáticas, Edificio Albert Einstein, Universidad de Córdoba, Spain)


We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which in turn have been intensely and fruitfully studied in the mathematical and physical literatures for over half a century. More concretely, we prove a restricted version of a conjectural analogue for Brinkmann spacetimes of a rigidity result obtained by M.T. Anderson for stationary spacetimes.We also highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt. Indeed, it turns out that the sub-class of Brinkmann spacetimes we consider in our main theorem is enough to settle an important special case of the Ehlers–Kundt conjecture in terms of the well known class of Cahen–Wallach spaces.

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The authors are partially supported by the Spanish Grant MTM2013-47828-C2-2-P (MINECO and FEDER funds). JLF is also supported by the Regional J. Andalucía grant P09-FQM-4496, with FEDER funds. IPCS wishes to acknowledge a research scholarship from CNPq, Brazil, Programa Ciência sem Fronteiras, process number 200428/2015-2.