JournalsrmiVol. 37, No. 1pp. 45–94

Structure of globally hyperbolic spacetimes-with-timelike-boundary

  • Luis Aké Hau

    Universidad Autónoma de Yucatán, Mérida, Mexico
  • José Luis Flores Dorado

    Universidad de Málaga, Spain
  • Miguel Sánchez Caja

    Universidad de Granada, Spain
Structure of globally hyperbolic spacetimes-with-timelike-boundary cover
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Abstract

Globally hyperbolic spacetimes-with-timelike-boundary (M=MM,g)(\overline{M} = M \cup \partial M, g) are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if M\partial M is obtained by means of a conformal embedding) can be posed. M\partial M represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of M\partial M, the splitting of any globally hyperbolic (M,g)(\overline{M},g) as an orthogonal product R×Σˉ\mathbb{R}\times \bar\Sigma with Cauchy slices-with-boundary {t}×Σˉ\{t\}\times \bar\Sigma is proved. This is obtained by constructing a Cauchy temporal function~τ\tau with gradient τ\nabla \tau tangent to M\partial M on the boundary. To construct such a~τ\tau, results on stability of both global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by M\partial M. The techniques also show that M\overline{M} is isometric to the closure of some open subset in a globally hyperbolic spacetime (without boundary). As a trivial consequence, the interior MM both splits orthogonally and can be embedded isometrically in some LN\mathbb{L}^N, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.

Cite this article

Luis Aké Hau, José Luis Flores Dorado, Miguel Sánchez Caja, Structure of globally hyperbolic spacetimes-with-timelike-boundary. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 45–94

DOI 10.4171/RMI/1201