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Globally hyperbolic spacetimes-with-timelike-boundary are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if is obtained by means of a conformal embedding) can be posed. represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of , the splitting of any globally hyperbolic as an orthogonal product with Cauchy slices-with-boundary is proved. This is obtained by constructing a Cauchy temporal function~ with gradient tangent to on the boundary. To construct such a~, 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 . The techniques also show that is isometric to the closure of some open subset in a globally hyperbolic spacetime (without boundary). As a trivial consequence, the interior both splits orthogonally and can be embedded isometrically in some , extending so properties of globally spacetimes without boundary to a class of causally continuous ones.
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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. 45–94DOI 10.4171/RMI/1201