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Register a new userGlobally hyperbolic spacetimes can be defined without the causal condition
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Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non $C^{1,1}$ metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.
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Globally hyperbolic spacetimes with timelike boundary $(overline{M} = M cup partial M, g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $overline{M}$ is obtained by means of a conformal embedding) can be posed. $partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $partial M$, the splitting of any globally hyperbolic $(overline{M},g)$ as an orthogonal product ${mathbb R}times bar{Sigma}$ with Cauchy slices with boundary ${t}times bar{Sigma}$ is proved. This is obtained by constructing a Cauchy temporal function $tau$ with gradient $ abla tau$ tangent to $partial M$ on the boundary. To construct such a $tau$, 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 $partial M$. As a consequence, the interior $M$ both, splits orthogonally and can be embedded isometrically in ${mathbb L}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.
In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold $M$ as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the underlying algorithmic construction in the language of ultrastatic Lorentzian manifolds. Subsequently we show that the construction carries over to the case of static backgrounds thanks to a suitable reduction to the ultrastatic scenario. Finally we prove that the overall procedure can be generalised to any globally hyperbolic spacetime with compact Cauchy surfaces. As an application, we discuss how, from our procedure, one can recover the local Hadamard expansion which plays a key role in all applications in quantum field theory on curved backgrounds.
The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $tau$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= R times {cal S}$, $g= - beta(tau,x) dtau^2 + bar g_tau $, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $tau$ with timelike gradient $ abla tau$ on all $M$.
Some recent results obtained by the author and collaborators about QFT in asymptotically flat spacetimes at null infinity are summarized and reviewed. In particular it is focused on the physical properties of ground states in the bulk induced by the BMS-invariant state defined at null infinity.
We consider pseudoconvexity properties in Lorentzian and Riemannian manifolds and their relationship in static spacetimes. We provide an example of a causally continuous and maximal null pseudoconvex spacetime that fails to be causally simple. Its Riemannian factor provides an analogous example of a manifold that is minimally pseudoconvex, but fails to be convex.
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