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Register a new userSmooth globally hyperbolic splittings and temporal functions
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Gerochs theorem about the splitting of globally hyperbolic spacetimes is a central result in global Lorentzian Geometry. Nevertheless, this result was obtained at a topological level, and the possibility to obtain a metric (or, at least, smooth) version has been controversial since its publication in 1970. In fact, this problem has remained open until a definitive proof, recently provided by the authors. Our purpose is to summarize the history of the problem, explain the smooth and metric splitting results (including smoothability of time functions in stably causal spacetimes), and sketch the ideas of the solution.
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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 $M$.
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.
Given a globally hyperbolic spacetime $M$, we show the existence of a {em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $R times S$.
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.
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.
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