# Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes

@article{Bernal2005SmoothnessOT, title={Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes}, author={Antonio N. Bernal and Miguel Grau S{\'a}nchez}, journal={Communications in Mathematical Physics}, year={2005}, volume={257}, pages={43-50} }

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 whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function with timelike gradient on all M.

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#### References

SHOWING 1-10 OF 12 REFERENCES

Global Lorentzian Geometry

- Mathematics
- 1981

Introduction - Riemannian themes in Lorentzian geometry connections and curvature Lorentzian manifolds and causality Lorentzian distance examples of space-times completness and extendibility… Expand

Causal hierarchy of spacetimes, temporal functions and smoothness of Geroch's splitting. A revision

- Physics, Mathematics
- 2004

After the heroic epoch of Causality Theory, problems concerning the smoothability of time functions and Cauchy hypersurfaces remained as unanswered folk questions. Just recently solved, our aim is to… Expand

On Smooth Cauchy Hypersurfaces and Geroch’s Splitting Theorem

- Mathematics, Physics
- 2003

Given a globally hyperbolic spacetime M, we show the existence of a smooth spacelike Cauchy hypersurface S and, thus, a global diffeomorphism between M and ℝ×S.

The domain of dependence

- Mathematics
- 1970

The various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for… Expand

The Large Scale Structure of Space-Time

- Physics, Computer Science
- 1973

The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. Expand

The existence of cosmic time functions

- Physics
- Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- 1969

It is shown that stable causality is the necessary and sufficient condition that there should exist a cosmic time function which increases along every future directed timelike or null curve. Stable… Expand

General relativity and cosmology

- Mathematics
- 1977

A broad survey of the mathematics of general relativity theory is presented. Among the topics covered are Einstein field equations, cosmological models, black holes, space-times, and… Expand

Global Lorentzian Geometry. Monographs Textbooks Pure Appl. Math

- Global Lorentzian Geometry. Monographs Textbooks Pure Appl. Math
- 1996

Global Lorentzian geometry, Monographs Textbooks

- Pure Appl. Math
- 1996