# The representation of spacetime through steep time functions

@article{Minguzzi2017TheRO, title={The representation of spacetime through steep time functions}, author={Ettore Minguzzi}, journal={Journal of Physics: Conference Series}, year={2017}, volume={968} }

In a recent work I showed that the family of smooth steep time functions can be used to recover the order, the topology and the (Lorentz-Finsler) distance of spacetime. In this work I present the main ideas entering the proof of the (smooth) distance formula, particularly the product trick which converts metric statements into causal ones. The paper ends with a second proof of the distance formula valid for globally hyperbolic Lorentzian spacetimes.

## 3 Citations

### Global hyperbolicity through the eyes of the null distance

- Mathematics
- 2022

No Hopf–Rinow Theorem is possible in Lorentzian Geometry. Nonethe-less, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of…

### The Lorentzian distance formula in noncommutative geometry

- Mathematics
- 2017

For almost twenty years, a search for a Lorentzian version of the well-known Connes’ distance formula has been undertaken. Several authors have contributed to this search, providing important…

### Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

- MathematicsJournal of Mathematical Physics
- 2018

An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space)…

## References

SHOWING 1-10 OF 22 REFERENCES

### On the existence of smooth Cauchy steep time functions

- Mathematics
- 2016

A simple proof (based on results in Chruściel et al 2015 Ann. Henri Poincaré arXiv:1301.2909) is given that every globally hyperbolic spacetime admits a smooth Cauchy steep time function. This result…

### Global Hyperbolicity for Spacetimes with Continuous Metrics

- Mathematics
- 2014

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while…

### Connes duality in pseudo-Riemannian geometry

- Mathematics
- 2000

The Connes formula giving the dual description for the distance between points of a Riemannian manifold is extended to the Lorentzian case. It resulted that its validity essentially depends on the…

### On smooth time functions

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2011

Abstract We are concerned with the existence of smooth time functions on connected time-oriented Lorentzian manifolds. The problem is tackled in a more general abstract setting, namely in a manifold…

### Global hyperbolicity is stable in the interval topology

- Mathematics
- 2011

We prove that global hyperbolicity is stable in the interval topology on the spacetime metrics. We also prove that every globally hyperbolic spacetime admits a Cauchy hypersurface which remains…

### ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES

- Mathematics
- 2003

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a…

### On Differentiability of Volume Time Functions

- Mathematics
- 2016

We show differentiability of a class of Geroch’s volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over…

### Causality theory for closed cone structures with applications

- MathematicsReviews in Mathematical Physics
- 2019

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular…

### The existence of cosmic time functions

- MathematicsProceedings 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…

### Noncommutative Geometry

- Mathematics
- 1997

Noncommutative Spaces It was noticed a long time ago that various properties of sets of points can be restated in terms of properties of certain commutative rings of functions over those sets. In…