A Lorentzian Gromov–Hausdorff notion of distance

@article{Noldus2003ALG,
  title={A Lorentzian Gromov–Hausdorff notion of distance},
  author={Johan Noldus},
  journal={Classical and Quantum Gravity},
  year={2003},
  volume={21},
  pages={839-850}
}
  • Johan Noldus
  • Published 22 August 2003
  • Mathematics
  • Classical and Quantum Gravity
This paper is the first of three in which I study the moduli space of isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I introduce a notion of Gromov?Hausdorff distance which makes this moduli space into a metric space. Further properties of this metric space are studied in the next two papers. The importance of the work is in fields such as cosmology, quantum gravity and?for the mathematicians?global Lorentzian geometry. 

The moduli space of isometry classes of globally hyperbolic spacetimes

This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov–Hausdorff theory based on the use of the Lorentz distance as

The moduli space of isometry classes of globally hyperbolic spacetimes

This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov–Hausdorff theory based on the use of the Lorentz distance as

On generalizations of the Lorentzian distance function Contents

One of the most relevant objects of my study was the Lorentzian distance and some related concepts. This is, however confusingly, not a distance. It’s usual definition in the setting of a smooth

The limit space of a Cauchy sequence of globally hyperbolic spacetimes

In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In section 2, I work gradually towards a construction of the limit space. I prove that the

Lorentzian metric spaces and their Gromov-Hausdorff convergence

We present an abstract approach to Lorentzian Gromov-Hausdorff distance and convergence. We begin by defining a notion of (abstract) bounded Lorentzian-metric space which is sufficiently general to

Properties of the Null Distance and Spacetime Convergence

The null distance for Lorentzian manifolds was recently introduced by Sormani and Vega. Under mild assumptions on the time function of the spacetime, the null distance gives rise to an intrinsic,

How Riemannian Manifolds Converge

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We

ON THE GEOMETRY OF LORENTZ SPACES AS A LIMIT SPACE

Abstract. In this paper, we prove that there is no branch point in theLorentz space (M,d) which is the limit space of a sequence {(M α ,d α )}ofcompact globally hyperbolic interpolating spacetimes

How Riemannian Manifolds Converge: A Survey

This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We

Minkowski space is locally the Noldus limit of Poisson process causets

A Poisson process P λ on R d with causal structure inherited from the the usual Minkowski metric on R d has a normalised discrete causal distance D λ ( x , y ) given by the height of the longest

References

SHOWING 1-10 OF 15 REFERENCES

The moduli space of isometry classes of globally hyperbolic spacetimes

This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov–Hausdorff theory based on the use of the Lorentz distance as

The origin of Lorentzian geometry

A new topology on the space of Lorentzian metrics on a fixed manifold

We give a covariant definition of closeness between (time-oriented) Lorentzian metrics on a manifold M, using a family of functions which measure the difference in volume form on one hand and, on the

The limit space of a Cauchy sequence of globally hyperbolic spacetimes

In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In section 2, I work gradually towards a construction of the limit space. I prove that the

Global Lorentzian Geometry

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

Statistical Lorentzian geometry and the closeness of Lorentzian manifolds

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with

Metric Structures for Riemannian and Non-Riemannian Spaces

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-

A new topology for curved space–time which incorporates the causal, differential, and conformal structures

A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal,

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper

Introduction to Modern Canonical Quantum General Relativity

This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the