On the space of Lorentz metrics

This research it is divided into 5 parts: Chapter 1, is the introduction and research purpose of this article. part 2 , Lorentz introduces the basic concept of space and The third part is the dual space and {ID}_1^3 dual Lorentz space defines it. In Chapter 4, {IR}_1^3 and {ID}_1^n n-dimensional dual Lorentz spaces are mentioned. In the fifth chapter, the space of Lorentz metrics, which is the main purpose of the thesis, is examined. For Example topics such as Einstein equations, space-tims, global properties, G-singularities are discussed. This study is based on David E. Lerner's work titled "Lorentz metric spaces". This study considers the set of all 𝐶² Lorentz metrics on the non-compact 4-manifold. These metrics are important elements that define the physical space-time structure of the manifold. The set of Lorentz metrics on the manifold, Whitney fine 𝐶² with the topology. This topology defines the topology of the sections between the vector fields defined on the manifold. This definition indicates that Lorentz metrics reflect the topological properties of the manifold and provide an accurate framework of the space-time structure we want to analyze. In particular, it provides an important basis for discussing the global properties and singularity theorems of space-like manifolds. Space-like manifolds are among the structures that describe the physical properties of space-time and play an important role in the general theory of relativity. This work provides an accurate framework for studying the general properties of these manifolds and for a deeper understanding of the singularity theorems. As a main conclusion, a theorem is presented which deals with the behavior of Lorentz metrics of the Robertson-Walker big bang (global infinite density singularity in the finite past). This theorem states that the metric tensor is sufficiently small, but otherwise random, finite 𝐶² It shows that it is stable under perturbations. This result is an important step towards understanding how the big bang affected the general behavior of the universe and how singularities were formed. It contributes to a better understanding of the space-time structure with analyzes and results in the space of Lorentz metrics. This topic inspires in-depth study and research in fields such as general relativity and cosmology.