## 2 The Admisssible Lorentzian Metric  
### The Underlying Hyperbolic Geometry  
As indicated in previous section, the spacetime manifold we denote by \(\mathcal{M}\) is an emergent geometric structure arising somehow \"naked\", i.e., neither time orientation nor any Lorentzian metric are really defined on the emergent spacetime manifold \(\mathcal{M}\) from localization; appart the fundamental, projective ground structure. Nevertheless, we can start with a few following geometric assumptions about the geometric structure of \(\mathcal{M}\)[22]: (1) The spacetime manifold \(\mathcal{M}\) is _time-orientable_ and _simply connected_, (2) we provide \(\mathcal{M}\) with an Euclidean metric \(ds^{2}\) such that \((\mathcal{M},ds^{2})\) is the four-dimensional _hyperbolic space_\(\mathcal{H}^{4}\subset P\mathbb{R}^{4}\), (3) to each event \(e\in\mathcal{M}\), a system of four _Riemann normal coordinates_\((u^{i}_{p})\equiv p\in\mathcal{M}\) can be attached such that \(u^{\alpha}_{e}=0\) (\(i=1,\ldots,4\))2, (4) the coordinates \(u^{i}\) are assumed to be the _inhomogeneous coordinates_ of the local projective geometry of \(\mathcal{H}^{4}\), and (5) to the inhomogeneous coordinates \(u^{i}\) there corresponds five _homogeneous coordinates_\(U^{\alpha}\) (\(\alpha=0,1,\ldots,4\)). Footnote 2: Considering the _cell decomposition_\(P\mathbb{R}^{4}\equiv\mathbb{R}^{4}\cup P\mathbb{R}^{3}\) of \(P\mathbb{R}^{4}\), then we identify \(\mathbb{R}^{4}\) with \(\mathcal{M}\) such that \(P\mathbb{R}^{4}\simeq\mathcal{M}\cup P\mathbb{R}^{3}\). Unlike what is usually done in such similar circumstances, it is an \"underlying\" _hyperbolic space_\(\mathcal{H}^{4}\) that is considered here because this space is simply connected contrarily to _anti-de Sitter spaces_\(AdS_{n}\) ; all the more so as anti-de Sitter spaces are not subspaces of projective spaces \(P\mathbb{R}^{n}\) but of Euclidean spaces \(\mathbb{R}^{n+1}\). We recall that \(\mathcal{H}^{4}\) is defined from a quadratic form \(Q\) and a metric \(dS^{2}\) on \(\mathbb{R}^{5}\) such that \(Q(U)\equiv(U^{0})^{2}-\sum_{i=1}^{4}(U^{i})^{2}\) and \(dS^{2}\equiv\sum_{i=1}^{4}(dU^{i})^{2}-(dU^{0})^{2}\). More precisely, we have the following. **Definition 1**: _The hyperbolic space \(\mathcal{H}^{4}\subset P\mathbb{R}^{4}\) is the open set of lines in \(\mathbb{R}^{5}\) generated by points \(U\in\mathbb{R}^{5}\) such that \(Q(U)>0\) and equiped with the metric \(ds^{2}\) (see below)._  
Futhermore, defining from \(Q\) the section \(\sigma:(u^{i})\in\mathbb{R}^{4}\longrightarrow(U^{\alpha})\in\mathbb{R}^{5}\) such that \(U^{0}=\sigma(u)^{0}\equiv\sqrt{1+\sum_{i=1}^{4}(u^{i})^{2}}\) and \(U^{i}=\sigma(u)^{i}\equiv u^{i}\) (\(i=1,\ldots,4\)), then, the Euclidean metric \(ds^{2}\) can be defined as the pull-back of \(dS^{2}\) by \(\sigma\), i.e., \(ds^{2}\equiv\sigma^{*}(dS^{2})\). Then, we obtain  
\[ds^{2}=\frac{\sum_{i=1}^{4}(du^{i})^{2}+\sum_{i=1<j}^{4}(u^{i}\,du^{i}-u^{i}\, du^{j})^{2}}{1+\sum_{i=1}^{4}(u^{i})^{2}}\, \tag{1}\]  
which is the Euclidean metric defined on \(\mathbb{R}^{4}\simeq\mathcal{M}\) and defining completely \(\mathcal{H}^{4}\) as a Riemannian manifold embedded in \(P\mathbb{R}^{4}\). To complete the description of \(\mathcal{H}^{4}\), we ought to mention that the _restricted Lorentz group_\(SO^{+}(1,4)\subset PGL(5,\mathbb{R})=SL(5,\mathbb{R})\) acts _transitively_ on \(\mathcal{H}^{4}\) via _homographies_\([\mathfrak{A}]\) defined from linear actions \(\mathfrak{A}\equiv(\mathfrak{A}^{\alpha}_{\beta})\) on \(\mathbb{R}^{5}\).