## 2 The Admisssible Lorentzian Metric
### The Pseudo-Hyperbolic Space and the Admissible Lorentzian Metric  
To recover an admissible Lorentzian metric \(d\tau^{2}\) instead of the Euclidean metric \(ds^{2}\) on \(\mathcal{M}\), we use the assumption that \(\mathcal{M}\) is time orientable, i.e., we can find a nowhere non-vanishing, everywhere defined vector field on \(\mathcal{M}\). We assume that this vector field exists and that it is the dual of the differential 1-form \(du^{1}\in T^{*}\mathcal{M}\). Then, \(\mathcal{M}\) becomes time-oriented and \(d\tau^{2}\) is defined by formula: \(d\tau^{2}\equiv(du^{1})^{2}-ds^{2}\). Also, \(d\tau^{2}\) is the pull-back by \(\sigma\) of a metric \(dT^{2}\) on \(\mathbb{R}^{5}\), i.e., we have \(d\tau^{2}\equiv\sigma^{*}(dT^{2})\) where \(dT^{2}\equiv 2(dU^{1})^{2}-dS^{2}=\sum_{k=0}^{1}(dU^{k})^{2}-\sum_{i=2}^{4}(dU^{i})^{2}\). Clearly, this last metric \(dT^{2}\) is the metric defined on the anti-de Sitter space \(AdS_{4}\subset\mathbb{R}^{5}\) of signature \((2,2)\) and which is defined from the quadratic form \(\Lambda\) on \(\mathbb{R}^{5}\) such that \(\Lambda(U)\equiv Q(U)+2(U^{1})^{2}=\sum_{k=0}^{1}(U^{k})^{2}-\sum_{i=2}^{4}(U^{ i})^{2}\). Then, \(AdS_{4}\) is the set of points \(U\) such that \(\Lambda(U)=-k^{2}=cst<0\). Nevertheless, we must not consider \(AdS_{4}\subset\mathbb{R}^{5}\) which is not simply connected but a _simply connected_ subspace of \(P\mathbb{R}^{4}\) that we call the _pseudo-hyperbolic space_\(\mathcal{H}^{1,3}\). Let us note also that if \(Q(U)>0\) then, necessarily, we have \(\Lambda(U)>0\) which forbids to consider \(AdS_{4}\).  
Definition 2: The pseudo-hyperbolic space \(\mathcal{H}^{1,3}\subset P\mathbb{R}^{4}\) 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 \(d\tau^{2}\).  
Then, the orthochronous pseudo-orthogonal group \(SO^{+}(2,3)\subset PGL(5,\mathbb{R})\) is the group of invariance of the quadratic form \(\Lambda\) and the metric \(dT^{2}\). This group \(SO^{+}(2,3)\) acts by linear maps \(\mathfrak{A}\) on \(\Lambda\) and \(dT^{2}\). In return, this involves to restrict \(SO^{+}(2,3)\) to the subgroup \(SO^{+}(1,4)\cap SO^{+}(2,3)\equiv[\widehat{G}_{\tau}^{+}]\) in order to simultaneously keep an invariance of \(Q\) defining \(\mathcal{H}^{4}\) and \(\mathcal{H}^{1,3}\). Nevertheless, the metric \(d\tau^{2}\) is no more invariant with respect to homographies \([\mathfrak{A}]\).  
Let \(\pi:\mathbb{R}^{5}\longrightarrow P\mathbb{R}^{4}\) be the _projective map_ such that, in particular, \(\pi(U)^{i}=U^{i}/U^{0}\) whenever \(U^{0}\
eq 0\), and \(\mathfrak{a}:\mathbb{R}^{4}\longrightarrow B^{4}\) the _embedding_ such that \(\mathfrak{a}\equiv\pi\circ\sigma\) and where \(B^{4}\subset\mathbb{R}^{4}\) is the four-dimensional open ball. Then, we define the representation \([\ ]\)  of homographies such that \([\mathfrak{A}]=\mathfrak{a}^{-1}\circ[\mathfrak{A}]\circ\mathfrak{a}\) where \([\mathfrak{A}]\in[\widehat{G}_{\tau}^{+}]\). Then, \(d\tau^{2}\) is invariant with respect to \([\mathfrak{A}]\). More precisely, it can be shown that \([\mathfrak{A}]^{*}(d\tau^{2})=d\tau^{2}\) where \(\mathfrak{A}\in\widehat{G}_{\tau}^{+}\).