# On Hyperbolic Complex Numbers
## 3 Spherical Hyperbolic Complex Numbers
### Four Dimensional Space-Time Structure  
Let now a hyperbolic half-sphere of hyperbolic radius \(r>0\) be defined as  
\[C_{h}^{+}(r)=\{(t,x,y,z)^{T}\in\mathbb{R}^{4}:t^{2}-x^{2}-y^{2}-z^{2}=r^{2},\ t>0\}\]and the event space or space-time cone as  
\[\mathfrak{x}=\bigcup_{r>0}C_{h}^{+}(r)=\{(t,x,y,z)^{T}\in\mathbb{R}^{4}:t^{2}-x^{2 }-y^{2}-z^{2}>0,\ t>0\}.\]  
We recall that a class of new analytical vector products in \(\mathbb{R}^{4}\) is introduced in [25]. Here, we consider a hyperbolic analog as follows. Definition 12: Let  
\[\zeta_{1,l}=\sqrt{x_{l}^{2}+y_{l}^{2}+z_{l}^{2}}\text{ and }\zeta_{2,l}=\sqrt{y_{l}^{ 2}+z_{l}^{2}},\ l=1,2.\]  
Unless for \(\zeta_{1,l}=0\) or \(\zeta_{2,l}=0\) for at least one value of \(l\in\{1,2\}\) the analytical hyperbolic vector product of \(\mathfrak{r}_{1}=(t_{1},x_{1},y_{1},z_{1})^{T}\) and \(\mathfrak{r}_{2}=(t_{2},x_{2},y_{2},z_{2})^{T}\) is defined as  
\[\left( \begin{array}{c}t_{1}\\ x_{1}\\ y_{1}\\ z_{1}\end{array} \right)\odot_{h}\left( \begin{array}{c}t_{2}\\ x_{2}\\ y_{2}\\ z_{2}\end{array} \right)=\left( \begin{array}{c}t_{1}t_{2}+\zeta_{1,1}\zeta_{1,2 }\\ \zeta_{2,1}\end{array} \right)\left( \begin{array}{c}x_{1}x_{2}-\zeta_{2,1} \zeta_{2,2}\\ \left[\begin{array}{c}y_{1}y_{2}-z_{1}z_{2}\\ y_{1}z_{2}+z_{1}y_{2}\end{array}\right)\end{array} \right). \tag{22}\]  
Moreover, we put  
\[(t,x,y,z)^{T}\odot_{h}(\lambda,0,0,0)^{T}=(\lambda t,\lambda x,\lambda y, \lambda z)^{T}\]  
and  
\[\left( \begin{array}{c}t\\ x\\ y\\ z\end{array} \right)\odot_{h}\left( \begin{array}{c}u\\ v\\ 0\\ 0\end{array} \right)=u\left( \begin{array}{c}t\\ x\\ y\\ z\end{array} \right)+v\left( \begin{array}{c}\varrho\\ xt/\varrho\\ yt/\varrho\\ zt/\varrho\end{array} \right) \tag{23}\]  
where \(\varrho=\sqrt{x^{2}+y^{2}+z^{2}}\). It should be noted that Formula (22) differs only slightly from Formula (12) in [25], namely with respect to two times the factor \(-1\). One of these factors is due to the circumstance that \(\mathfrak{x}\) is a one-sided cone and the other factor is caused by the fact that instead of a norm used in [25] we use an antinorm according to [27], here. To better understand this vector product, we first introduce a new coordinate system. Definition 13: With the notation \(\Theta=\mathbb{R}^{+}\times[0,\pi)\times[0,2\pi)\times\mathbb{R}^{+}\), we define the hyperbolic coordinate transformation \(Hyp:\Theta\to\mathfrak{x},(t,x,y,z)^{T}=Hyp(r,\varphi_{1},\varphi_{2},\psi)\), by  
\[ \begin{array}{l}t=r\cosh\psi,\hskip 56.905512 ptx=r\cos\varphi_{1}\sinh\psi,\\ y=r\sin\varphi_{1}\cos\varphi_{2}\sinh\psi\hskip 56.905512 ptand\hskip 56.905512 ptz=r\sin \varphi_{1}\sin\varphi_{2}\sinh\psi.\end{array}  \tag{24}\]  
With \(\varrho=\sqrt{x^{2}+y^{2}+z^{2}}\), the a.e. uniquely defined inverse map \(Hyp^{-1}:\mathfrak{x}\to\Theta\) is given by  
\[r=\sqrt{t^{2}-\varrho^{2}},\ \varphi_{1}=\arcsin\frac{\sqrt{y^{2}+z^{2}}}{ \varrho},\ \varphi_{2}=\arctan\frac{z}{y},\ \psi=\ln\sqrt{\frac{t+\varrho}{t-\varrho}}\.\]  
The event space \(\mathfrak{x}\) is a subset of the vector space \(\mathbb{R}^{4}\). A basis of the latter consists of the vectors \(\mathfrak{e}=(1,0,0,0)^{T},\mathfrak{j}_{1}=(0,1,0,0)^{T}\), \(\mathfrak{j}_{2}=(0,0,1,0)^{T}\) and \(\mathfrak{j}_{3}=(0,0,0,1)^{T}\). The subspaces spanned by \(\mathfrak{j}_{1},\mathfrak{j}_{2}\) and \(\mathfrak{j}_{3}\) or \(\mathfrak{e}\) are, respectively, denoted by \(\mathfrak{M}\) and \(\mathfrak{M}\) and the orthogonal projections of the vector \(\mathfrak{x}=(t,x,y,z)^{T}\) onto the spaces \(\mathfrak{M}\) and \(\mathfrak{M}\) by \(\Pi_{1}\) and \(\Pi_{2}\), respectively. The meaning of \(r\) and \(\varphi_{1},\varphi_{2}\) lacking as a hyperbolic radius and two angles, and \(\psi\) as an area content motivates us for the following definition. **Definition 14**.