Einstein Velocity Addition/Subtraction Formula for Special Relativity

1. Einstein Velocity Addition/Subtraction Formula is used for Addition and Subtraction of Velocities whoes Magnitudes are comparable to the Speed of Light (i.e. 299,792,458 m/s).
2. Following are the formulae for Addition/Subtraction of 2 Velocities \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\)
   
   \(\vec{\mathbf{u}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{v}}=\frac{\vec{\mathbf{u}}\hspace{.1cm}-\hspace{.1cm}{\vec{\mathbf{u}}}_{||}\hspace{.1cm}+\hspace{.1cm}{\gamma}_\mathbf{v}\hspace{.1cm}({\vec{\mathbf{u}}}_{||}\hspace{.1cm}\pm\hspace{.1cm}\vec{\mathbf{v}})}{{\gamma}_\mathbf{v}\hspace{.1cm}(1\hspace{.1cm}\pm\hspace{.1cm}\frac{\vec{\mathbf{u}}.\vec{\mathbf{v}}}{c^2})}\)
   
   \(\vec{\mathbf{v}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{u}}=\frac{\vec{\mathbf{v}}\hspace{.1cm}-\hspace{.1cm}{\vec{\mathbf{v}}}_{||}\hspace{.1cm}+\hspace{.1cm}{\gamma}_\mathbf{u}\hspace{.1cm}({\vec{\mathbf{v}}}_{||}\hspace{.1cm}\pm\hspace{.1cm}\vec{\mathbf{u}})}{{\gamma}_\mathbf{u}\hspace{.1cm}(1\hspace{.1cm}\pm\hspace{.1cm}\frac{\vec{\mathbf{v}}.\vec{\mathbf{u}}}{c^2})}\)
   
   where \({\gamma}_\mathbf{u}=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}\),   \({\gamma}_\mathbf{v}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\),   \(\vec{\mathbf{u}}_{||}=\)  Projection of \(\vec{\mathbf{u}}\) on \(\vec{\mathbf{v}}\),   \(\vec{\mathbf{v}}_{||}=\)  Projection of \(\vec{\mathbf{v}}\) on \(\vec{\mathbf{u}}\)
3. If the 2 Velocities \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\) are Parallel, which implies \(\vec{\mathbf{u}}=\vec{\mathbf{u}}_{||}\) and \(\vec{\mathbf{v}}=\vec{\mathbf{v}}_{||}\), then the Velocity Addition/Subtraction formula reduces to
   
   \(\vec{\mathbf{u}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{v}}=\frac{\vec{\mathbf{u}}\hspace{.1cm}\pm\hspace{.1cm}\vec{\mathbf{v}}}{1\hspace{.1cm}\pm\hspace{.1cm}\frac{\vec{\mathbf{u}}.\vec{\mathbf{v}}}{c^2}}\)
   
   \(\vec{\mathbf{v}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{u}}=\frac{\vec{\mathbf{v}}\hspace{.1cm}\pm\hspace{.1cm}\vec{\mathbf{u}}}{1\hspace{.1cm}\pm\hspace{.1cm}\frac{\vec{\mathbf{v}}.\vec{\mathbf{u}}}{c^2}}\)
4. If the 2 Velocities \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\) are Perpendicular, which implies \(\vec{\mathbf{u}}_{||}=0\) and \(\vec{\mathbf{v}}_{||}=0\) and \(\vec{\mathbf{u}} \cdot \vec{\mathbf{v}}=0\), then the Velocity Addition/Subtraction formula reduces to
   
   \(\vec{\mathbf{u}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{v}}=\frac{\vec{\mathbf{u}}\hspace{.1cm}\pm\hspace{.1cm}{\gamma}_\mathbf{v}\hspace{.1cm}\vec{\mathbf{v}}}{{\gamma}_\mathbf{v}}\)
   
   \(\vec{\mathbf{v}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{u}}=\frac{\vec{\mathbf{v}}\hspace{.1cm}\pm\hspace{.1cm}{\gamma}_\mathbf{u}\hspace{.1cm}\vec{\mathbf{u}}}{{\gamma}_\mathbf{u}}\)
5. \({\gamma}_{(\vec{\mathbf{u}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{v}})}={\gamma}_{(\vec{\mathbf{v}}\hspace{.1cm}{}^\oplus_\ominus\hspace{.1cm}\vec{\mathbf{u}})}={\gamma}_{\mathbf{u}}{\gamma}_{\mathbf{v}}\hspace{.1cm}(1 \pm \frac{\vec{\mathbf{u}}.\vec{\mathbf{v}}}{c^2})\)
6. Relative Velocity of \(\vec{\mathbf{u}}\) with respect to \(\vec{\mathbf{v}}\) is calculated as \(\vec{\mathbf{u}}\ominus\vec{\mathbf{v}}\). Similarly, Relative Velocity of \(\vec{\mathbf{v}}\) with respect to \(\vec{\mathbf{u}}\) is calculated as \(\vec{\mathbf{v}}\ominus\vec{\mathbf{u}}\).
7. Unless \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\) are Parallel    \(\vec{\mathbf{u}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{v}} \neq \vec{\mathbf{v}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{u}}\)    and   \(\vec{\mathbf{u}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{v}} \neq - (\vec{\mathbf{v}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{u}})\)
8. \(|\vec{\mathbf{u}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{v}}| = |\vec{\mathbf{v}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{u}}|\)    and   \(|\vec{\mathbf{u}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{v}}| = |\vec{\mathbf{v}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{u}}|\)
9. The angle \(\theta\) between the Vectors \(\vec{\mathbf{u}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{v}}\) and \(\vec{\mathbf{v}}\hspace{.1cm}\oplus\hspace{.1cm}\vec{\mathbf{u}}\) is calculated as
   
   \(\theta = \cos ^{-1} (\frac{ {(\gamma_{\mathbf{u}}\hspace{.1cm}+\hspace{.1cm}\gamma_{\mathbf{v}}\hspace{.1cm}+\hspace{.1cm}\gamma_{(\mathbf{u}+\mathbf{v})}\hspace{.1cm}+\hspace{.1cm}1)}^2}{(\gamma_{\mathbf{u}} + 1) (\gamma_{\mathbf{v}} + 1) (\gamma_{(\mathbf{u} + \mathbf{v})} + 1) } -1) = \sin ^{-1} (\frac{ \beta_{\mathbf{u}}\beta_{\mathbf{v}}\gamma_{\mathbf{u}}\gamma_{\mathbf{v}}(\gamma_{\mathbf{u}}\hspace{.1cm}+\hspace{.1cm}\gamma_{\mathbf{v}}\hspace{.1cm}+\hspace{.1cm}\gamma_{(\mathbf{u}+\mathbf{v})}\hspace{.1cm}+\hspace{.1cm}1)}{(\gamma_{\mathbf{u}} + 1) (\gamma_{\mathbf{v}} + 1) (\gamma_{(\mathbf{u} + \mathbf{v})} + 1) }\sin \psi) \)
   
   The angle \(\psi\) is the angle between Vectors \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\). The angle \(\theta\) is known as the Wigner Rotation Angle.
10. The angle \(\phi\) between the Vectors \(\vec{\mathbf{u}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{v}}\) and \(\vec{\mathbf{v}}\hspace{.1cm}\ominus\hspace{.1cm}\vec{\mathbf{u}}\) is calculated as
    
    \(\phi = \pi - \cos ^{-1} (\frac{ {(\gamma_{\mathbf{u}}\hspace{.1cm}+\hspace{.1cm}\gamma_{\mathbf{v}}\hspace{.1cm}+\hspace{.1cm}\gamma_{(\mathbf{u}-\mathbf{v})}\hspace{.1cm}+\hspace{.1cm}1)}^2}{(\gamma_{\mathbf{u}} + 1) (\gamma_{\mathbf{v}} + 1) (\gamma_{(\mathbf{u} - \mathbf{v})} + 1) } -1) = \pi - \sin ^{-1} (\frac{ \beta_{\mathbf{u}}\beta_{\mathbf{v}}\gamma_{\mathbf{u}}\gamma_{\mathbf{v}}(\gamma_{\mathbf{u}}\hspace{.1cm}+\hspace{.1cm}\gamma_{\mathbf{v}}\hspace{.1cm}+\hspace{.1cm}\gamma_{(\mathbf{u}-\mathbf{v})}\hspace{.1cm}+\hspace{.1cm}1)}{(\gamma_{\mathbf{u}} + 1) (\gamma_{\mathbf{v}} + 1) (\gamma_{(\mathbf{u} - \mathbf{v})} + 1) }\sin \psi)\)
    
    The angle \(\psi\) is the angle between Vectors \(\vec{\mathbf{u}}\) and \(\vec{\mathbf{v}}\).