Lorentz Transformation of Space-Time Coordinates

1. Lorentz Transformation is used for Velocity Transformation of Space-Time Coordinates in Special Relativity
2. Lorentz Transformation of Space-Time Coordinate (\(x\), \(y\), \(z\) , \(t\)) (or (\(t\), \(x\), \(y\), \(z\)) ), on application of Velocity \(\vec{V}\) is given as
   
   \(\begin{bmatrix}ct'\\ x' \\ y' \\ z'\end{bmatrix}=\begin{bmatrix}\gamma & \gamma {\Large \frac{v_x}{c}} & \gamma {\Large \frac{v_y}{c}} & \gamma {\Large \frac{v_z}{c}} \\ \gamma{\Large \frac{v_x}{c}} & 1+(\gamma-1) {n_x}^2 & (\gamma-1) n_xn_y & (\gamma-1) n_xn_z \\ \gamma{\Large \frac{v_y}{c}} & (\gamma-1) n_yn_x & 1 + (\gamma-1) {n_y}^2 & (\gamma-1) n_yn_z \\ \gamma{\Large \frac{v_z}{c}} & (\gamma-1) n_zn_x & (\gamma-1) n_zn_y & 1+(\gamma-1) {n_z}^2\end{bmatrix}\begin{bmatrix}ct \\ x \\ y \\ z\end{bmatrix}\)   ...(1)
   
   \(\Rightarrow \begin{bmatrix}ct'\\ \vec{R'}\end{bmatrix}=\begin{bmatrix}\gamma & \gamma {\Large \frac{\vec{V}^T}{c}} \\ \gamma{\Large \frac{\vec{V}}{c}} & I+(\gamma-1) \hat{n}{\hat{n}}^T\end{bmatrix}\begin{bmatrix}ct\\ \vec{R}\end{bmatrix}\)   ...(2)
   
   which implies
   
   \(ct'=\gamma(ct + {\Large \frac{\vec{V}\cdot\vec{R}}{c} })\hspace{2mm}\Rightarrow t'= \gamma(t + {\Large \frac{\vec{V}\cdot\vec{R}}{c^2} })\)   ...(3)
   
   \(\vec{R'}= \vec{R} + (\gamma - 1) (\vec{R}.\hat{n})\hat{n} + \gamma\vec{V}t\)   ...(4)
   
   where
   
   \(c = \) Speed of Light in Vacuum
   
   \(t = \) Time Value of the Given Space-Time Coordinate
   
   \(t' =\) Time Value of the Transformed Space-Time Coordinate
   
   \(\vec{R} =\) Position Vector corresponding to Variables \(x\), \(y\) and \(z\) of Given Space-Time Coordinate
   
   \(\vec{R}' =\) Position Vector corresponding to Transformed Variables \(x'\), \(y'\) and \(z'\) of Transformed Space-Time Coordinate
   
   \(\vec{V} =\) Applied Velocity Vector
   
   \(v_x, v_y, v_z =\) Components of Vector \(\vec{V}\)
   
   \(\hat{n} =\) Unit Vector in the Direction of \(\vec{V}\)
   
   \(n_x, n_y, n_z =\) Components of Unit Vector \(\hat{n}\)
   
   \(|\vec{V}| =\) Speed corresponding to Applied Velocity Vector
   
   \(\gamma =\) Lorentz factor given by \(\sqrt{{\Large \frac{1}{1-\frac{|\vec{V}|^2}{c^2}}}}\). \(\gamma \approx 1\) when \(|\vec{V}| \lll c\) and \( > 1\) when \(|\vec{V}|\) is Near to \(c\)
   
   
3. When \(|\vec{V}| \lll c\), \(\gamma \approx 1\), and Lorentz Transformation defaults to Galilean Transformation as follows
   
   \(\begin{bmatrix}ct'\\ x' \\ y' \\ z'\end{bmatrix}=\begin{bmatrix}1 & {\Large \frac{v_x}{c}} & {\Large \frac{v_y}{c}} & {\Large \frac{v_z}{c}} \\ {\Large \frac{v_x}{c}} & 1 & 0 & 0 \\ {\Large \frac{v_y}{c}} & 0 & 1 & 0 \\ {\Large \frac{v_z}{c}} & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}ct \\ x \\ y \\ z\end{bmatrix}\)   ...(5)
   
   \(\Rightarrow \begin{bmatrix}ct'\\ \vec{R'}\end{bmatrix}=\begin{bmatrix}1 & {\Large \frac{\vec{V}^T}{c}} \\ {\Large \frac{\vec{V}}{c}} & I\end{bmatrix}\begin{bmatrix}ct\\ \vec{R}\end{bmatrix}\)   ...(6)
   
   which implies
   
   \(ct'=(ct + {\Large \frac{\vec{V}\cdot\vec{R}}{c} })\hspace{2mm}\Rightarrow t'= (t + {\Large \frac{\vec{V}\cdot\vec{R}}{c^2} })\)   ...(4)
   
   Since \(|\vec{V}| \lll c\), \({\Large \frac{\vec{V}\cdot\vec{R}}{c} } \approx 0\) and hence,
   
   \(ct'=ct \hspace{2mm}\Rightarrow t'= t\)   ...(6)
   
   \(\vec{R'}= \vec{R} + \vec{V}t\)   ...(7)
   
   
4. The Inverse of Lorentz Transformation Matrix for Velocity \(\vec{V}\) (as given in equation (1) and (2) above) is Lorentz Transformation Matrix for Velocity \(-\vec{V}\) as given below
   
   \({\begin{bmatrix}\gamma & \gamma {\Large \frac{v_x}{c}} & \gamma {\Large \frac{v_y}{c}} & \gamma {\Large \frac{v_z}{c}} \\ \gamma{\Large \frac{v_x}{c}} & 1+(\gamma-1) {n_x}^2 & (\gamma-1) n_xn_y & (\gamma-1) n_xn_z \\ \gamma{\Large \frac{v_y}{c}} & (\gamma-1) n_yn_x & 1 + (\gamma-1) {n_y}^2 & (\gamma-1) n_yn_z \\ \gamma{\Large \frac{v_z}{c}} & (\gamma-1) n_zn_x & (\gamma-1) n_zn_y & 1+(\gamma-1) {n_z}^2\end{bmatrix}}^{-1} = \begin{bmatrix}\gamma & -\gamma {\Large \frac{v_x}{c}} & -\gamma {\Large \frac{v_y}{c}} & -\gamma {\Large \frac{v_z}{c}} \\ -\gamma{\Large \frac{v_x}{c}} & 1+(\gamma-1) {n_x}^2 & (\gamma-1) n_xn_y & (\gamma-1) n_xn_z \\ -\gamma{\Large \frac{v_y}{c}} & (\gamma-1) n_yn_x & 1 + (\gamma-1) {n_y}^2 & (\gamma-1) n_yn_z \\ -\gamma{\Large \frac{v_z}{c}} & (\gamma-1) n_zn_x & (\gamma-1) n_zn_y & 1+(\gamma-1) {n_z}^2\end{bmatrix} \)   ...(8)
   
   \(\Rightarrow {\begin{bmatrix}\gamma & \gamma {\Large \frac{\vec{V}^T}{c}} \\ \gamma{\Large \frac{\vec{V}}{c}} & I+(\gamma-1) \hat{n}{\hat{n}}^T\end{bmatrix}}^{-1}= \begin{bmatrix}\gamma & -\gamma {\Large \frac{\vec{V}^T}{c}} \\ -\gamma{\Large \frac{\vec{V}}{c}} & I+(\gamma-1) \hat{n}{\hat{n}}^T\end{bmatrix} \)   ...(9)
   
   Lorentz Transformation of Space-Time Coordinate (\(x\), \(y\), \(z\) , \(t\)) (or (\(t\), \(x\), \(y\), \(z\)) ) on application of Velocity \(-\vec{V}\) is given as
   
   \(ct'=\gamma(ct - {\Large \frac{\vec{V}\cdot\vec{R}}{c} })\hspace{2mm}\Rightarrow t'= \gamma(t - {\Large \frac{\vec{V}\cdot\vec{R}}{c^2} })\)   ...(10)
   
   \(\vec{R'}= \vec{R} + (\gamma - 1) (\vec{R}.\hat{n})\hat{n} - \gamma\vec{V}t\)   ...(11)