91Ó°ÊÓ

Consider a algebraic identity is${\mathrm{γ}}_{u\text{'}}=\left(1-\frac{u_{x}v}{c^2}\right){\mathrm{γ}}_v{\mathrm{γ}}_u$.

Write the formula of algebraic equation of relativistic velocity transformation. $\underset{\mathbf{i}}{\mathbf{∑}}{\mathit{γ}}_{{\mathbf{u}}_{\mathbf{i}}^{\mathbf{\text{'}}}}{\mathit{m}}_{\mathbf{i}}{\mathit{u}}_{\mathbf{i}}^{\mathbf{\text{'}}}\mathbf{=}{\mathbf{γ}}_{\mathbf{u}}\underset{\mathbf{i}}{\mathbf{∑}}{\mathit{γ}}_{{\mathbf{u}}_{\mathbf{i}}^{\mathbf{\text{'}}}}{\mathit{m}}_{\mathbf{i}}{\mathit{u}}_{\mathbf{i}}\mathbf{-}\mathit{u}{\mathbf{γ}}_{\mathbf{u}}\underset{\mathbf{i}}{\mathbf{∑}}{\mathit{γ}}_{\mathbf{u}\mathbf{i}}{\mathit{m}}_{\mathbf{i}}$ …… (1)

Here, ${\mathit{γ}}_{{\mathbf{u}}_{\mathbf{i}}^{\mathbf{\text{'}}}}$is relativistic velocity transformation, ${\mathbf{m}}_{\mathbf{i}}$is mass and ${\mathbf{u}}_{\mathbf{i}}$ is velocity transformation.

As reference of equation 2.23.

Determine the algebraic equation of relativistic velocity transformation.

Consider the LHS of the equation (1).

$\underset{i}{∑}{\mathrm{γ}}_{u_i^{\text{'}}}{m}_i{u}_i^{\text{'}}$

First we solve it without summation.

$$\begin{gathered} {\mathrm{γ}}_{\mathrm{u}\text{'}}{m}_{\mathrm{i}}u\text{'}=\left(1-\frac{u_{x}v}{c^2}\right){\mathrm{γ}}_{\mathrm{u}}{\mathrm{γ}}_{\mathrm{u}}m·\frac{u_x-v}{\left(1-\frac{u_{x}v}{c^2}\right)} \\ ={\mathrm{γ}}_{\mathrm{u}}{\mathrm{γ}}_{\mathrm{u}}m{u}_x-{\mathrm{γ}}_{\mathrm{u}}{\mathrm{γ}}_{\mathrm{u}}vm \end{gathered}$$

Now summing over whole system’s particles:

$\underset{i}{∑}{\mathrm{γ}}_{u_i^{\text{'}}}{m}_i{u}_i^{\text{'}}={\mathrm{γ}}_u\underset{i}{∑}{\mathrm{γ}}_{u_i^{\text{'}}}{m}_i{u}_i-u{\mathrm{γ}}_u\underset{i}{∑}{\mathrm{γ}}_{ui}{m}_i$

Therefore, the value of algebraic equation of relativistic velocity transformation is $\underset{\mathrm{i}}{∑}{\mathrm{γ}}_{{\mathrm{u}}_{\mathrm{i}}^{\text{'}}}{m}_{\mathrm{i}}{u}_{\mathrm{i}}^{\text{'}}={\mathrm{γ}}_{\mathrm{u}}\underset{\mathrm{i}}{∑}{\mathrm{γ}}_{{\mathrm{u}}_{\mathrm{i}}^{\text{'}}}{m}_{\mathrm{i}}{u}_{\mathrm{i}}-u{\mathrm{γ}}_{\mathrm{u}}\underset{\mathrm{i}}{∑}{\mathrm{γ}}_{\mathrm{ui}}{m}_{\mathrm{i}}$.