Question: The Lorentz transformation equations have x and t and x^' and t^'. Why no v and v^' ?

The Lorentz equation defines the relation of the two different frames at the specific position and time. The equations for the Lorentz transformation are given by,

$$\begin{gathered} \mathit{t}\mathbf{\text{'}}\mathbf{=}{\mathit{\gamma }}_{\mathbf{v}}\left(t-\frac{v}{c^2}x\right) \\ \mathit{x}\mathbf{\text{'}}\mathbf{=}{\mathit{\gamma }}_{\mathbf{v}}\left(x-vt\right) \end{gathered}$$ (1)

Here,t^' is the time in the frame s^' , t is the time in the frame S, is the Lorentz factor for the velocity, c is the speed of the light.

^'

The Lorentz transformation equation relates the position and time of one frame of the position and time to another frame: that is why the Lorentz transformation equation has the time , position in the frame , and time t , position x in the frame S. But it is not specified at the particular velocity. Therefore, equations don’t have v and v^' .

Hence the event does not occur at a particular velocity: therefore, Lorentz transformation doesn’t have v and v^'.

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