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Two different inertial systems (primed and unprimed) with a common origin *X _{0}*

*=*

*X*

_{0}

**= 0 assign**

*'*different labels

*X*′ =

*ct*′+

**x**′ and

*X*=

*ct*+

**x**

**to each spacetime event. If**

*V*=

*γ*(1+

**v**/

*c*) is the proper velocity of the primed frame with respect to the unprimed frame, then, in the timelike plane containing both time axes, the labels are related by the

**:**

*Lorentz transformation* *X *= *VX*′ or *X*′ = *ṼX .*

This is called the ** passive interpretation** of a Lorentz transformation (as opposed to the

**--- see link at the right). There is just one event in the diagram,**

*active interpretation**X*. Each system reads its own coordinates for that event. The equations above are simply expressing a procedure for how to translate coordinates in the primed system to coordinates in the unprimed system or vice versa.

**Exercise: **Derive therefrom the standard relations between times and positions:

(Click* *__here__ for a solution to this exercise.)

*(Follow links on the right to subtopics for this section.)*