Lorentz Transformations

Previous section: Doppler Shift.

Two different inertial systems (primed and unprimed) with a common origin  X0 = X0' = 0  assign
different labels  
X′ = ct′+x  and  ct+x  to each spacetime event. If = 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 .

x ct ct' X₀ = X₀' = 0 x-axis (ct=0) ct-axis (x=0) x'-axis (ct'=0) ct'-axis (x'=0) Lightcone x' X β β V γ γ v

This is called the passive interpretation of a Lorentz transformation (as opposed to the active interpretation --- see link at the right). There is just one event in the diagram, 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:

\[\Large{t'\,\,\, = \,\,\,\gamma \,\left( {t\,\, - \,\,\frac{{{\bf{v}} \cdot {\bf{x}}}}{{{c^2}}}} \right)\,\,,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x'}}\,\,\, = \,\,\,\gamma \,\left( {{\bf{x}}\,\, - \,\,{\bf{v}}t} \right)}\]

Click here for a solution to this exercise.)

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© David Hestenes 2005, 2014