## Solution: Time & Position Transformations

Exercise: Derive from  X′ = ṼX  the standard relations between times and positions:

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

Keeping in mind that we are considering motion in one dimension (our x-axis), vx = 0, so simply multiply the terms and collect scalar and vector parts separately:

$\Large{\begin{array}{l}X'\,\,\, = \,\,\,ct'\,\, + \,\,{\bf{x'}}\,\,\, = \,\,\,\widetilde V\,X\,\,\, = \,\,\,\gamma \,\left( {1\,\, - \,\,{\bf{v}}/c} \right)\,\left( {ct\,\, + \,\,{\bf{x}}} \right)\\\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~~ct'\,\,\, = \,\,\,\gamma \,\left( {ct\,\, - \,\,{\bf{v}} \cdot {\bf{x}}/c} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~{\rm{and}}~~~~~~\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x'}}\,\,\, = \,\,\,\gamma \,\left( {{\bf{x}}\,\, - \,\,{\bf{v}}t} \right)\end{array}}$

Similarly,

$\Large{\begin{array}{l}X\,\,\, = \,\,\,ct\,\, + \,\,{\bf{x}}\,\,\, = \,\,\,V\,X'\,\,\, = \,\,\,\gamma \,\left( {1\,\, + \,\,{\bf{v}}/c} \right)\,\left( {ct'\,\, + \,\,{\bf{x'}}} \right)\\\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~ct\,\,\, = \,\,\,\gamma \,\left( {ct'\,\, + \,\,{\bf{v}} \cdot {\bf{x'}}/c} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~{\rm{and}}~~~~~~\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x}}\,\,\, = \,\,\,\gamma \,\left( {{\bf{x'}}\,\, + \,\,{\bf{v}}t'} \right)\end{array}}$

We can check these against previous spacetime maps and conclusions, such as on the
Solution: Lorentz Contraction Formula page. For example, if x' = 0, we're on the time axis of the primed system and both sets of these formulas correctly say x = vt and ct = γct':

$\Large{\begin{array}{l}{\rm{In}}\,\,{\rm{the}}\,\,{\rm{case}}\,\,{\bf{x'}}\,\, = \,\,0\,:\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\\\\\,\,{{\rm{1}}^{{\rm{st}}}}\,\,{\rm{set}}:\,\,\,\,\,\,\,\,\,\,\,\,ct'\,\,\, = \,\,\,\gamma \,\left( {ct\,\, - \,\,{\bf{v}} \cdot {\bf{x}}/c} \right)\,\,\,\,\,\,\,\,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x'}}\,\,\, = \,\,\,\gamma \,\left( {{\bf{x}}\,\, - \,\,{\bf{v}}t} \right)\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~~~~{\bf{x}}\,\,\, = \,\,\,{\bf{v}}t\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~{\rm{and}}~~~~~~\,\,\,\,\,\,\,\,\,\,\,\,ct'\,\,\, = \,\,\,\gamma \,\left( {ct\,\, - \,\,{\bf{v}} \cdot {\bf{v}}t/c} \right)\,\,\, = \,\,\,ct\,/\,\gamma \\\\{2^{{\rm{nd}}}}\,\,{\rm{set}}:\,\,\,\,\,\,\,\,\,\,\,\,ct\,\,\,\, = \,\,\,\gamma \,\left( {ct'\,\, + \,\,{\bf{v}} \cdot {\bf{x'}}/c} \right)\,\,\,\,\,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x}}\,\,\,\,\, = \,\,\,\gamma \,\left( {{\bf{x'}}\,\, + \,\,{\bf{v}}t'} \right)\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~~~ct\,\,\, = \,\,\,\gamma \,ct'\,\,\,\,\,\,\,\,\,\,\,\,~~~~~~{\rm{and}}~~~~~~\,\,\,\,\,\,\,\,\,\,\,\,{\bf{x}}\,\,\,\,\, = \,\,\,\gamma \,{\bf{v}}t'\,\,\, = \,\,\,{\bf{v}}t\end{array}}$