Proposition: Time Dilation, Lorentz Factor

Let \(\mathcal I_1\) and \(\mathcal I_2\) be two inertial frames of reference moving relatively to each other with the constant speed \(v\) obeying the inequation \(0\le v < c\), where \(c\) denotes the constant speed of light1. If an observer inside the reference frame \(\mathcal I_1\) measures the time \(t_1\) elapsed in \(\mathcal I_1\) and compares it to the time \(t_2\) elapsed he observes in the inertial frame of reference \(\mathcal I_2\), then both times elapsed will show a difference called time dilation, which is given by the formula

\[t_1=t_2\cdot \gamma.\]

The factor \[\gamma:=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}},\quad\quad (0\le v < c)\]

is called the Lorentz factor2.

Proofs: 1 Explanations: 1

Explanations: 1

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  1. Weingärtner, Andreas: "Spezielle Relativitätstheorie - ganz einfach", Books On Demand, 2016


  1. If both inertial frames of reference are in a vacuum, this speed equals \(c=299\,792\,458~m/s.\) 

  2. Named so after Hendrik Antoon Lorentz (1853 - 1928).