#### Postulates and their consequences
red.sr.p

Einstein's postulates:

- Postulate 1: All intertial frames are equivalent with respect to all the laws of physics
- Postulate 2: The speed of light in empty space always has the same value \(c\).

This has a number of important consequences.

Relativity of simultaneity: two events which are simultaneous in one reference frame, are not necessarily simultaneous in another one.

Time dilation: example of a light ray in a travelling train car. For the observer inside the car: \(\Delta t_{\mbox{car}} = h/c\). For an observer on the ground, if the train is moving at velocity \(v\), then \(\Delta t_{\mbox{gr}} = \sqrt{h^2 + v^2 \Delta t_{\mbox{gr}}^2}/c\) so \[ \Delta t_{\mbox{gr}} = \frac{h}{c} \frac{1}{\sqrt{1 - v^2/c^2}} \] and we get \[ \Delta t_{\mbox{tr}} = \sqrt{1 - v^2/c^2}~ \Delta t_{\mbox{gr}} \] so the time interval in the train is shorter, namely there is a

Time dilation factor

\[ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \tag{gamma}\label{gamma} \]

Lorentz contraction: lengths are also modified. Back to our train, with a mirror on one end. A light signal is sent from the opposite end, and the time for the round-trip of the light is measured. For the observer on the train, the time is \(\Delta t_{\mbox{tr}} = 2 \Delta x_{\mbox{tr}}/c\) with \(\Delta x_{\mbox{tr}}\) being the length of the train car. For the observer on the ground, the total time is made up of the back and forth journey of the light, with times \[ \Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}} + v \Delta t_{\mbox{gr,1}}}{c}, \hspace{10mm} \Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}} - v \Delta t_{\mbox{gr,2}}}{c} \] so \[ \Delta t_{\mbox{gr,1}} = \frac{\Delta x_{\mbox{gr}}}{c-v}, \hspace{10mm} \Delta t_{\mbox{gr,2}} = \frac{\Delta x_{\mbox{gr}}}{c+v} \] and thus \[ \Delta t_{\mbox{gr}} = \Delta t_{\mbox{gr,1}} + \Delta t_{\mbox{gr,2}} = \frac{2 \Delta x_{\mbox{gr}}}{c} \frac{1}{1 - v^2/c^2}. \] Using the time dilation relation then gives

Lorentz contraction \[ \Delta x_{\mbox{tr}} = \frac{1}{\sqrt{1 - v^2/c^2}} \Delta x_{\mbox{gr}} \]

Note that a moving object is only contracted in its direction of motion.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42