Quick questions on Special relativity: postulates, time dilation and length contraction, HSC Physics Module 7

By the late nineteenth century, light was understood as a wave. Waves needed a medium, so physicists postulated the luminiferous aether: a hypothetical, all-pervading substance through which light propagated. The aether was assumed stationary (or close to it), so the Earth must move through it at orbital speed (about $30$ km/s).

Special relativity rests on just two postulates:

Most relativistic formulas use:

A clock moving with speed $v$ relative to an observer ticks slowly compared to a clock at rest with that observer. If the moving clock measures proper time $t_0$ (time between two events at the same place in its own frame), the time interval measured by the stationary observer is:

An object moving at speed $v$ along its length is measured to be shorter than its proper length $L_0$ (the length in its rest frame) by:

Two events that are simultaneous in one inertial frame are generally not simultaneous in another moving relative to the first.

A muon created at the top of the atmosphere ($15$ km altitude) travels at $0.998c$ toward the ground. Its proper lifetime is $2.2$ $\mu$s.

The moving clock measures the proper time $t_0$ (the shorter interval). The observer who sees the clock moving measures $t = \gamma t_0$ (longer interval).

The object at rest in some frame has the proper length $L_0$ in that frame. Any frame that sees the object moving measures a shorter length.

Proper time is between two events at the same place in some frame. Coordinate time depends on the synchronisation of clocks at different places.

$\gamma \geq 1$ always, equal to 1 only when $v = 0$.