Apply special relativity to time dilation and length contraction at high relative speeds

What this dot point is asking

WACE wants you to state the postulates, use the Lorentz factor, and apply time dilation and length contraction, identifying which quantity is the proper one. These effects are negligible at everyday speeds and only matter near the speed of light.

The two postulates

Einstein built special relativity on two ideas. First, the principle of relativity: the laws of physics are identical in all inertial (non-accelerating) frames, so no experiment can tell you that you are "really" moving. Second, the constancy of light speed: light travels at $c$ in a vacuum for every observer, regardless of the motion of the source or observer. The strange consequences follow from taking these two seriously together.

The Lorentz factor

Both effects are governed by

At low speeds $\gamma\approx1$ and nothing measurable changes. As $v$ approaches $c$, $\gamma$ grows without bound, which is why effects become extreme only near light speed and why no massive object can reach $c$.

Time dilation

A clock measures the shortest time, the proper time $t_0$, in the frame where it is at rest. An observer moving relative to that clock measures a longer time

So a moving clock appears to run slow. The effect is reciprocal: each observer sees the other's clock running slow, because each considers themselves at rest.

Length contraction

An object has its greatest length, the proper length $L_0$, in its own rest frame. An observer past whom it moves measures a contracted length along the direction of motion

Only the dimension along the motion contracts; dimensions perpendicular to the motion are unchanged.

Evidence from muons

Muons created high in the atmosphere have a short half-life and should mostly decay before reaching the ground at their speed. Yet far more arrive than classical physics predicts. From the ground frame, the muons' internal clocks run slow (time dilation), so they live longer. From the muon frame, the atmosphere is thin (length contraction), so there is less distance to cross. Both views agree the muons reach the surface, confirming the theory.

When relativistic effects matter

A useful skill is judging when relativity is needed at all. The Lorentz factor stays within $1\%$ of $1$ for speeds below about $0.14c$, which is still over $4\times10^{7}\ \text{m s}^{-1}$, far faster than any everyday object. This is why time dilation and length contraction are completely undetectable for cars, planes or even satellites in ordinary problems, and Newtonian physics is an excellent approximation. The effects only become significant for particles in accelerators, cosmic-ray muons, and precision systems such as GPS satellites, where uncorrected relativistic clock differences would accumulate into navigation errors of kilometres per day. Being able to say why a given scenario does or does not need relativity, by estimating $v/c$, is a frequent conceptual mark.

Spotting the proper quantity

The proper time is measured by a clock present at both events (the moving object's clock); the proper length is measured in the object's rest frame. Time dilation multiplies the proper time by $\gamma$; length contraction divides the proper length by $\gamma$. Choosing the wrong frame for the proper value inverts the result.