Apply Einstein's postulates to time dilation, length contraction and mass-energy equivalence.

What this dot point is asking

Special relativity, published by Einstein in 1905, overturned the everyday assumption that time and length are absolute. It is a cornerstone of modern physics and a key topic in Unit 4.

The two postulates

1. The principle of relativity: the laws of physics are identical in all inertial (non-accelerating) reference frames. There is no experiment that can tell you whether you are at rest or moving at constant velocity.
2. The constancy of the speed of light: the speed of light in a vacuum, $c = 3.0 \times 10^8\ \text{m s}^{-1}$, is the same for all observers, regardless of the motion of the source or the observer.

The second postulate is the radical one. It contradicts our intuition that speeds simply add together, and it forces space and time themselves to adjust so that everyone measures the same speed of light.

The Lorentz factor

The consequences of the postulates are captured by the Lorentz factor:

When $v$ is much less than $c$, $\gamma \approx 1$ and relativity reduces to everyday physics. As $v$ approaches $c$, $\gamma$ grows without limit, which is why relativistic effects only become noticeable at very high speeds.

Time dilation

A moving clock runs slow as measured by a stationary observer. If $t_0$ is the proper time (the interval measured in the frame where the two events happen at the same place), then the dilated time measured in a frame moving relative to it is:

This is not an illusion or a fault in the clock; time genuinely passes more slowly in the moving frame. Fast-moving muons created in the upper atmosphere reach the ground only because their decay clocks run slow from our point of view.

Length contraction

An object moving relative to an observer is measured to be shorter along its direction of motion. If $L_0$ is the proper length (measured at rest with the object), the contracted length is:

Only the dimension parallel to the motion contracts; lengths perpendicular to the motion are unchanged.

Mass-energy equivalence

Einstein's most famous result is that mass and energy are two forms of the same thing. The rest energy of a mass $m$ is:

Because $c^2$ is enormous, a tiny amount of mass corresponds to a huge amount of energy. This relationship explains the energy released in nuclear reactions, where a small mass difference becomes a large energy output.

For exam success, identify which observer measures the proper quantity, calculate $\gamma$ carefully, and remember that moving clocks run slow and moving lengths shrink. State whether each effect makes a quantity larger or smaller before you compute, as a sanity check on your arithmetic.