Revise Galilean and special relativity for IB Physics (A.5)

A reference frame is simply a coordinate system, together with a clock, used to record where and when events happen. An inertial frame is one that is not accelerating, so that Newton's first law holds within it. Picture a train gliding along a straight track at constant velocity: drop a ball inside and it falls straight down to your feet, exactly as it would on the platform. The principle of Galilean relativity says the laws of mechanics are identical in all such inertial frames — no mechanical experiment done inside the smoothly moving carriage can tell you whether you are moving or standing still.

To compare measurements between a frame on the platform and a frame on the train moving at speed $v$, classical physics uses the Galilean transformations. They assume, above all, that time is absolute — one universal clock ticks the same for every observer.

$\text{Position: } x' = x - vt$ $\text{Time: } t' = t$ $\text{Velocity: } u' = u - v$

The velocity rule is classical (Galilean) velocity addition: to convert an object's velocity from the platform frame to the train frame, subtract the train's velocity. Throw a ball forwards at $8$ m s⁻¹ inside a train moving at $30$ m s⁻¹, and the platform sees it move at $38$ m s⁻¹. For everyday speeds this is indistinguishable from reality.

The trouble appears when the moving object is light itself. Applied to a torch beam on the train, Galilean addition predicts the platform should measure the light at $c + v$. Yet Maxwell's equations of electromagnetism give a single fixed speed, $c$, for light in a vacuum, with no reference to any source's motion — and every experiment has confirmed it. Galilean velocity addition and the behaviour of light cannot both be right.

Rather than patch the theory, Einstein founded a new one on two deceptively simple statements. Everything that follows — the Lorentz factor, time dilation, length contraction, the relativity of simultaneity — is a logical consequence of these two postulates.

The second postulate is the radical one. If light comes out at $c$ for everyone no matter how they move, then two observers in relative motion cannot agree on all time intervals and all distances, because speed is distance over time. Keeping $c$ fixed forces time and space to flex. That flexing is captured by a single quantity, the Lorentz factor.

The Lorentz factor $\gamma$ measures how strongly relativistic effects apply at a given relative speed. It depends only on the ratio $v/c$.

$\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

Because $v < c$ for anything with mass, the fraction $v^2/c^2$ lies between 0 and 1, the square root is less than 1, and so $\gamma$ is always greater than or equal to 1. At everyday speeds $v/c$ is tiny and $\gamma$ is indistinguishable from 1, which is why we never notice relativity in daily life. As $v$ climbs towards $c$, the denominator shrinks towards zero and $\gamma$ grows without bound. A quick feel for the numbers: at $0.10c$, $\gamma \approx 1.005$; at $0.60c$, $\gamma = 1.25$; at $0.80c$, $\gamma \approx 1.67$; at $0.995c$, $\gamma \approx 10.0$.

The first great consequence is time dilation: a clock moving relative to an observer is measured to run slow. If a process takes a time $\Delta t_0$ in the frame where it happens at a single place, then an observer who sees that frame move past measures a longer time $\Delta t$.

$\Delta t = \gamma \, \Delta t_0$

The quantity $\Delta t_0$ is the proper time: the interval between two events measured in the frame where those events occur at the same location — for instance, the reading on the travelling clock itself, or the lifetime of a particle in its own rest frame. Because $\gamma \ge 1$, the proper time is always the shortest interval; every other observer measures more. Identifying which interval is the proper time is the single most important decision in a time-dilation problem.

Space contracts just as time dilates. An object moving relative to an observer is measured to be shorter along its direction of motion than it is in its own rest frame.

$L = \dfrac{L_0}{\gamma}$

Here $L_0$ is the proper length: the length measured in the frame where the object is at rest, and the largest length any observer measures. A moving observer measures the smaller $L = L_0/\gamma$. Crucially, only the dimension along the direction of motion contracts — widths and heights perpendicular to the motion are unchanged. A rocket flying past nose-first is measured shorter from nose to tail but exactly as wide as ever.

Galilean velocity addition ($u' = u - v$) cannot be right at high speeds, because it would let two velocities combine to exceed $c$. The correct relativistic rule keeps every result at or below $c$.

u' = \dfrac{u - v}{1 - \dfrac{uv}{c^2}}

In this form, $u$ is the object's velocity measured in one frame, $v$ is the velocity of a second frame relative to the first, and $u'$ is the object's velocity measured in that second frame. When $u$ and $v$ are both far smaller than $c$, the term $uv/c^2$ is negligible and the formula collapses back to the familiar Galilean $u' = u - v$. But feed in $u = c$ and the result is always exactly $c$, for any $v$ — the invariance of the speed of light is built into the algebra.

Perhaps the strangest consequence of the two postulates is that observers in relative motion cannot even agree on whether two separated events happened at the same time. Imagine a train carriage moving fast past a platform, with a lamp exactly at its centre. The lamp flashes, and light spreads out towards the front and back walls. In the train's own frame the two walls are equal distances away and the light, travelling at $c$ in every direction, reaches both walls at the same instant — the two arrival events are simultaneous.

Now watch the same flash from the platform. The light still travels at $c$ (second postulate), but during its flight the rear wall rushes towards the point where the flash was emitted while the front wall races away from it. So the platform observer sees the light reach the rear wall first and the front wall later — the two events are not simultaneous. Neither observer is mistaken: simultaneity is relative, a property of the chosen inertial frame rather than of the events themselves. There is no universal 'now' shared by all observers.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and error-carried-forward (ECF) means a wrong number early on need not cost you the marks that follow. An accuracy mark is dependent on its method: it is only awarded if the method that earns it is present and correct, but through ECF it is judged against your earlier figure, not only the official one. That protection exists only if your method is written down. Study how each mark below is earned by a specific line.

The Lorentz factor introduced here is the gateway to the rest of relativity. The same $\gamma$ that stretches time and shrinks length also governs relativistic momentum and total energy, and it underlies the spacetime-diagram methods used to visualise the relativity of simultaneity more precisely. Master the habits now — identify the proper time and proper length before touching a formula, compute $v/c$ first, keep $\gamma$ to full precision, and show every line — and the relativistic mechanics that follows becomes variations on a method you already own.