Galilean and Special Relativity 🚀

Introduction

students, imagine you are on a smoothly moving train and you toss a ball straight up. To you, the ball comes back into your hand. To someone standing on the ground, the ball follows a curved path because it keeps moving forward with the train. This simple idea leads to one of the biggest questions in physics: how do we describe motion when observers are moving relative to each other?

In this lesson, you will learn the main ideas behind Galilean relativity and special relativity, two important ways physicists describe motion in different reference frames. By the end, you should be able to explain the key terminology, compare the two ideas, and recognize when each is useful in IB Physics HL. You will also see how these ideas connect to motion, forces, momentum, energy, and the broader study of space and time.

Objectives

• Explain the main ideas and terminology behind Galilean and special relativity.
• Apply IB Physics HL reasoning to situations involving moving observers.
• Recognize extension concepts connected to relativity.
• Summarize how relativity fits into the study of space, time, and motion.
• Use evidence and examples to support explanations of relativistic ideas.

Galilean Relativity: Motion Depends on the Observer

Galilean relativity is the classical idea that the laws of mechanics are the same in all inertial reference frames. An inertial frame is one that is not accelerating. In simple terms, if you are in a car moving at constant velocity, and you drop a coin, the coin behaves according to the same laws as if you were standing still—provided the car is not speeding up or turning.

The key idea is that velocity is relative. If students is on a bus moving at $20\,\text{m s}^{-1}$ east and throws a ball forward at $5\,\text{m s}^{-1}$ relative to the bus, then a person standing on the road sees the ball move at $25\,\text{m s}^{-1}$ east. This uses the Galilean velocity transformation:

$$

$\vec{u}' = \vec{u} - \vec{v}$

$$

where $\vec{u}$ is the velocity in one frame, $\vec{v}$ is the velocity of the moving frame relative to the first, and $\vec{u}'$ is the velocity measured in the moving frame.

This works well for everyday motion because speeds are usually much smaller than the speed of light. For example, when a cyclist passes another cyclist, simple addition of velocities gives accurate results. Galilean relativity is the foundation of Newtonian mechanics, including Newton’s laws of motion and conservation of momentum.

Example: Moving Train 🚆

Suppose a train moves east at $30\,\text{m s}^{-1}$. A student walks forward inside the train at $2\,\text{m s}^{-1}$ relative to the train. The ground observer measures the student’s speed as

$$

30\,$\text{m s}^{-1}$ + 2\,$\text{m s}^{-1}$ = 32\,$\text{m s}^{-1}$

$$

east. This is a Galilean transformation in action.

Galilean relativity also assumes that time is absolute. That means all observers agree on the same time intervals. In classical physics, $t' = t$. This idea seems natural in daily life, but it breaks down at extremely high speeds.

Why Galilean Relativity Is Not Enough

In the late 1800s, experiments showed that light does not behave like ordinary objects. According to classical ideas, if a light source moves, its light should travel faster or slower depending on the observer’s motion, just as a thrown ball does. However, experiments such as the Michelson–Morley experiment found no evidence for the expected “ether wind” effect. In other words, the speed of light appeared to be the same in all directions and for all inertial observers.

This created a problem for Galilean relativity, because if velocities simply add, then different observers should measure different speeds for light. But nature does not work that way.

The result was a new theory: special relativity, developed by Albert Einstein in 1905. Special relativity applies to inertial reference frames and to objects moving at speeds close to the speed of light, $c$.

Special Relativity: Space and Time Are Linked

Special relativity is built on two postulates:

1. The laws of physics are the same in all inertial reference frames.
2. The speed of light in a vacuum is constant for all inertial observers, with value $c \approx 3.00 \times 10^8\,\text{m s}^{-1}$.

These two statements have dramatic consequences. They show that measurements of time, distance, and simultaneity depend on the observer’s frame of reference.

One of the most important ideas is that time is not universal. A moving clock is measured to tick more slowly than a clock at rest relative to the observer. This is called time dilation:

$$

$\Delta t = \gamma \Delta t_0$

$$

where $\Delta t_0$ is the proper time, $\Delta t$ is the dilated time, and

$$

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

$$

is the Lorentz factor.

If $v$ is much smaller than $c$, then $\gamma \approx 1$, so special relativity gives nearly the same results as classical physics. But when $v$ becomes a significant fraction of $c$, the difference matters.

Example: Fast Muons 🌌

Muons are unstable particles created high in Earth’s atmosphere. They should decay before reaching the ground, but many are detected at sea level. From Earth’s frame, the muons travel very fast, so their lifetime is stretched by time dilation. From the muons’ frame, the atmosphere is length-contracted, so they have less distance to travel. Both descriptions are correct because special relativity changes how space and time are measured.

Length Contraction and Simultaneity

Special relativity also predicts length contraction. If an object moves relative to an observer, its length along the direction of motion is measured to be shorter:

$$

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

$$

where $L_0$ is the proper length.

This does not mean the object is “physically squashed” in its own rest frame. It means measurements of length depend on the observer’s motion.

Another subtle idea is relativity of simultaneity. Two events that happen at the same time in one frame may not happen at the same time in another moving frame. This is very different from classical physics, where everyone assumes one shared clock for the universe.

Example: Lightning on a Moving Train ⚡

Imagine lightning strikes the front and back of a moving train at the same time according to a person standing on the platform. A passenger in the middle of the train is moving toward one flash and away from the other. Because light speed is constant for all observers, the passenger does not agree that the strikes were simultaneous. This shows that simultaneity depends on the reference frame.

Momentum, Energy, and Relativistic Thinking

IB Physics HL connects relativity to momentum and energy. In special relativity, momentum is given by

$$

$\vec{p} = \gamma m \vec{v}$

$$

where $m$ is the rest mass. As $v$ approaches $c$, $\gamma$ increases a lot, so momentum grows very large. This is one reason why accelerating a massive object to the speed of light is impossible.

The total energy is

$$

$E = \gamma m c^2$

$$

and the rest energy is

$$

$E_0 = m c^2$

$$

This famous equation shows that mass itself is a form of energy. The kinetic energy is the difference between total and rest energy:

$$

K = ($\gamma$ - 1) m c^2

$$

At low speeds, this reduces approximately to the classical result

$$

$K \approx \frac{1}{2}mv^2$

$$

so classical mechanics is still useful for everyday situations.

Relativistic energy and momentum satisfy

$$

$E^2 = (pc)^2 + (mc^2)^2$

$$

This equation is important because it combines motion, mass, and energy into one relationship.

Comparing Galilean and Special Relativity

The difference between the two theories can be summarized simply:

• Galilean relativity assumes time is the same for everyone and velocities add normally.
• Special relativity keeps the speed of light the same for all inertial observers and changes how we measure time and space.

Galilean relativity works very well for everyday speeds, such as cars, airplanes, and sports. Special relativity is needed for particles in accelerators, satellites using precise timing systems, and high-speed cosmic particles.

In IB Physics HL, you should recognize that special relativity is an extension of classical ideas, not a complete replacement in all cases. When $v \ll c$, the relativistic equations reduce to classical ones. That is why Newtonian mechanics remains accurate in most ordinary situations.

Conclusion

Galilean and special relativity are both about how motion looks from different reference frames, but they make very different assumptions. Galilean relativity says time is absolute and velocities add simply. Special relativity shows that space and time are linked, the speed of light is constant, and observers may disagree about time intervals, lengths, and simultaneity.

For IB Physics HL, the main skill is knowing when classical reasoning is enough and when relativistic reasoning is required. students, if the speeds involved are tiny compared with $c$, use Galilean ideas. If speeds are very high or if light is involved, special relativity becomes essential. These ideas help explain real experiments, particle behavior, and modern technology such as satellite timing and particle accelerators.

Study Notes

• Galilean relativity applies to inertial frames and everyday speeds.
• In Galilean physics, time is absolute: $t' = t$.
• Velocities add normally in classical physics: $\vec{u}' = \vec{u} - \vec{v}$.
• Special relativity is based on two postulates: the laws of physics are the same in all inertial frames, and the speed of light in vacuum is constant.
• The Lorentz factor is $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.
• Time dilation: $\Delta t = \gamma \Delta t_0$.
• Length contraction: $L = \frac{L_0}{\gamma}$.
• Relativity of simultaneity means two observers may disagree about whether events happened at the same time.
• Relativistic momentum is $\vec{p} = \gamma m \vec{v}$.
• Total energy is $E = \gamma m c^2$ and rest energy is $E_0 = mc^2$.
• Special relativity becomes important when $v$ is a significant fraction of $c$.
• In most everyday situations, classical mechanics remains an excellent approximation.