Apply Galilean & Special Relativity — Apply A.5 Galilean...

SECTION 1

Historical Context & Motivation

If you throw a ball on a moving train, how fast is the ball moving relative to the ground? For centuries, physicists answered this question using Galilean relativity — simply add the velocities together. This approach works perfectly for cars, planes, and baseballs. But when scientists tried to apply the same idea to light, something strange happened: the speed of light refused to follow the addition rule. This mystery eventually led Albert Einstein to construct an entirely new framework called special relativity, which replaced Galilean ideas for objects moving at significant fractions of the speed of light.

1632

Galileo's Ship Analogy

In his Dialogue Concerning the Two Chief World Systems, Galileo argued that the laws of physics are the same inside a ship moving at constant velocity as on land. This principle of Galilean invariance became the first formal statement of relativity.

1687

Newton's Principia

Isaac Newton codified mechanics with his three laws, all built on the assumption that time and space are absolute. Galilean transformations provided the mathematical tool to relate positions and velocities between different inertial frames.

1887

Michelson–Morley Experiment

Albert Michelson and Edward Morley attempted to detect Earth's motion through the luminiferous aether by measuring differences in the speed of light. They found no difference at all, suggesting the speed of light is constant in all directions.

1905

Einstein's Special Relativity

Einstein published 'On the Electrodynamics of Moving Bodies', replacing Galilean transformations with Lorentz transformations. He showed that time dilation, length contraction, and the constancy of the speed of light are natural consequences of two simple postulates.

The key question this lesson addresses is: When should you use Galilean transformations, and when must you switch to special relativity? Understanding the boundary between these two frameworks is essential for IB Physics, and for solving problems involving reference frames, velocities, time, and length.

SECTION 2

Core Principles & Definitions

Before diving into equations, you need a solid grasp of the foundational ideas that underpin both Galilean and special relativity. Both frameworks deal with inertial reference frames — coordinate systems that move at constant velocity (no acceleration). The fundamental difference is how each framework treats time, length, and the speed of light.

Inertial Reference Frame

A frame of reference in which Newton's first law holds: an object at rest stays at rest, and an object in motion continues at constant velocity unless acted on by a force. Examples include a train moving at constant speed or a lab on Earth's surface (approximately).

Galilean Relativity

The classical principle stating that the laws of mechanics are the same in all inertial frames. Time is absolute (identical for all observers), and velocities add linearly: v_total = v₁ + v₂. Valid when speeds are much less than c.

Einstein's Two Postulates

(1) The laws of physics are the same in all inertial reference frames. (2) The speed of light in a vacuum, c ≈ 3.00 × 10⁸ m/s, is the same for all observers regardless of their motion or the source's motion.

Lorentz Factor (γ)

The factor γ = 1 / √(1 − v²/c²) appears throughout special relativity. At low speeds γ ≈ 1 and relativistic effects vanish. As v approaches c, γ increases without bound, producing dramatic time dilation and length contraction.

Proper Time & Proper Length

Proper time (Δt₀) is measured by a clock present at both events. Proper length (L₀) is measured in the frame where the object is at rest. All other observers measure dilated times and contracted lengths.

✦ KEY TAKEAWAY

Think of Galilean relativity as the 'common-sense' version of physics — it works great for everyday speeds, like adding your walking speed to the speed of a moving sidewalk. Special relativity is the 'high-speed upgrade' that kicks in when things move at a significant fraction of the speed of light. Just as your phone switches from Wi-Fi to cellular when you leave the house, physics 'switches' frameworks when velocities become extreme.

SECTION 3

Visualizing Reference Frames & Velocity Addition

The diagram below illustrates the contrast between Galilean and relativistic velocity addition. A spaceship travels at velocity v relative to Earth, and it fires a probe forward at velocity u′ relative to itself. The question is: how fast does Earth see the probe moving?

Top panel: Galilean addition gives a total speed exceeding c, which is physically impossible. Bottom panel: the relativistic velocity addition formula automatically keeps the result below c. Notice that both panels use the same input values (v = 0.6c, u′ = 0.5c) but produce very different results.

In the upper panel, the Galilean approach simply adds the two speeds to get 1.1c — faster than light. Einstein's second postulate forbids this, so the lower panel uses the relativistic formula. The denominator (1 + vu′/c²) acts as a correction factor that prevents the total from ever reaching or exceeding c. At low speeds this denominator is essentially 1, and both formulas give nearly the same answer. That is why Galilean relativity worked perfectly for centuries of everyday physics.

SECTION 4

Mathematical Framework

Galilean Transformations

Suppose frame S′ moves at constant velocity v along the x-axis relative to frame S. The Galilean transformation equations connect coordinates in S to those in S′. They assume absolute time, meaning clocks in both frames always read the same value.

GALILEAN POSITION TRANSFORMATION

x′ = x − vt

x′ = position in the moving frame, x = position in the stationary frame, v = relative velocity of frames, t = time (same in both frames).

GALILEAN VELOCITY ADDITION

u = u′ + v

u = velocity of object in frame S, u′ = velocity of object in frame S′, v = velocity of S′ relative to S. This is simple vector addition.

Special Relativity — Lorentz Transformations

When speeds approach c, you must replace Galilean transformations with Lorentz transformations. These account for the fact that time and space are intertwined and that different observers can disagree on simultaneity, time intervals, and lengths.

LORENTZ FACTOR

γ = 1 / √(1 − v²/c²)

γ (gamma) ≥ 1 always. At v = 0, γ = 1. At v = 0.87c, γ ≈ 2. As v → c, γ → ∞. This factor governs how strongly relativistic effects appear.

TIME DILATION

Δt = γΔt₀

Δt₀ = proper time (measured by a clock at rest relative to the events), Δt = dilated time (measured by an observer who sees the clock moving). Moving clocks run slow.

LENGTH CONTRACTION

L = L₀ / γ

L₀ = proper length (measured in the rest frame of the object), L = contracted length (measured by an observer who sees the object moving). Moving objects appear shorter along the direction of motion.

RELATIVISTIC VELOCITY ADDITION

u = (u′ + v) / (1 + u′v/c²)

u = velocity in frame S, u′ = velocity in frame S′, v = relative frame velocity. The denominator ensures u never exceeds c, even when u′ and v are each close to c.

💡 IB Exam Tip

Always check the ratio v/c first. If v/c < 0.1, Galilean transformations are perfectly fine and much simpler. If v/c ≥ 0.1, you should use the Lorentz equations. The IB data booklet provides all the formulas above — your job is to know when and how to apply each one.

SECTION 5

Time Dilation & Length Contraction in Detail

The two most dramatic consequences of special relativity are time dilation and length contraction. These effects are not optical illusions — they are real, measurable, and experimentally confirmed. The table below shows how γ grows with speed and the magnitude of each effect.

How the Lorentz factor γ scales with speed

Speed (v)	v / c	γ	Time Dilation Factor	Length (% of L₀)
30 km/s	0.0001	≈ 1.000000005	negligible	≈ 100%
0.10c	0.10	1.005	× 1.005	99.5%
0.50c	0.50	1.155	× 1.155	86.6%
0.80c	0.80	1.667	× 1.667	60.0%
0.90c	0.90	2.294	× 2.294	43.6%
0.99c	0.99	7.089	× 7.089	14.1%

The Lorentz factor curve remains nearly flat (γ ≈ 1) up to about 0.4c, meaning relativistic effects are negligible at everyday speeds. Beyond 0.8c the curve rises steeply, producing large time dilation and length contraction.

The key feature of the graph is the dramatic upturn near v = c. At 50% the speed of light, the Lorentz factor is only 1.15 — a 15% correction. But at 90% it has already doubled, and at 99% it reaches about 7. This non-linear behavior is why relativistic effects seem to appear 'suddenly' — they were always there, but they become large enough to notice only at high speeds.

SECTION 6

Worked Example — Muon Decay

One of the best real-world confirmations of time dilation involves muons — subatomic particles created when cosmic rays hit the upper atmosphere at about 10 km altitude. Muons have a half-life of 1.56 μs in their rest frame, so classically they should decay long before reaching Earth's surface. Yet ground-level detectors observe far more muons than expected. Let's see why.

How do muons reach Earth's surface?

Step 1 — State the problemA muon is created at 10 km altitude and travels toward Earth at v = 0.98c. Its rest-frame half-life is Δt₀ = 1.56 μs. Determine (a) the Galilean prediction of the distance it should travel in one half-life, and (b) the dilated half-life and actual distance using special relativity.

Step 2 — Galilean predictionUsing Galilean mechanics, the muon travels a distance d = vΔt₀ in one half-life. Substituting: d = (0.98 × 3.00 × 10⁸ m/s)(1.56 × 10⁻⁶ s) = 458.6 m. This is far less than 10 km, so classically most muons should never reach the ground.

Galilean distance per half-life ≈ 459 m

Step 3 — Calculate the Lorentz factorγ = 1 / √(1 − v²/c²) = 1 / √(1 − 0.98²) = 1 / √(1 − 0.9604) = 1 / √(0.0396) = 1 / 0.1990 ≈ 5.025.

γ ≈ 5.03

Step 4 — Time dilation approachFrom Earth's perspective, the muon's half-life is dilated: Δt = γΔt₀ = 5.03 × 1.56 μs = 7.84 μs. The distance it travels in this dilated half-life is d = vΔt = (0.98 × 3.00 × 10⁸)(7.84 × 10⁻⁶) = 2305 m ≈ 2.3 km. With several half-lives available over a 10 km descent, a measurable fraction survive to ground level.

Dilated half-life ≈ 7.84 μs, distance per half-life ≈ 2.3 km

Step 5 — Length contraction approach (muon's frame)Alternatively, from the muon's perspective its own clock runs normally (Δt₀ = 1.56 μs), but the atmosphere is length-contracted: L = L₀/γ = 10 000 m / 5.03 = 1989 m ≈ 2.0 km. The muon only needs to travel about 2 km — well within reach during its 1.56 μs lifetime — explaining why so many survive to ground level.

Contracted atmosphere thickness ≈ 2.0 km. Both frames agree: muons reach the surface.

✓ CONSISTENCY CHECK

Time dilation and length contraction are two sides of the same coin. Earth observers say 'the muon's clock runs slow, giving it more time.' The muon (if it could talk) would say 'the atmosphere is shorter, so I don't need as much time.' Both perspectives yield the same prediction — the muon reaches the ground.

SECTION 7

Galilean vs. Special Relativity — Side by Side

The table below summarizes the key differences between the two frameworks. Knowing these distinctions is essential for choosing the correct approach on IB exam problems.

Galilean vs. Special Relativity — Key Contrasts

Feature	Galilean Relativity	Special Relativity
Time	Absolute — the same for all observers	Relative — depends on the observer's motion (time dilation)
Length	Absolute — the same for all observers	Relative — contracts along the direction of motion
Velocity addition	u = u′ + v (simple addition)	u = (u′ + v) / (1 + u′v/c²)
Speed of light	Can be exceeded by adding velocities	c is an absolute upper limit in all frames
Mass	Constant regardless of speed	Rest mass is invariant; relativistic momentum increases with γ
Valid regime	v ≪ c (everyday speeds)	All speeds; reduces to Galilean when v ≪ c

✦ KEY TAKEAWAY

Galilean relativity is not 'wrong' — it is the low-speed approximation of special relativity. Just as rounding 1.00003 to 1 is perfectly acceptable in everyday arithmetic, treating γ as 1 is perfectly acceptable at everyday speeds. Special relativity is the more general, more accurate theory, and Galilean relativity is the special case you recover when v/c is very small.

SECTION 8

Connection to General Relativity & Modern Physics

Special relativity handles inertial (non-accelerating) frames. But what about gravity and acceleration? In 1915, Einstein extended his theory into general relativity, which describes gravity not as a force but as the curvature of spacetime caused by mass and energy. The table below outlines how our IB-level special relativity connects to these broader ideas.

Special vs. General Relativity

Concept	Special Relativity (IB Level)	General Relativity (Beyond IB)
Reference frames	Inertial frames only (constant velocity)	All frames, including accelerating and gravitational
Gravity	Not addressed (flat spacetime)	Described as curvature of spacetime
Time dilation	Due to relative velocity (Δt = γΔt₀)	Also due to gravitational fields (clocks near massive objects run slower)
Applications	Particle physics, cosmic ray muons, fast spacecraft	GPS satellites, black holes, gravitational waves, cosmology

A practical example of general relativity you use every day is the Global Positioning System (GPS). GPS satellites orbit at about 14 000 km/h and experience both special-relativistic time dilation (their clocks run slightly slow due to their speed) and general-relativistic time dilation (their clocks run slightly fast because they are farther from Earth's gravitational field). Without correcting for both effects, GPS positions would drift by roughly 10 km per day — making your phone's map completely useless.

🔭 Looking Ahead

If you continue into university physics, you will encounter the full mathematical machinery of Lorentz transformations in matrix form, four-vectors, spacetime intervals, and the equivalence principle. The IB A.5 content gives you the conceptual and computational foundation for all of these advanced topics.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A passenger on a train moving at 30 m/s rolls a ball forward at 2 m/s relative to the train floor. An observer on the platform measures the ball's speed. Would the Galilean or the Lorentz transformation give a noticeably different answer in this scenario? Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

A spaceship travels at v = 0.60c relative to Earth. Calculate the Lorentz factor γ for this speed.

PROBLEM 3 — INTERMEDIATE

A rocket moves at 0.80c relative to Earth. An astronaut on the rocket measures the length of the rocket as 120 m. (a) What length does an Earth observer measure? (b) The astronaut measures 5.0 s on her clock between two events that occur at the same location on the rocket. What time interval does the Earth observer measure?

PROBLEM 4 — APPLIED

Two spaceships approach each other, each traveling at 0.70c relative to a space station. Using the relativistic velocity addition formula, determine the speed of one spaceship as measured by an observer on the other spaceship.

PROBLEM 5 — CRITICAL THINKING

A particle accelerator boosts a proton to 0.995c. In the lab frame, the proton travels through a 2.00 km long straight section of the accelerator. (a) How long is this section in the proton's rest frame? (b) How long does the proton take to traverse the section according to a lab clock? (c) How long does the traversal take according to the proton's own clock? (d) Verify that your answers to (b) and (c) are consistent by using the relationship Δt = γΔt₀.

SUMMARY

Lesson Summary

Galilean relativity uses simple velocity addition (u = u′ + v) and assumes absolute time and absolute length. It works perfectly for everyday speeds where v ≪ c. Einstein's special relativity replaces these ideas with Lorentz transformations, introducing the Lorentz factor γ = 1/√(1 − v²/c²) to account for the constancy of the speed of light. The key results are time dilation (Δt = γΔt₀), length contraction (L = L₀/γ), and relativistic velocity addition u = (u′ + v)/(1 + u′v/c²).

When solving IB problems, always identify proper time (the clock present at both events) and proper length (measured in the object's rest frame). Calculate γ first, then apply the appropriate equation. Remember that time dilation and length contraction are complementary perspectives — different observers disagree about which effect they see, but they always agree on physical outcomes, such as whether a muon reaches the ground.