Understand Galilean & Special Relativity — Understand A.5 Galilean and special relativity - IB Physics

SECTION 1

Historical Context & Motivation

Imagine you are sitting in a perfectly smooth train carriage with the blinds drawn. Could you tell whether the train is moving at a constant speed or standing still? This deceptively simple question has driven physicists for over four centuries. The search for an answer led from Galileo Galilei's thought experiments aboard sailing ships all the way to Albert Einstein's revolutionary 1905 paper on special relativity. Understanding this journey is essential for grasping how we describe motion, measure time, and define simultaneity.

1632 • Galileo's Ship

In his Dialogue Concerning the Two Chief World Systems, Galileo argued that the laws of mechanics are identical in any uniformly moving cabin—butterflies fly the same way whether the ship sails or sits in port.

1687 • Newton's Principia

Isaac Newton formalised mechanics with his three laws and introduced the concept of absolute space and absolute time as a fixed stage on which events unfold.

1865 • Maxwell's Equations

James Clerk Maxwell unified electricity and magnetism and showed that light is an electromagnetic wave travelling at a fixed speed c ≈ 3 × 10⁸ m s⁻¹. This speed appeared to be independent of the observer's motion—an unsettling clash with Galilean ideas.

1887 • Michelson–Morley Experiment

Albert Michelson and Edward Morley tried to detect Earth's motion through the supposed 'luminiferous aether.' Their null result suggested that the speed of light really is the same in all directions, regardless of the observer's motion.

1905 • Einstein's Special Relativity

Albert Einstein published 'On the Electrodynamics of Moving Bodies,' establishing that the laws of physics (including the speed of light) are identical in all inertial frames—forever changing our understanding of space and time.

The central question that links all of these milestones is: How do we translate the description of an event from one observer's viewpoint to another's? Galilean relativity gave an intuitive answer that works brilliantly at everyday speeds. But when objects approach the speed of light, that intuitive answer breaks down, and special relativity takes over.

SECTION 2

Core Principles & Definitions

Before diving into equations, you need a firm grip on a few foundational ideas. An inertial reference frame is any viewpoint (frame) that moves at a constant velocity—including zero velocity. Newton's first law holds in every inertial frame: an object free of net force moves in a straight line at constant speed (or stays at rest). Any frame that accelerates, rotates, or vibrates is non-inertial and requires extra 'fictitious' forces (like centrifugal force) to describe motion.

Galilean Relativity Principle

The laws of mechanics are the same in every inertial frame. No mechanical experiment performed inside a closed lab can reveal whether the lab is stationary or moving at constant velocity.

Galilean Velocity Addition

If you walk at 2 m s⁻¹ on a train moving at 30 m s⁻¹, a person on the platform sees you move at 32 m s⁻¹. Velocities simply add: v′ = v + u.

Einstein's First Postulate

The laws of physics—all of them, including electromagnetism—are the same in every inertial reference frame. This extends Galileo's principle beyond mechanics.

Einstein's Second Postulate

The speed of light in a vacuum, c = 3.00 × 10⁸ m s⁻¹, is the same for all inertial observers, regardless of the motion of the source or the observer.

Invariant Speed Limit

Nothing with mass can reach or exceed the speed of light. As an object's speed approaches c, its Lorentz factor (γ) grows without bound, making further acceleration require ever more energy.

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KEY TAKEAWAY

Think of the speed of light like a cosmic speed limit posted on every road in the universe. No matter how fast your car (reference frame) is already going, when you point your headlights forward, the photons still leave at exactly c relative to you. That's very different from throwing a ball on a moving train—in that everyday case, velocities just add up. Einstein's insight was that nature plays by different rules at extreme speeds, and the 'speed-limit sign' never changes for any observer.

SECTION 3

Visualising Reference Frames

The diagram below illustrates two inertial reference frames: frame S (a platform observer) and frame S′ (an observer on a train moving to the right at velocity v). An event P is described by coordinates (x, t) in S and (x′, t′) in S′. Under Galilean transformation, x′ = x − vt and t′ = t (time is absolute). Under the Lorentz transformation of special relativity, both space and time coordinates mix together, and the simple subtraction no longer works.

Frame S is at rest (platform observer). Frame S′ moves at velocity v to the right. Both frames record the same event P using their own coordinate axes. The Galilean transformation keeps time universal (t′ = t), while the Lorentz transformation mixes space and time.

Notice how the Galilean equations simply slide the x-axis by the amount vt. Time is shared—every clock in the universe ticks at the same rate. The Lorentz transformation, by contrast, introduces the Lorentz factor γ (gamma) and a term −vx/c² inside the time equation. This means two events that are simultaneous in S may not be simultaneous in S′. That shocking conclusion—called the relativity of simultaneity—has been confirmed by countless experiments.

SECTION 4

Mathematical Framework

Galilean Transformation Equations

Suppose frame S′ moves at constant velocity v along the x-direction relative to frame S, and both origins coincide at time t = 0. The Galilean transformation relates the coordinates of an event as seen in the two frames.

GALILEAN POSITION TRANSFORM

x′ = x − vt

x = position in frame S, x′ = position in frame S′, v = relative velocity of S′ along x, t = time (same in both frames).

GALILEAN TIME TRANSFORM

t′ = t

Time is absolute in Galilean relativity—all observers share one universal clock.

GALILEAN VELOCITY ADDITION

u′ = u − v

u = velocity of an object in S, u′ = velocity of that object in S′. For motion along the x-axis, velocities simply subtract (or add).

Lorentz Transformation Equations

Einstein replaced the Galilean equations with transformations that keep the speed of light invariant. The key ingredient is the Lorentz factor, γ (gamma).

LORENTZ FACTOR

γ = 1 / √(1 − v²/c²)

v = relative speed between frames, c = 3.00 × 10⁸ m s⁻¹. When v ≪ c, γ ≈ 1 and the Lorentz equations reduce to the Galilean ones.

LORENTZ POSITION TRANSFORM

x′ = γ(x − vt)

The factor γ stretches or compresses spatial separations depending on relative speed.

LORENTZ TIME TRANSFORM

t′ = γ(t − vx / c²)

Time is no longer absolute. The term −vx/c² means that the time reading depends on where the event occurs—this is the origin of the relativity of simultaneity.

RELATIVISTIC VELOCITY ADDITION

u′ = (u − v) / (1 − uv/c²)

u = velocity of an object in S, u′ = velocity in S′. The denominator prevents the result from ever exceeding c.

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Low-Speed Limit

When v ≪ c, the factor v²/c² is negligibly small, so γ → 1 and the denominators in the velocity addition formula approach 1. Every Lorentz equation then collapses back to its Galilean counterpart. This is why classical mechanics works perfectly for cars, aeroplanes, and baseballs.

SECTION 5

Key Consequences — Time Dilation & Length Contraction

Two of the most dramatic predictions of special relativity follow directly from the Lorentz transformations: time dilation and length contraction. Both effects are negligible at everyday speeds but become enormous as v approaches c. A proper time interval (Δt₀) is the time between two events measured in the frame where they happen at the same location. Any other inertial observer measures a longer time interval Δt = γΔt₀. Similarly, proper length (L₀) is the length of an object in its rest frame. A moving observer measures a shorter length L = L₀/γ.

TIME DILATION

Δt = γ Δt₀

Δt₀ = proper time (clock at rest relative to events), Δt = dilated time measured by moving observer, γ = Lorentz factor.

LENGTH CONTRACTION

L = L₀ / γ

L₀ = proper length (measured in the object's rest frame), L = contracted length measured by observer who sees the object moving.

The graph shows how the Lorentz factor γ varies with speed. Below about 0.1c the curve is nearly flat at γ ≈ 1, meaning relativistic effects are negligible. Above 0.9c the curve climbs steeply—at v = 0.99c, γ ≈ 7.1, meaning a moving clock ticks more than seven times slower than a stationary one.

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KEY TAKEAWAY

Time dilation and length contraction are not optical illusions—they are real physical effects. Muons created in cosmic-ray collisions high in the atmosphere survive long enough to reach Earth's surface precisely because their internal clocks run slowly from our perspective (time dilation), and they 'see' the atmosphere as compressed in length (length contraction). Both pictures are consistent and confirmed by experiment.

SECTION 6

Worked Example — Velocity Addition

A spacecraft (S′) travels at 0.80c relative to Earth (S). A probe is launched from the spacecraft at 0.60c relative to the spacecraft, in the same direction of travel. What speed does an Earth observer measure for the probe?

Relativistic Velocity Addition

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Step 1 — Identify Given Values

v = 0.80c (spacecraft speed relative to Earth), u′ = 0.60c (probe speed relative to the spacecraft). We want u, the probe speed as measured from Earth.
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Step 2 — Write the Relativistic Velocity Addition Formula

The relativistic velocity addition formula (rearranged for u from the observer's frame) is: u = (u′ + v) / (1 + u′v/c²)
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Step 3 — Substitute Values

u = (0.60c + 0.80c) / (1 + (0.60c)(0.80c)/c²) u = 1.40c / (1 + 0.48) u = 1.40c / 1.48

u ≈ 0.946c
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Step 4 — Compare with Galilean Prediction

Under Galilean addition, u = u′ + v = 0.60c + 0.80c = 1.40c, which exceeds the speed of light—an impossibility. The relativistic formula keeps the answer safely below c.
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Step 5 — Interpret

The Earth observer measures the probe moving at about 0.946c. Even though the spacecraft and probe each move at large fractions of c, the combined speed cannot reach c. This is a direct consequence of Einstein's second postulate.

SECTION 7

Galilean vs Special Relativity — Side by Side

Feature	Galilean Relativity	Special Relativity
Time	Absolute — same for all observers	Relative — depends on frame (time dilation)
Length	Same for all observers	Contracts along direction of motion
Simultaneity	Preserved — if simultaneous in one frame, simultaneous in all	Relative — events simultaneous in one frame may not be in another
Velocity addition	u′ = u − v (simple subtraction)	u′ = (u − v) / (1 − uv/c²)
Speed of light	Not invariant — changes with observer motion	Invariant — c in all inertial frames
Valid regime	v ≪ c (everyday speeds)	All speeds from 0 to just below c

Comparison of Galilean and Special Relativity

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KEY TAKEAWAY

Galilean relativity is not 'wrong'—it is a highly accurate approximation for the world we experience daily. Special relativity is the more complete theory that reduces to the Galilean version when speeds are small compared to c. This is an example of the correspondence principle: a new theory must reproduce the predictions of the older theory in the domain where the older theory already works.

SECTION 8

Connection to General Relativity & Modern Physics

Special relativity deals exclusively with inertial (non-accelerating) frames. When gravity enters the picture, or when observers accelerate, we need Einstein's 1915 General Theory of Relativity. General relativity describes gravity not as a force, but as the curvature of spacetime caused by mass and energy. The table below contrasts special and general relativity.

Aspect	Special Relativity (1905)	General Relativity (1915)
Frames covered	Inertial (constant velocity) frames only	All frames, including accelerating and gravitational fields
Gravity	Not included	Described as curvature of spacetime
Geometry	Flat (Minkowski) spacetime	Curved spacetime
Key equation	Lorentz transformations	Einstein field equations
Applications	Particle accelerators, nuclear energy, GPS clock corrections	Black holes, gravitational waves, cosmology

Special vs General Relativity

One practical application you encounter every day is the Global Positioning System (GPS). GPS satellites orbit at high speed (requiring special-relativistic time dilation corrections) and at high altitude where gravity is weaker (requiring general-relativistic corrections). Without both corrections, GPS positions would drift by roughly 10 km per day. So, relativity is not just abstract theory—it keeps your map app working.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A passenger on a smoothly moving train drops a ball straight down. The ball falls vertically in the train's frame. Describe the path of the ball as seen by an observer standing on the platform. Does this observation violate the principle of relativity? Explain.

Show answer

The platform observer sees the ball follow a parabolic (projectile) path because the ball has the train's horizontal velocity in addition to its downward acceleration due to gravity. This does not violate the principle of relativity. Both observers apply Newton's second law correctly in their own inertial frames. The trajectory looks different because they use different reference frames, but the physical laws governing the motion are identical. This is precisely Galileo's point: no mechanical experiment performed inside the train can reveal whether it is moving or stationary.

Question 2: BASIC CALCULATION

A car moves at 25 m s⁻¹ relative to the road. A passenger inside throws a ball forward at 10 m s⁻¹ relative to the car. Using Galilean velocity addition, what is the ball's speed relative to the road?

Show answer

Using the Galilean velocity addition u = u′ + v, we have u = 10 m s⁻¹ + 25 m s⁻¹ = 35 m s⁻¹. The ball moves at 35 m s⁻¹ relative to the road. At these everyday speeds (far below the speed of light), the Galilean formula is indistinguishable from the relativistic one because the correction term uv/c² is negligibly small—about 2.8 × 10⁻¹⁵ in this case.

Question 3: INTERMEDIATE

Rocket A moves at 0.70c relative to Earth. Rocket A fires a missile forward at 0.50c relative to itself. Calculate the speed of the missile as observed from Earth, using (a) Galilean addition and (b) relativistic velocity addition. Comment on the difference.

Show answer

(a) Galilean: u = 0.50c + 0.70c = 1.20c. This exceeds c, violating the speed-of-light limit. (b) Relativistic: u = (0.50c + 0.70c) / (1 + (0.50)(0.70)) = 1.20c / (1 + 0.35) = 1.20c / 1.35 ≈ 0.889c. The relativistic result is safely below c. The Galilean answer over-predicts the speed by about 35% at these relativistic velocities, clearly demonstrating why the classical formula fails when v is a significant fraction of c. The denominator (1 + u′v/c²) always keeps the result below c, consistent with Einstein's second postulate.

Question 4: APPLIED

Muons are created in the upper atmosphere at about 10 km altitude and travel toward Earth at 0.998c. In the muon's rest frame, its mean lifetime is 2.2 μs. (a) Calculate the Lorentz factor γ. (b) Determine the dilated lifetime as measured by an Earth observer. (c) Show that the muon can indeed reach the ground.

Show answer

(a) γ = 1/√(1 − v²/c²) = 1/√(1 − 0.998²) = 1/√(1 − 0.996004) = 1/√(0.003996) ≈ 1/0.06322 ≈ 15.8. (b) Dilated lifetime Δt = γΔt₀ = 15.8 × 2.2 μs ≈ 34.8 μs. (c) In the Earth frame, the muon travels d = vΔt = 0.998 × 3.00 × 10⁸ × 34.8 × 10⁻⁶ ≈ 10,400 m ≈ 10.4 km. This exceeds the 10 km altitude, so the muon reaches the ground. Without time dilation, the muon would only travel about 660 m before decaying (0.998c × 2.2 μs), far short of 10 km. This real-world observation is one of the strongest confirmations of special relativity.

Question 5: CRITICAL THINKING

A spaceship shines a laser beam forward. The ship moves at 0.90c relative to Earth. Using the relativistic velocity addition formula, show that the Earth observer also measures the light speed as c. Then explain in your own words why this result is philosophically significant—what does it reveal about the nature of space and time?

Show answer

Let v = 0.90c and u′ = c (the laser light in the ship's frame). Substituting into the relativistic velocity addition formula: u = (c + 0.90c) / (1 + (c)(0.90c)/c²) = 1.90c / (1 + 0.90) = 1.90c / 1.90 = c. The result is exactly c, as predicted by Einstein's second postulate. This works for any value of v because the algebra always simplifies to c when u′ = c. Philosophically, this result forces us to abandon the intuitive notion that time flows at the same rate for everyone and that space is a fixed, rigid stage. If light speed is invariant, then the 'cost' is paid by space and time themselves: they stretch, compress, and interleave depending on the observer's motion. Simultaneity, duration, and distance are all frame-dependent. This shows that our everyday intuition about time and space is not a fundamental truth but merely an approximation valid at low speeds.

SUMMARY

Lesson Summary

Summary

Galilean relativity states that the laws of mechanics are the same in all inertial reference frames (frames moving at constant velocity). Under the Galilean transformation, positions shift by x′ = x − vt while time stays absolute (t′ = t), and velocities add by simple arithmetic: u′ = u − v. This works perfectly at everyday speeds, but predicts absurdities (speeds exceeding c) for objects near the speed of light.

Einstein's special relativity resolves this by postulating that (1) all laws of physics hold in every inertial frame, and (2) the speed of light c is invariant. The resulting Lorentz transformations mix space and time via the Lorentz factor γ = 1/√(1 − v²/c²), producing time dilation (moving clocks run slow) and length contraction (moving objects shorten). Relativistic velocity addition ensures that combined speeds never exceed c. At low speeds (v ≪ c), all relativistic formulas reduce to their Galilean counterparts—an example of the correspondence principle.