Galilean and Special Relativity

Introduction

Galilean relativity works beautifully for everyday speeds: a ball thrown inside a smoothly moving train behaves as if the train were at rest. Special relativity begins when that comforting picture meets the experimental fact that light in vacuum has the same speed for every inertial observer. The IB guide places this topic in the HL extension because it requires a new way of linking measurements of space and time.

Guide Focus

• Compare Galilean and special relativity for inertial reference frames.
• Use Lorentz transformations, time dilation, length contraction, and velocity addition.
• Interpret space-time diagrams and experimental evidence such as muon decay.

Key Concepts

1. Reference frames and Galilean relativity

An inertial reference frame is non-accelerating. In Galilean relativity, Newton’s laws have the same form in all inertial frames and events transform as x’ = x - vt and t’ = t. This leads to the everyday velocity transformation u’ = u - v.

2. Postulates of special relativity

Special relativity is built on two statements: the laws of physics are identical in all inertial frames, and the speed of light in vacuum is the same for all inertial observers. These postulates replace absolute time with measurements that depend on the observer’s frame.

3. Lorentz factor and transformations

For relative speed v, gamma = 1 / sqrt(1 - v^2 / c^2). Coordinates transform using x’ = gamma(x - vt) and t’ = gamma(t - vx / c^2). The relativistic velocity transformation is u’ = (u - v) / (1 - uv / c^2).

4. Time dilation and length contraction

Proper time is measured in the frame where the two events occur at the same place, and moving observers measure a longer interval: delta t = gamma delta t0. Proper length is measured in the object’s rest frame, and a moving observer measures L = L0 / gamma along the direction of motion.

5. Space-time interval and diagrams

The interval (delta s)^2 = (c delta t)^2 - (delta x)^2 is invariant. On a space-time diagram, the angle between a particle’s world line and the time axis is linked to speed by tan(theta) = v / c.

Common Mistakes

• Using Galilean velocity addition when speeds are a significant fraction of c.
• Applying length contraction to directions perpendicular to motion.
• Forgetting that gamma is always greater than or equal to 1.

Exam Tips

• Identify the proper quantity first: proper time belongs to one clock; proper length belongs to the object’s rest frame.
• Check the limit v << c. Relativistic equations should reduce approximately to familiar Galilean results.
• Use units consistently in space-time interval questions; c delta t and delta x must both be lengths.

Practice Questions

Question 1 (Multiple Choice)

A spacecraft moves past Earth at 0.80c. Which statement is correct according to an observer on Earth?

A. The spacecraft length parallel to its motion is shorter than its proper length. B. The spacecraft clocks run faster than Earth clocks. C. The spacecraft length perpendicular to its motion is contracted. D. The speed of light emitted from the spacecraft is c + 0.80c.

Question 2 (Structured Paper 2 Style)

A muon has a proper lifetime of 2.2 microseconds and moves at 0.98c relative to a laboratory.

(a) Calculate gamma. [2 marks]

(b) Calculate the lifetime measured in the laboratory. [2 marks]