05Foundations of Special Relativity
Explore Einstein's postulates of special relativity, time dilation, length contraction, relativistic momentum, and mass-energy equivalence.
Special relativity overturned classical assumptions about space and time. It underpins GPS technology, nuclear energy, and modern particle physics.
How does motion at speeds approaching the speed of light change our understanding of time, space, mass, and energy?
In 1905, Albert Einstein published his special theory of relativity, fundamentally reshaping physics. Classical (Newtonian) mechanics works perfectly at everyday speeds, but breaks down when objects travel at a significant fraction of the speed of light (c = 3 × 10⁸ m/s). Special relativity introduces two simple postulates from which remarkable consequences follow: moving clocks run slow (time dilation), moving rulers shrink (length contraction), mass and energy are interchangeable (E = mc²), and velocities cannot simply be added together once they approach c. These effects are not illusions — they have been confirmed by countless experiments, from muon detection in cosmic rays to the precision of GPS satellites.
Einstein's Two Postulates
Postulate 1 — Principle of Relativity
The laws of physics are the same in all inertial (non-accelerating) reference frames. No experiment can distinguish between being at rest and moving at constant velocity.
Postulate 2 — Constancy of the Speed of Light
The speed of light in a vacuum is c = 3 × 10⁸ m/s for all observers in all inertial frames, regardless of the motion of the source or the observer.
The Lorentz Factor (γ)
| Speed (v) | v/c | γ (Lorentz factor) |
|---|---|---|
| 0.1c | 0.10 | 1.005 |
| 0.5c | 0.50 | 1.155 |
| 0.8c | 0.80 | 1.667 |
| 0.9c | 0.90 | 2.294 |
| 0.99c | 0.99 | 7.089 |
Key Equations
Lorentz factor
γ = 1 / √(1 − v²/c²)
Time dilation
Δt = γ Δt₀
Length contraction
L = L₀ / γ
Relativistic momentum
p = γmv
Relativistic energy
E = γmc²
Rest energy
E₀ = mc²
Δt₀ = proper time (measured in the rest frame of the event); L₀ = proper length (rest length); m = rest mass; c = 3 × 10⁸ m/s.
Worked Examples
A spaceship travels at v = 0.6c relative to Earth. Calculate the Lorentz factor γ.
Write the formula: γ = 1 / √(1 − v²/c²)
Substitute v = 0.6c: v²/c² = (0.6)² = 0.36
Compute the radicand: 1 − 0.36 = 0.64
√0.64 = 0.8
γ = 1 / 0.8 = 1.25
A muon has a proper lifetime of Δt₀ = 2.2 μs and travels at v = 0.98c relative to Earth. What is its observed lifetime as measured by an Earth observer?
Find γ: v²/c² = (0.98)² = 0.9604
1 − 0.9604 = 0.0396
√0.0396 ≈ 0.1990
γ = 1 / 0.1990 ≈ 5.025
Apply time dilation: Δt = γ Δt₀ = 5.025 × 2.2 μs
Δt ≈ 11.06 μs
A spaceship has a rest length of L₀ = 100 m. It travels at v = 0.8c relative to Earth. What length does an Earth observer measure?
From the Lorentz factor table (or calculation): at v = 0.8c, γ = 1.667
Apply length contraction: L = L₀ / γ
L = 100 m / 1.667
L ≈ 60.0 m
Calculate the rest energy of a proton (mass m = 1.67 × 10⁻²⁷ kg). Express your answer in joules and in MeV.
Use the rest energy formula: E₀ = mc²
E₀ = (1.67 × 10⁻²⁷ kg) × (3 × 10⁸ m/s)²
E₀ = (1.67 × 10⁻²⁷) × (9 × 10¹⁶)
E₀ = 1.503 × 10⁻¹⁰ J
Convert to MeV: 1 MeV = 1.602 × 10⁻¹³ J
E₀ = (1.503 × 10⁻¹⁰) / (1.602 × 10⁻¹³) ≈ 938 MeV
Rocket A moves at 0.6c relative to Earth. It fires a probe forward at 0.5c relative to the rocket. What is the probe's speed relative to Earth? (Use relativistic velocity addition.)
Classical addition would give 0.6c + 0.5c = 1.1c — impossible!
Relativistic velocity addition: u = (v + u′) / (1 + v·u′/c²)
Here v = 0.6c (rocket relative to Earth), u′ = 0.5c (probe relative to rocket)
Numerator: v + u′ = 0.6c + 0.5c = 1.1c
Denominator: 1 + (0.6)(0.5) = 1 + 0.30 = 1.30
u = 1.1c / 1.30 ≈ 0.846c
Guided Problems
A particle travels at v = 0.5c. Calculate its Lorentz factor γ.
Hint: Square v/c first: (0.5)² = 0.25. Then compute 1 − 0.25 = 0.75, take the square root, and divide 1 by that result.
An astronaut's heart beats once every Δt₀ = 1.0 s in her own frame. Her rocket travels at v = 0.9c relative to Earth. How many seconds pass on Earth between her heartbeats?
Hint: Use Δt = γ Δt₀. First find γ at v = 0.9c (γ ≈ 2.294 from the table), then multiply by 1.0 s.
A train at rest is L₀ = 200 m long. It travels through a tunnel at v = 0.6c. What length does a stationary observer at the tunnel entrance measure?
Hint: Use L = L₀ / γ. You already know γ = 1.25 at v = 0.6c (from Worked Example 1). Divide 200 m by 1.25.
An electron has rest mass m = 9.11 × 10⁻³¹ kg. Calculate its rest energy in joules.
Hint: Apply E₀ = mc². Use c = 3 × 10⁸ m/s, so c² = 9 × 10¹⁶ m²/s². Multiply m × c².
Spaceship X moves at 0.7c relative to Earth. It launches a shuttle backward at 0.4c relative to itself. What is the shuttle's speed relative to Earth?
Hint: Use relativistic velocity addition with a negative u′ (backward): u = (v − u′) / (1 − v·u′/c²). Here v = 0.7c and u′ = 0.4c (backward, so subtract).
Key Vocabulary
Inertial reference frame
A reference frame that is not accelerating — one in which Newton's first law holds. Objects at rest stay at rest and objects in motion continue at constant velocity unless acted on by a net force.
Example: A train moving at constant speed is an inertial frame; a braking train is not.
Speed of light (c)
The universal speed limit: c = 3 × 10⁸ m/s in a vacuum. According to Einstein's second postulate, this value is the same for all observers regardless of relative motion.
Example: Light from a moving flashlight travels at c relative to you, not c + v.
Lorentz factor (γ)
The factor γ = 1/√(1 − v²/c²) that quantifies how much time, length, momentum, and energy differ between a moving frame and a rest frame. γ ≥ 1 always.
Example: At v = 0.8c, γ = 1.667, so a moving clock ticks 1.667× slower.
Time dilation
The phenomenon by which a clock moving relative to an observer ticks more slowly than an identical clock at rest. Δt = γ Δt₀, where Δt₀ is the proper time (measured in the rest frame of the event).
Example: Cosmic-ray muons survive long enough to reach Earth's surface because their internal 'clocks' run slow relative to Earth.
Length contraction
The shortening of an object's measured length along its direction of motion. L = L₀/γ, where L₀ is the proper (rest) length. Only the dimension parallel to motion contracts.
Example: A 100 m spaceship traveling at 0.8c appears only 60 m long to a stationary observer.
Rest energy
The energy equivalent of an object's mass when it is at rest: E₀ = mc². Even a stationary object contains an enormous amount of energy locked in its mass.
Example: The rest energy of 1 gram of matter is 9 × 10¹³ J — equivalent to about 21 kilotons of TNT.
Mass-energy equivalence
Einstein's famous result E = mc² (or more generally E = γmc²), showing that mass and energy are two forms of the same quantity and can be converted into each other.
Example: Nuclear fission converts a tiny fraction of uranium's mass into enormous amounts of energy.
Relativistic momentum
The momentum of a moving object corrected for relativistic effects: p = γmv. At low speeds γ ≈ 1 and p ≈ mv (classical). As v → c, p → ∞.
Example: Particle accelerators must account for relativistic momentum when steering protons near the speed of light.
Workbook Check — Multiple Choice
Interactive Practice — 5 Questions
According to Einstein's second postulate, the speed of light in a vacuum is:
A spaceship moves at v = 0.8c. Its Lorentz factor γ is approximately:
Time dilation means that a moving clock, compared to a stationary clock:
A rod of proper length L₀ = 50 m moves at v = 0.6c (γ = 1.25). Its contracted length is:
The rest energy of an object with mass m is given by:
Independent Practice
A spaceship moves at 0.60c relative to Earth. A clock on the ship ticks off 10 s. How much time passes on Earth? (γ = 1.25)
A 100 m long spaceship travels at 0.80c relative to an observer on Earth. What length does the Earth observer measure? (γ = 1.67)
Calculate the rest energy (in joules) of a proton. (m_p = 1.67 × 10⁻²⁷ kg, c = 3.0 × 10⁸ m/s)
A particle moves at 0.90c. Calculate its Lorentz factor γ and its relativistic momentum if its rest mass is 1.0 × 10⁻²⁷ kg.
★ A muon created in the upper atmosphere at 15 km altitude travels toward Earth at 0.998c. (a) What is the muon's lifetime as measured in Earth's frame? (b) What is the distance to Earth as measured in the muon's frame? (c) Without time dilation, would the muon reach Earth if its proper lifetime is 2.2 μs? (c = 3.0 × 10⁸ m/s)
ChallengeCommon Mistakes
Thinking time dilation means the moving clock is broken or malfunctioning
Time dilation is a real physical effect — time genuinely passes more slowly in a moving frame. Both observers are correct in their own frames
Applying time dilation to the frame that is moving relative to itself
Time dilation: Δt = γΔt₀ where Δt₀ is the PROPER time (measured in the rest frame of the event). The dilated time Δt is always longer
Forgetting that length contraction only occurs along the direction of motion
Only the dimension parallel to the velocity is contracted: L = L₀/γ. Dimensions perpendicular to motion are unchanged
Using classical momentum p = mv at relativistic speeds
Relativistic momentum: p = γmv where γ = 1/√(1−v²/c²). Classical formula is only valid for v ≪ c
Math Tips
Lorentz factor: γ = 1/√(1−β²) where β = v/c. At v = 0.6c, γ = 1.25; at v = 0.8c, γ = 1.67; at v = 0.99c, γ ≈ 7.09. γ is always ≥ 1
Mass-energy: E = mc² (rest energy) and E_total = γmc². Kinetic energy in relativity: KE = (γ−1)mc². Use classical KE = ½mv² only when v ≪ c
Proper time and proper length: proper time Δt₀ is measured by a clock that is present at both events; proper length L₀ is measured in the object's rest frame
Relativistic velocity addition: u' = (u − v)/(1 − uv/c²). Even if u = c and v = 0.9c, the result is still c — nothing exceeds the speed of light