Basics of relativity

At everyday speeds Newtonian mechanics works perfectly. But at speeds close to the speed of light, time, length, mass and the rule for adding velocities all change. Einstein’s special theory of relativity (1905) describes these cases. Its predictions are confirmed every day in particle accelerators, in cosmic rays and in GPS satellites.

§1. The postulates of relativity

Galileo’s principle of relativity: mechanical phenomena proceed identically in all inertial frames, and velocities simply add (v = v₁ + v₂). However, the Michelson experiment showed that the speed of light does not depend on the Earth’s motion. Einstein put forward two postulates:

1. All laws of physics (mechanics and electrodynamics) are the same in all inertial frames.
2. The speed of light in vacuum, c = 3·10⁸ m/s, is the same in all inertial frames and does not depend on the speed of the source or the observer.

An important consequence of the second postulate: no body or signal can travel faster than c. Events that are simultaneous in one frame may not be simultaneous in another — simultaneity is relative.

A thought experiment makes this clear. A lamp flashes in the middle of a fast-moving carriage. For a passenger, the light reaches the front and back walls at the same time: the distances are equal and the speed is the same. For an observer on the platform, the back wall moves towards the light and the front wall moves away; since the speed of light does not depend on the motion of the source, the light reaches the back wall first. The same two events are simultaneous in one frame and happen one after the other in another. That is why time intervals and lengths also turn out to depend on the frame. The failure of the Michelson–Morley experiment (1887) to find the expected difference in the speed of light led physicists to abandon the idea of an “ether”.

The relativistic factor appears in the formulas below:

§2. Time dilation

A moving clock runs slow relative to a stationary observer:

t₀ is the proper time (measured in the frame moving with the clock), t is the time measured by the observer; always t > t₀.

Worked example

A process on a spaceship lasts 4 s. The ship moves at 0.6c relative to the Earth. How long does it last for an Earth observer?

1. √(1 − 0.36) = 0.8.
2. t = 4/0.8 = 5 s.

Worked example

A muon’s proper lifetime is 2.2 μs. It moves through the atmosphere at 0.99c. How long does it live in the Earth’s frame, and how far does it travel?

1. √(1 − 0.99²) = √0.0199 ≈ 0.141.
2. t = 2.2/0.141 ≈ 15.6 μs.
3. s = 0.99·3·10⁸·1.56·10⁻⁵ ≈ 4.6·10³ m — instead of 650 m by the classical estimate. That is why muons reach the Earth’s surface.

§3. Length contraction

A moving body’s size along the direction of motion contracts:

l₀ is the proper length (in the body’s own frame). Dimensions perpendicular to the motion do not change.

Worked example

A 1 m rod moves along its length at 0.8c. Its length?

1. √(1 − 0.64) = 0.6.
2. l = 1·0.6 = 0.6 m.

§4. Relativistic addition of velocities

• At low speeds the denominator ≈ 1 and the formula becomes the classical v₁ + v₂.
• If v₁ = c, then v = c — the speed of light is the same in every frame.

Worked example

Two particles fly towards each other, each at 0.6c. Speed of one relative to the other?

1. v = (0.6c + 0.6c)/(1 + 0.36) = 1.2c/1.36.
2. v ≈ 0.88c (not 1.2c!).

§5. Mass–energy relation

• Every body, even at rest, has enormous energy: 1 g of matter corresponds to 10⁻³·9·10¹⁶ = 9·10¹³ J.
• Mass defect, nuclear power and annihilation are based on this formula.
• For v ≪ c, E_k ≈ m₀v²/2, but near c that formula fails.

Worked example

A particle’s mass is 5/3 of its rest mass. Its speed and kinetic energy?

1. √(1 − v²/c²) = 3/5 → v²/c² = 1 − 9/25 = 16/25 → v = 0.8c.
2. E_k = (5/3 − 1)m₀c² = (2/3)m₀c².

Worked example

An electron (m₀ = 9.1·10⁻³¹ kg, E₀ ≈ 0.511 MeV) moves at 0.6c. Its momentum and kinetic energy?

1. m = m₀/0.8 = 1.25m₀.
2. p = 1.25·9.1·10⁻³¹·1.8·10⁸ ≈ 2.05·10⁻²² kg·m/s.
3. E_k = (1.25 − 1)·0.511 ≈ 0.128 MeV.

§6. Energy and momentum; evidence for the theory

The relation between total energy, momentum and rest energy holds at any speed:

For a particle with zero rest mass, a photon, E = pc: it always moves at c and is never at rest.

Relativity is not just theory; everyday technology has to allow for it.

• GPS satellite clocks run slow by about 7 μs per day because of their speed, and fast by about 45 μs per day because gravity is weaker up there (general relativity); without these corrections position errors would build up by kilometres per day.
• In particle accelerators protons move extremely close to c but never exceed it; adding energy mainly increases their momentum and energy.
• The Sun converts about 4·10⁹ kg of mass into radiated energy every second in fusion reactions (Δm = P/c²).

Worked example

For an electron moving at 0.6c (E₀ = 0.511 MeV) find the total energy and pc, and check E² = (pc)² + E₀².

1. E = E₀/0.8 = 1.25·0.511 ≈ 0.639 MeV.
2. pc = mvc = 1.25·0.6·0.511 ≈ 0.383 MeV.
3. Check: √(0.639² − 0.511²) ≈ 0.383 MeV, which agrees.

Check yourself

1. A clock on a ship moving at 0.8c shows 3 s. How much time passes on Earth clocks?

2. A ship with proper length 10 m flies at 0.6c. Its length for an Earth observer?

3. Add the speeds 0.5c and 0.5c relativistically.

4. Rest energy of 1 kg of matter?

5. At what speed does the kinetic energy equal the rest energy?

6. A rocket moving at 0.9c emits light forwards. The light’s speed relative to the Earth?

7. The Sun radiates 3.8·10²⁶ W. How much mass does it lose each second?

8. At what speed does a particle with zero rest mass move?

Common mistakes

• Adding relativistic velocities the ordinary way (the result exceeds c).
• Using E_k = mv²/2 at speeds close to the speed of light.
• Swapping t and t₀: proper time is always the smaller one.
• Thinking length contracts in all directions: only along the motion.
• Forgetting to square v/c: it is √(1 − v²/c²), not √(1 − v/c).