Limits of Newtonian mechanics and the Michelson–Morley experiment
Newton's mechanics assumes time and length are the same for everyone, and speeds simply add. That works for cars and planets, but light broke the rule.
Scientists thought light travelled through an invisible 'aether'. If Earth moved through it, light should go faster one way than the other. In 1887 Michelson and Morley split a light beam into two paths at right angles and joined them again to look for a shift in the interference fringes as the apparatus was turned. They found no shift. The speed of light did not depend on Earth's motion. There was no aether to detect.
Einstein's two postulates
- Principle of relativity: the laws of physics are the same in every inertial frame (a frame moving at constant velocity). No experiment can tell whether you are 'really' moving.
- Constancy of c: light in a vacuum travels at c ≈ 3.00 × 10⁸ m/s for all observers, whatever the motion of the source or observer.
If c is fixed, then time and space must adjust. The factor that tells us how much is the Lorentz factor γ = 1/√(1 − v²/c²).
Time dilation
In the light clock, the mirrors are distance L apart. At rest, one trip takes t₀ = 2L/c. This is the proper time: time measured by a clock at rest with the event.
Seen from the ground, the moving clock's light goes along a slanted path. Using Pythagoras, (ct/2)² = L² + (vt/2)², which gives t = γ t₀. A moving clock runs slow.
Evidence: muons live about 2.2 μs at rest. Even at 0.99c they would travel only about 650 m, yet many reach the ground from 10 km up, because their lifetime is dilated by γ ≈ 7.
Length contraction
The proper length L₀ is measured by someone at rest with the object. An observer who sees it move at v measures L = L₀ / γ, shorter only along the direction of motion. From the muon's view, the 10 km of air is contracted to about 1.4 km, so it gets through in its short life. Both views agree on what happens.
Mass and energy: E = mc²
Mass is a form of energy. The rest energy of a mass m is E₀ = mc². The total energy of a moving particle is E = γmc², and its kinetic energy is (γ − 1)mc². As v → c, γ → ∞, so an infinite energy would be needed: nothing with mass can reach c.
In the Sun and in nuclear reactors a small loss of mass becomes a large energy output. Units: in particle physics energies are often given in MeV (1 eV = 1.60 × 10⁻¹⁹ J).
Beyond: a glimpse of general relativity
Special relativity deals with frames moving at constant velocity. Einstein's general relativity (1915) adds gravity: mass curves space-time, so light bends near the Sun and clocks run slower in stronger gravity. GPS must correct for both effects.
Try it: on the slider, find the speed where γ = 2. (Answer: v ≈ 0.87c.) Then check it with the formula.
Key formulas and definitions
- γ = 1 / √(1 − v²/c²)
- t = γ t₀ (time dilation, t₀ = proper time)
- L = L₀ / γ (length contraction, L₀ = proper length)
- E₀ = mc²; E = γmc²; Eₖ = (γ − 1)mc²
- c ≈ 3.00 × 10⁸ m/s
Worked examples
1. Find γ for v = 0.6c.
v²/c² = 0.36; 1 − 0.36 = 0.64; √0.64 = 0.8; γ = 1/0.8 = 1.25.
2. A spaceship moves at 0.8c. A clock on board ticks 1.0 s. How long is that tick seen from Earth?
γ = 1/√(1 − 0.64) = 1/0.6 = 1.67. t = γt₀ = 1.67 × 1.0 = 1.67 s.
3. A rod is 5.0 m long at rest. How long is it measured when moving along its length at 0.8c?
γ = 1.67. L = L₀/γ = 5.0 / 1.667 = 3.0 m.
4. How much energy is in 1.0 g of mass?
E = mc² = 0.001 × (3.0 × 10⁸)² = 0.001 × 9.0 × 10¹⁶ = 9.0 × 10¹³ J.
5. Muons have a proper lifetime of 2.2 μs and move at 0.995c. How far do they travel in one lifetime as seen from the ground?
γ = 1/√(1 − 0.990) = 1/√0.00998 ≈ 10.0. Dilated lifetime = 22 μs. Distance = 0.995 × 3.0 × 10⁸ × 22 × 10⁻⁶ ≈ 6.6 km (versus only 0.66 km without relativity).
6. An electron (m = 9.11 × 10⁻³¹ kg) moves with γ = 3. Find its kinetic energy.
Eₖ = (γ − 1)mc² = 2 × 9.11 × 10⁻³¹ × 9.0 × 10¹⁶ = 1.64 × 10⁻¹³ J (about 1.02 MeV).
Common mistakes
- Putting v in m/s and c as 1 in the same formula. Use v/c as a ratio, or both in m/s.
- Swapping proper and dilated time. Proper time t₀ is on the clock that moves with the event, and it is the shortest.
- Thinking length shrinks in all directions. Only the length along the motion shrinks.
- Adding speeds like Newton: two beams of light approaching each other do not close at 2c. Every observer still measures c.