This unit explains what happens to time, length, momentum, and energy when light in a vacuum has the same speed in every inertial frame. By the end, you should be able to state Einstein's two postulates, explain why simultaneity is relative, and calculate time dilation, length contraction, and mass-energy conversions using the Lorentz factor.
Read the big ideas first, then work through each topic's notes and try the worked examples and checkpoints before looking at the answers.
Special relativity describes space and time in inertial frames, replacing Newton's idea of a shared time and fixed lengths. Its effects become most noticeable at speeds close to the speed of light. The unit has two parts. First come the postulates and why they were needed, then the measurable consequences.
Honors Physics Unit 10 covers special relativity. It starts with Einstein's 1905 postulates, which say the laws of physics are identical in all inertial frames and that light in a vacuum moves at c ≈ 3.0 × 10⁸ m/s for every observer. From there you learn why the Michelson-Morley experiment and Maxwell's equations pushed physics toward these ideas, and why two events can be simultaneous in one frame but not another. The second half applies the Lorentz factor γ = 1/√(1 − v²/c²) to time dilation, length contraction, and relativistic momentum. It also covers mass-energy equivalence, including rest energy (E₀ = mc²) and the mass defect in fission and fusion.
Classical velocity addition says speeds stack, so chasing a light beam should make it look slower. Experiments say otherwise. Every inertial observer measures light in a vacuum moving at about 3.0 × 10⁸ m/s. Special relativity starts with that result and the principle that physics has the same laws in every inertial frame.
If speed stays fixed, then distance and time have to adjust. A moving clock runs slow by a factor of γ, and a moving object is shorter along its motion by the same factor. At everyday speeds γ is so close to 1 that you never notice.
An object at rest still has energy E₀ = mc². Because c² is about 9 × 10¹⁶ m²/s², a tiny mass loss releases a huge amount of energy. That is what powers nuclear reactors and the Sun.
The results in this unit follow from the two postulates about the laws of physics and the speed of light in a vacuum. Simultaneity, clock rates, lengths, and momentum all adjust so that every inertial observer measures c and sees the same laws of physics. The Lorentz factor γ tells you how big those adjustments are at any speed.
Special relativity rests on two postulates Einstein published in 1905.
The postulates lead to measurable effects, and the Lorentz factor \gamma = 1/\sqrt{1 - v^2/c^2} sets their size.
Special relativity rests on two postulates Einstein published in 1905. The first, the principle of relativity, says the laws of physics are the same in every inertial reference frame. No experiment inside a lab moving at constant velocity can tell you whether it is moving relative to an absolute rest frame. The second says light in a vacuum travels at m/s for every observer, no matter how the source or observer moves.
The second postulate is the surprising one. Galilean velocity addition says speeds just add, so running toward a flashlight should make its light faster. Maxwell's equations disagreed. So did the 1887 Michelson-Morley experiment, which found no detectable motion of Earth through the proposed light-carrying "aether."
If light's speed can't change, then time and distance measurements have to. Simultaneity changes first. In Einstein's train thought experiment, lightning strikes both ends of a moving train at the same moment for a ground observer. A rider at the train's center moves toward the front flash and gets its light first. Since she sits at the middle of the train, she concludes the front strike happened first. Each observer is correct in their own frame.
Worked example. A spaceship moves toward Earth at and fires a laser pulse ahead. Galilean addition predicts Earth measures . Relativity says Earth measures exactly . So in Earth's frame the pulse crosses m in s. The crew also measures the pulse moving at .
Which of Einstein's two postulates would Galileo and Newton have already accepted, and which one broke classical physics? (Answer: They would have accepted the principle of relativity. The constant speed of light for all observers is the one that broke classical physics, because it contradicts simple velocity addition.)
The postulates lead to measurable effects, and the Lorentz factor sets their size. At everyday speeds is so close to 1 that Newtonian physics works fine. At , , and at , .
Time dilation means an inertial clock moving relative to you runs slow. For two events at that clock, , where is the proper time measured in the clock's rest frame. It is the shortest time between those events measured in an inertial frame. Length contraction means a moving object is shorter along its direction of motion, so . The proper length is measured in the object's rest frame. Dimensions perpendicular to the motion don't change.
Momentum becomes , which grows without limit as approaches . That's why no object with mass can reach light speed. Mass also holds energy. An object at rest has , and its total energy is . In energy-releasing fission and fusion reactions, the products have less total rest mass than the reactants. The mass defect corresponds to energy carried by the products, including their kinetic energy and radiation.
Worked example. A muon has a mean lifetime of in its own frame. If it moves at , then , and Earth measures a mean lifetime of . In that time it travels about km, compared with about 0.66 km without time dilation. Individual muons decay at different times, but time dilation helps explain why some reach the ground from high in the atmosphere.
Worked example. Converting 1.0 g of mass releases J.
A spaceship has a proper length of 100 m and flies past you along its length at 0.60c. How long do you measure it to be? (Answer: γ = 1.25, so L = 100 m / 1.25 = 80 m.)
| Quantity | Classical physics (v much less than c) | Special relativity |
|---|---|---|
| Time between two events | Same for every observer | Δt = γΔt₀, so moving clocks run slow |
| Length of an object | Same for every observer | L = L₀/γ along the direction of motion |
| Momentum | p = mv | p = γmv, which approaches infinity as v approaches c |
| Kinetic energy | KE = ½mv² | KE = (γ − 1)mc² |
| Energy at rest | Zero kinetic energy | Rest energy E₀ = mc² |
| Speed of light | Depends on observer's motion | c in a vacuum for every inertial observer |
| Term | Definition |
|---|---|
| Special relativity | Einstein's 1905 theory describing how measurements of space, time, momentum, and energy change between inertial frames moving at constant velocity. |
| Michelson-Morley experiment | An 1887 experiment that looked for an effect of Earth's motion through a proposed light-carrying aether. It found no detectable aether wind, a result consistent with a constant speed of light. |
| Luminiferous aether | A medium that 19th-century physicists believed light traveled through, which experiments failed to detect and relativity made unnecessary. |
| Time dilation | The effect where a clock moving relative to an observer is measured to tick more slowly, given by Δt = γΔt₀. |
| Length contraction | The effect where a moving object is measured to be shorter along its direction of motion, given by L = L₀/γ. |
| Twin paradox | A thought experiment where a twin who travels near light speed and returns home has aged less than the twin who stayed on Earth. |
| Relativistic momentum | Momentum at high speed, p = γmv, which reduces to p = mv when v is much less than c. |
| Rest energy | The energy an object has because of its mass alone, E₀ = mc², even when it is not moving. |
| Spacetime | The combination of three space dimensions and one time dimension into a single framework, where different observers can split space and time differently. |
Proper time is measured by a clock present at both events, such as a muon's clock at its creation and decay. To use Δt = γΔt₀, identify an inertial frame in which both events happen at the same place. Its measured time is Δt₀, which is no greater than the time measured between those events in another inertial frame.
These effects are real measurements, not tricks of light delay. Moving muons have a longer measured mean lifetime. Atomic clocks flown on jets also show predicted differences from ground clocks, with the direction of the difference depending on the route and gravitational effects. Each observer's measurement is correct in their own frame.
Simple addition like 0.6c + 0.6c = 1.2c only works at low speeds. Relativistic momentum p = γmv blows up as v approaches c, so nothing with mass can reach or pass c. Older books describe this with a growing "relativistic mass" γm, but most physicists now treat mass m as the same in every frame and put the γ in momentum and energy instead.
Length contraction only happens along the direction of motion. A rocket moving horizontally gets shorter front to back, but its height and width stay the same. Check which dimension lines up with the velocity before dividing by γ.