Postulates of Special Relativity and Reference Frames
Special relativity rests on two postulates: the laws of physics are the same in all inertial frames of reference (frames in uniform motion), and the speed of light in a vacuum has the same value c = 299,792,458 m/s in every inertial frame, regardless of the motion of the source or the observer. In brief: the second postulate contradicts the Galilean rule for adding velocities and leads to the relativity of simultaneity, time dilation and length contraction. Velocities add according to u = (v + u’)/(1 + vu’/c2), so nothing can be made to exceed c. At speeds much smaller than c, special relativity reduces to Newtonian mechanics.
In 1905, Albert Einstein resolved a conflict between Newtonian mechanics and Maxwell’s electromagnetism by changing the concepts of space and time. This guide explains inertial reference frames and the principle of relativity, the experimental background and the postulates, the relativity of simultaneity and the velocity-addition formula, compares Galilean and Einsteinian relativity in a table, works through numerical examples and corrects common mistakes.
What is an inertial reference frame?
A reference frame is a coordinate system with a clock for measuring the positions and times of events. An inertial frame is one in which Newton’s first law holds: a body with no net force moves at constant velocity. Any frame moving at constant velocity relative to an inertial frame is also inertial. The Earth’s surface is approximately inertial for short experiments; a car that is accelerating, braking or turning is not. Galileo noted in 1632 that no mechanical experiment inside a closed cabin of a ship can tell whether the ship is at rest or moving uniformly (the principle of relativity). Einstein extended this principle to all the laws of physics, including electromagnetism and optics.
What problem did Einstein solve?
Maxwell’s equations predict that electromagnetic waves, including light, travel at c = 1/√(μ0ε0) = 3.00 × 108 m/s. According to Galilean relativity, speeds are measured relative to a medium, so it was assumed that light travels at c relative to a luminiferous ether, and that moving observers measure different speeds. The Michelson-Morley experiment (1887), which compared the speed of light along and across the Earth’s motion, found no difference (a null result within an error of a few km/s, much smaller than the 30 km/s orbital speed of the Earth). Lorentz and FitzGerald proposed ad hoc length contractions to explain it. Einstein took a more radical approach: the null result is a consequence of a fundamental principle, and the ether is unnecessary. Another problem was the asymmetry in the explanation of electromagnetic induction between a magnet moving relative to a coil, and a coil moving relative to a magnet, which have the same observable result.
What are the two postulates?
- The principle of relativity: the laws of physics have the same form in all inertial frames. There is no absolute rest frame.
- The constancy of the speed of light: the speed of light in a vacuum is c in all inertial frames, independent of the motion of the source and of the observer.
The second postulate is the surprising one: a light beam from a moving spaceship travels at c relative to the ship and at the same c relative to a station, not c plus the ship’s speed. To keep it consistent, the concepts of time and length must depend on the observer. The transformation between frames, the Lorentz transformation, x’ = γ(x − vt), t’ = γ(t − vx/c2), with γ = 1/√(1 − v2/c2), replaces the Galilean transformation (x’ = x − vt, t’ = t). For v ≪ c, γ ≈ 1 and the Galilean result is recovered. The speed c is the maximum speed at which energy or information can be transmitted.
Why is simultaneity relative?
Consider a train moving at high speed, and two lightning bolts that strike the front and the back of the train. An observer standing on the platform, midway between the strikes, receives light from both at the same moment, and concludes that they were simultaneous. An observer in the middle of the train is moving toward the light from the front strike and away from the light from the back strike, and receives the front flash first. Since both observers agree that the light travels at c, the train observer concludes that the front strike happened first. Both are right in their own frame: simultaneity is relative. Events that are simultaneous but at different positions in one frame are not simultaneous in another. Cause and effect are preserved: if an event A can influence an event B (the separation is “timelike”, light could travel between them), all observers agree on the order.
Galilean and Einsteinian relativity compared
| Feature | Galilean (Newtonian) | Einsteinian (special relativity) |
|---|---|---|
| Time | Absolute, same for all | Depends on the frame |
| Lengths | Same for all | Depend on the frame |
| Speed of light | Depends on the observer’s motion | c for all inertial observers |
| Velocity addition | u = v + u’ | u = (v + u’)/(1 + vu’/c2) |
| Valid for | Speeds much less than c | All speeds |
Worked example: adding velocities
- Galilean prediction. A spaceship moves at v = 0.8c relative to the Earth and fires a probe forward at u’ = 0.8c relative to the ship. The Galilean sum would be 1.6c, faster than light.
- Relativistic result: u = (v + u’) ÷ (1 + vu’/c2) = (0.8c + 0.8c) ÷ (1 + 0.64) = 1.6c ÷ 1.64 = 0.976c, less than c.
- Two speeds of 0.5c: u = c ÷ (1 + 0.25) = 0.80c. Everyday speeds: a train at 50 m/s and a ball thrown at 30 m/s: the relativistic correction factor 1 + (50 × 30)/(3 × 108)2 = 1 + 1.7 × 10−14, negligible.
- Light. A ship at 0.9c fires a light beam forward: u = (0.9c + c) ÷ (1 + 0.9) = 1.9c ÷ 1.9 = c. The light speed is c in both frames, as the second postulate requires.
- Gamma factor. For v = 0.8c: γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 1.667. For v = 30 km/s (the Earth’s orbital speed, v/c = 10−4): γ = 1 + 5 × 10−9, indistinguishable from 1 in everyday life.
- Simultaneity in numbers. A train of length 100 m (in its own frame) moves at 0.6c past a platform. Two events at the ends of the train, such as the two flashes of light, are simultaneous in the train frame. In the platform frame, the time difference between them is Δt = γvL0 ÷ c2 = 1.25 × 0.6 × 3.0 × 108 × 100 ÷ (3.0 × 108)2 = 2.5 × 10−7 s = 250 ns, with the event at the rear of the train happening first in the platform frame.
How did relativity develop?
Galileo described the principle of relativity in 1632. Maxwell’s equations (1865) predicted c; Michelson and Morley’s experiment (1887) failed to detect the ether. Hendrik Lorentz (1904) and Henri Poincaré developed the transformations that bear Lorentz’s name. Einstein published “On the Electrodynamics of Moving Bodies” in 1905, deriving them from the two postulates; Hermann Minkowski introduced spacetime in 1908. Einstein’s general theory of relativity (1915) extended the theory to accelerated frames and gravity. Special relativity has been confirmed in countless experiments, such as particle accelerators and GPS.
Which mistakes are common?
Mistake 1: Thinking that relativity says “everything is relative”
The speed of light and the laws of physics are the same for all inertial observers. Quantities such as the spacetime interval are invariant.
Mistake 2: Believing that the speed of light depends on the speed of the source
It does not. This is the second postulate.
Mistake 3: Using special relativity for accelerating frames without care
The postulates concern inertial frames. Accelerated motion can be analyzed from an inertial frame, or by general relativity.
Mistake 4: Thinking that objects cannot be accelerated to c because of lack of power
The energy needed increases without limit as v approaches c. A massive object cannot reach c.