On this page:
Introduction Reference Frames Galilean Relativity The Problem with Light Einstein's Postulates The Lorentz Factor Time Dilation Length Contraction Relativity of Simultaneity Relativistic Momentum Mass-Energy Equivalence Energy-Momentum Relation Relativistic Velocity Addition Spacetime Why Don't We Notice? Evidence for Special Relativity The Light Speed Limit Relationship to Galilean Relativity ConclusionReference frame (frame of reference): A coordinate system plus a clock for measuring positions and times of events. All measurements are made relative to a chosen reference frame.
Inertial reference frame: A reference frame in which Newton's first law holds—an object with no net force acting on it either remains at rest or moves at constant velocity. Inertial frames are non-accelerating.
Examples: A stationary laboratory, a car moving at constant velocity on a straight road, a spacecraft drifting in deep space with engines off.
Non-inertial reference frame: An accelerating reference frame where Newton's laws don't hold in their usual form. Pseudo-forces (fictitious forces) appear.
Examples: A car accelerating or braking, a rotating carousel, an airplane during takeoff.
Galilean principle of relativity: The laws of mechanics are the same in all inertial reference frames. No mechanical experiment performed inside an inertial frame can determine whether the frame is at rest or moving at constant velocity.
Galilean transformation equations:
x' = x - vt y' = y z' = z t' = t (absolute time)
Galilean velocity addition: u = u' + v
Example: You walk at 2 m/s forward in a train moving at 20 m/s relative to the ground. An observer on the ground sees you moving at u = 2 + 20 = 22 m/s.
In the late 1800s, Maxwell's equations predicted that electromagnetic waves (light) should travel at the speed c = 3.00 × 10⁸ m/s. But in what frame?
The Michelson-Morley experiment (1887) attempted to detect Earth's motion through the hypothetical "luminiferous ether" by measuring differences in light speed in different directions.
Result: No difference detected. Light speed appeared constant regardless of Earth's motion.
This contradicts Galilean velocity addition. If you shine light from a moving train, both you (on the train) and a ground observer should measure the same speed c, not c + v or c - v.
The laws of physics are the same in all inertial reference frames. No experiment can determine absolute motion.
This extends Galileo's principle from mechanics to all physics.
The speed of light in vacuum has the same value c = 3.00 × 10⁸ m/s in all inertial reference frames, regardless of the motion of the source or observer.
γ = 1/√(1 - v²/c²)
Δt = γΔt₀
Δt₀ = proper time (time in rest frame)
Δt = dilated time (moving frame)
Example: A spacecraft travels at v = 0.8c relative to Earth. A clock on the spacecraft measures 1 hour. How much time elapses on Earth?
γ = 1/√(1 - 0.8²) = 1/0.6 ≈ 1.67
Δt = (1.67)(1 hour) = 1.67 hours
Evidence: Muons created in the upper atmosphere (altitude ≈15 km) have a half-life of 2.2 μs in their rest frame. At v ≈ 0.98c, they should decay before traveling more than ≈ 600 m. Yet many reach Earth's surface. Time dilation explains this.
L = L₀/γ = L₀√(1 - v²/c²)
L₀ = proper length (length in rest frame)
Example: A spacecraft 100 m long (in its rest frame) travels at v = 0.6c. What length do Earth observers measure?
γ = 1/√(1 - 0.6²) = 1/0.8 = 1.25
L = 100/1.25 = 80 m
Important: Contraction occurs only in the direction of motion. Perpendicular dimensions are unchanged.
Simultaneity is relative: Events that are simultaneous in one reference frame are not necessarily simultaneous in another frame moving relative to the first.
There is no absolute universal "now" across space. Observers in different frames disagree about whether spatially separated events occur at the same time.
p = γmv
As v → c, γ → ∞, so p → ∞. Infinite momentum would require infinite force, making it impossible to accelerate any massive object to light speed.
At low speeds (v << c), γ ≈ 1, and p ≈ mv (classical momentum).
E = mc²
Example: An electron (m₀ = 9.11 × 10⁻³¹ kg) at rest. Rest energy:
E₀ = (9.11 × 10⁻³¹)(3.00 × 10⁸)² = 8.2 × 10⁻¹⁴ J ≈ 0.511 MeV
E² = (pc)² + (m₀c²)²
For photons (m₀ = 0): E = pc
Photons have zero rest mass but carry momentum and energy.
u = (u' + v)/(1 + u'v/c²)
Example: A spacecraft travels at 0.8c relative to Earth. It fires a missile forward at 0.9c relative to the spacecraft. How fast does Earth observe the missile?
u = (0.9c + 0.8c)/(1 + (0.9)(0.8)) = 1.7c/1.72 ≈ 0.988c
(Not 1.7c, which would exceed light speed.)
Δs² = c²Δt² - Δx² - Δy² - Δz²
Special relativity unites space and time into a four-dimensional continuum called spacetime.
For a car at 30 m/s (108 km/h):
v/c = 30/(3 × 10⁸) = 10⁻⁷
γ ≈ 1 + 5 × 10⁻¹⁵
Deviation from 1 is about 0.0000000000000005 — completely unmeasurable.
Relativistic effects are negligible until v exceeds roughly 10% of c (3 × 10⁷ m/s).
Nothing with mass can reach or exceed the speed of light.
As v → c: γ → ∞, KE → ∞, momentum → ∞, length → 0
Massless particles (photons, gluons) must always travel at exactly c in vacuum.
Causality protection: The light speed limit protects causality (cause preceding effect).
Galilean relativity is the low-velocity limit of special relativity.
When v << c:
This explains why Newton's laws work so well for non-relativistic situations.
Special relativity revolutionized our understanding of space, time, and motion. Its predictions, initially shocking and counterintuitive, have been verified in countless experiments.
These principles are essential for understanding high-energy physics, astrophysics, cosmology, and modern technology like particle accelerators and GPS. Special relativity, along with quantum mechanics, forms the foundation of 20th and 21st-century physics.