Relativity as a theory of spacetime
Special relativity describes measurements made in inertial frames, where objects not acted on by a net force move at constant velocity. It begins with two postulates: the laws of physics have the same form in every inertial frame, and light in vacuum has the same speed c for every inertial observer, independent of source motion. Space and time therefore cannot transform separately as in classical mechanics; they form spacetime.
γ = 1/√(1 − v²/c²), always at least 1.
Δt = γΔτ, where Δτ is proper time measured by one clock at one place.
L = L₀/γ, along the direction of motion only.
Proper length L₀ is measured in the rest frame of an object. Proper time Δτ is measured between two events occurring at the same position in a clock's rest frame. Measurements in another frame require a Lorentz transformation. Events simultaneous in one frame need not be simultaneous in another, because synchronizing distant clocks depends on the frame.
The diagonal light path is longer, yet every inertial observer measures the same c. The time interval between ticks must therefore be larger in the frame where the clock moves. This is not an optical delay or a faulty clock; it is a geometric property of spacetime.
Relativistic energy and momentum
For a particle of invariant rest mass m, momentum and total energy are p = γmv and E = γmc². Kinetic energy is K = E − mc² = (γ − 1)mc². They satisfy the frame-independent relation E² = (pc)² + (mc²)². For a photon, m = 0, so E = pc. At speeds much smaller than c, γ is nearly 1 and the expressions approach p ≈ mv and K ≈ ½mv².
Invariant core: observers may disagree about time, distance, energy and momentum separately, but the laws and spacetime or energy-momentum invariant relations remain consistent.