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Relativity

An observer's clock and ruler are not absolute. Special relativity starts from two postulates — the laws of physics look the same in every inertial frame, and light travels at the same speed c for everyone — and follows them without flinching to conclusions that overturn Newton's separate space and time.

FIG_002
rest framemoving frame
Δt=γΔt\Delta t' = \gamma\,\Delta t
A moving light clock runs slow: the photon's diagonal path is longer, so each tick takes longer.

Space and time merge into a single four-dimensional spacetime. The one quantity every observer agrees on is the interval between events, and the light cone it defines fixes what can cause what — sorting every event into an absolute future, past, or the spacelike elsewhere no signal can reach.

The bridge between frames is the Lorentz transformation. It mixes space and time while leaving the speed of light invariant, and its consequences — time dilation, length contraction, the relativity of simultaneity — are geometry, not illusion.

FIG_003
ctxx'ct'
A Lorentz boost tilts the moving frame's space and time axes toward the light line — which is why simultaneity is relative.

Dynamics is rebuilt to match. Momentum and energy join into a single four-vector, mass reveals itself as a form of energy through E = mc², and Maxwell's electromagnetism turns out to have been relativistic all along.

Then gravity. The equivalence principle — that free fall is indistinguishable from floating in empty space — says gravity is not a force but the curvature of spacetime, and matter is what does the curving.

FIG_001
ctxfuturepastelsewhereelsewhereworldline
The light cone through an event splits spacetime into causal future, past, and an unreachable elsewhere; a worldline threads the interior.
FIG_004
Mass curves the spacetime grid, and a free body orbits by following the straightest available path through it.

The Einstein field equation ties that curvature to energy and momentum. Its first exact solution, the Schwarzschild metric, describes spacetime around a star and predicts the bending of light, the precession of Mercury, and clocks that tick faster with altitude.

FIG_005
h+ polarizationh_+ \ \text{polarization}
A gravitational wave stretches and squeezes a ring of free test masses as it passes.

Push a mass small enough and an event horizon forms: a black hole, a region from which not even light escapes. Ripples in the curvature — gravitational waves — travel outward at c and were first caught stretching a detector in 2015.

Turn the same equations on the universe as a whole and they give an expanding cosmos with a finite past. Relativity is the geometry underneath all of it.

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