The Maths of General Relativity (2/8) - Spacetime velocity

Objects moving through spacetime follow worldlines described by coordinates that evolve as proper time advances, creating trajectories whose velocities can be represented as geometric arrows.

All objects in the universe—massive particles, photons, planets, and satellites—move through spacetime with identical speed regardless of their spatial motion relative to observers.

Physicists using coherent units—measuring time in seconds and distance in light-seconds—discover that proper time directly quantifies spacetime distance without additional conversion factors.

Relativists describing velocity vectors need coordinate-based representations, decomposing vectors into components along basis vectors that define coordinate directions and scales at each spacetime point.

Einstein introduced this notation to simplify general relativity’s equations by eliminating explicit summation symbols, making complex tensor expressions readable and manipulation straightforward.

Observers measuring a satellite’s orbit decompose its four-velocity into temporal component (relating proper time to coordinate time) and spatial component (describing angular motion around Earth).

Two satellites orbiting Earth at different radii can have identical angular velocity components despite moving at vastly different physical speeds, revealing coordinate values don’t directly represent physical quantities.

Relativists confronting coordinate arbitrariness cannot use the Pythagorean theorem to calculate physical distances from components because coordinates don’t represent real distances requiring a generalized distance measure.