The Maths of General Relativity (3/8) - Geodesics

Physicists studying motion in general relativity track how velocity vectors change as proper time passes, enabling prediction of object trajectories through curved spacetime.

All objects in the universe naturally follow geodesic trajectories when no external forces act upon them, from apples to planets to light rays.

Mathematicians applying calculus to curved spaces extend the familiar product rule from scalar functions to vector decompositions into components and bases.

Differential geometers recognize that basis vectors defining coordinate directions can change throughout spacetime, unlike the constant basis vectors of Euclidean geometry.

Mathematicians and physicists use Christoffel symbols, denoted by capital gamma, to quantify how coordinate grids deviate from perfect rectilinearity throughout spacetime.

Einstein’s general relativity uses the geodesic equation to predict how freely-falling objects move through curved spacetime without invoking gravitational forces.

Cartographers and physicists encounter coordinate irregularity when representing curved surfaces like Earth using latitude-longitude systems that seem natural but contain inherent distortions.

Navigators discovered that shortest paths on Earth’s surface are great circles - the sphere’s geodesics - long before mathematicians formalized the concept for arbitrary curved spaces.