If geodesics are the straightest paths, why do orbits look curved?

Orbits are straight in the sense of being geodesics of curved four-dimensional spacetime; their apparent curvature in space arises because spacetime itself is curved by mass, so the locally straightest worldline projects onto a curved spatial path.

How is curvature distinguished from a mere choice of coordinates?

Coordinate effects can be removed by changing coordinates, but genuine curvature shows up in the Riemann tensor and in tidal geodesic deviation, which cannot be transformed away and are present wherever real gravity acts.

Spacetime Curvature and Geodesics

In general relativity matter curves spacetime, and free particles and light rays follow geodesics, the straightest possible paths through that curved geometry; the relative bending of nearby geodesics is what we perceive as gravitational tidal forces.

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Definition

Spacetime curvature is the deviation of spacetime geometry from flatness, quantified by the Riemann curvature tensor, and a geodesic is the worldline of a freely falling particle, obtained by parallel-transporting its own tangent vector and extremizing proper time.

Scope

This topic covers geodesics as extremal-length worldlines and the geodesic equation, parallel transport and the connection, the Riemann curvature tensor and its contractions, geodesic deviation as the measure of tidal effects, and the way curvature reproduces and corrects Newtonian gravitational attraction in the weak-field limit.

Core questions

What does it mean for spacetime to be curved rather than flat?
Why do freely falling bodies move along geodesics?
How does geodesic deviation account for tidal gravitational forces?

Key concepts

Geodesic
Affine connection and Christoffel symbols
Parallel transport
Riemann curvature tensor
Geodesic deviation
Tidal forces

Key theories

Geodesic equation: A freely falling particle follows a geodesic that extremizes its proper time, satisfying an equation in which the connection coefficients encode the gravitational field, so that gravity becomes inertial motion in curved spacetime.
Riemann curvature and geodesic deviation: The Riemann tensor measures the failure of parallel transport around a loop and governs how neighboring geodesics accelerate toward or away from each other, identifying curvature with the observable tidal forces of gravity.

Clinical relevance

Geodesics determine the orbits of planets and spacecraft in relativistic gravitational fields, the paths of light producing gravitational lensing, and the precession of orbits such as Mercury's perihelion; curvature also describes the tidal stretching experienced near compact objects.

History

The geometry of curved spaces was created by Gauss and Riemann in the nineteenth century; Levi-Civita and Ricci developed the tensor calculus and parallel transport in the 1900s, and Einstein adopted these tools to express gravity as curvature, with geodesics replacing Newton's force trajectories.

Key figures

Bernhard Riemann
Albert Einstein
Tullio Levi-Civita

Seminal works

wald1984
mtw1973

Frequently asked questions

If geodesics are the straightest paths, why do orbits look curved?: Orbits are straight in the sense of being geodesics of curved four-dimensional spacetime; their apparent curvature in space arises because spacetime itself is curved by mass, so the locally straightest worldline projects onto a curved spatial path.
How is curvature distinguished from a mere choice of coordinates?: Coordinate effects can be removed by changing coordinates, but genuine curvature shows up in the Riemann tensor and in tidal geodesic deviation, which cannot be transformed away and are present wherever real gravity acts.