Geodesic (general Relativity)

Geodesic (general Relativity)

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress-energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.

In theories such as special and general relativity, spacetime is treated as a Lorentzian manifold. Geodesics on a Lorentzian manifold fall into three classes according to the sign of the norm of their tangent vector. With a metric signature of (-+++) being used,

• timelike geodesics have a tangent vector whose norm is negative;
• null geodesics have a tangent vector whose norm is zero;
• spacelike geodesics have a tangent vector whose norm is positive.

Note that a geodesic cannot be spacelike at one point and timelike at another.

An ideal particle (ones whose gravitational field and size are ignored) not subject to electromagnetic forces (or any other non-gravitational force) will always follow timelike geodesics. Note that not all particles follow geodesics, as they may experience external forces, for example, a charged particle may experience an electric field — in such cases, the worldline of the particle will still be timelike, as the tangent vector at any point of a particle's worldline will always be timelike.

Massless particles like the photon follow null geodesics. Spacelike geodesics do exist, though they do not correspond to the path of any physical particle; in a space that has space-sections orthogonal to a timelike Killing vector a spacelike geodesic (with its affine parameter) within such a space section represents the graph of a tightly stretched, massless filament.