Geodesic (general relativity)

In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. In this theory, gravity is not a force but is instead a curved spacetime geometry where the source of curvature is the stress-energy tensor. Thus, for example, the orbital path of a planet around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.

Related Topics:
Physics - General relativity - Geodesics - World line - Stress-energy tensor

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Recall that spacetime in general relativity is 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,

Related Topics:
Spacetime - Lorentzian manifold - Metric signature

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

Note that a geodesic cannot be spacelike at one point and timelike at another since parallel transport preserves the norm of the vector (since the metric is parallel transported along any curve).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Ideal particles (ones whose gravitational field is ignored) in free fall and any particle not subject to electromagnetic or pressure forces (or the like) 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 will follow null geodesics. Spacelike geodesics exist. They do not correspond to the path of any physical particle, but 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.

Related Topics:
Photon - Killing vector - Graph

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

 
Geodesic (general relativity): Mathematical expression ►