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. Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... Two-dimensional visualization of space-time distortion. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... The stress-energy tensor is a tensor quantity in relativity. ...

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, World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

• 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. For the Science Fiction weapon, as seen in Star Trek, see Photon torpedo. ... In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ...

Mathematical expression

A timelike geodesic is a worldline which parallel transports its own tangent. If a worldline has tangent then this can be expressed as A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

which says that the covariant derivative of the tangent in the direction of the tangent is zero. The above equation can be restated in terms of components of : In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... Proof 1 , , , , , (return to article) Proof 2 The goal being to extremize the value of where such goal can be accomplished by calculating the Euler-Lagrange equation for f, which is . Substituting the expression of f into the Euler-Lagrange equation (which extremizes the value of the integral l), gives...

where

and

,

τ being proper time (an affine parameter which makes the curve a unit-speed curve). Proper time is time as measured by the clock for an observer who is traveling through spacetime. ...

Geodesic as maximal curve

A geodesic between two events could also be described as the curve joining those two events which has the maximum possible length. The four-length of a curve in spacetime is

where

.

Then the Euler-Lagrange equation, In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ...

becomes, after some calculation, Proof 1 , , , , , (return to article) Proof 2 The goal being to extremize the value of where such goal can be accomplished by calculating the Euler-Lagrange equation for f, which is . Substituting the expression of f into the Euler-Lagrange equation (which extremizes the value of the integral l), gives...

If parameter τ is chosen to be affine (so that the tangent with respect to it has constant magnitude), then the right side the above equation vanishes (because is constant, thereby its derivative is zero),

An affine parameter is directly proportional to proper time, and the above equation remains true (geodesically) for any linear reparameterization

.

Geodesic incompleteness and singularities

The notion of geodesic incompleteness is used in the study of gravitational singularities. A gravitational singularity occurs when an astrophysical model, typically based on general relativity, predicts a point of infinite space-time curvature. ...

References

• Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, (1972) John Wiley & Sons, New York ISBN 0-471-92567-5 See chapter 3.
• Lev D. Landau and Evgenii M. Lifschitz, The Classical Theory of Fields, (1973) Pergammon Press, Oxford ISBN 0-08-018176-7 See section 87.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.
• Bernard F. Schutz, A first course in general relativity, (1985; 2002) Cambridge University Press: Cambridge, UK; ISBN 0-521-27703-5. See chapter 6.