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Geodesic

In mathematics, a geodesic is a generalization of the notion of a "staight line" to curved spaces. It takes its name from the science of geodesy of measuring the size and shape of the earth, and was originally the shortest route between two points on the surface of the earth. For example the great circle path between points on the Earth, idealised as a sphere, is a geodesic. A small circle path is not. In intuitive terms, an elastic band stretched along a path that is not geodesic would contract its length for energy reasons to a nearby shorter path — this though only serves to explain that a geodesic is a local minimum for length.

Contents

1 Metric geometry

2 Riemannian geometry

3 Geodesics in general relativity

4 See also

Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, if $M$ is a metric space, a curve $\gamma:I\to M$ is a geodesic if there is a constant $v\ge 0$ such that for any $t\in I$ there is a neighborhood $J$ of $t$ in $I$ such that for any $t_1,t_2\in J$ we have

d (γ(t 1),γ(t 2)) = v | t 1 - t 2 |

This notion generalizes notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered almost always equipped with natural parametrization, i.e. in the above identity v=1 and

d (γ(t 1),γ(t 2)) = | t 1 - t 2 |

If the last equality is satisfied on all $I$ , the geodesic is called a minimizing geodesic or shortest path.

In general, a metric space may have no geodesics, except constant curves.

Riemannian geometry

On a (pseudo-)Riemannian manifold M a geodesic can be defined as a smooth curve γ(t) that parallel transports its own tangent vector. That is,

$\frac{D}{dt}\dot\gamma(t) = \nabla_{\dot\gamma(t)}\dot\gamma(t) = 0$ .

where ∇ stands for Levi-Civita connection on M.

In terms of local coordinates on M the geodesic equation can be written (using the summation convention):

$\frac{d^2x^i}{dt^2} + \Gamma^{i}_{jk}\frac{dx^j}{dt}\frac{dx^k}{dt} = 0$

where xⁱ(t) are the coordinates of the curve γ(t) and $\Gamma^{i}_{jk}$ are the Christoffel symbols.

Equivalently, geodesics can be defined as extremal curves for the following energy functional

$E(\gamma)=\int g(\dot\gamma(t),\dot\gamma(t))\,dt,$

where $g$ is Riemannian (or pseudo-Riemannian) metric. (In fact, this "energy functional" should be called action, but nobody in mathematics does so.)

The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. Note that if A and B are antipodal points (like the North pole and the South pole), then there are many shortest paths between them.

There is a local existence and uniqueness theorem for geodesics which follows from the theory of ordinary differential equations. This theorem states that for any point p in M and for any vector V in T_pM (the tangent space to M at p) there exists a unique geodesic γ : I → M such that $γ(0) = p$ and $\dot\gamma(0) = V$ . Here I is a maximal open interval in R containing 0. Note that I may not be all of R since the geodesic may terminate somewhere on the manifold.

The differential geometry of curves provides definitions and methods for analyzing smooth curves in Riemannian manifolds using the tools of calculus.

Geodesics in general relativity

Geodesics are important in the theory of general relativity where they represent the paths of particles in free fall, that is, particles moving under the influence of no forces except gravity. For example, the orbit of a planet around a star follows a geodesic path.

Recall that spacetime in general relativity is a Lorentzian manifold with signature (− + + +). Geodesics on a Lorentzian manifold fall into three classes according to the sign of the norm of their tangent vector:

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).

Massive particles and objects will always follow timelike geodesics, while massless particles like the photon will follow null geodesics. Spacelike geodesics exist, but do not correspond to the path of any physical particle.

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Geodesic

Metric geometry

Riemannian geometry

Geodesics in general relativity

See also

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