Geodesic

In mathematics, a geodesic /ˌdʒiəˈdɛsɪk, -ˈdisɪk/[jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line" to "curved spaces". In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it.

The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

Geodesics are of particular importance in general relativity.

Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve, and then minimizing this length using standard techniques of calculus and differential equations. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic. Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy; the resulting shape of the band is a geodesic.

Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In relativistic physics, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all described by geodesics in the theory of general relativity. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds. The article geodesic (general relativity) discusses the special case of general relativity in greater detail.

Examples

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. If A and B are antipodal points (like the North pole and the South pole), then there are infinitely many shortest paths between them.

Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ: I → M from the unit interval I to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is a neighborhood J of t in I such that for any t₁, t₂ ∈ J we have

Failed to parse (Missing texvc executable; please see math/README to configure.): d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|.\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|.\,

If the last equality is satisfied for all t₁, t₂ ∈I, the geodesic is called a minimizing geodesic or shortest path.

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

(Pseudo-)Riemannian geometry

Just as in a standard metric space, a geodesic on a (pseudo-)Riemannian manifold M is defined as a curve γ(t) minimizes the length of the curve. Explicitly, we can write the length of any curve as

Failed to parse (Missing texvc executable; please see math/README to configure.): l(\gamma)=\int_\gamma \sqrt{ \pm g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \dot\gamma

represents the derivative with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): t

, and is a vector. The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves. A geodesic, then, is the curve which extremizes this quantity (locally).

In the case of a manifold with torsion-free and metric-compatible connection (which is almost always assumed to be the case in Relativity, for example), a geodesic curve is also an autoparallel curve. That is, the curve parallel transports its own tangent vector, so

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot\gamma^\mu \nabla_\mu \dot\gamma^\nu = 0

at each point along the curve. Here, ∇ stands for the Levi-Civita connection on M.

In this case, using local coordinates on M, we can write the geodesic equation (using the summation convention) as

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{~\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0\ ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): x^\mu (t)

are the coordinates of the curve γ(t) and Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma^{\lambda }_{~\mu \nu }
are the Christoffel symbols.  This is just an ordinary differential equation for the coordinates.  It has a unique solution, given an initial position and an initial velocity.  Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold.

Geodesics can also be defined as extremal curves for the following action functional

Failed to parse (Missing texvc executable; please see math/README to configure.): S(\gamma)=\frac{1}{2}\int g(\dot\gamma(t),\dot\gamma(t))\,dt,

where Failed to parse (Missing texvc executable; please see math/README to configure.): g

is a Riemannian (or pseudo-Riemannian) metric.  In pure mathematics, this quantity would generally be referred to as an energy.  The geodesic equation can then be obtained as the Euler-Lagrange equations of motion for this action.

In a similar manner, one can obtain geodesics as a solution of the Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian. See Riemannian manifolds in Hamiltonian mechanics for further details.

Existence and uniqueness

The local existence and uniqueness theorem for geodesics states that geodesics exist, and are unique; this is a variant of the Frobenius theorem. More precisely:

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma \,

: I → M such that 

Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma(0) = p \,

and 

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot\gamma(0) = V

,

where I is a maximal open interval in R containing 0.

In general, I may not be all of R as for example for an open disc in R². The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the Picard-Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and V.

Geodesic flow

Geodesic flow is an Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R -action on tangent bundle Failed to parse (Missing texvc executable; please see math/README to configure.): T(M)

of a manifold Failed to parse (Missing texvc executable; please see math/README to configure.): M
defined in the following way 

Failed to parse (Missing texvc executable; please see math/README to configure.): G^t(V)=\dot\gamma_V(t)

where Failed to parse (Missing texvc executable; please see math/README to configure.): t\in \mathbb R , Failed to parse (Missing texvc executable; please see math/README to configure.): V\in T(M)

and Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma_V
denotes the geodesic with initial data Failed to parse (Missing texvc executable; please see math/README to configure.): \dot\gamma_V(0)=V

.

It defines a Hamiltonian flow on (co)tangent bundle with the (pseudo-)Riemannian metric as the Hamiltonian. In particular it preserves the (pseudo-)Riemannian metric Failed to parse (Missing texvc executable; please see math/README to configure.): g , i.e.

Failed to parse (Missing texvc executable; please see math/README to configure.): g(G^t(V),G^t(V))=g(V,V)

. That makes possible to define geodesic flow on unit tangent bundle Failed to parse (Missing texvc executable; please see math/README to configure.): UT(M)

of the Riemannian manifold Failed to parse (Missing texvc executable; please see math/README to configure.): M

.

Geodesic spray

The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray.

Affine and projective geodesics

In the presence of a metric, geodesics are (locally) the length-minimizing curves. However, even if a manifold lacks a metric, geodesics are still well-defined in the presence of an affine connection. A curve in such a manifold is a geodesic if its tangent vector remains parallel to the curve when it is transported along it.

See also

References

• Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1. See section 1.4.
• Adler, Ronald; Bazin, Maurice & Schiffer, Menahem (1975), Introduction to General Relativity (2nd ed.), New York: McGraw-Hill, ISBN 978-0-07-000423-8. See chapter 2.
• Abraham, Ralph H. & Marsden, Jerrold E. (1978), Foundations of mechanics, London: Benjamin-Cummings, ISBN 978-0-8053-0102-1. See section 2.7.
• Weinberg, Steven (1972), Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, New York: John Wiley & Sons, ISBN 978-0-471-92567-5. See chapter 3.
• Landau, L. D. & Lifshitz, E. M. (1975), Classical Theory of Fields, Oxford: Pergamon, ISBN 978-0-08-018176-9. See section 87.
• Misner, Charles W.; Thorne, Kip & Wheeler, John Archibald (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0
• Ortín, Tomás (2004), Gravity and strings, Cambridge University Press, ISBN 978-0-521-82475-0. Note especially pages 7 and 10.

External links

• Caltech Tutorial on Relativity — A nice, simple explanation of geodesics with accompanying animation.
• Geomathbn:জিওডেসিক

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