Geodesic

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In mathematics, a geodesic is a generalization of the notion of a "straight line" to curved spaces. More precisely, geodesics are defined to be (locally) the shortest path between two points on manifolds which have a metric that defines a notion of distance. This name is taken from geodesy, the science of measuring the size and shape of the earth; in the original sense, a geodesic was the shortest route between two points on the surface of the earth, namely, a segment of a great circle.

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 physics, geodesics describe the motion of point particles; 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 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: IM from the unit interval I to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any tI there is a neighborhood J of t in I such that for any t1, t2J we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\gamma(t_1),\gamma(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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|.\,}

If the last equality is satisfied for all t1, t2I, the geodesic is called a minimizing geodesic or shortest path (although in geometric group theory the word geodesic is usually used exclusively in this latter sense).

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

(pseudo-)Riemannian geometry

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{D}{dt}\dot\gamma(t) = \nabla_{\dot\gamma(t)}\dot\gamma(t) = 0} .

where ∇ stands for Levi-Civita connection on M.

In case of Riemannian manifold, the geodesic that one obtains this way are identical to geodesics for the induced metric space.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2x^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} = 0}

where xa(t) are the coordinates of the curve γ(t) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma^{a}_{bc}} are the Christoffel symbols.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E(\gamma)=\frac{1}{2}\int g(\dot\gamma(t),\dot\gamma(t))\,dt,}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is Riemannian (or pseudo-Riemannian) metric. This "energy functional" should be called action, but only few in mathematics use this term; the geodesic equation can then be obtained as the Euler-Lagrange equations of motion for this action.

Therefore, from mechanical point of view, geodesics can be thought as trajectories of free particles inside of manifold (at least in Riemannian case).

In a similar maner, one can obtain geodesics as a solution of 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 TpM (the tangent space to M at p) there exists a unique geodesic γ : IM such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(0) = p} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot\gamma(0) = V} . Here 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 ODE's with prescribed initial conditions. γ depends smoothly on both p and V.

Geodesic flow

Geodesic flow is an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R} -action on tangent bundle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(M)} of a manifold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} defined in the following way

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^t(V)=\dot\gamma_V(t)}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t\in \mathbb R} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\in T(M)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_V} denotes geodesic with initial data Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot\gamma_V(0)=V} .

It defines a Hamiltonian flow on (co)tangent bundle with (pseudo-)Riemannian metric as Hamiltonian. In particular it preserves the (pseudo-)Riemannian metric Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} , i.e.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(G^t(V),G^t(V))=g(V,V).}

That makes possible to define geodesic flow on unit tangent bundle of Riemannian manifold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle UT(M)} .

Geodesic spray

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

Formally, the tangent bundle to the tangent manifold is known as a jet bundle; thus the geodesic spray is a vector field in the (first) jet bundle of the manifold.

See also

References

  • Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 See section 1.4.
  • Ronald Adler, Maurice Bazin, Menahem Schiffer, Introductin to General Relativity (Second Edition), (1975) McGraw-Hill New York, ISBN 0-07-000423-4 See chapter 2.
  • Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 2.7.
  • 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.

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