Geodesic
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In mathematics, a geodesic is a generalization of the notion of a "straight 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. Geodesics play an important role in the theory of general relativity, where they are the world lines of a particle free from all external force; see the main article geodesic (general relativity) for details.
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Metric geometry
In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, if <math>M<math> is a metric space, a curve <math>\gamma:I\to M<math> is a geodesic if there is a constant <math>v\ge 0<math> such that for any <math>t\in I<math> there is a neighborhood <math>J<math> of <math>t<math> in <math>I<math> such that for any <math>t_1,t_2\in J<math> we have
- <math>d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|<math>.
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
- <math>d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|<math>.
If the last equality is satisfied on all <math>I<math>, the geodesic is called a minimizing geodesic or shortest path.
In general, a metric space may have no geodesics, except constant curves.
Riemannian geometry and the geodesic equation
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,
- <math>\frac{D}{dt}\dot\gamma(t) = \nabla_{\dot\gamma(t)}\dot\gamma(t) = 0<math>.
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):
- <math>\frac{d^2x^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} = 0<math>
where xa(t) are the coordinates of the curve γ(t) and <math>\Gamma^{a}_{bc}<math> are the Christoffel symbols.
Equivalently, geodesics can be defined as extremal curves for the following energy functional
- <math>E(\gamma)=\frac{1}{2}\int g(\dot\gamma(t),\dot\gamma(t))\,dt,<math>
where <math>g<math> 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. The geodesic that one obtains by extremizing the energy functional is identical to the geodesic obtained by extremizing the length functional; both are given by the geodesic equations.
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. 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.
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 : I → M such that <math>\gamma(0) = p<math> and <math>\dot\gamma(0) = V<math>. 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 follows from the Picard-Lindelöf theorem for the solutions of ODE's with prescribed initial conditions. Note that γ depends smoothly on both p and V.
Completeness
Given a point p in M and a vector V in TpM, the exponential map will map the vector tV to a geodesic in M, where t is a real number, scaling the vector V. The Hopf-Rinow theorem states, among other things, that any two points on a Riemannian manifold are joined by a geodesic.
Geodesic flow
Geodesics can also be understood to be the Hamiltonian flows of a very special Hamiltonian defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
The geodesic equations are second order differential equations; they can be re-expressed as first-order ordinary differential equations taking the form of Hamiltonian equations by introducing additional independent variables, as shown below. Start by finding a chart that trivializes the cotangent bundle <math>T^*M<math> (i.e. a local trivialization):
- <math>T^*M|_{U}\simeq U \times \mathbb{R}^n<math>
where U is an open subset of the manifold M, and the tangent space is of rank n. Label the coordinates of the chart as <math>(x^1,x^2,...,x^n,p_1,p_2,...,p_n)<math>. Then introduce the Hamiltonian as
- <math>H(x,p)=\frac{1}{2}g^{ab}(x)p_a p_b.<math>
Here, <math>g^{ab}(x)<math> is the inverse of the metric tensor: <math>g^{ab}(x)g_{bc}(x)=\delta^a_c<math>. This inverse almost always exists for a broad class of metric manifolds. The behaviour of the metric tensor under coordinate transformations implies that H is invariant under a change of variable. The geodesic equations can then be written as
- <math>\dot{x}^a = \frac{\partial H}{\partial p_a} = g^{ab}(x) p_b<math>
and
- <math>\dot{p}_a = - \frac {\partial H}{\partial x^a} =
-\frac{1}{2} \frac {\partial g^{bc}(x)}{\partial x^a} p_b p_c.<math>
The second order geodesic equations are easily obtained by substitution of one into the other. The flow determined by these equations is called the cogeodesic flow. The first of the two equations gives the flow on the tangent bundle TM, the geodesic flow. Thus, the geodesic lines are the integral curves of the geodesic flow onto the manifold M. Note that this is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics:
- <math>\frac{dH}{dt} = \frac {\partial H}{\partial x^a} \dot{x}^a +
\frac{\partial H}{\partial p_a} \dot{p}_a = - \dot{p}_a \dot{x}^a + \dot{x}^a \dot{p}_a = 0.<math> Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy
- <math>M_E = \{ (x,p) \in T^*M : H(x,p)=E \}<math>
for each energy <math>E \ge 0<math>, so that
- <math>T^*M=\bigcup_{E \ge 0} M_E<math>.
The Hopf-Rinow theorem guarantees the completeness of the manifold. Note that the positivity of the energy follows from the positivity of the metric tensor; this analysis is modified on pseudo-Riemannian manifolds.
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 Cosomology: 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.ca:Geodèsica