Geodesic

In mathematics, particularly differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be (locally) the shortest path between points in 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, as they describe the motion of inertial test particles.

Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to demand not only that the curve locally minimize length but also that it is parameterized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to |s−t|. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimisation). 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.

In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t2 from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In general relativity, 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 geodesics in curved space-time. 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 images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc 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 an interval I of the reals 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

\( d(\gamma(t_1),\gamma(t_2))=v|t_1-t_2|.\, \)

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

\( d(\gamma(t_1),\gamma(t_2))=|t_1-t_2|.\,\)

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. At the other extreme, any two points in a length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic.
Riemannian geometry

In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ : [a,b] → M is defined by

\( L(\gamma)=\int_a^b \sqrt{ g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) }\,dt. \)

The distance d(p, q) between two points p and q of M is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ : [a,b] → M such that γ(a) = p and γ(b) = q. With this definition of distance, geodesics in a Riemannian manifold are then the locally distance-minimizing paths, in the above sense.

The minimizing curves of L in a small enough open set of M can be obtained by techniques of calculus of variations. Typically, one introduces the following action or energy functional

\( E(\gamma)=\frac{1}{2}\int g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,dt. \)

It is then enough to minimize the functional E, owing to the Cauchy–Schwarz inequality

\( L(\gamma)^2\le 2(b-a)E(\gamma) \)

with equality if and only if |dγ/dt| is constant.

The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by

\( \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0, \)

where\( \Gamma^\lambda_{\mu\nu} \) are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.
Calculus of variations

Techniques of the classical calculus of variations can be applied to examine the energy functional E. The first variation of energy is defined in local coordinates by

\( \delta E(\gamma)(\varphi) = \left.\frac{\partial}{\partial t}\right|_{t=0} E(\gamma + t\varphi). \)

The critical points of the first variation are precisely the geodesics. The second variation is defined by

\( \delta^2 E(\gamma)(\varphi,\psi) = \left.\frac{\partial^2}{\partial s\partial t}\right|_{s=t=0}E(\gamma + t\varphi + s\psi). \)

In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields. Jacobi fields are thus regarded as variations through geodesics.

By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows. They are solutions of the associated Hamilton–Jacobi equations, with (pseudo-)Riemannian metric taken as Hamiltonian.
Affine geodesics

A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so

\( \nabla_{\dot\gamma} \dot\gamma= 0 (1) \)

at each point along the curve, where \dot\gamma is the derivative with respect to t. More precisely, in order to define the covariant derivative of \dot\gamma it is necessary first to extend \dot\gamma to a continuously differentiable vector field in an open set. However, the resulting value of (1) is independent of the choice of extension.

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

\( \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0\ , \)

where \( x^\mu (t) \) are the coordinates of the curve γ(t) and \( \Gamma^{\lambda }_{\mu \nu } \) are the Christoffel symbols of the connection ∇. 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. Indeed, the equation \nabla_{\dot\gamma} \dot\gamma= 0 means that the acceleration of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by the gravity.
Existence and uniqueness

The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique. 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 \( \gamma \, : I → M \) such that

\( \gamma(0) = p \, \) and
\( \dot\gamma(0) = V, \)

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 a local R-action on tangent bundle TM of a manifold M defined in the following way

\( G^t(V)=\dot\gamma_V(t) \)

where t ∈ R, V ∈ TM and\( \gamma_V \) denotes the geodesic with initial data\( \dot\gamma_V(0)=V \). Thus, Gt(V) = exp(tV) is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M.

On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical one-form. In particular the flow preserves the (pseudo-)Riemannian metric g, i.e.

\( g(G^t(V),G^t(V))=g(V,V). \)

In particular, when V is a unit vector, \( \gamma_V \) remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

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.

More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles:

\( TTM = H\oplus V. \)

The geodesic spray is the unique horizontal vector field W satisfying

\( \pi_* W_v = v\, \)

at each point v ∈ TM; here π∗ : TTM → TM denotes the pushforward (differential) along the projection π : TM → M associated to the tangent bundle.

More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle \( TM \ {0}) \) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf. Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy

\( H_{\lambda X} = d(S_\lambda)_X H_X\, \)

for every X ∈ \( TM \ {0} \) and λ > 0. Here d(Sλ) is the pushforward along the scalar homothety \( S_\lambda: X\mapsto \lambda X \). A particular case of a non-linear connection arising in this manner is that associated to a Finsler manifold.
Affine and projective geodesics

Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form

\( t\mapsto at+b \)

where a and b are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of (1) are called geodesics with affine parameter.

An affine connection is determined by its family of affinely parameterized geodesics, up to torsion (Spivak 1999, Chapter 6, Addendum I). The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if \( \nabla, \bar{\nabla} \) are two connections such that the difference tensor

\( D(X,Y) = \nabla_XY-\bar{\nabla}_XY \)

is skew-symmetric, then \( \nabla \) and \( \bar{\nabla} \) have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as \nabla, but with vanishing torsion.

Geodesics without a particular parameterization are described by a projective connection.
See also

Basic introduction to the mathematics of curved spacetime
Clairaut's relation
Closed geodesic
Complex geodesic
Differential geometry of curves
Exponential map
Geodesic dome
Geodesic (general relativity)
Geodesics as Hamiltonian flows
Hopf–Rinow theorem
Intrinsic metric
Jacobi field
Quasigeodesic
Solving the geodesic equations
Zoll surface
Nautical chart
Rhumb line (loxodrome)
Great circle (earth geodesic)
Meridian arc

References

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.
Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1. See section 1.4.
Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Vol. 1 (New ed.), Wiley-Interscience, ISBN 0-471-15733-3.
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.
Spivak, Michael (1999), A Comprehensive introduction to differential geometry (Volume 2), Houston, TX: Publish or Perish, ISBN 978-0-914098-71-3
Volkov, Yu.A. (2001), "Geodesic line", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104.
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.

External links

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