Geodesic

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. ==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

t

I

such that for any

t_1,t_2\in J

we have :

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 Curve#Length_of_curves, 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 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 and the geodesic equation== On a (pseudo-Riemannian manifold-)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^a}{dt^2} + \Gamma^{a}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt} = 0

where ''x''^''a''(''t'') are the coordinates of the curve γ(''t'') and

\Gamma^{a}_{bc}

are the Christoffel symbols. Equivalently, geodesics can be defined as extremal curves for the following energy functional :

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

where

g

is Riemannian (or pseudo-Riemannian) metric. This "energy functional" should be called action (physics), 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 ''T''_''p''''M'' (the tangent space to ''M'' at ''p'') there exists a unique geodesic γ : ''I'' → ''M'' such that

\gamma(0) = p

and

\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 R�. 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-Lindel�f theorem for the solutions of ODE's with prescribed initial conditions. Note that γ depends smooth function on both ''p'' and ''V''. ===Completeness=== Given a point ''p'' in ''M'' and a vector ''V'' in ''T''_''p''''M'', 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 vector field 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 mechanics by introducing additional independent variables, as shown below. Start by finding a chart (topology) that trivializes the cotangent bundle

T^*M

(''i.e.'' a ''local trivialization''): :

T^*M|_{U}\simeq U \times \mathbb{R}^n

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

(x^1,x^2,...,x^n,p_1,p_2,...,p_n)

. Then introduce the Hamiltonian vector field as :

H(x,p)=\frac{1}{2}g^{ab}(x)p_a p_b.

Here,

g^{ab}(x)

is the inverse of the metric tensor:

g^{ab}(x)g_{bc}(x)=\delta^a_c

. 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 :

\dot{x}^a = \frac{\partial H}{\partial p_a} = g^{ab}(x) p_b

and :

\dot{p}_a = - \frac {\partial H}{\partial x^a} = 
-\frac{1}{2} \frac {\partial g^{bc}(x)}{\partial x^a} p_b p_c.

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: :

\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.

Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy :

M_E = \{ (x,p) \in T^*M : H(x,p)=E \}

for each energy

E \ge 0

, so that :

T^*M=\bigcup_{E \ge 0} M_E

. 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== * differential geometry of curves * geodesic dome * geodesic (general relativity) ==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. Riemannian geometry Metric geometry Hamiltonian mechanics

Geodesic

"In Euclidean geometry, the geodesic are the straight line, but in more general spaces they need not be" -- not sure about this. A geodesic is what we ''mean'' by a "straight line" -- what else can a straight line be if not a geodesic? The fact that great circles don't appear straight is just a nasty side-effect of the Mercator projection mindset. -- User:Tarquin 14:54 Oct 28, 2002 (UTC) Yes, I agree; I reformulated a bit. Also, geodesics by definition really only give ''locally'' the shortest paths, not necessarily globally. User:AxelBoldt 19:26 Oct 28, 2002 (UTC) It's been a (somewhat) long time since I studied that, but aren't geodesics straight with respect to the local curvature in some sense ? That should be added to the article. User:FvdP == Einstein Equation == Article said: :This curvature is in turn determined by the energy and mass distribution; this is the content of the Einstein equation. Moved from article: : ; this is the content of the Einstein equation * If that refers to E=Mc**2, that is part of special rel & curvature of space-time is general rel . (If eqn implies this phenom, say more how and abt why, and explain at least here on talk in what sense it is "the content", a metaphysical-sounding expression that doesn't belong in physics w/o much explanation.) * If not, needs clarification of the term Einstein equation, i.e. write a stub for your dead link. --User:Jerzy 17:34, 2004 Jan 26 (UTC) ==Geodesic are straight== I removed "In mathematics, a geodesic is a curve which is everywhere locally straight." I belive it can not be usefull, it pretends to be little mathematical, but does not have more sense than the old formulation, contrary "Geodesic stay for the curves which are "straight" in a sense." at least explaign the meaning of the word and no mathwords used. User:Tosha 22:39, 22 Feb 2004 (UTC) I reverted that, because it makes absolutely no sense as a statement in the English language. User:Anthony DiPierro 22:54, 22 Feb 2004 (UTC) I do not know English, its true. Ok, is it better now? User:Tosha 23:18, 22 Feb 2004 (UTC) Not much. What does "stay for the curves" mean? I'll let someone else, who knows more about geodesics than I do, fix it. User:Anthony DiPierro 23:32, 22 Feb 2004 (UTC) :I rewrote it before, as the simplest expression that implies in the general case. I recognize some of what User:Tosha is trying to say. I've worked with a great number of Russian scientists who are very good and know the difficult some have of learning English. But the way it is written is simply jibberish right now. Tosha, if you feel the definition needs clarifying, please discuss it here. I would be more than happy to collobarate with you on parts of the article. In the meantime, I am going to change it back. -- User:Decumanus 23:36, 22 Feb 2004 (UTC) Ok, I hope now everybody happy(?) I wanted to get rid from word locally here (see above) User:Tosha 01:24, 23 Feb 2004 (UTC) :Well the phrase "in some sense" is just too vague. It means nothing here and is misleading. I'm curious: why do you object to the word "locally". It is the correct term here. It does not "pretend" to be mathematical. It is a standard way of stating this property. -- User:Decumanus 02:01, 23 Feb 2004 (UTC) What you do is much worse, if you want to grab an idea of geodesic then it is "straight in some sense" to do math you should define what is the "sense" and infect there are many different senses for this on the same space depending on structure you have/choose, now you have this strange curvature ... what does it mean for metric space for instance... I will not change it back, I'm tired (hope someone will do it) User:Tosha 03:13, 23 Feb 2004 (UTC) :Are you telling me you are not aware of the concept of geodesic curvature? -- User:Decumanus There is no meaning for geodesic curvature in metric space, but even if you make one the curves with zero geod.curvature might not be geodesic. User:Tosha 03:22, 23 Feb 2004 (UTC) :Your statement "there is no meaning for geodesic curvature in metric space" tells me a lot about your background in mathematics. Up until now I was prepared to give you the benefit of the doubt. No I am not so sure. -- User:Decumanus BTW geodesic curvature is nearly defined in the article, so one could just add one line in subsection "Riemannian and pseudo-Riemannian manifolds" instead of giving ref. User:Tosha 04:22, 23 Feb 2004 (UTC) Tosha, i can't tell whether you or Decumanus is the more authoritative editor for this article, but i'd urge you to work out the wording you have in mind on this page. For instance, after staring for a while at the first paragraph of your first edit of Geodesic, which reads : In general geodesic stay for the curves which are "straight" in a sense , i begin to suspect that you intended the meaning of that sentence to be close to : In each of the various contexts in which the term Geodesic is used, it stands for any one of the curves which "straight", in the sense of "straight" that is appropriate to that context. In proposing that reading, i note that the concepts of "stay" and "stand" are related (in fact, the probably come from the same ancient word) even tho a native speaker would never consider using "stay" to cover the metaphorical sense of "stand" that occurs in my suggested interpretation. (And to me, that sounds like a good introductory "motivating the idea" approach for an opening paragraph, and one that is consistent with shifting to a much more rigorous approach, such as Decumanus seems to be pushing for, in discussing the specific meaning of geodesic in the various contexts. (By the way, i also note that your use of "in a sense", which has caused some objections here, is not something you suddenly added to the article, but rather a variation or elaboration of : In more general spaces the geodesics can be more complicated, but one often still thinks of them as "straight" in a sense. which was introduced by User:AxelBoldt in an edit of 18:55, 2002 Oct 28, as far as i so far notice, remaaining in the article without objection for over a year. And part of my point to you is that fact does not seem to have come out here. IMO you're going to have to work hard not just at the technical content, but also both in communicating it clearly in this weird language of English (that you so bravely have undertaken to learn to an impressive degree), and finding out why other editors are being so seemingly stupid in not following your reasoning.) You may have things to add to the article that no one else on WP is prepared to contribute, but right now, we can't tell whether or not that's the case. I don't want to try and address the question of whether Decumanus has tried as hard as ''they'' should to understand what you're saying, but IMO there will be no hope of finding that out without your taking time to make sure that the ideas you are bringing forward are clearly understood. IMO, that will take a lot of patience on your part to help Decumanus and others understand your meanings. --User:Jerzy 05:24, 2004 Feb 23 (UTC) I've done a copy edit here. I'm with User:Tosha on this - he's a valuable contributor in this field, and I believe he knows exactly what he is talking about. I have similarly copy edited other pages of his. On the geodesic curvature matter; the definition now standing on that page needs work; for one thing it isn't obvious to me that it is compatible with the link from the Gauss-Bonnet theorem page (though it may be in fact). I think that point could be addressed there. For the time being, I felt linking geodesic to geodesic curvature wasn't clarifying, and I took out the link. User:Charles Matthews 08:53, 23 Feb 2004 (UTC) :It is the same as in the Gauss-Bonnet theorem, integrated along the curve. I have no problem with User:Tosha (and anyone else), as per my original offer.. -- User:Decumanus 20:53, 23 Feb 2004 (UTC) ::Forgive me. It is of course the scalar magnitude that appears in the Gauss-Bonnet theorem. This may have been the source of a little confusion. The statement about it vanishing along a ''geodesic'' of course applies to either the scalar or the vector. ::Also I was in the midst of doing battle with a certain user I won't name over at Talk:Quasar, and was abnormally testy. Please accept my apologies if I impugned your mathematical knowledge, User:Tosha,-- User:Decumanus 21:53, 23 Feb 2004 (UTC) Many things come to my mind when I think of geodesics, and "straight" is definitely not one of them. As discussed above, "straight" could be taken to depend on context, but the most intuitive notion of "straight" to me comes from lines in Euclidean space. These are in direct contrast with the mental image of a 2-sphere embedded in Euclidean 3-space in the natural way with spherical geodesics drawn upon it. Granted, these lines on the sphere only appear to be curved as an artifact of the Euclidean viewpoint, but I would argue that this is the most intuitive viewpoint for the reader new to the material. What do you all think about this? - User:Gauge 03:18, 15 Oct 2004 (UTC) ==Hilbert problem== I removed 4th Hilbert problem from this page, it is clearly relevant, but not at all the first problem which should be mentioned here, also the note after that is not quite correct, it was solved at least in dimension 2. User:Tosha 17:04, 8 Jun 2004 (UTC) == higher dimensions == The article applies "geodesic" only to lines. It can be also be applied to higher-dimensional submanifolds. I could write something on this, but I'm not sure whether there's a difference between geodesic surfaces and ''totally'' geodesic surfaces (see e.g. [http://www.physto.se/~ingemar/Kurs.ps], p.6). Does anyone know? User:Fpahl 15:48, 14 Sep 2004 (UTC) == more precisely? == The second sentence under "Geodesic#For metric spaces" begins with "More precisely". It seems to me that the precise definition is actually something new, and that it is a non-trivial statement that a curve satisfying that definition "is everywhere locally a distance minimizer". Also, the definition seems to be too narrow in that it requires the curve parameter to be proportional to the arc length; wouldn't any reparametrization of such a geodesic still count as a geodesic? User:Fpahl 15:56, 14 Sep 2004 (UTC) == Revert by Linas of edits by pdn == Linas, you make the statement: "masses follow timelike geodesics, period. no ifs-ands or buts". It is my understanding that only the world lines of ''free'' particles are geodesics of spacetime. Thus, here on the surface of the Earth, I feel the 'weight' of acceleration and my world line(s) is certainly not a geodesic. User:Alfred Centauri 14:27, 6 Jun 2005 (UTC) : Isn't the problem one of distinguishing between paths and geodesics ? Free particles (and photons) definitely follow geodesics (and they are timelike (null for photons) geodesics), but if the particle is not free, then the concept of geodesic doesn't apply - although the concept of path (or worldline) still does. My conclusion: *Worldlines are paths in spacetime. *Worldlines of free particles are called geodesics of the spacetime. *Particles that are not free obviously still have wordlines (but they are not geodesics), but one can still calculate tangent vectors at any point on the worldline and determine if they are spacelike, timelike or null (assuming we know the metric). For material particles, they are timelike at each point of the worldline, for photons etc. they are null at each point. Suggestions/comments ? User:Mpatel 15:58, 6 Jun 2005 (UTC) :Sorry about the revert, maybe I was having a bad day. I admit I misread the statements about timelike; (I misread them as saying that worldines of massive particles aren't timelike...oops). On the other hand, I completely failed to understand what was being said about the spacelike geodesics .. that they're "graphs of filaments"? My brain fritzed at that point, so I just reverted. :Maybe its time to start a new article, say, geodesic (general relativity) ? The current geodesic article is getting long, and the GR stuff is kind of buried at the bottom. There's lots of interesting things that can be said about GR geodesics, and that article isn't really structured to say them. The current article could continue to stand as the "math formalities" of geodesics. User:Linas 23:51, 6 Jun 2005 (UTC) ::Acted upon Linas' suggestion by creating the suggested new article: geodesic (general relativity). ---User:Mpatel 16:56, 13 Jun 2005 (UTC)

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Words begining with Geodesic:

Geodesic
Geodesic
Geodesics
Geodesic_(general_relativity)
Geodesic_Airship_Design
Geodesic_Airship_Design
Geodesic_curvature
Geodesic_curvature_vector
Geodesic_dome
Geodesic_dome
Geodesic_domes
Geodesic_equation
Geodesic_flow
Geodesic_flow
Geodesic_structures