Talk:Geodesic

message: 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.
--Jerzy 17:34, 2004 Jan 26 (UTC)
::Presumably that should have been Einstein field equations, which is relevant, although it would have been more accurate to write  Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:53, 6 September 2020 (UTC)
thread_title:  Einstein Equation 

message: I removed 4th Hilbert problem (link to Hilbert Problems: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html) 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.
 Tosha 17:04, 8 Jun 2004 (UTC)
thread_title: Hilbert problem

message: 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. , p.6). Does anyone know? Fpahl 15:48, 14 Sep 2004 (UTC)
thread_title:  higher dimensions 

message: The second sentence under "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? Fpahl 15:56, 14 Sep 2004 (UTC)
thread_title:  more precisely? 

message: Domain I is not introdcued before it is used in the first definition.  It should be introduced to make the article self-contained and accessible to people who know mathematics but do not use geometry daily. Wazow 10:54, 11 August 2005 (UTC)
replies: text: Fixed. linas 22:16, 15 August 2005 (UTC)
thread_title:  I is not defined before it is used. 

message: It would be nice to have, in this article, or in a related one,
Further Edification:
* Discussion of exponential map and its use for solving geodesic equation, in general, and not just on Lie groups. Also, curves of "constant speed".
* Discussion of straightness in this context and very very breif mention of parallel transport with regards to straightness.
Exotic geometry:
* There are scads of metric spaces. Which ones have bizzare geodesics?  Or are otherwise geodesically patholigical in interesting ways? 
* Examples of metric manifolds that are not geodesically complete (there are examples in GR). 
* Examples of manifolds for which geodesics terminate. Schwarzchild singularities provide a 1-D example, Kerr provides a 2D.  (Point reader to article geodesic (general relativity)) A slightly more exotic example is the three-body problem in Newtonian gravity. How about examples where the terminus is fractal, e.g. the Hopf whatever-its-called-disaster from chaos theory? Cauchy horizon.
* Examples of manifolds for for which shortest geodesics are not unique. (Is there a name for this class of manifold??)
* Mention of Finsler manifold.
* Discussion of geodesy on infinite-dimensinal manifolds, e.g. topological vector spaces.
* Vastly expanded coverage of sub-Riemanian geometry.
* geodesics in algebra.
Mention of whiz-bang uses and applications:
* prime geodesic
* Selberg zeta function
* Selberg trace formula
* ergodic theory and ergodicity on hyperbolic surfaces. (the article on ergodic theory already discusses this, need a reciprocal mention here only).
* relationship to quantum mechanics.
linas 04:45, 30 September 2005 (UTC)
replies: text: I think we should not expand it too much,  Finsler and sub-Riemannian cases are too special and might be done in a separate article (as geodesic (general relativity)) Tosha 05:19, 20 October 2005 (UTC)
thread_title:  Wishlist, todo list 

message: On intro: the curent intro is has partly wrong def, it is not sutabale for all cases, in particular it is very wrong for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. It was a long discussion before we converge to the one I like and I think we have to keep it. 
:What was wrong with the current intro? Several people thanked me for replacing the old intro with the current one; the old intro was poorly written and did not make much sense.
Did you read what I wrote? Tosha 03:22, 21 October 2005 (UTC)
replies: text: Yes, but you did not answer my question. What was wrong with the current intro? I did read the old discussion about the old intro, and it was clear that several people in that conversation did not know what they were talking about.  
Main problem: it does not work for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. Note that I did not change it so far, but I will do it later Tosha 15:30, 21 October 2005 (UTC)
thread_title: Intro

message: On Riemannian geometry and the geodesic equation One can not define it trough lenght, at least yoyu should ask for constant speed and it does not work for pseudorieamannian case.
:I don't think I put anything in there about constant speed, that may have been left over from the old version of the article.  Lets fix that.
I do not blame you, I'm  just correcting mistake Tosha 03:22, 21 October 2005 (UTC)
replies: text: What was the mistake? 09:02, 21 October 2005 (UTC)
thread_title: (pseudo-)Riemannian geometry

message: In addition, the geodesic flow section can be removed almost completely, and refer to Hamiltonian mechanics
:No,  because the geodesic flow is fundamental to what it means to be geodesic. It is a much better and more modern presentation than the Euler-Lagrange equations, (which is the old-fashioned approach). If one wanted to be absurd, one could just say that geodesics are a special case of Lagrangian mechanics, and just redirect this article to that.  But that would be wrong. 
So why not leave all technical stuff there there is no need to repeat it here Tosha
:What technical stuff, where? The definition that a geodesic is a curve that parallel transports its own vector is a terrible definition and is very difficult to understand unless one already knows this topic very very well. I would like to remove that section entirely. By contrast, a derivation of the geodesic equation from either lagrangian or hamiltonian principles is the usual way of introducing the concept, and is simple and easy to understand.  The hamiltonian method is simpler than the Euler-Lagrange equations because it gives a pair of first order differential equations which can actually be solved simply, for real-life mainfolds. You can actually write down something that is solvable, e.g. on a computer, by working with these.  It is much much harder to convert the parallel-transport equation into something useful for practical calculations.
the Hamiltonian can be mentioned in def of geodesic (I'm not against anything which is correct) note that there is no technical detales about lagrangian as well.
There is reason to do this def for Riemannian geometry and most books do it this way (and it is not becouse they are stupid, main thing: it is more usefull).
Also, as it is stated in the end metric def coinsided with Riemaniann
This section had many things not really related to the topic now at least it is clear what geodesic flow is Tosha 15:30, 21 October 2005 (UTC)
thread_title: geodesic flow

message: I changed 
:E(\gamma)=\frac{1}{2}\int_I g(\dot\gamma(t),\dot\gamma(t))\,dt,
to 
:E(\gamma)=\frac{1}{2}\int_I g(\gamma(t),\dot\gamma(t))\,dt,
which I believe to be a correct.  But as I am no expert of on Riemannian manifolds, I figured I would alert others to the change. 
Jrohrs 18:04, 1 August 2006 (UTC)
thread_title:  Correction to energy functional 

message: Tosha, you also removed the following section:
: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.  
 This is a strage statement, it does not define exp.map, or at least give very non-complete def.
The Hopf-Rinow theorem states, among other things, that any two points on a Riemannian manifold are joined by a geodesic.''
 I guess one should add "complete"
:In general, for manifolds that are not Riemannian, geodesics need not exist, need not be unique, and the manifold may not be geodesically complete.
 Well if not Riemannian then geodesics might be not defined, which spaces are ment here? pseudo-Riemanian? Finsler? metric spaces? what is the meaning of non-uniqueness and so on, it is just too foggy...
I am guessing that you removed this because the article failed to define the meaning of the exponential map in the context of a Riemann surface? linas 13:13, 26 October 2005 (UTC)
replies: text: Robot arms and satellite orbits are given by sub-Riemannian manifolds.  The metric only works in some directions, not all. (one of the prototype problem is the holonomy (Berry phase) for the "falling cat" which can turn around while falling to land on its feet). The variational equations aren't always for "shortest energy/length", they're often for "shortest time from here to there, as limited by maximum torque of the motors", so instead of being constant-velocity, they try to be constant-acceleration.  I've never studied this stuff closely, I know that there are books on this stuff, and I know that questions of existance are important.  Its more or less that same kind of math, w/ metrics, connections, parallel transport, geodesics, slightly modified, since its usually easier to work in the tangent space and the hamiltonian formulation. Not sure why, it has something to do with the incomplete metric. (whereas the inner product on the tangent space is still well defined). linas 00:57, 27 October 2005 (UTC)
thread_title: Completeness

message: There should be a section (or at least a subsection) on the geodesic equation, as this is certainly something that a general reader should see stand out. Looking at the table of contents, the concepts mentioned are important, but geodesic equation should be there too. MP   (talk) 09:39, 19 February 2006 (UTC)
replies: text: The derivation of the geodesic equation in relativity theory is already presented at geodesics in general relativity.  The present article already does describe how to obtain the geodesic equation from the calculus of variations, although it is generally expected that someone looking for detailed derivations of results would consult any of a number of standard textbooks on the subject, rather than relying on a general-purpose encyclopedia.  Sławomir Biały (talk) 10:58, 15 August 2014 (UTC)
thread_title:  Geodesic equation  

message: In a general space, there is a difference between an autoparallel curve (which parallel transports its tangent vector) and a geodesic (which is defined as a curve which extremizes length, regardless of which type of space you're working in).  In the specific case of a (pseudo-)Riemannian manifold with torsion-free and metric-compatible connection, the difference vanishes.  I've tried to establish this distinction more clearly.  For a nice treatment, I recommend the first dozen pages of Ortin's "Gravity and Strings".  Note that Wald assumes torsion-free and metric-compatible very early on, so he never separates the two cases. --MOBle 21:26, 1 August 2006 (UTC)
thread_title:  Autoparallel versus Geodesic 

message: I find this article to be too focussed on mathematics.  If you are mentioning "metrics" and the 
"affine connection" in the first paragraph you are already in trouble.  While I can appreciate Riemannian geometry and so forth, I remind you all that "geodesic" has "geo" for "earth" in front of it, hence remind you of its origins.  That is, in geodesy.  I suggest two changes to the article:  (1) that the introduction be reordered so that the simpler "path over the earth" definition/discusson is given as the first paragraph, followed by the ramp up to the mathematics (you are scaring people away now), and (2) the inclusion of a new section, one coming right after the introduction, on earth geodesics.  The new section would talk about such things as geodesics on a sphere, and spheroids such as WGS84, with some simple examples, perhaps even a figure...  In short...you math people have to back off! :)  The article might even benefit from a split into two articles, one being the simple, earth-centric discussion, the other being the more complex math-centric discussion.   
(Consider the poor high-schooler writing his report the night before it is due who just wants to know what geodesic means...and he goes to his favorite reference wikipedia, only to find this article...Yikes!)
As an example of a figure, see:  http://909ers.apl.washington.edu/~dushaw/perth_bermuda2.jpg
The upper path is the geodesic on a sphere, the lower path is a geodesic on the wgs84 ellipsoid.  I am the author of the figure and could release it to wikipedia if you thought it useful.  The path goes from Perth, Australia to Bermuda.   Bdushaw 21:35, 6 October 2007 (UTC)
thread_title:  A comment...this article is too mathematics-centric 

message: "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."
Strictly speaking, this isn't quite right.  The Earth is not a perfect sphere (it's an oblate spheroid or ellipsoid - i.e slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle.  But I'm not sure how best to explain this in the intro without over-complicating it. Wardog (talk) 11:15, 21 February 2008 (UTC)
thread_title:  Geodesic =! Great Circle 

message: I don't really see why there should be a separate geodesic (general relativity) article. The existence of two articles suggest that they are different concepts, while they are exactly the same concept. (TimothyRias (talk) 15:25, 17 July 2008 (UTC)
replies: text: Not exactly the same.  In GR, there are timelike, spacelike, and null geodesics, and associated questions of causality.  Strictly speaking this is separate from the structure of an affine geodesic.  Furthermore, the present article focuses on geodesics in Riemannian geometry, which are strong minimizers of the energy functional, rather than just extremals as is the case in GR, and the related notion of affine geodesics.  Anyway, I oppose the merge, since much of the material in Geodesic (general relativity) is far too application-specific to be included here (discussion of timelike Killing fields, etc.)  siℓℓy rabbit (talk) 15:39, 17 July 2008 (UTC)
replies: text: There being time and spacelike geodesics is not particular to GR but to the case of a Lorentzian metric, to be comprehensive this article needs to discuss the pseudo-Riemannian case and some applications any way. So, again I really don't see the need for a separate GR article. (TimothyRias (talk) 08:06, 18 July 2008 (UTC)

text: Since geodesics are relevant to subjects other than relativity---e.g., geodesy---it would seem strange to have this merged with a GR article.  —Preceding unsigned comment added by 131.107.0.70 (talk) 19:37, 27 May 2009 (UTC)

text: I think it's very useful to have a separate article specifically on the use of the term geodesic in general relativity.  Having to wade through a mathematical article when one is looking for a physical concept would be a pain. Warren Dew (talk) 05:40, 28 June 2009 (UTC)

text: I agree with Warren Dew. I was glad that I didn't have to wade through a mathematical article when I was simply trying to define or understand a physical concept (Geodesic). I prefer this article's particular relevance to GR and or other physics phenomena. Specifically this is what I was looking for. Furthermore, looking at the discussion on this talk page, it looks as though this article can be expanded. I would prefer that this article be expanded rather than merged. Therefore, I don't agree with merging this article. Ti-30X (talk) 02:08, 22 September 2009 (UTC)
thread_title:  Merger proposal 

message: \Gamma^{\lambda}_{~\mu \nu } should be \Gamma^{\lambda}_{\mu \nu } as the Gamma symbols are not tensors.  Therefore there is no need to stagger the indices: they can't be lowered anyway.  If everyone agrees I will delete the confusing space before mu. Tkuvho (talk) 13:03, 24 February 2010 (UTC)
replies: text: A better place to raise this issue is at Christoffel symbols, which uses the same, staggered, convention.  Personally, I don't stagger the indices on the Christoffel symbols, but some authors do and I don't feel very strongly one way or the other about it.  Certainly, if a metric is available then one can lower the index on the Christoffel symbols, and this is usually known as a Christoffel symbol of the first kind: they are more computationally stable than the usual Christoffel symbols.  Also, the assertion that the "Gamma symbols are not tensors" is a bit of dogma that essentially depends on what one means by the notation: as the difference between a pair of affine connections, the Gamma symbols certainly do transform tensorially, but this is not usually what is meant by the assertion.  Anyway, both of these rejoinders are peripheral to the stylistic issue of whether the indices should be staggered.  Just on aesthetic grounds, I would have no objection to changing it to a non-staggered configuration if the change is applied consistently, provided no one makes a compelling case for having staggered indices. Sławomir Biały (talk) 16:46, 24 February 2010 (UTC)
replies: text: We should first check if this is a math vs physics thing.  If physics textbooks tend to stagger these indices there is no point stirring up a tempest in a kettle.  Tkuvho (talk) 17:04, 24 February 2010 (UTC)

text: I tracked down the original discussion four years ago, and copied it at the talk page of Gamma symbols.  You can comment there.  It seems to me that the reason mentioned there does not apply. Tkuvho (talk) 15:14, 25 February 2010 (UTC)
thread_title:  staggering the gamma symbols 

message: Obviously a lot of technically skilled people have edited this article. But that seems to have blinded them to the obvious. For the majority of readers coming to this article all they are interested is great the circle idea of a geodesic. They just need a simple sentence to inform or confirm their notion of what a geodesic is.
The don't need or want a topic sentence like this : "...In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space....". This sentence contains a raft of references to concepts and words that a general reader will be unfamiliar with. A general reader is likely to either skip the article or start on a trip through the internet trying to figure out what the heck this sentence means.
What the general reader is probably interested in is contained nicely in this paragraph: "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."
It seems to this reader that the problems of the lead section could be fixed easily by just reversing the order of the two paragraphs. The technical reader can easily skip past the opening section that he probably is well aware of and begin to try to understand the more technical uses of the word in mathematics and physics with the second paragraph of the lead section serving as a summary of the technical information to follow. — Preceding unsigned comment added by Davefoc (talk • contribs)  05:47, 15 April 2016‎ (UTC)
JRSpriggs made the edit suggested above. Thank you. Davefoc (talk) 05:01, 20 September 2016 (UTC)
thread_title:  Change the sentence order in the lead paragraph 

message: Across, much of the article "t" is used as the affine parameter along a geodesic curve. This is a rather unfortunate choice as to many readers it will suggest a relation with time, where no such connection need to exist (or even make sense in the case or Riemannian geometry). I would suggest changing it to something more "neutral" such as λ.TR 11:26, 29 September 2016 (UTC)
thread_title:  Use of t as an affine parameter 

message: The current short description is misleading in several regards. First, in a lorentzian manifold a geodesic maximizes the interval rather than minimizing it. Second, in a manifold with a connection but no metric, geodesics are defined but path length is not. Can anybody come up with wording that is more accurate but still concise?     Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:01, 4 September 2020 (UTC)
thread_title:  Need more accurate short description 

message: The article has the short title . That is correct for a manifold with a positive definite metric, but it is incorrect for a manifold with a Lorentzian metric. I've considered changing it to "Extremal path", but that's not quite right either. Your thoughts? Shmuel (Seymour J.) Metz Username:Chatul (talk) 01:10, 28 January 2021 (UTC)
replies: text: It is clear that the article as it stands is solely concerned with the notion in Riemannian geometry so the original version seems fine (up to some points of detail that should be relegated to the body of the article). I think the notion in Lorentzian geometry (or in more general settings) is very much secondary compared to the metric notion so might without harm be discussed in the articles about these notions (with perhaps a link in the "see also" section at the bottom). jraimbau (talk) 12:58, 28 January 2021 (UTC)
replies: text: That is not clear; the article has references to, e.g., General Relativity as a special case. If the article is solely concerned with the notion in Riemannian geometry then the , lede and Introduction should be modified accordingly, e.g., , remove "". Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:33, 28 January 2021 (UTC)
replies: text:  Yes. jraimbau (talk) 06:17, 29 January 2021 (UTC)
thread_title:  Short description 

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