Talk:Geodesics in general relativity

message: There is a sentence: A geodesic between two events could also be described as the curve joining those two events which has the maximum possible length.
From other sources I've heard, that the geodesic is actually a minimum possible length.
(The maximum-length curve goes arround the whole universe before getting to the target...)
Could you, please, verify that?
: In GR, there is an "invariant interval" which for timelike paths is the proper time experienced as one travels along that path.  This proper time acts as a length in relativity and is maximized for inertial motion (which is also geodesic motion in relativity).  So it is correct that geodesics in realtivity maximize the length between events, while purely spatial geodesics minimize the length between positions.  (For an example of how acceleration decreases the elapsed peoper time between events, see the twin paradox article.) --EMS | Talk 22:50, 24 October 2006 (UTC)
replies: text:  As noted in the leader, geodesics can be spacelike or timelike or null.  Timelike geodesics are particularly interesting because they are the paths that free massive particles follow, but they are not the only geodesics.  Also, I think it's good policy to try to incorporate the answers to legitimate questions (like this one) into the text of the article itself.  I'm trying to do this, and brush up a few other points.  --MOBle 03:44, 4 December 2006 (UTC)

text: Timelike curves that are locally maximal are geodesics, but I believe that the claim above that spacelike curves have local extremal length ("shortest") is incorrect. Penrose (in The Road to Reality) describes geodesics as "stationary", not "extremal". Variations to a spacelike geodesic in a timelike direction will shorten it, even locally, and variations in a spacelike direction will lengthen it.  Hence it is not locally extremal, but it is stationary in the sense that it is a saddle point in any smoothly parameterized variation.  One can also vary the geodesic in a way in which the length remains exactly constant.  —Quondum 11:33, 11 March 2014 (UTC)
thread_title:  Really maximum length? 

message: The article says
{{\rm \ddot x^i = -\sum _{j=1}^4 \sum _{k=1}^4 \ \Gamma^i_{j k} \ {\dot x^j} \ {\dot x^k} +q \ {F^{i k}} \ {\dot x^j}} \ {g_{\rm j k}}}
but shouldn't it be
{{\rm \ddot x^i = -\sum _{j=1}^4 \sum _{k=1}^4 \ \Gamma^i_{j k} \ {\dot x^j} \ {\dot x^k} -q \ {F_{i k}} \ {\dot x^j}} \ {g^{\rm j k}}} ? 
The version as it is given in the article right now gives some nasty Csc[2θ] terms in the equation for φ" in the Kerr-Newman-metric, which lead to infinite acceleration at the equator, while the shifted indicies (version below) reduces to the Coulomb-law at far distances. So it seems to me we need the covariant electromagnetic tensor multiplied with the contravariant metric tensor, not the other way round. If no objections come, I'll change that in the next few days.--Yukterez (talk) 03:47, 24 September 2017 (UTC)
replies: text: What happened to Wikipedia's requirement of wp:verifiability and wp:reliable sources? If you can find a reliable source for your version (or for the original version) then feel free to make the change, referring to the source (or leave it as it was, and add the source). Thanks! - DVdm (talk) 19:41, 24 September 2017 (UTC)

text: Note also that the version that you seem to prefer violates basic conservation of indices.TR 21:56, 24 September 2017 (UTC)
replies: text: Since I don't have a source for the version which gives the Coulomb force at large distances, nor one for the present version which seems to give the wrong output I won't change it, but as I said, the present version gives an infinite acceleration at the equator: csc(2θ)-term, while the shifted indicies give a meaningful output: φ"-solution. Since the present version follows from a cartesian Minkowski-solution, where the indicies don't matter, maybe the contravariant Maxwell tensor times the covariant metric tensor only works in flat Minkowski, while the general solution needs to be the other way around? Maybe I made some error here, but since there is no source for any of the wo possibilieties, maybe some expert can shine a light on that. I don't even see how one should invert F_μν in the case of the Reissner-Nordström metric where the inversion to F^μν would give a singular matrix... Maybe I missed he point that explains this, but whatever the point is, it doesn't seem to be in the article so other readers might miss it too. Maybe a "citation needed"-tag would fit the present version, since it isn't really clear who derived that solution. --Yukterez (talk) 02:20, 25 September 2017 (UTC)
replies: text: Speaking of the Kerr–Newman metric#Equations of motion, there seems to be an unnecessary factor of \csc ^2 \theta  in the formula for the derivative of time. It cancels out with factors of the sine which are scattered throughout the numerator. I do not know if that is related to this issue or not. JRSpriggs (talk) 02:56, 25 September 2017 (UTC)

text: I added a source for the formula. Moreover, you do realize that indices are not raised and lowered through matrix inversion, do you? If not, it is probably time to pick up a textbook on GR. TR 07:27, 25 September 2017 (UTC)
replies: text: No need for a textbook, the rule F^{\mu\nu}=g^{\mu\sigma} \ g^{\nu\kappa} \ F_{\sigma \kappa} can also be found online. With that everything becomes clear now. Nevertheless, thanks for the hint; although you comment seems rather snootily, I appreciate it.  --Yukterez (talk) 09:02, 25 September 2017 (UTC)
thread_title:  Motion of a charged particle 

message: Please see section Geodesics in general relativity#Deriving the geodesic equation via an action. Although the derivation there gives the correct result, it appears to me that it fails to be mathematically correct. Specifically, one is taking square roots of things which might be negative and dividing by things which might be zero.
The variation of a time-like path might include portions which are space-like or light-like even when shrunk down to an infinitesimal amount. Just imagine adding a multiple of the cube root of t minus tmidpoint to x in a neighborhood of the midpoint of the path. Similarly, a space-like path might gain time-like or light-like parts when varied. A light-like path is even worse since, even without variation, one cannot divide by the time derivative of the proper-time or proper-distance since they are zero.
It appears to me that there is no way to get around these problems without avoiding using the square-root altogether. But how can one talk about the interval (distance or duration) without using the integral of the square-root? JRSpriggs (talk) 20:26, 28 June 2018 (UTC)
replies: text: OK, by restricting variations to C2 functions of time, we should be able to handle the time-like geodesics and the space-like geodesics since I think that the sets of C2 time-like paths and C2 space-like paths are open sets. But, the similar derivation for light-like geodesics still appears to be invalid. JRSpriggs (talk) 04:26, 2 July 2018 (UTC)
thread_title:  Can the invalid derivation from action be fixed? 

message: At the end of the section Geodesics as curves of stationary interval the text says "If the parameter s is chosen to be affine...", but doesn't define "affine" in this context, nor does it provide a link where you can find out. -- Dr Greg  talk  16:47, 4 July 2021 (UTC)
replies: text: The full sentence is "If the parameter s is chosen to be affine, then the right side of the above equation vanishes (because U_\nu U^\nu is constant).". I think that the parenthetical expression defines "affine". In other words, the time parameter is affine if speed (magnitude of velocity) is constant. JRSpriggs (talk) 06:07, 5 July 2021 (UTC)
replies: text: Thanks. If what you say is correct, the sentence as it stands doesn't really make that clear, as it suggests the constant length is a consequence of having an affine parameter (a one-way implication) rather than a definition (a two-way implication). -- Dr Greg  talk  10:48, 5 July 2021 (UTC)
thread_title:  What is an affine parameter?