Timeline for Primary definition of a geodesic
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2017 at 16:23 | comment | added | Sebastian Goette | In the "géométricon", one of the adventures of Anselme Lanturlu by Jean-Pierre Petit, there is indeed a "handwaving description" of geodesics by autoparallelism. At least on surfaces, this works well: just try to follow a geodesic by gluing a very long piece of sticky tape on your surface. | |
| May 27, 2017 at 19:09 | comment | added | Deane Yang | Perhaps "primary" is in the eye of the beholder. I think for a geometer, "local minimizing" is the fundamental motivation. However, for a physicist, who is more used to working with non-positive-definite Hamiltonians, "stationary point of an energy functional" is perhaps the fundamental concept. And, given their past experience, the fact that this leads to constant acceleration curves is not a surprise. | |
| May 27, 2017 at 13:05 | comment | added | Willie Wong | @VítTuček I don't think EIH does both at the same time (the curve is fixed). In the context of (pseudo)Riemannian geometry, all three descriptions of the geodesics are transformable to each other with a few steps, so I don't really know whether anything is new. // On the other hand, in spite of my short interpretation above, the formulation is not as simple as "first variation of the length functional with respect to local diffeomorphisms"; there are a few subtleties involved to get an if and only if statement. | |
| May 27, 2017 at 7:30 | comment | added | Vít Tuček | @WillieWong That sounds really interesting. Is something new gained when considering both variations at the same time? | |
| May 27, 2017 at 2:27 | comment | added | Willie Wong | @JosephO'Rourke: Sternberg's write up is much clearer than EIH's original. The characterization is indeed general. The characterization can be described as what happens if in the "calculus of variations" description of geodesic, instead of varying the curve by actually moving the curve, you vary the curve by "moving the manifold" instead (in the sense of one parameter family of local diffeomorphisms generated by a vector field of compact support). | |
| May 27, 2017 at 2:02 | comment | added | Joseph O'Rourke | @WillieWong: You are well ahead of me on the EIH paper. This is far from my expertise, and I defer to your recollections. | |
| May 27, 2017 at 2:00 | comment | added | Willie Wong | @JosephO'Rourke Is that geodesics in general or time like geodesics in a Lorentzian manifold under assumptions? The EIH paper if I remember right is attacking the problem of whether a "test particle travels along a time-like geodesic"; the problem in GR being that the theory is nonlinear so you cannot just "add" test particles. | |
| May 27, 2017 at 1:58 | comment | added | Joseph O'Rourke | I did not emphasize the historical dimension in my question, so I appreciate that you did. | |
| May 27, 2017 at 1:44 | comment | added | Joseph O'Rourke | The alternative characterization involves a "nowhere vanishing symmetric tensor field" along the curve $\gamma$, with an associated linear function on the tensor that satisfies a particular zero-equation. I do not understand this aspect except that it is (according to Sternberg) a characterization of geodesics. | |
| May 27, 2017 at 1:32 | history | answered | Rodrigo A. Pérez | CC BY-SA 3.0 |