Timeline for Affine and metric geodesics
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 15, 2021 at 1:34 | comment | added | Eletie | @Mozinurlah Yes, history I'm familiar with (although not the specific dialogue) but thanks for sharing. | |
| Mar 15, 2021 at 1:07 | comment | added | Mozibur Ullah | @Eletie: ... and that those constructions were very natural once you understood the geometry. | |
| Mar 15, 2021 at 1:07 | comment | added | Mozibur Ullah | @Eletie: I updated my answer because it wasn't a full answer - that was prompted by your remark that it wasn't a full answer. It wasn't updated with your comments, it was independently done. The mathematical definitions and theory are important as they were for QFT theorists when they realised in 70's that their notion of a gauge theory was exactly that of connections on a fibre bundle independently discovered by geometers. In fact, Yang told Chern then that he found it remarkable that geometers had independtly come up with the same construction and he replied that it wasn't remarkable ... | |
| Mar 15, 2021 at 0:48 | comment | added | Eletie | Anyway, I'm glad to see you updated your answer with the comments I made, although I'm not sure repeating mathematical definitions are overly helpful as an answer. | |
| Mar 15, 2021 at 0:35 | comment | added | Eletie | Also we're dealing with GR and a lorentzian manifold, it's strange to try and change the setting once again | |
| Mar 15, 2021 at 0:34 | comment | added | Mozibur Ullah | @Eletie: Like the Palantini action instead of the Hilbert-Einstein action where only the metric is used. I know. | |
| Mar 15, 2021 at 0:33 | comment | added | Eletie | @MozinburUllah in metric-affine theories we have both an independent metric and connection. Please see the cited papers of my answer if you're unfamiliar with this, as it's quite a well known topic in these theories of gravitation (as is the question of which geodesic equation should be seen as physical) | |
| Mar 15, 2021 at 0:33 | comment | added | Mozibur Ullah | @Eletie: You can delete what ever you like. There's no clutter in my answer. | |
| Mar 15, 2021 at 0:30 | comment | added | Mozibur Ullah | @Eletie: Thats not correct. An affine connection on a vector bundle doesn't imply the existance of a metric so there isn't a way of defining a geodesic by extremising the distance. You're thinking of a general affine connection on a Lorentzian manifold and which isn't metric. | |
| Mar 15, 2021 at 0:27 | comment | added | Eletie | It's not about the differing connections that give rise to different geodesic equations, that'd be trivial true (well not trivial because connections that differ by torsion give the same autoparallel equation). It's that you always have a 'metric' one from extremising the action, and whether this fact itself has any physical reason why one should choose the Levi-Civita connection. Does this make sense? | |
| Mar 15, 2021 at 0:24 | comment | added | Eletie | To be even more clear, even if one picks an arbitrary affine connection, the extremising of (2) still gives rise to the Levi-Civita type geodesic equation. For this reason different authors have disagreed a lot in the past in the context of metric-affine gravity about which geodesic equation should properly describe a particles inertial motion (e.g. see Heyl). Hope this helps! | |
| Mar 15, 2021 at 0:22 | comment | added | Mozibur Ullah | @Eletie: I've said that geodesics for general affine connections don't coincide for those of a metric connection as they're different connections. What exactly is your point? | |
| Mar 15, 2021 at 0:19 | comment | added | Eletie | OP's question is about the autoparallel curves of a general affine connection not coinciding with the geodesic equation obtained from extremising the action (2). I.e. two types of "geodesics" which go by differing names in the literature but are distinct for a general affine connection. I see you've edited your answer, but unfortunately this is still missing the point of the question and just giving definitions that OP appears to already understand. See my answer for an explanation of these differing concepts. | |
| Mar 15, 2021 at 0:18 | comment | added | Mozibur Ullah | @Eletie: I know that, which you can see from my updated answer. Like I said, I prefer the term affine connection for connections on affine bundles and vector connections for connections on vector bundles - it causes less confusion. | |
| Mar 15, 2021 at 0:08 | history | edited | Mozibur Ullah | CC BY-SA 4.0 | added 1312 characters in body |
| Mar 14, 2021 at 23:53 | comment | added | Mozibur Ullah | @Eletie: No, I'm not. The affine connection is also referred to as the linear connection which is a general connection on a vector bundle with a linear connector. There are also connections on affine bundles which can also be termed affine connections. The affine connection that you're thinkingbof is probably the canonical Levi-Civita connection on a pseudo-Riemannian manifold. | |
| Mar 14, 2021 at 19:35 | comment | added | Eletie | This doesn't address the questions OP has asked. You're probably confusing affine geodesics with the affine connection. | |
| Mar 14, 2021 at 15:53 | history | answered | Mozibur Ullah | CC BY-SA 4.0 |