Timeline for Confusion re: geodesics, connections, and straightness
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 20, 2019 at 10:45 | comment | added | user87745 | An old question of mine is also slightly related: physics.stackexchange.com/q/318200 | |
| Sep 20, 2019 at 7:45 | comment | added | Qmechanic♦ | Related: physics.stackexchange.com/q/342821/2451 and links therein. | |
| Sep 20, 2019 at 7:44 | history | edited | Qmechanic♦ |
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| Sep 20, 2019 at 7:34 | answer | added | TimRias | timeline score: 1 | |
| Sep 20, 2019 at 2:48 | answer | added | J. Murray | timeline score: 1 | |
| Sep 20, 2019 at 1:49 | comment | added | nothingIsMere | I'm familiar with the connection and not confused about it in particular. My question is specifically how the notion of 'straightest possible curve' can be formalized in terms of metrics, connections, etc. I think my question boils down to whether there 'straightness' can be defined apart from a connection. Perhaps using the variational approach with a metric is the way to do that? I just wonder how there can be a 'straightest possible curve' if 'straightness' is ultimately arbitrary. | |
| Sep 20, 2019 at 1:27 | comment | added | Cinaed Simson | Start with $3$ dimensional Euclidean space - or classical differential geometry. In particular the Frenet–Serret formulas - which calculates the frame in terms of the derivatives of frame's vector fields. In essence, the connection is a generalization of this technique, i.e., calculating the derivative of the basis vectors. | |
| Sep 20, 2019 at 1:14 | history | edited | Qmechanic♦ |
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| Sep 20, 2019 at 0:58 | comment | added | nothingIsMere | This is the variational approach to defining geodesic trajectories, yes? I'm aware of it but haven't studied with care. Does that approach require a metric? I am trying to steer away from definitions of these things that require a metric for now. Maybe that is part of my confusion, i.e. that a great circle is the straightest possible trajectory only when we judge straightness using a connection derived from a particular metric? | |
| Sep 20, 2019 at 0:29 | comment | added | G. Smith | Your understanding that a geodesic parallel-transports its tangent vector is correct. As far as I’m concerned, “straightest possible” is a vague way of saying this when talking to people who have no idea what a tangent vector or parallel transport are. Do you know about the “stationary length” definition? | |
| Sep 20, 2019 at 0:14 | history | asked | nothingIsMere | CC BY-SA 4.0 |