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# Geodesics

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1. Hi

I'm trying to understand different types of geodesics. I understand that a geodesic is the shortest distance between two points on a manifold, but what's are these types:

affine
metric - I assume this is just the distance as calculated from the metric tensor
null
spacelike
timelike

Can someone explain these?
2. It's not necessarily the path of shortest distance. For example, if you have two points on a sphere, there are two (non-self-intersecting) geodesics joining them, one of which goes 'the short way round' the sphere and the other of which goes the long way round.

You're asking quite a big question though. If you're just after definitions then why not look them up?
3. Because i'm struggling to find a definition I fully understand
4. (Original post by shawrie777)
Because i'm struggling to find a definition I fully understand
What level are you working at? And what perspective are you coming from?
5. (Original post by shawrie777)
Hi

I'm trying to understand different types of geodesics. I understand that a geodesic is the shortest distance between two points on a manifold, but what's are these types:

affine
metric - I assume this is just the distance as calculated from the metric tensor
null
spacelike
timelike

Can someone explain these?
The wikipedia isn't too bad: let me summarize briefly what I learned from 5 minutes reading having previously known dirt all about (Psuedo)-Riemannian geometry:

A (metric) geodesic on a Riemannian manifold is loosely a curve that is locally the shortest distance between two points.

If you have a smooth manifold with an affine connection then one can define a geodesic of a curve in terms of parallel transport. On a Riemannian manifold, if you take the Levi-Civita connection and solve the geodesic equations then you can show that a geodesic in this sense is the same as a geodesic in the sense of locally minimising the metric.

The last three adjectives, as one would expect, refer to geodesics on a Lorentzian manifold. One may also equip a Lorentzian manifold with the Levi-Civita connection which is the unique connection satisfying the same properties as in the Riemannian case. Thus again there is no distinction between metric geodesics and affine geodesics w.r.t Levi-Civita. Then (assuming the positive part of the metric is space and the negative part is time) a geodesic is space like if the norm of its tangent vector is positive, timelike if the norm of its tangent vector is negative and null if it is zero.

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