Just one sec...
Hey! Sign in to get help with your study questionsNew here? Join for free to post

Geodesics

Announcements Posted on
    • Thread Starter
    Offline

    ReputationRep:
    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?
    Offline

    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?
    • Thread Starter
    Offline

    ReputationRep:
    Because i'm struggling to find a definition I fully understand
    Offline

    (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?
    Offline

    ReputationRep:
    (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.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. Oops, you need to agree to our Ts&Cs to register
  2. Slide to join now Processing…

Updated: April 17, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

Poll
How do you sleep?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read here first

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups
Study resources

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22

Registered Office: International House, Queens Road, Brighton, BN1 3XE

Quick reply
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.