About null and timelike geodesics

[Astronomer107]

Can you explain a little more about null and timelike geodesics (I think that's how you spell it)? I was reading Hawking and Penrose's The Nature of Space and Time, but it got a little technical. I would really like to know more about these though... thanks!

 

[Hurkyl]

To put it simply, a null geodesic is a path light can take, a timelike geodesic is a path everything else can take.

 

[Stranger]

it does not completely answers your question but helps...

http://www.mrao.cam.ac.uk/~cjld1/pages/geometryandcosmology.pdf

 

[thankqwerty]

a geodesic is a the shortest path from point to point in a specific space, i.e. straight line on a piece of paper, a curve on a sphere
timelike means future pointing.
timelike geodeise is the shortest path an object can travel from event A to event B in spacetime.

hope this is right or i'll fail my exam in 2 weeks time...

 

[selfAdjoint]

You're right, but don't forget Null Geodesics. These are the ones that only light (and massless particles) can follow. It was a null geodesic that the light followed in Eddington's 1919 observation of light shifting near the Sun. Also "shortest time" for a geodesic should be replaced with "stationary action" - but perhaps your course hasn't got to that yet.

 

[Arcon]

Originally posted by thankqwerty
a geodesic is a the shortest path from point to point in a specific space, i.e. straight line on a piece of paper, a curve on a sphere

I wrote up a web page on this a while back. Its located here
http://www.geocities.com/physics_world/ma/geodesic.htm

There are three basic ways to obtain the geodesic equation that I know of and so I posted those three derivations.

Please note that a geodesic is not defined as the shortest path from one point to another. It's a path of extremal length (where length has a meaning defined by the metric). There may be multiple paths between the same two points which are geodesics and each may have a different length.

Consider the cylinder r = R. Let the z-axis be the axis of the cylinder. consider the two points.

Point 1: r = R, theta = 0, z = 0
Point 2: r = R, theta = 0, z = b

The straight line from Point 1 to Point 2 is a geodesic. However the helix

x(t) = R cos(t) i + R sin(t) j + (b/2*pi)t k

is also a geodesic. See Figure 4 at
http://www.geocities.com/physics_world/euclid_vs_flat.htm

Notice that there are an infinite number of geodesics between those two points. You can define a helix which has one end at Point 1 and hich coils around the cylinder N times before passing through Point 2 where N is an arbitrary integer. And of course there are two helices for each end which differ only in the direction that it winds.

If you were to draw each of these curves onto the clyinder and cut the cylinder along its length then lay it out flat then each curve would be a straight line.

Think of a geodesic as the straightest possible curve.