Geodesics between two points ?

[smallphi]

How many geodesics (up to affine reparametrization) connect two arbitrary points in arbitrary spacetime ?

On the 2D sphere for example, you have two geodesics which are the parts of a the great circle connecting the two points but circulating in opposite directions. I think one of them minimizes the distance between the two points, the other maximizes it.

 

[HallsofIvy]

Taking the surface of a sphere in 3d as an example, there exist an infinite number of geodesics connecting the "north pole" and the "south pole".

 

[smallphi]

I wonder if the usual signature of spacetime (-, +, +, +) restricts in some way the possible number of geodesics ?

The sphere is a good example but doesn't have such signature.

 

[smallphi]

I think the sphere example works even in Schwartzchild spacetime due to its spherical symmetry:

One can take two antipodal spatial points in Sch. metric (points with the same r, diametrically opposite across the black hole) but with different coordinate times. If there is one geodesic connecting them, then there are infinite images of that geodesic obtained by rotating it in 'space' around the spatial axis of the two points.

It seems like the more symmetry the spacetime has (the more Killing vectors), the more geodesics.

 

[Stingray]

In general, there's no limit to the number of geodesics connecting two points. There is, however, a theorem proving that there exists a neighborhood around every point for which geodesics are unique. That neighborhood is the entire manifold in Minkowski spacetime.

 

[Chris Hillman]

One easy way to see that there are potentially infinitely many null geodesic paths between two events in curved Lorentzian manifolds is to study the Einstein ring configuration as treated in weak-field gravitational lensing.

 

[pervect]

If you want some realistic examples of multiple geodesics connecting two events in space-time, you need only consider orbits.

How many different orbits have the same period around a central body? An infinite number - as long as the semi-major axis is the same, two bodies will follow the same orbit and "meet up again" at the same event (in the Newtonian approximation, anyway).

Pericentron effects (like the shifting of the perihelion of mercury) will complicate this simple analysis, but I don't believe it changes the result. I'm not positive, because I haven't worked it out in sufficient detail.

 

[smallphi]

Ok I think we resolved the original question.

Here is a little twist: It it always possible to find at least one geodesic that connects two spacetime points? I don't mean necessarily a timelike geodesic.

 

[pervect]

It probably depends on the topology. For instance, if you take a plane (R^2), you have a single straight line between any two points. Now remove a point at the origin - any pair of points that has a straight line that originally passed through the origin won't have a geodesic connecting them after you remove the origin.

 

[smallphi]

Can such a 'defect' in the manifold (the removed point) be 'felt' by calculating something within the manifold instead of embedding in a bigger manifold (the full plane) and seeing that there is a hole.

 

[robphy]

Possibly interesting reading:
http://philsci-archive.pitt.edu/archive/00002354/
Observational Indistinguishability and Geodesic Incompleteness (Manchak)

http://www.lps.uci.edu/home/fac-staff/faculty/malament/papers/ObservIndisting Spacetime.pdf
Observationally Indistinguishable Spacetimes (Malament)

 

[pmb_phy]

I wonder if the usual signature of spacetime (-, +, +, +) restricts in some way the possible number of geodesics ?

The sphere is a good example but doesn't have such signature.

One need not have a metric defined in order to define a geodesic. If there exists an affine connection (which does not require a metric to be defined at all) on the manifold then geodesics become possible.

Pete