Number of Geodesics Between Two Points: Topology vs. Homeomorphism

[Pencilvester]

In general, is the number of distinct geodesics between two fixed points purely a feature of the topology of the manifold? i.e. Can there ever exist two topologically equivalent (I can’t remember the proper word right now- is homeomorphic the right one?) manifolds, $\mathcal M$ and $\mathcal N$, related by a bijective mapping, $\phi : \mathcal M \to \mathcal N$, in which there exists a pair of points in $\mathcal M$, $\mathbf a$ and $\mathbf b$ where the number of geodesics between $\mathbf a$ and $\mathbf b$ differs from the number of geodesics between $\phi (\mathbf a )$ and $\phi (\mathbf b )$?

 

[PeterDonis]

Can there ever exist two topologically equivalent (I can’t remember the proper word right now- is homeomorphic the right one?) manifolds, $\mathcal M$ and $\mathcal N$, related by a bijective mapping, $\phi : \mathcal M \to \mathcal N$, in which there exists a pair of points in $\mathcal M$, $\mathbf a$ and $\mathbf b$ where the number of geodesics between $\mathbf a$ and $\mathbf b$ differs from the number of geodesics between $\phi (\mathbf a )$ and $\phi (\mathbf b )$?

Yes. Any curved manifold with topology $R^4$ will, in general, have more than one geodesic between at least some pairs of points; but flat Minkowski spacetime, which has topology $R^4$, has one and only one geodesic between any pair of points. An example of a curved manifold with topology $R^4$ and (AFAIK) multiple geodesics between some points is Godel spacetime.

A more pedestrian example would be any curved spacetime describing an isolated gravitating mass like a planet or star (but not a black hole, since we want the topology to be $R^4$); in general, given any pair of events on a circular geodesic orbit around the mass, there will be another geodesic between the same pair of events that describes the trajectory of an object thrown upward from the first event with just the right initial velocity to fall back down and cross the orbit again at the second event. Such a spacetime will not be describable by a single solution of the EFE, since the stress-energy tensor is not the same everywhere; but it can be described by joining a Schwarzschild solution describing the vacuum region outside the mass with an appropriate solution describing the interior of the mass. The joining can be continuous, so that the manifold as a whole is still a valid manifold with topology $R^4$.

 

[Pencilvester]

Yes.

Thanks! What is the loosest word in math vocab that would imply that all corresponding pairs of points between two manifolds have equal number of distinct geodesics? By “loosest” I mean the least limiting, e.g. whatever the word is that implies both manifolds are equivalent in every meaningful way (topologically, geometrically, etc.) would be the most limiting.

 

[PeterDonis]

What is the loosest word in math vocab that would imply that all corresponding pairs of points between two manifolds have equal number of distinct geodesics?

As far as I know the manifolds must have the same geometry (i.e., same topology plus same curvature at every point) for this to be true. But I am not an expert in this area.

 

[Pencilvester]

As far as I know the manifolds must have the same geometry.

Ok, thanks!

 

[pervect]

This is sort of backwards to the original question, but around a point on a manifold, there is always a neighborhood, called the local convex neighborhood ((IIRC)), in which there is only one geodesic connecting two points in said neighborhood.

 

[Pencilvester]

This is sort of backwards to the original question, but around a point on a manifold, there is always a neighborhood, called the local convex neighborhood ((IIRC)), in which there is only one geodesic connecting two points in said neighborhood.

I guess this goes hand-in-hand with saying that a curved spacetime locally looks Lorentzian?

 

[PAllen]

I guess this goes hand-in-hand with saying that a curved spacetime locally looks Lorentzian?

Yes, but I think it is a nontrivial result, and the region can be quite large. Interestingly, this fact is the basis of the two point World Function introduced by J. L. Synge. This function gives the metric interval along the connecting geodesic between pairs of points, and its domain is effectively all pairs of points in each other’s local convex neighborhood.

 

[Mentz114]

I guess this goes hand-in-hand with saying that a curved spacetime locally looks Lorentzian?

Yes. I think this is true for a region around a point on a geodesic worldline. Here one can always find a region that is Lorentzian in the limit of zero extension. For instance Fermi normal coordinates. Generally proper accelertion cannot be transformed away.

 

[PeterDonis]

I guess this goes hand-in-hand with saying that a curved spacetime locally looks Lorentzian?

The local convex neighborhood can actually be significantly larger than a single patch in which curvature is negligible. For example, consider the case I described earlier, with a geodesic circular orbit around a planet and a second geodesic that is launched radially upward at one event on the orbit, with just the right velocity to come back down one orbit later. For this case, the local convex neighborhood of, say, the first event extends, heuristically, at least halfway around the orbit (note that the orbit, in spacetime, is a helix, not a circle), which is well beyond the patch of spacetime centered on the first event in which spacetime curvature (tidal gravity) is negligible; there is no local Lorentz frame that covers half of the orbit.

 

[Pencilvester]

For this case, the local convex neighborhood of, say, the first event extends, heuristically, at least halfway around the orbit

My intuition is telling me it should extend up to halfway around the orbit because if you include the halfway point, wouldn’t there be infinite radially symmetric, but distinct geodesics connecting those two points?

 

[Pencilvester]

For this case, the local convex neighborhood of, say, the first event extends, heuristically, at least halfway around the orbit

My intuition is telling me it should extend up to halfway around the orbit because if you include the halfway point, wouldn’t there be infinite radially symmetric, but distinct geodesics connecting those two points?

I mean, in either case, your point is valid and noted.

 

[PeterDonis]

if you include the halfway point, wouldn’t there be infinite radially symmetric, but distinct geodesics connecting those two points?

Not infinitely many, but if we imagine a tunnel through the center of the planet, then yes, you're right, there would be a radial geodesic connecting an event on the orbit with a second event halfway around the orbit; at least to a first approximation, it would take the same time to travel through the radial tunnel as it would to go halfway around the orbit.

 

[George Jones]

Yes. Any curved manifold with topology $R^4$ will, in general, have more than one geodesic between at least some pairs of points

I haven't tried to work it out, but isn't a flat FRLW universe a counterexample to this?

 

[PeterDonis]

isn't a flat FRLW universe a counterexample to this?

Hm. I think you're right. It might be that the condition is actually more strict than the one I gave: possibly "a manifold with topology $R^4$ that has nonzero Weyl curvature" (since the flat FLRW spacetime's curvature is all Ricci) would be required to ensure that there are pairs of points with more than one geodesic between them. Or I might be misremembering; I thought I had seen a theorem to this effect somewhere, but I could be wrong.

 

[Pencilvester]

Not infinitely many, but if we imagine a tunnel through the center of the planet, then yes, you're right, there would be a radial geodesic connecting an event on the orbit with a second event halfway around the orbit

What about the other orbital geodesics that look exactly like the first one, except that they start with a different direction? e.g. Let’s say the 3-velocity vector of the particle at the initial point A lies in the tangent plane to the 2-sphere that includes point A and is centered on the massive body’s center of mass. Can’t we take that initial velocity vector and point it in any direction in T_A and have the resulting geodesic connect A to the fixed event B on the opposite side of the massive body? Wouldn’t each set of initial velocity vectors that have equal length and equal inclination above (or declination below) T_A create geodesics that connect A to B? (Of course B would be different for each set)

 

[PeterDonis]

What about the other orbital geodesics that look exactly like the first one, except that they start with a different direction?

Yes, there will be an infinite number of orbital geodesics that all pass through the same event on the other side of the planet (since there are an infinite number of possible orbital planes). But there won't be an infinite number of radial geodesics connecting those two events; there will only be one.

 

[Pencilvester]

Yes, there will be an infinite number of orbital geodesics that all pass through the same event on the other side of the planet (since there are an infinite number of possible orbital planes). But there won't be an infinite number of radial geodesics connecting those two events; there will only be one.

Gotcha. I think I said “radially symmetric,” but I wouldn’t have even considered the radial geodesic if you hadn’t mentioned it.