Stationary/extremal L for timelike/null/spacelike geodesics

[bcrowell]

I would be interested in knowing if others think I have the correct analysis of whether length is stationary and/or extremal in the cases of geodesics that are timelike, null, and spacelike.

Timelike

In Minkowski space, the proper time $\tau=\int \sqrt{g_{ij}dx^i dx^j}$ (+--- metric) is both maximized and stationary. Apparently this need not hold in some cases in GR, if there are multiple geodesics connecting two events; WP references MTW pp 316-319 on this, but I don't have my copy handy. Is there a simple example?

Spacelike

In Minkowski space, with n+1 dimensions for $n\ge 2$, the proper length $\sigma=\int \sqrt{-g_{ij}dx^i dx^j}$ is stationary and is a saddle point. To see that it's a saddle point, pick a frame in which the two events are simultaneous and lie on the x axis. Deforming the geodesic in the xy plane does what we expect according to Euclidean geometry: it increases the length. Deforming the geodesic in the xt plane, however, reduces the length (as becomes obvious when you consider the case of a large deformation that turns the geodesic into a curve of length zero, consisting of two null line segments).

I don't know how much this has to be weakened for GR.

Null

This was discussed here last year: https://www.physicsforums.com/threads/null-geodesic-definition-by-extremisation.768196/ . I agree with some of the posts and disagree with others, and there didn't seem to be a consensus reached at the end.

In this case you have to define what "length" is. You can either take an absolute value, $L=\int \sqrt{|g_{ij}dx^i dx^j|}$, or not, $L=\int \sqrt{g_{ij}dx^i dx^j}$.

If you don't take the absolute value, L need not be real for small variations of the curve, and therefore you don't have a well-defined ordering, and can't say whether L is a max or min or neither.

Regardless of whether you take the absolute value, L doesn't have differentiable behavior for small variations around a null geodesic, so you can't say whether it's stationary; see https://www.physicsforums.com/threads/null-geodesic-definition-by-extremisation.768196/#post-4844366 , which seems to verify this aspect of my analysis.

If you do take the absolute value, then for the geodesic curve, the length is zero, which is the shortest possible. However, one can have nongeodesic curves of zero length, such as a lightlike helical curve about the t axis.

 

[PeterDonis]

Apparently this need not hold in some cases in GR, if there are multiple geodesics connecting two events; WP references MTW pp 316-319 on this, but I don't have my copy handy. Is there a simple example?

Sure, just compare a radial geodesic in Schwarzschild spacetime that starts upward from some $r$, rises, then falls back to the same $r$, with a circular orbit geodesic at that same $r$. All you need is to adjust the initial upward velocity of the radial geodesic so that it intersects the circular orbit geodesic again after one orbit. Then the radial geodesic will have a larger elapsed proper time than the circular orbit one between the same two events; so the circular orbit geodesic is not a global maximum of proper time between those events (though it is a local extremum).

 

[bcrowell]

Sure, just compare a radial geodesic in Schwarzschild spacetime that starts upward from some $r$, rises, then falls back to the same $r$, with a circular orbit geodesic at that same $r$. All you need is to adjust the initial upward velocity of the radial geodesic so that it intersects the circular orbit geodesic again after one orbit. Then the radial geodesic will have a larger elapsed proper time than the circular orbit one between the same two events; so the circular orbit geodesic is not a global maximum of proper time between those events (though it is a local extremum).

Hmm...thanks, that's a helpful example. Well, it's obvious that in such a case the two geodesics can't both be the global extremum, since their proper times are unequal. Maybe it will clarify things if I give the text from the WP article:

And for some geodesics in such instances, it's possible for a curve that connects the two events and is nearby to the geodesic to have either a longer or a shorter proper time than the geodesic.

I think what they're saying is that such a geodesic may not even be a local extremum.

It's not obvious to me how to tweak your example to prove this. The two orbits will in general have different periods, so to get them to coincide at a later event in spacetime, we'd probably have to choose a case in which one of them did m orbits while the other did n orbits. But then it's not a family of geodesics that's a function of a continuous parameter.

 

[bcrowell]

Here's an example that is not a local extremum but may be stationary. In a Schwarzschild spacetime, you launch a projectile from a spatial point P and it ends up at P's antipodal spatial point Q. The azimuthal angle $\phi$ about the line PQ was arbitrary, so we have a family of geodesics that are a function of one continuous parameter and that all take the same amount of time to get from P to Q, and by symmetry all of them have the same proper time.

But that still doesn't seem to be an example quite as strong as what the WP article suggests.

 

[PeterDonis]

I think what they're saying is that such a geodesic may not even be a local extremum.

If "extremum" is interpreted so as not to include "saddle point", then any null geodesic would qualify, since a saddle point is a stationary point but not an extremum.

It's not obvious to me how to tweak your example to prove this.

I don't see how one could.

In a Schwarzschild spacetime, you launch a projectile from a spatial point P and it ends up at P's antipodal spatial point Q. The azimuthal angle $\phi$ about the line PQ was arbitrary

I'm not sure I understand. If the spatial points P and Q are both fixed, and the time it takes the projectile to travel between them is fixed, doesn't that fix a unique geodesic? It doesn't seem like there are any free parameters left.

 

[bcrowell]

If "extremum" is interpreted so as not to include "saddle point", then any null geodesic would qualify, since a saddle point is a stationary point but not an extremum.

They're talking about timelike geodesics. (And as described in my #1, I don't think a null geodesic is stationary.)

I'm not sure I understand. If the spatial points P and Q are both fixed, and the time it takes the projectile to travel between them is fixed, doesn't that fix a unique geodesic? It doesn't seem like there are any free parameters left.

Let the Schwarzschild spatial coordinates be $(r,\theta,\phi)$. Spatial point P is at $r=r_0$, $\theta=0$. Q is at $r=r_0$, $\theta=\pi$. Pick some geodesic from P to Q that lies in the $\phi=0$ half-plane. In any other half-plane defined by another choice of $\phi$, there is a geodesic from P to Q that takes the same time.

 

[PeterDonis]

as described in my #1, I don't think a null geodesic is stationary

I read through the discussion you linked to, which does raise an issue I don't have an immediate answer to; I'll have to check further into this when I get a chance.

In any other half-plane defined by another choice of $\phi$, there is a geodesic from P to Q that takes the same time.

Ah, ok got it--because of the rotational symmetry there will be a one-parameter family of geodesics connecting the same two events. The same would be true for the circular orbit geodesics I described in my earlier example.

I'm not sure that these geodesics (either "half-circle" or full circle) would be an example of what's mentioned in the Wikipedia article, because I don't think there are any "nearby" geodesics to any of these that connect the same two events and have a longer proper time. The article references MTW, so I'll have to check it when I get a chance to see what example they had in mind.

 

[bcrowell]

I'm not sure that these geodesics (either "half-circle" or full circle) would be an example of what's mentioned in the Wikipedia article, because I don't think there are any "nearby" geodesics to any of these that connect the same two events and have a longer proper time.

Right, I agree.

Here's a possible way to modify it so it does make a full example of what they're describing. We have some rotating body, and the surrounding spacetime is described by a Kerr metric. The axis of rotation is $\theta=\{0,\pi\}$. Let spatial point P be at $r=r_0$, $\theta=\pi/2$, $\phi=0$, and let Q be similar but with $\phi=\pi$. If the body's rotation was zero, then we would have a family of semicircular trajectories from P to Q that would all have equal proper times. If we turn on the rotation, each of these orbits can be adjusted within its own plane so as to cause all of them to arrive at the same time, but now the proper times will all be different.

Maybe MTW have a simpler example or one that's more obviously correct than mine.

 

[PAllen]

What I've seen claimed (e.g. in Synge's GR textbook) is that for any timelike geodesic, if you pick points on it sufficiently close to each other, the given geodesic between them is unique (and maximal, both locally and globally). Further, in any non-singular region, a sufficiently small ball around a given event will have unique timelike geodesics between the given event any any other events in the ball that can be connected by timelike curves. These features are used to make his 'Two Point World Function' well defined, which he uses heavily to do geometric series expansions without coordinates. However, for arbitrary events in arbitrary spacetimes that can be connected by a timelike world line, there is no guarantee that the geodesic bertween them is maximal, even locally.

This seems consistent with consistent with the analysis above.

 

[PeterDonis]

If we turn on the rotation, each of these orbits can be adjusted within its own plane so as to cause all of them to arrive at the same time

I'm probably being dense, but I don't see how this adjustment can be made. It seems to me that, if the radial coordinate of the orbit is constant, and we pick a particular plane (value of $\theta$), then the orbit is fully determined. In Kerr spacetime, as you say, the proper time as a function of $\theta$ will not be constant; but I don't see that the coordinate time can be made to be constant either.

 

[PAllen]

It seems there is a much simpler example of case a timelike geodesic not being a local maximum. Consider one complete non-circular (not quiet closed) orbit in SC spacetime. Forget about the near radial geodesic that happens to be the global maximum in this case. Consider local deformations of the orbit (non-inertial) such that they (a) go a bit faster at aphelion and slower at peri-helion and (b) the reverse [both (a) and (b) deformations adjusted to meet the geodesic after one 'orbit']. While I haven't calculated it explicitly, it seems clear that (a) deformations may elapse less proper time than the geodesic, while (b) deformations may elapse more. Thus, the geoesic is a saddle point.

 

[bcrowell]

I'm probably being dense, but I don't see how this adjustment can be made. It seems to me that, if the radial coordinate of the orbit is constant, and we pick a particular plane (value of $\theta$), then the orbit is fully determined. In Kerr spacetime, as you say, the proper time as a function of $\theta$ will not be constant; but I don't see that the coordinate time can be made to be constant either.

I'm talking about adjusting the orbit so it becomes non-circular. If you aim low and shoot at high speed, you can get to the other side faster. If you aim high and shoot at low speed, you can get there after more time.

 

[bcrowell]

It seems there is a much simpler example of case a timelike geodesic not being a local maximum. Consider one complete non-circular (not quiet closed) orbit in SC spacetime. Forget about the near radial geodesic that happens to be the global maximum in this case. Consider local deformations of the orbit (non-inertial) such that they (a) go a bit faster at aphelion and slower at peri-helion and (b) the reverse [both (a) and (b) deformations adjusted to meet the geodesic after one 'orbit']. While I haven't calculated it explicitly, it seems clear that (a) deformations may elapse less proper time than the geodesic, while (b) deformations may elapse more. Thus, the geoesic is a saddle point.

What is SC spacetime? Does SC mean Schwarzschild?

I'm not convinced at all that this example works. We do normally expect timelike geodesics to be local maxima of the proper time. Something special has to be going on for this not to be the case, and from the WP article, it sounds like the something special is for "a pair of widely-separated events to have more than one time-like geodesic that connects them." Since you haven't invoked any such property in your example, it seems unlikely to me that it works out.

 

[PeterDonis]

I'm talking about adjusting the orbit so it becomes non-circular.

Non-circular, but still passing through two antipodal points at the same radial coordinate? (In that case, the "orbits" would not be geodesic.)

 

[Orodruin]

think what they're saying is that such a geodesic may not even be a local extremum.

I do not have a direct example in the case of an indefinite metric, but the standard example for a Riemannian metric would be the longer great circle between two points on a sphere, which is a local saddle point.

We do normally expect timelike geodesics to be local maxima of the proper time.

This depends on what you mean by "local". If you with "local" mean "proper time between two nearby points", then yes. If you mean "local" in the space of all world-lines, then no. The issue is the same as with the great circle above, the great circle minimises the distance between nearby points, but in the space of curves on the sphere, it is a saddle point.

 

[bcrowell]

Non-circular, but still passing through two antipodal points at the same radial coordinate? (In that case, the "orbits" would not be geodesic.)

What makes you think that? You have two degrees of freedom to play with on the initial conditions, so I don't see why it should be a problem to hit a target.

 

[bcrowell]

This depends on what you mean by "local". If you with "local" mean "proper time between two nearby points", then yes. If you mean "local" in the space of all world-lines, then no. The issue is the same as with the great circle above, the great circle minimises the distance between nearby points, but in the space of curves on the sphere, it is a saddle point.

Local as in the calculus of variations. Local means that you can change the curve, but you can't move any point on the curve by more than $\epsilon$ (as measured, for example, by coordinate changes). We then expect that the limit of small $\epsilon$, the proper time will change by $O(\epsilon^2)$.

 

[PeterDonis]

What makes you think that?

Because I don't think a geodesic that has a varying radial coordinate can have two antipodal points where the radial coordinate is the same--unless perhaps it is so close to the central body that its perihelion precesses by $\pi$ on every orbit.

 

[PAllen]

What is SC spacetime? Does SC mean Schwarzschild?

I'm not convinced at all that this example works. We do normally expect timelike geodesics to be local maxima of the proper time. Something special has to be going on for this not to be the case, and from the WP article, it sounds like the something special is for "a pair of widely-separated events to have more than one time-like geodesic that connects them." Since you haven't invoked any such property in your example, it seems unlikely to me that it works out.

What's special is that the given geodesic is not a global maximum, by construction. We have an elliptical orbit, not quite circular (and not elliptical in that it doesn't quite close due to perihelion shift). We know the nearly radial geodesic (as in Peter's example, which I've used many times as well) is the global geodesic. MTW makes the statement that whenever more than one timelike geodesic connects two events, then a 'typical one' is 'usually' a saddle point.

However, I agree the disagreement can't be resolved without a calculation or more formal mathematical argument. Otherwise, we just have my intuition versus yours.

 

[bcrowell]

Because I don't think a geodesic that has a varying radial coordinate can have two antipodal points where the radial coordinate is the same--unless perhaps it is so close to the central body that its perihelion precesses by $\pi$ on every orbit.

I don't see why you would think that. Say I position myself at P and raise my cannon to 3 degrees above the horizontal. If my muzzle velocity $v$ is too low, my cannonball will pass under the antipode Q. If too high, it passes over. By the intermediate value theorem, there is guaranteed to be some value of $v$ for which it hits Q.

 

[bcrowell]

MTW makes the statement that whenever more than one timelike geodesic connects two events, then a 'typical one' is 'usually' a saddle point.

Do you have MTW in front of you? Maybe we could save ourselves a lot of grief by seeing exactly what it says. My copy is at work.

 

[PAllen]

MTW page 316 says:

"When several distinct geodesics connect two events, the typical one is not a local maximum ("mountain peak"), but a saddle point ("mountain pass") ..."

However, they don't use my example as an illustration, so unless one of us calculates it, we won't know for sure. [And so far, I'm not seeing a reliable order of magnitude type calculation because you have competing effects, and the ability to try to adjust them at different points in the orbit.]

 

[Orodruin]

Local as in the calculus of variations. Local means that you can change the curve, but you can't move any point on the curve by more than $\epsilon$ (as measured, for example, by coordinate changes). We then expect that the limit of small $\epsilon$, the proper time will change by $O(\epsilon^2)$.

But this was my point exactly, in this sense geodesics in Riemannian manifolds are not local minima, so we should not be surprised if (time-like) geodesics in Lorentzian manifolds are not local maxima. You do not need to restrict yourself to geodesics to consider this. In fact, for the Riemannian example I gave, you get a shorter path by deforming the geodesic to a non-geodesic.

 

[PeterDonis]

I don't see why you would think that.

Ah, I see where my mistake was; I was assuming that the given radius $r_0$ was the radius of the orbit's perihelion. You are describing orbits where it isn't.

 

[PAllen]

I think my orbit example is too simple, but not because you need any exotic metric (The MTW example that is more complex uses only weak quasi-Newtonian gravity). My counter-argument to my example is that the same argument could apply to any section of an elliptical orbit. Yet we know that a small section must be a global maximum. What I think might work is two events far enough apart in time that there are both a single revolution a double revolution 'orbit' connecting them (as well as a radial trajectory that will be the global maximum). Then, I believe the double revolution orbit will be a saddle point where you exploit both small radial as well as speed deformations from the double orbit geodesic, and that such local deformation need not include the exact 'first time around' event.

 

[George Jones]

But this was my point exactly, in this sense geodesics in Riemannian manifolds are not local minima, so we should not be surprised if (time-like) geodesics in Lorentzian manifolds are not local maxima.

Yes, the same concept is involved, conjugate points.

 

[PAllen]

I think my orbit example is too simple, but not because you need any exotic metric (The MTW example that is more complex uses only weak quasi-Newtonian gravity). My counter-argument to my example is that the same argument could apply to any section of an elliptical orbit. Yet we know that a small section must be a global maximum. What I think might work is two events far enough apart in time that there are both a single revolution a double revolution 'orbit' connecting them (as well as a radial trajectory that will be the global maximum). Then, I believe the double revolution orbit will be a saddle point where you exploit both small radial as well as speed deformations from the double orbit geodesic, and that such local deformation need not include the exact 'first time around' event.

A key reference on this is (unfortunately, behind a pay wall):

http://scitation.aip.org/content/aapt/journal/ajp/79/1/10.1119/1.3488986

However, looking at several discussions referencing this, it appears that my revised statement above is correct: a 'long', twice around geodesic between events in Schwarzschild geometry would be a saddle point, not a local maximum. Quite generally (no need for exotic spacetimes), 'long enough' geodesics in GR are typically saddle points. In contrast, and as I should have expected, any stable orbit is a local (but not usually global) maximum over one period, and is a global maximum over shorter arcs.

[In the above, of course, I refer to timelike geodesics. The situation for spacelike and null geodesics is covered in the OP.]

 

[bcrowell]

MTW gives an explicit example on pp. 318-319. They have a particle moving in a potential $\Phi(z)=\frac{1}{2} |z|$. They write down trial solutions of the form $z(t)=a_1\sin(\pi t/2)+a_2\sin(\pi t)$ for motion from $(t,z)=(0,0)$ to $(2,0)$. They explicitly calculate the Newtonian action, and show that $(a_1,a_2)=(0,0.129)$ is a saddle point rather than an extremum. The action is locally maximized (proper time minimized) with $(a_1,a_2)=(0,0)$ and locally minimized (proper time maximized) with $(a_1,a_2)=(0.516,0)$

 

[George Jones]

A very nice reference for all of this is the 1964 Nuovo Cimento article "The Clock Paradox in General Relativity" by Robert Boyer (also Boyer-Lindquist coordinates), who, along with 14 other people, was assassinated in 1966 by a gunman.

Boyer considered a frictionless tunnel through the centre of a uniform density spherical object and two clocks, one that hovers at the start of the tunnel while the other clock is dropped from rest through the tunnel. The clock that moves through the tunnel travels on a spacetime geodesic, and the hovering clock has non-geodesic motion, yet when the clocks periodically meet, the hovering clock has recorded the greater elapsed time.

 

[bcrowell]

With hindsight, it's probably not surprising that a geodesic doesn't even have to be a local extremum of the action. The same thing happens in optics and mechanics. A simple example in optics would be a ray of light reflected from a saddle point on a mirror. If you replace the ray of light with a material particle, and replace the mirror with some suitably contrived gravitational field, you probably get the same result.

 

[A.T.]

a 'long', twice around geodesic between events in Schwarzschild geometry would be a saddle point, not a local maximum.

Is this similar to how the shorter arc of a great circle is a minimum for the path length between the two points (any perturbation makes it longer), but the longer arc is merely a saddle point (some perturbations make it shorter, some longer).

 

[PAllen]

Is this similar to how the shorter arc of a great circle is a minimum for the path length between the two points (any perturbation makes it longer), but the longer arc is merely a saddle point (some perturbations makes it shorter, some longer).

Exactly.

 

[bcrowell]

It seems like we have a menagerie of examples, but no general description of the conditions under which saddle points occur. Here is a list of conditions that don't seem to be necessary:

existence of conjugate points
counterexample: spacelike geodesics in Minkowski space

semi-Riemannian signature
counterexample: "long" geodesics on the sphere

curvature
counterexample: spacelike geodesics in Minkowski space

non-timelike geodesic
counterexample: Boyer's clock paradox (see #29)

Probably in order to clarify this we would need to see the paper by Gray and Poisson. Annoying that they never posted it on arxiv. Wald has a discussion of this kind of thing on p. 223, and there's also some relevant material in Winitzki's section 4.1, https://sites.google.com/site/winitzki/index/topics-in-general-relativity . Wald says that existence of conjugate points is the condition for timelike geodesics not to be local maxima of proper time, and similarly for length in the Euclidean case. I guess that's OK, if you want to ignore spacelike geodesics. Winitzki proves a theorem saying that a geodesic extremizes the action between p and q if and only if there is no conjugate point to p along the geodesic. I believe he intends this to hold for the Riemannian case, and also for the semi-Riemannian case if the geodesic is timelike, but his discussion isn't particularly clear on this point.

 

[pervect]

Geodesics between two points are guaranteed to be unique in the convex normal neighborhood of a point, if that helps. See Wald, pg 191, for instance, which I believe also implies that at any point in an arbitrary space-time, a convex normal neighborhood exists, if I'm getting all the details right (I might be mangling it a bit).

 

[PAllen]

Geodesics between two points are guaranteed to be unique in the convex normal neighborhood of a point, if that helps. See Wald, pg 191, for instance, which I believe also implies that at any point in an arbitrary space-time, a convex normal neighborhood exists, if I'm getting all the details right (I might be mangling it a bit).

That seems to be what Synge relies on to make his world function well defined.

 

[bcrowell]

The theorem Winitzki gives (see #33) as an if-and-only-if seems pretty obvious to me in one direction, at least in a restricted form where the endpoints p and q are the conjugate points. If p and q are conjugate, then we have geodesics from p to q that differ only infinitesimally from one another. Clearly if two geodesics differ infinitesimally, then they can't both be local extrema of the action.

But what about the other direction, which states that if a geodesic is a saddle, then there are conjugate points? This seems less obvious, and in fact some of our examples seem like counterexamples. For example, I summarized an example from MTW in #28. In that example, we have three geodesics, one that's a local maximum of proper time, one that's a local minimum, and one that's a saddle. And yet I don't think there are any conjugate points in that example at all...?

 

[pervect]

I haven't had time to read the whole thread properly, but has anyone mentioned the null zig-zag path (I forget the german name) that connects any two points in Minkowskii space?

If you have a space-time diagram, and two space-like separated points on it, you can connect them by a geodesic, or you can connect them by a zig-zag line composed of null segments. You can do this for timelike paths too, but at the moment I'm not thinking (or writing) about them.

So even in Minkowskii space,with no conjugate points, it seems to me that a spacelike geodesic must neither be a global maximum or minimum. You'll always have a shoreter zig-zag path of null sgements, and it's easy to imagine an arbitrary long meandering path.

 

[bcrowell]

So even in Minkowskii space,with no conjugate points, it seems to me that a spacelike geodesic must neither be a global maximum or minimum. You'll always have a shoreter zig-zag path of null sgements, and it's easy to imagine an arbitrary long meandering path.

This is a nice example, although it seems "fancier" than necessary. Cf. the example I gave in #1. Yes, I think the theorem Winitzki states definitely requires that the geodesic be timelike in the semi-Riemannian case, although as noted in #33 I don't think he states his assumptions very clearly.

I'm intrigued by trying to figure out whether or why not the example by MTW that I summarized in #28 is a counterexample to the theorem in Winitzki. Given a metric, is there some straightforward way that you can test whether two given points are conjugate? I set up a calculation in Maxima to find stuff like the Christoffel symbols and Einstein tensor:

load(ctensor);
ct_coords:[t,x,y,z];
lg:matrix([exp(abs(z)),0,0,0],
  [0,-1,0,0],
  [0,0,-1,0],
  [0,0,0,-1]
);
cmetric();
christof(all);
lriemann(true);
uriemann(true);
einstein(true);
scurvature(); /* scalar curvature */
rinvariant (); /* Kretchmann */

(Maxima seems not to realize that the metric isn't differentiable at z=0.)

I guess once you have the Riemann tensor, you can write down the geodesic deviation equation, and then you have a differential equation with certain boundary conditions, and you want to check whether there are solutions...?

 

[PAllen]

I think the MTW (two oscillations through a potential well) example and my two period orbit example are both consistent with what I learned as the Jacobi (necessary) condition in calculus of variations (for a local minimum (maximum in the Lorentzian case)) there can be no point conjugate to one of the ends between them (along the arc satisfying Euler-Lagrange). That is, I think they both have such a conjugate point, and that is why they are saddle points. Formally, conjugate points have a global definition in terms of the envelope of all the geodesics through a given point.

The following, by one of the co-authors of the pay-walled paper I gave earlier, has discussion of this (and it was from here that I borrowed the terminology of "long" versus "short" geodesics; "long" meaning long enough to have a conjugate point).

http://www.scholarpedia.org/article/Principle_of_least_action

In the above article, "kinetic focus" corresponds to what is called a conjugate point in the variational literature, and the "caustic" is the envelope as used in the variational literature.

 

[PAllen]

I thought it would be worth relating this conjugate point analysis to the case mentioned by AT and others of great circles on a 2-sphere. In this case, the envelope of the geodesics through a point is a single point, instead of a curve: the opposite pole. A geodesic through a point then contains a conjugate to that point if it contains the opposite pole. Thus, by the Jacobi conditions, a geodesic less than a semicircle is a local (in this case, global as well) minimum, otherwise it is a saddle point.

 

[PAllen]

MTW gives an explicit example on pp. 318-319. They have a particle moving in a potential $\Phi(z)=\frac{1}{2} |z|$. They write down trial solutions of the form $z(t)=a_1\sin(\pi t/2)+a_2\sin(\pi t)$ for motion from $(t,z)=(0,0)$ to $(2,0)$. They explicitly calculate the Newtonian action, and show that $(a_1,a_2)=(0,0.129)$ is a saddle point rather than an extremum. The action is locally maximized (proper time minimized) with $(a_1,a_2)=(0,0)$ and locally minimized (proper time maximized) with $(a_1,a_2)=(0.516,0)$

The (0,0) case is not a local minimum of proper time. There is no such thing as a local minimum of proper time because you can add as many small close to lightlike excursions as you want, that are 'small' in the variational sense. The (0,0) being a proper time minimum is just a statement about the particular 2 parameter family of world lines considered. Variationally, the (0,0) geodesic is another saddle point.

[edit: More interesting about the (0,0) world line is that, if a geodesic, it would seem to violate the idea that a short enough arc of a geodesic must be proper time maximum. The answer, I believe, is provided by the observation that there is no pure one dimensional gravitational well in GR. But this is not obvious. More thought needed. ]

 

[PAllen]

The (0,0) case is not a local minimum of proper time. There is no such thing as a local minimum of proper time because you can add as many small close to lightlike excursions as you want, that are 'small' in the variational sense. The (0,0) being a proper time minimum is just a statement about the particular 2 parameter family of world lines considered. Variationally, the (0,0) geodesic is another saddle point.

[edit: More interesting about the (0,0) world line is that, if a geodesic, it would seem to violate the idea that a short enough arc of a geodesic must be proper time maximum. The answer, I believe, is provided by the observation that there is no pure one dimensional gravitational well in GR. But this is not obvious. More thought needed. ]

Further supporting both points above:

1) That the z=0 geodesic is saddle point when a more complete family of curves is considered. Simply add a very high period term e.g. a₃ sin (kπt/2),
with k very large. This makes dz/dt large, with proper time approaching zero (or, within bounds of his Newtonian action, the KE term can be made arbitrarily large no matter
how small the amplitude). Thus, this geodesic is definitely NOT really a local maximum of his action, it is a saddle point when a full variation is considered.

2) The scenario must not really be consistent with GR for z=0 vicinity, or the Newtonian approximation breaks down in some relevant way. Pervect gave a reference from Wald on geodesic uniqueness within some convex neighborhood of a point ; Synge makes a similar claim without proof in his GR text. Yet, along the z=0 line, no matter how small a range of t is considered, there will always be a single period excursion geodesic connecting the same end points. Note also that such a neighborhood theorem being true seems required by local Lorentz equivalence, in that two points determine a unique geodesic in Minkowski space. An example of a possible required correction is that the effective potential is slightly rounded at the bottom of the V. Then, for small enough t range, the z=0 geodesic would be unique and and a proper time maximum.

 

[PAllen]

Supporting the above, I recall papers showing that to achieve gz potential in GR requires a an infinite singularity, and this oscillator would be crossing the singularity. In any real matter source (like the disc galaxy), the potential would, indeed, be rounded at the bottom. So the z=0 case of the MTW simplified example should just be ignored. The rest is typical of general situations in GR (including, that the two period geodesic is expected to include a point conjugate to either end).