Stationary/extremal L for timelike/null/spacelike geodesics

[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).