A Minimal property of Spacelike geodesics in GR/curved spacetime?

[Kostik]

TL;DR Summary

For a spacelike geodesic, it is easy to see that $\int \sqrt{-ds^2}$ is a minimum by considering nearby paths that make the integral arbitrarily large in special relativity. It is not so obvious how to extend this argument to curved spacetime.

Notation: The flat spacetime metric $\eta_\mu\nu = \text{diag}(1,-1,-1,-1)$. Thus, the proper time element for a timelike path is $ds$, and the proper distance element for a spacelike path is $\sqrt{-ds^2}$.

It is well known that $\int ds$ is stationary if and only if the path of integration is a geodesic, except for a null geodesic.

For a timelike geodesic, in the case of special relativity, it is easy to see that $\int ds$ is a maximum by considering nearby paths with zig-zag trajectories resembling forward light cones, whose proper time can be made arbitrarily small; see the picture below (left). This argument generalizes instantly to general relativity since, again, any trajectory has an arbitrarily close variation approximated by a light-like path, with arbitrarily small $\int ds$.

For a spacelike geodesic, in the case of special relativity, it is again easy to see that $\int \sqrt{-ds^2}$ is a minimum by considering nearby paths that make the integral arbitrarily large. This is obvious in special relativity where (in Cartesian coordinates) $g_{mn} = -\delta_{mn}$ since then $$\sqrt{-ds^2} = \sqrt{dx^2 + dy^2 +dz^2 - dt^2} \,\, .$$ The picture below (right) illustrates a variation that increases $\int \sqrt{-ds^2}$.

The question is: how does this generalize to curved spacetime? Now we have $$\sqrt{-ds^2} = \sqrt{-g_{\mu\nu} dx^\mu dx^\nu}$$ and it is not immediately obvious that if $z(\lambda)$ is a geodesic between $P$ and $Q$, then any variation $\bar{z}(\lambda)$ of the geodesic will have a larger value of $\sqrt{-ds^2} = \sqrt{-g_{\mu\nu} dx^\mu dx^\nu}$.

 

[PAllen]

Spacelike geodesics are a saddle point of the variation. You can see that they are neither minimum nor maximum quite trivially in SR. Consider a geodesic in xy plane in standard Minkowski coordinates. Specifically take it as a segment of the x axis. Then a nearby spacelike path in the xy plane is longer in interval, but a nearby spacelike path in the xt plane will be shorter in interval.

 

[martinbn]

All of this is standard and can be found in many books on geometry or relativity. For example O'Neill's "Semi-Riemannian Geometry With Applications to Relativity" has all the details, including why there are no extremal properties in signatures with at least two pluses and at least two minuses.

 

[Kostik]

Spacelike geodesics are a saddle point of the variation. You can see that they are neither minimum nor maximum quite trivially in SR. Consider a geodesic in xy plane in standard Minkowski coordinates. Specifically take it as a segment of the x axis. Then a nearby spacelike path in the xy plane is longer in interval, but a nearby spacelike path in the xt plane will be shorter in interval.

Thank you. So my premise was completely wrong, in fact. At least in 2 dimensions; what about in 1 dimension? (Edit: Yes, I have a simple example in 1 dimension.)

One more question: can you specify a path and call it a geodesic? (Clearly you cannot do this on the surface of a sphere.) Is it valid to say "let a segment of the x-axis be a geodesic"? (Edit: The argument can be made to work with any spacelike geodesic.)

 

[Kostik]

Thank you. So my premise was completely wrong, in fact. At least in 2 dimensions; what about in 1 dimension? (Edit: Yes, I have a simple example in 1 dimension.)

One more question: can you specify a path and call it a geodesic? (Clearly you cannot do this on the surface of a sphere.) Is it valid to say "let a segment of the x-axis be a geodesic"? (Edit: The argument can be made to work with any spacelike geodesic.)

Let the displacement A be along a spacelike geodesic. Then $-ds^2_A=dx^2-dt^2$. Note that $-ds^2_B=(dx-dt)^2$ and $-ds^2_C=(dx+dt)^2$. Hence, $-ds^2_A$ is not extremal.

 

[Dale]

You cannot have a 1D spacetime. I think you mean 1+1D spacetime. A 2D manifold with (-+) signature.

 

[Kostik]

You cannot have a 1D spacetime. I think you mean 1+1D spacetime. A 2D manifold with (-+) signature.

Yes of course I meant 1 spatial dimension. This is clear from the picture I posted.

 

[Orodruin]

The premise is incorrect as has already been established. The argument fails because it is not sufficient to find some paths that make the distance larger - in order to show minimality you need to show that all path variations have larger distance.

As has already been mentioned, some geodesics in curved space will correspond to saddle points. This is true also in purely Riemannian geometry.

But it is also a subtle point. I have found this error (misidendtifying a saddle point as a maximum) in a … well-known … text on mathematical methods.

 

[Kostik]

The premise is incorrect as has already been established. The argument fails because it is not sufficient to find some paths that make the distance larger - in order to show minimality you need to show that all path variations have larger distance.

Agreed! What that in mind, how do you show maximality of the proper time $\int ds$ for a timelike geodesic? The light-cone argument clearly shows that $\int ds$ is not minimal. But can there be a varied path with a longer proper time (in which case the geodesic would be a saddle point of the integral $\int ds$)?

 

[martinbn]

Agreed! What that in mind, how do you show maximality of the proper time $\int ds$ for a timelike geodesic? The light-cone argument clearly shows that $\int ds$ is not minimal. But can there be a varied path with a longer proper time (in which case the geodesic would be a saddle point of the integral $\int ds$)?

That is not true in full generality. Timelike geodesics are not maximal if there are conjugate points along the way. (for example Wald theorem 9.3.3)

It is easier if look up the Riemmanian case first. Lee's book on Riemanian Manifolds is really good.

The usual example is the sphere. If you go along a geodesic (a great circle) but you go further than the antipodal point it is not going to be the shortest path.

 

[PAllen]

That is not true in full generality. Timelike geodesics are not maximal if there are conjugate points along the way. (for example Wald theorem 9.3.3)

It is easier if look up the Riemmanian case first. Lee's book on Riemanian Manifolds is really good.

The usual example is the sphere. If you go along a geodesic (a great circle) but you go further than the antipodal point it is not going to be the shortest path.

Right, but in SR, at least, a time like geodesic is always maximal, among timelike paths. In GR, it remains true that if there is a maximal timelike path, then it is a geodesic. The typical situation is that there are multiple timelike geodesics connecting two points, and only one is maximal.

With space like paths, even in SR, there is no extremal property at all.

I always liked the treatment in G. A. Bliss 1926 monograph on calculus of variations for detailed treatment of necessary and sufficient conditions for both local and global extremal reoperties.

 

[PAllen]

Note that even in SR, between two points connected by a timelike geodesic, there are spiral space like paths of arbitrary large spacelike interval connecting them. Thus, the extremal property only holds among all causal paths.

 

[martinbn]

I always liked the treatment in G. A. Bliss 1926 monograph on calculus of variations for detailed treatment of necessary and sufficient conditions for both local and global extremal reoperties.

Is it this one?
https://www.ams.org/journals/bull/1926-32-04/S0002-9904-1926-04208-7/

 

[JimWhoKnew]

As has already been mentioned, some geodesics in curved space will correspond to saddle points. This is true also in purely Riemannian geometry.

By "purely Riemannian geometry", do you mean positive definite metric? If so, can you please provide an example to a saddle geodesic?

 

[Kostik]

That is not true in full generality. Timelike geodesics are not maximal if there are conjugate points along the way. (for example Wald theorem 9.3.3)

It is easier if look up the Riemmanian case first. Lee's book on Riemanian Manifolds is really good.

The usual example is the sphere. If you go along a geodesic (a great circle) but you go further than the antipodal point it is not going to be the shortest path.

Are you saying that (in SR or GR) a timelike geodesic is not necessarily a maximum of $\int ds$?

 

[PAllen]

Is it this one?
https://www.ams.org/journals/bull/1926-32-04/S0002-9904-1926-04208-7/

No, but that presumably has some similar material. The one I have is a short book (189 pages) simply titled “calculus of variations”, first published in 1925. It’s the same author, obviously. It was the first of series of “Carus mathematical monographs”.

 

[PAllen]

Are you saying that (in SR or GR) a timelike geodesic is not necessarily a maximum of $\int ds$?

In GR, consider the geodesic path representing one orbit around a massive body. There is also a geodesic connecting the same two events representing a radial out and back free fall path (a ballistic path, as it were). The orbital path is not extremal, while the ballistic path is. This lack of exremality is present, as you see, even among just causal paths. Note that in this case, each geodesic represents a local maximum among causal paths, but only one is a global maximum.

In SR, while a global extremal property holds for timelike geodesics among causal paths, I already gave a counter example if you consider all paths.

 

[PAllen]

By "purely Riemannian geometry", do you mean positive definite metric? If so, can you please provide an example to a saddle geodesic?

@Orodruin already gave an example on a sphere where there are two geodesic paths between two points, only one of which is minimal. But guess this isn’t really a saddle point in that both are local minima. Whereas spacelike paths in SR lack even a local extremal property, thus are saddle points of the variation.

 

[martinbn]

No, but that presumably has some similar material. The one I have is a short book (189 pages) simply titled “calculus of variations”, first published in 1925. It’s the same author, obviously. It was the first of series of “Carus mathematical monographs”.

Right, I found it. It looks very nice.

 

[JimWhoKnew]

Are you saying that (in SR or GR) a timelike geodesic is not necessarily a maximum of ∫ds?

Feynman defines timelike geodesics in spacetime (either flat or curved) as paths for which $\int d\tau$ is locally maximal (Lectures on Physics Vol. 2 sections 42.8 , 42.9).

 

[Orodruin]

By "purely Riemannian geometry", do you mean positive definite metric? If so, can you please provide an example to a saddle geodesic?

The longer arc of the great circle connecting two points on a sphere.

@Orodruin already gave an example on a sphere where there are two geodesic paths between two points, only one of which is minimal. But guess this isn’t really a saddle point in that both are local minima. Whereas spacelike paths in SR lack even a local extremal property, thus are saddle points of the variation.

This is actually where Arfken was wrong. The long arc is not a local (as pertains to the set of possible paths) minima - it is actually a saddle point. You can make a small perturbation to the path and it becomes shorter.

It is only a local minimum between nearby points.

 

[Orodruin]

Digging out a post from my first year at PF, now 11 years old:

Post in thread 'Understand the major arc connecting two points on a sphere'
https://www.physicsforums.com/threa...ng-two-points-on-a-sphere.767025/post-4829841

In that post I showed explicitly that the major arc is indeed a saddle point.

 

[PAllen]

In GR, consider the geodesic path representing one orbit around a massive body. There is also a geodesic connecting the same two events representing a radial out and back free fall path (a ballistic path, as it were). The orbital path is not extremal, while the ballistic path is. This lack of exremality is present, as you see, even among just causal paths. Note that in this case, each geodesic represents a local maximum among causal paths, but only one is a global maximum.

By an argument similar to @Orodruin’s sphere case, the orbital geodesic is actually a saddle point, not even a local maximum.

 

[JimWhoKnew]

Digging out a post from my first year at PF, now 11 years old:

Post in thread 'Understand the major arc connecting two points on a sphere'
https://www.physicsforums.com/threa...ng-two-points-on-a-sphere.767025/post-4829841

In that post I showed explicitly that the major arc is indeed a saddle point.

Thanks. Very nice.

 

[Orodruin]

Feynman defines timelike geodesics in spacetime (either flat or curved) as paths for which $\int d\tau$ is locally maximal (Lectures on Physics Vol. 2 sections 42.8 , 42.9).

This is a different concept of locality though iirc. What is being referred to is being local in the sense of sufficiently small spacetime regions - not in terms of nearby curves in the space of all possible curves.

 

[PAllen]

This is a different concept of locality though iirc. What is being referred to is being local in the sense of sufficiently small spacetime regions - not in terms of nearby curves in the space of all possible curves.

and this gets at the difference between cases like the long great circle arc on a sphere or the orbital geodesic in GR - these are local minima/maxima in the sense that for any two sufficiently close points on them the geodesic is extremal in a neighborhood of those two points, versus spacelike geodesics. For spacelike geodesics, there is no extremal property no matter how small a region you consider.

 

[JimWhoKnew]

This is a different concept of locality though iirc. What is being referred to is being local in the sense of sufficiently small spacetime regions - not in terms of nearby curves in the space of all possible curves.

The full text is accessible by Caltech:

"An object always moves from one place to another so that a clock carried on it gives a longer time than it would on any other possible trajectory—with, of course, the same starting and finishing conditions."

And on the footnote:

"Strictly speaking it is only a local maximum. We should have said that the proper time is larger than for any nearby path..."

 

[PAllen]

The full text is accessible by Caltech:

"An object always moves from one place to another so that a clock carried on it gives a longer time than it would on any other possible trajectory—with, of course, the same starting and finishing conditions."

And on the footnote:

"Strictly speaking it is only a local maximum. We should have said that the proper time is larger than for any nearby path..."

And this is not quite true. You also need to restrict to sufficiently close pairs of points along the trajectory, or equivalently, within small neighborhoods along the trajectory.

 

[Kostik]

Agreed! What that in mind, how do you show maximality of the proper time $\int ds$ for a timelike geodesic? The light-cone argument clearly shows that $\int ds$ is not minimal. But can there be a varied path with a longer proper time (in which case the geodesic would be a saddle point of the integral $\int ds$)?

Sorry, I was not clear -- the responses below this question refer to local versus global maximality. I am only talking about a local maximum, as one expects in a variation problem.

Once again: a spacetime geodesic is a saddle-point of the integral $\int \sqrt{-ds^2}$ because we can easily construct variations which make the integral larger or small. The "light cone" argument shows that a timelike geodesic is definitely not a minimum of the integral $\int ds$, since there are timelike (but quasi-lightlike) variations which make the integral arbitrarily small. But this argument does not show that a timelike geodesic is a local maximum of $\int ds$. How do we know that there are no variations which make the integral larger?

 

[Orodruin]

Sorry, I was not clear -- the responses below this question refer to local versus global maximality. I am only talking about a local maximum, as one expects in a variation problem.

Once again: a spacetime geodesic is a saddle-point of the integral $\int \sqrt{-ds^2}$ because we can easily construct variations which make the integral larger or small. The "light cone" argument shows that a timelike geodesic is definitely not a minimum of the integral $\int ds$, since there are timelike (but quasi-lightlike) variations which make the integral arbitrarily small. But this argument does not show that a timelike geodesic is a local maximum of $\int ds$. How do we know that there are no variations which make the integral larger?

You have to look at the general form of the variation to conclude this. As has already been indicated, it is only true in some cases. Other cases will be saddle points.

 

[Kostik]

You have to look at the general form of the variation to conclude this. As has already been indicated, it is only true in some cases. Other cases will be saddle points.

In SR, it's clear that a timelike geodesic maximizes $\int ds = \int \sqrt{dt^2 - dx^2 - dy^2 - dz^2}$. Can you give a concrete example in GR of a gravitational field and a timelike geodesic where the geodesic is a saddle point of the integral $\int ds$?

 

[Orodruin]

In SR, it's clear that a timelike geodesic maximizes $\int ds = \int \sqrt{dt^2 - dx^2 - dy^2 - dz^2}$. Can you give a concrete example in GR of a gravitational field and a timelike geodesic where the geodesic is a saddle point of the integral $\int ds$?

@PAllen already did in post #23

 

[Orodruin]

He said both geodesics were local maxima. Neither one is a local saddle point.

Read again.

 

[Kostik]

@PAllen already did in post #23

He said both geodesics were local maxima. Neither one is a local saddle point.

 

[Kostik]

Read again.

Sorry, I see now. So, in GR, neither timelike nor spacelike geodesics are local extrema. That's interesting.

 

[Kostik]

By an argument similar to @Orodruin’s sphere case, the orbital geodesic is actually a saddle point, not even a local maximum.

Can you explain descriptively how to vary these two timelike geodesics (the orbital and the ballistic ones) to make $\int ds = \int \sqrt{g_{\mu\nu}dx^\mu dx^\nu}$ larger and smaller? It's hard to visualize.

More specifically, we can assume the metric is $$ds^2 = (1-2m/r)dt^2 - (1-2m/r)^{-1}dr^2 - r^2d\theta^2 - r^2\sin^2\theta d\phi^2 \,.$$ Can you describe how to vary the geodesics (orbits) to make $\int ds$ larger and smaller?

 

[A.T.]

By an argument similar to @Orodruin’s sphere case, the orbital geodesic is actually a saddle point, not even a local maximum.

What about the geodesic worldline for the center of a spherical mass? Is it a local minimum or a saddle point?

 

[Orodruin]

What about the geodesic worldline for the center of a spherical mass? Is it a local minimum or a saddle point?

Geodesics are always a local maximum or saddle point (remember that the geodesic in SR maximizes proper time).

In this case it would depend on the time interval. If events are close enough it is always a max.

However, for longer times I believe it will be a saddle point. The center of the spherical mass has the lowest gravitational potential and therefore the maximal time dilation for a stationary observer. Perturb the stationary geodesic by going slightly off-center where time dilation is lower, stay there until you accumulate enough proper time and then go back and this will result in a larger proper time.

 

[Kostik]

@Orodruin How do we know that the geodesic in SR maximizes proper time? The zig-zag argument shows that there are always variations with arbitrarily small proper time. But how do we know in SR that a timelike geodesic is NOT a saddle point?

EDIT: I think I have it. The geodesic equation implies $v^\mu$ is constant. Any variation from a straight line timelike path increases the proper time.

 

[PeterDonis]

Can you explain descriptively how to vary these two timelike geodesics (the orbital and the ballistic ones) to make $\int ds = \int \sqrt{g_{\mu\nu}dx^\mu dx^\nu}$ larger and smaller?

The ballistic geodesic is a local (and global) maximum. Only the orbital one is a saddle point. Even in curved spacetime there will be some timelike geodesics which are maxima. Just not all of them.

 

[PeterDonis]

for longer times I believe it will be a saddle point

Not always. For example, the radial ballistic geodesic in the example @PAllen gave in post #23 is a local maximum.

 

[PAllen]

Not always. For example, the radial ballistic geodesic in the example @PAllen gave in post #23 is a local maximum.

Actually, it is a global maximum, but that is quite a chore to prove. You have to bring in global topology, but in this case, assuming standard Kruskal topology, it is true.

 

[Orodruin]

Not always. For example, the radial ballistic geodesic in the example @PAllen gave in post #23 is a local maximum.

The discussion was about the geodesic at the center of the spherical body, not about the ballistic geodesic. I was replying to and quoted @A.T. ‘s post that specified this.

 

[PAllen]

Not always. For example, the radial ballistic geodesic in the example @PAllen gave in post #23 is a local maximum.

I think @Orodruin assumed that, and was only referring to the orbital geodesic, which is analogous to the larger great circle path on a sphere. @Orodruin described something similar to what I had in mind to argue for a saddle point. My version was to transfer at near c to a slightly larger radius, then non-inertially move in a circle there at the speed (in coordinate terms; so a slightly lower ancular speed than the orbit) of the orbit for most of the way, than, just in time to meet that orbital closure event, move at near c back. This procedure will fail for small segments of the orbit, but for some large enough segment, it will work; in particular, for the whole orbit.

 

[Orodruin]

@Orodruin How do we know that the geodesic in SR maximizes proper time? The zig-zag argument shows that there are always variations with arbitrarily small proper time. But how do we know in SR that a timelike geodesic is NOT a saddle point?

Without loss of generality, go to the frame where the events occur at the same spatial point. The proper time for any curve between the points is then
$$
\tau = \int_{t_0}^{t_1} \sqrt{1 - \dot x(t)^2} dt \leq t_1 - t_0$$ with equality if and only if $\dot x(t) = 0$ for all $t$.

Edit: The further restriction is that $x(t_1) = x(t_0)$.

Edit 2: Fixed typo, ”ane” -> ”and”

 

[Orodruin]

I think @Orodruin assumed that, and was only referring to the orbital geodesic, which is analogous to the larger great circle path on a sphere. @Orodruin described something similar to what I had in mind to argue for a saddle point.

I was referring to the geodesic at the center of the body, which is what @A.T. asked about in the post I quoted.

 

[PAllen]

Sorry, I see now. So, in GR, neither timelike nor spacelike geodesics are local extrema. That's interesting.

No, in GR, some timelike geodesics are saddle points and some are local maxima. The usual case is that for sufficiently close points, where one is in the future of the other, there will be a unique timelike geodesic that is a local maximum. For points not so close, there will often be more than one geodesic, and one of them will be local maximum (in realistic cases, it will also be a global maximum).

 

[cianfa72]

@Orodruin already gave an example on a sphere where there are two geodesic paths between two points, only one of which is minimal.

Actually I believe it was @martinbn who gave the example of the sphere in post #10.

 

[PAllen]

Without loss of generality, go to the frame where the events occur at the same spatial point. The proper time for any curve between the points is then
$$
\tau = \int_{t_0}^{t_1} \sqrt{1 - \dot x(t)^2} dt \leq t_1 - t_0$$ with equality if ane only if $\dot x(t) = 0$ for all $t$.

Edit: The further restriction is that $x(t_1) = x(t_0)$.

Simpler than what I was starting to write up - what I know of as the Legendre condition distinguishing whether you have a possible local minimum versus a possible local maximum for a variation.

 

[JimWhoKnew]

Can you explain descriptively how to vary these two timelike geodesics (the orbital and the ballistic ones) to make $\int ds = \int \sqrt{g_{\mu\nu}dx^\mu dx^\nu}$ larger and smaller? It's hard to visualize.

More specifically, we can assume the metric is $$ds^2 = (1-2m/r)dt^2 - (1-2m/r)^{-1}dr^2 - r^2d\theta^2 - r^2\sin^2\theta d\phi^2 \,.$$ Can you describe how to vary the geodesics (orbits) to make $\int ds$ larger and smaller?

In accord with @PAllen's observation in #23 , I think we can slightly generalize @Orodruin's example here to show that the orbital geodesic is a saddle for angles larger than $\pi$.

To do so, note that the orbital geodesic at the equator $\theta=\frac \pi 2$ and at constant $r=R$ can be parametrized by $\phi$. So if we take varied paths with the same form $$\theta(\phi) = \frac \pi 2 + \eta f(\phi)$$we get $$\tau[\theta(\phi)] = \int_0^{a\pi} \sqrt{\left(1-\frac{2m}{R}\right)\left(\frac{dt}{d\phi}\right)^2 -R^2 \eta^2 f'(\phi)^2-R^2 \cos^2(\eta f(\phi))} ~d\phi$$Since $\frac{dt}{d\phi}$ is constant, the first and second derivatives of $\tau[\theta(\phi)]$ wrt $\eta$ when evaluated at $\eta=0$, are proportional to @Orodruin's, and the same further reasoning applies.

Edit: corrected typo in equation