Trying to understand what extremal proper time means.

[zeromodz]

General relativity states that all free falling bodies follow spacetime geodesics. A geodesic is a path of extremal proper time.

My intuition tells me that extremal proper time is the maximum amount of real time. So does this mean that all objects will take the path of most time to get from point A to B. Then why is it, if I drop my pencil, it travels in a straight line down to earth. I thought a geodesic was defined as the shortest path between 2 points, but how can that be if it is also defined as the path of most time?

Thanks.

 

[JesseM]

Well, in the case of flat spacetime just think of the twin paradox, where the twin that takes the "straight line" (inertial) path between two encounters always ages more than the twin that accelerates at some point between the meetings. For the case of geodesics in curved spacetime, some of the illustrations in post #4 of this thread might help...

 

[Dale]

I thought a geodesic was defined as the shortest path between 2 points, but how can that be if it is also defined as the path of most time?

In relativity we speak of spacetime, which has the following metric (in local inertial coordinates):

ds² = -c²dt² + dx² + dy² + dz²

Note that this metric differs from a normal Euclidean metric by the presence of a term with a negative sign. This term can make ds² be positive (aka spacelike), negative (aka timelike), or zero (aka lightlike). Geodesics typically minimize ds², so spacelike geodesics are paths of minimum proper distance, and timelike geodesics are paths of maximum proper time.

 

[JesseM]

Geodesics typically minimize ds², so spacelike geodesics are paths of minimum proper distance, and timelike geodesics are paths of maximum proper time.

For spacelike geodesics it's only a "local" minimum of proper distance, even in Minkowski spacetime, right? (whereas in Minkowski spacetime a timelike geodesic is a global maximum of proper time) Seems like it can't be a global minimum, since between any two spacelike-separated events A and B you could draw two light-like worldlines that meet at some third event C, and then you could draw spacelike worldlines that hug arbitrarily close to the path ACB and thus have a proper distance arbitrarily close to zero.

 

[Dale]

it's only a "local" minimum

Correct, technically they are stationary curves meaning that they are a minimum, maximum, or "saddle point" with respect to infinitesimal variations of the curve. The change you are talking about is definitely a finite variation, not an infinitesmial variation.

 

[jfy4]

To be extremal it need only satisfy Lagrange's equations, which geodesics do.

To be a path of shortest length it needs to satisfy

$$d(p,q)=\inf\int_{\gamma}|\dot{\gamma}(t)|dt.$$

If it is the least upper-bound of these paths, it is the path of shortest length. You can find a proof in Riemannian Geometry by Postnikov.

 

[zeromodz]

In relativity we speak of spacetime, which has the following metric (in local inertial coordinates):

ds² = -c²dt² + dx² + dy² + dz²

Note that this metric differs from a normal Euclidean metric by the presence of a term with a negative sign. This term can make ds² be positive (aka spacelike), negative (aka timelike), or zero (aka lightlike). Geodesics typically minimize ds², so spacelike geodesics are paths of minimum proper distance, and timelike geodesics are paths of maximum proper time.

I still don't understand. If you maximize dt then your going to get a really large negative number for a timelike geodesic. If you want to minimize ds, then you want a small dt because the only large term in this metric is c^2, where dx,dy,dz are really nothing.

 

[IsometricPion]

If you maximize dt then your going to get a really large negative number for a timelike geodesic. If you want to minimize ds, then you want a small dt because the only large term in this metric is c^2, where dx,dy,dz are really nothing.

The idea is to extremize the interval (s) in both cases. The only difference between proper time and proper length is that the interval has a positive (in a -+++ metric) value for proper lengths and a negative value for proper times.

 

[Dale]

I still don't understand. If you maximize dt then your going to get a really large negative number for a timelike geodesic. If you want to minimize ds, then you want a small dt because the only large term in this metric is c^2, where dx,dy,dz are really nothing.

No, a large negative number is a minimum, not a small positive number (assuming the function has both). However, as JesseM and jfy4 both correctly pointed out, I was being a little sloppy. The correct term is "stationary" or "extremal". Some will be local minima, some will be local maxima, and many will be saddle points. The important thing is that a geodesic is a stationary curve meaning that small variations do not change the length of the path to first order.

Does that make sense now? Do you understand the concepts of saddle points, local, and global stationary points, etc. from normal optimization?

 

[zeromodz]

No, a large negative number is a minimum, not a small positive number (assuming the function has both). However, as JesseM and jfy4 both correctly pointed out, I was being a little sloppy. The correct term is "stationary" or "extremal". Some will be local minima, some will be local maxima, and many will be saddle points. The important thing is that a geodesic is a stationary curve meaning that small variations do not change the length of the path to first order.

Does that make sense now? Do you understand the concepts of saddle points, local, and global stationary points, etc. from normal optimization?

I have a concept and definitions for saddle points, local, and global stationary points from normal optimization. So let me get this straight. We want to minimize the line element. So we want to maximize the time because its multiplied times negative c^2. We also want to minimize the other spatial components. Is this what a geodesic would typically follow?

Because Isometric Pion states that.
"The idea is to extremize the interval (s) in both cases."

 

[Dale]

We want to minimize the line element.

We want to minimize, maximize, or find a saddle point. This is what Isometric Pion refers to by the word "extremize" or what I referred to by finding a "stationary point". All three types of curves (minima, maxima, saddle points) are geodesics.

 

[JesseM]

I have a concept and definitions for saddle points, local, and global stationary points from normal optimization. So let me get this straight. We want to minimize the line element. So we want to maximize the time because its multiplied times negative c^2. We also want to minimize the other spatial components. Is this what a geodesic would typically follow?

You talk about "spatial components" suggesting a coordinate-dependent notion of distance involving multiple space coordinates, but the only case where geodesics have anything to do with distance is for spacelike geodesics (which no particle moving at or below the speed of light could have as a worldline), where integrating ds along them gives a single frame-independent "proper distance" analogous to the proper time along timelike geodesics. A single geodesic doesn't both maximize the time and minimize distance, rather timelike geodesics have extremal proper time and spacelike geodesics have extremal proper distance.

 

[pervect]

If you consider a simple curve - y = f(x) - some of the technicalities might become more clear.

An extremal point on the curve y=f(x) is just a point where the slope is zero. There are several possible types of extremal point:

a local maximum
a local minimum
a point of inflection.

Furthermore, just because you have a "local" minimum of the cure, around some specific point, doesn't mean that it's an absolute minimum.

We can extend the same concepts up to curves of more and more variables - functions of up to 2 variables are easy to visualzie, after that it gets hard.Unlike the problem of finding the critical points on a function of a single variable, the problem of finding the optimum path requires an inifinte number of variables. So it's the same idea, in an infinite dimensional space.

Going back to the original question:

If you consider a path through space, and not a path through space-time, the geodesic path locally minimizes distance. But it's only a sort of local minimum. This can be important in cases where you have multiple geodesics connecting two points, which is actually fairly common. They're all "local" minimums, even though only one particular path can claim to be the "shortest".

If you consider a path through space-time, the geodesic path locally minimizes the lorentz interval, and not the "Euclidean distance". (But only when you chose the right domain of paths - they have to be paths that are timelike everywhere!).

Minimizing the Lorentz interval is done by maximizing proper time, as the interval is defined by x^2 + y^2 + z^2 - t^2. The minus sign in front of the t makes all the difference.

Hopefully this wil help - some...