Why extremal aging, not just maximal? (for an inertial body in spacetime)

[Colin Mitch]

One can use the Principle of Extremal Aging to calculate the path of a freely moving body (in an inertial frame not subject to any forces) in spacetime, curved or flat. Why extremal? Why not just maximal? All the examples I know of involve maximum proper time for a freely moving body. For example the twin paradox. The Earth is the freely moving body and the stay at home twin ages the maximum (ignore gravitational time dilation).
Can anyone give me an example of minimal aging and explain the circumstances a freely moving body experiences this? How does minimal aging occur?

 

[A.T.]

Can anyone give me an example of minimal aging and explain the circumstances a freely moving body experiences this?

Photons

 

[PAllen]

Yes, photons is one example. There are null paths (general) and null geodesics. Light follows null geodesics. This is clearly extremal, neither minimum nor maximum, since all such paths have zero interval along them.

However, in general relativity, there are examples for material bodies. Consider an orbiting body. Consider the event of passing a certain point on the orbit, and the event of passing the same point one orbit later. The orbit is an an inertial path between these events. However, it will entail less proper time than the path representing hovering stationary at that point (which is a non-inertial world line).

The issue here is that there are multiple geodesics between this pair of events. There is one that maximizes proper time. That is the one that shoots out radially from the position of interest, falling back in one orbital period. This particular geodesic will have maximum proper time.

The upshot: to define geodesics and inertial motion you do need 'extremal' not minimum or maximum. (Alternatively, my preference is to use parallel transport of tangent as the definition; then derive under what conditions you have or don't have maximum/minimum properties).

 

[pervect]

One good example of why you need the principle of extremal aging is the interesting fact that you can create a null worldline that's arbitrarily close to any timelike one.

Just have a "photon" or light beam "zig-zag" around the actual timelike path you want to take. As the zigs and zags become small, the "distance" of this null path from the desired path becomes zero, yet the length of the null path is always zero.

You might try to get around this by ruling out null paths, but I'm not sure there is an approach of this sort that is rigorous. It's better to just say that the path is in general extremal and not maximal, then you don't have to worry about the issue.

 

[PAllen]

One good example of why you need the principle of extremal aging is the interesting fact that you can create a null worldline that's arbitrarily close to any timelike one.

Just have a "photon" or light beam "zig-zag" around the actual timelike path you want to take. As the zigs and zags become small, the "distance" of this null path from the desired path becomes zero, yet the length of the null path is always zero.

You might try to get around this by ruling out null paths, but I'm not sure there is an approach of this sort that is rigorous. It's better to just say that the path is in general extremal and not maximal, then you don't have to worry about the issue.

I think this particular argument is not so good. For a timelike geodesic, you are maximizing proper time, so these null paths would be irrelevant. What you would need to find is a path of greater proper time than the geodesic. That you can do in the scenario I gave.

However, I should also point out that a lot of complexity goes away if you ask about 'sufficiently close' events. I believe the following are true:

1) Pick a timelike geodesic. Pick a point on it (A). Pick another (B), and consider moving (B) closer to A along the geodesic (e.g. via arbitrary affine parameter). Sufficiently close, the time like geodesic will be an absolute maximum proper time path between A and B.

2) Pick a null geodesic. Proceed as above. At sufficient proximity, the null geodesic will be the only null path between A and B.

Unfortunately, for spacelike geodesics, you can't get such a nice property even locally. There is not simple minimum property you can give for semi-riemannien metrics.

 

[Colin Mitch]

Quoting PAllan "However, in general relativity, there are examples for material bodies. Consider an orbiting body. Consider the event of passing a certain point on the orbit, and the event of passing the same point one orbit later. The orbit is an an inertial path between these events. However, it will entail less proper time than the path representing hovering stationary at that point (which is a non-inertial world line)."

Sorry, I don't understand this. To 'hover stationary' at the point on the orbit would entail using a force such as a constant rocket thrust to prevent falling towards the sun, so as you say it is a non-inertial world line. It involves acceleration. Isn't this acceleration OFF the geodesic in spacetime? How would this be a geodesic? Why would the proper time be more than that of the orbiting body?

 

[PAllen]

Quoting PAllan "However, in general relativity, there are examples for material bodies. Consider an orbiting body. Consider the event of passing a certain point on the orbit, and the event of passing the same point one orbit later. The orbit is an an inertial path between these events. However, it will entail less proper time than the path representing hovering stationary at that point (which is a non-inertial world line)."

Sorry, I don't understand this. To 'hover stationary' at the point on the orbit would entail using a force such as a constant rocket thrust to prevent falling towards the sun, so as you say it is a non-inertial world line. It involves acceleration. Isn't this acceleration OFF the geodesic in spacetime? How would this be a geodesic? Why would the proper time be more than that of the orbiting body?

What was unclear? I said the orbital path was inertial. I said the hovering path was non-inertial. It's right there in what you quoted! Then I said the non-inertial (I thought it was self evident by definition that this means non-geodesic) path has longer proper time. Thus this example clearly shows a violation of the dictum that a geodesic path between two events maximizes proper time. The rest of the example goes on to show that between the particular chosen events we can display two geodesics G1 (orbital), G2(ballistic), and a non-geodesic path H (hovering), such that the proper time along the paths is ordered as follows: G1 < H < G2. Clearly, this shows that in GR, geodesics do not (in general) maximize proper time. They are just extremal (stationary with respect to variation). G1 is not even a local maximum among nearby paths between the chosen events, yet it is still, without any doubt, a geodesic.

 

[pervect]

I don't currently see any obvious flaw in Pallen's earlier comments about my example being bad, though I need to think about them a bit more, and I'm not sure when I'll have the time.

I definitely agree with Pallen's point about the orbital path. You might actually have to calculate the numbers to convince yourself it's true.

If you consider (for convenience) a non-rotating Earth with no other gravity sources nearby you can do the caclulations by assuming a Schwarzschild metric and by integrating the proper time along the path.

You'll find that tossing a clock straight upwards generates a path of maximal proper time (one that always exists) between two points at the same altitude some delta-t apart.

The orbital path doesn't even exist unless delta-t is at least one orbital period.

Direct calculation also shows the remarks about the hovering clock are true.