# Geodesic applied to twins paradox

1. Dec 1, 2007

### Loren Booda

Regarding the twins paradox: do all closed trajectories require nonzero acceleration at some point, or can a closed geodesic fulfill overall the special relativistic requirement of constant velocity?

2. Dec 1, 2007

### mendocino

I don't understand what's "closed geodesic"?
Does "closed geodesic" mean you can travel backward in time?

3. Dec 1, 2007

### Chris Hillman

Huh?

Are you by any chance referring to accelerating observers who move in circular orbits in flat spacetime, and whose world lines can be visualized as helices on a "cylinder"?

My guess is that he meant "closed trajectory", as in, the projection of a helical world line to a spatial hyperslice giving a circular trajectory.

Last edited: Dec 1, 2007
4. Dec 1, 2007

### pervect

Staff Emeritus
I take it you want to find the path of maximum aging between two events in space-time?

It is necessary that such a path be a geodesic, but it is not sufficient. It may help to consider the example of finding the shortest distance between two points on a curved surface, which is mathematically rather similar to finding the path of longest time in relativity.

Such a "shortest distance" path will also be a geodesic, but it's possible to find examples in which a given geodesic between two points is not the shortest path connecting them. For instance, consider two towns with a mountain between them. There is a geodesic path that connects the two towns that goes over the mountain top, but it is not the shortest distance between the two towns. In this example, there is a shorter path that goes around the mountain. The absolute shortest path will be a geodesic path that goes around the mountain, but a non-geodesic path going around the mountain can still be shorter than the geodesic path that goes over the top of the mountain.

5. Dec 2, 2007

### Loren Booda

Thank you for your examples, pervect. Actually, I am asking whether any twins paradox can be considered using special relativity alone. All closed Euclidean paths, including a trip to "planet X" and back, somewhere require relative acceleration of twin A's trajectory to that of twin B. Unless, perhaps, where the twins remain on a common spacetime geodesic, which represents free fall (zero relative acceleration?) and can satisfy the parameters given by the problem. But in this latter, general relativistic case, would there remain any paradox?

6. Dec 3, 2007

### pervect

Staff Emeritus
If your space-time is essentially flat, then with proper care you can use SR to find the answer to any twin paradox problems.

However, if your space-time is not flat (for instance, if you have gravitational fields), then in general you can't use SR but must use GR to deal with the twin paradox.

The fact that space-time is not in general flat is what motivated my example of distances on a curved surfaces. Flat space-time is equivalent to SR, and non-flat space-time must be treated by GR.

7. Dec 3, 2007

### Loren Booda

Can you think of a closed path in flat spacetime on which a traveler would not undergo acceleration - or can all accelerations in flat spacetime be treated by SR?

8. Dec 3, 2007

### pervect

Staff Emeritus
If you have the right (or wrong) sort of global topology, it is possible to have a closed geodesic path even in a flat space-time (one with a zero curvature tensor).

I believe there are some past threads written about the twin paradox in this sort of topology, but I don't recall at the moment where they are. Garth has posted on the topic before, IIRC, that might help find the previous threads (or perhaps he'll read this and give us the info).

9. Dec 3, 2007

### Staff: Mentor

Accelerations in flat spacetime can easily be handled by SR. In the case of the twin paradox, however, you don't even have to explicitly handle the acceleration. All you need to do is to calculate the spacetime interval along the path. For timelike spacetime intervals the interval is equal to the proper time of a clock traveling along the interval.

10. Dec 8, 2007

### KingOrdo

Perhaps this is relevant to the conversation: Some time ago I posted the following question:

The thread was quickly filled with a lot of confused responses. And indeed, upon looking into the literature it became apparent that the issue remains unresolved. In fact, I e-mailed a few physicists (one a prominent relativist), and got totally different "answers". So it seems the Twin Paradox remains an open and confounding problem in (at least) the complex cases.

11. Dec 8, 2007

### Staff: Mentor

On a 2-sphere the path-length between two points is the same for any geodesic that contains those two points. I would assume that the same is true for a 3-sphere. That would imply that they would be the same age when they met up again.

More interesting would be a torus where different geodesics between the same point can have different path lengths. In any case, as long as you can calculate the spacetime interval traversed for each twin you can unambiguously determine which is older.

12. Dec 9, 2007

### jcsd

For two non-antipodal points on a sphere (i.e. a 2-sphere) there is a unique great circle on which both points sit. The two points divide the great circle into two segments of differing lengths, both these segments represent different geodesic paths of different path lenghts between the two points.

Of course our we must consider a geodesic in 4 dimenisonal spacetime of which our 3-sphere is only a spatial slice, but the 2-sphere analogy hints correctly at the answer. The two twins who are both free-falling observers can experince different proper times between two events that lie on both their worldlines.

Though from a local point of view the twins seem to be symmetric (i.e. they are both free-falling obserevers), there is a global assymetry between them that produces the difference in the proper time they observe.

At a guess, one way in which the global assymetry might manifest itself is that if the two twins would construct different spatial slices for themsleves with different spatial geometries even if they used the same technique to construct them. Another way is that twins may experince different anistropies in the distribution of mass-energy.

Last edited: Dec 9, 2007
13. Dec 9, 2007

### Staff: Mentor

Doh! You are absolutely correct. I was only thinking of antipodal points, I was not thinking about going "the long way around" with non-antipodal points.

14. Dec 9, 2007

### pervect

Staff Emeritus
This general topic has been discussed before, for one example see https://www.physicsforums.com/showthread.php?t=51197

A quote which I think sums up the affair nicely is due to robphy:

So the basis of SR is still the same - it's still a local statement, and in a local environment (which one can think of as a small lab for a short time) the principles of SR apply just as they have always done.

There may be other threads here on PF and elsewhere (which one can find by searchig for "cosmological twin paradox") but this one has a lot of links to the literature, which in my mind is a definite plus. This may take more work to read, of course.

15. Dec 10, 2007

### jcsd

I think the cosmological twins pardox is quite easy to resolve as long as you recognize the follwing:

a) there can be geodesic paths of different lengths between two points as seen from the simple examples of the sphere and pervect's 'moutnain' example.

b) the conditions for there to be geodesic paths of different length between two points are actually very weak as again illusrated by the simple examples. These conditons are certainly too weak to be imposed on GR.

c) free falling observers have worldines which are geodesic paths in spacetime.

d) the path length of an obsevers worldline between two events is the proper time experinced by that observer.

As long as you recognise these four relatively simple statements then soemthing like the cosmological twin paradox is not a problem.

I think I first heard about the cosmological twin paradox in the very thread that Pervect has linked to above, but the explanations were clear enough for me to see immediately that it isjust another one of those apparent but false paradoxes that occur in relativity.

16. Dec 12, 2007

### jcsd

My last post was about the cosmological twin paradox which is a general relativistic varaition on the twin paradox.

In special relativity there is only one geodesic path between any two events, so what I've said simply does not apply. The resolution of the vanilla (that is the original special relativistic version) twin paradox comes from recognising the worldine of one of the twins is not a geodesic path in spacetime.

The difference then between the vanilla and cosmological twin paradox is that in the vanilla twin paradox it is the curvature of one of the twins paths that breaks the symmetry between them, in the cosmological twin paradox it is the curvature of spacetime that breaks the symmetry between the twins.

17. Dec 12, 2007

### KingOrdo

There is a thread, about a year old, that discusses the twin paradox in weird geometries. There are a lot of links to the arXiv there, and in fact the papers cited have some excellent precis of the problem.

Some people may tell you to take it on faith that the paradox can be resolved without difficulty in these geometries. But I'm not one for faith-based arguments. I did some digging into the literature and e-mailed several physicists (one a prominent relativist who got his Ph.D. under Wheeler). They each gave me a different "answer" to the problem (though one admittedly flatly he had no idea how to resolve it). Objectively, it appears the paradox persists and is an outstanding and important question.

18. Dec 12, 2007

### Staff: Mentor

As long as you can compute the spacetime interval along the various geodesics connecting the two events I don't see what could possibly be paradoxical. The one with the larger spacetime interval will have experienced the greatest proper time.

I'm not one for faith-based assertions that there is a paradox either.

EDIT: IMO the easiest way to resolve the twin paradox in SR is not by virtue of the acceleration but by virtue of the spacetime interval. Although the acceleration breaks the symmetry, acceleration by itself doesn't cause time dilation, so it gets confusing to students.

Last edited: Dec 12, 2007
19. Dec 12, 2007

### Garth

The paradox is that according to the principle of 'non-preferred frames of reference' it is impossible to decide between the two geodesic paths from one event to another which one it is that has the longer elapsed proper time without referring to the 'outside' distribution of matter that determines the space-time in which the geodesics lie.

The paradox is real in GR IMHO and can only be resolved by recognition of the Machian concept.

Garth

20. Dec 12, 2007

### Staff: Mentor

Really? On a 2-sphere it is unambiguous which great-circle (geodesic) between any two non-antipodal points is the longest. No reference to anything external is required, it is inherent in the geometry on the 2-sphere. Why is it different in GR?