# A question on a tweek of the Twin Paradox

1. Jul 1, 2013

### Zeno Marx

now the resolution of the twin paradox as put down in textbooks i have read relies on one observer being accelerated and one not so one can claim to be inertial and thus priveledged - this i always felt to be a cheat - even edward teller uses it in conversations on the dark secrets of physics - but what if you had the two twins both accelerated but at different times one accelrated then de-accelerated then brought back to earth while the other one travels a lot longer at near the speed of light - i think the twin who travels the much longer round trip will still age less though they have both experienced the same accelerations - if you think about this this implies some kind of notion of absolute velocity unless you disagree with the conclusion the one who travelled further at near c will be younger in this scenario?

2. Jul 1, 2013

### Mentz114

There is a simple formula to calculate the ageing along a 4-dimensional path in spacetime. So if we know the details of the trips ( an exact itinerary ) we can calculate what their clocks will read. The ad hoc rule you suggest won't always hold, but the proper time calculation always does.

http://en.wikipedia.org/wiki/Proper_time

Last edited: Jul 1, 2013
3. Jul 1, 2013

### tom.stoer

Let's do some math.

Assume we have two twins located at (t,x) = (0,0) in one specific coordinate system. They will meet again at a later time T but at the same location x=0, i.e. at (T,0). The question now is "what are T and T' prime in which coordinate system?".

Now let's avoid coordinates.

Assume one twin is traveling along a curve C from point A to point B in spacetime. The second twin is traveling along a different curve C' from point A to point B in spacetime. Of course we could introduce the coordinates for A and B, but that is not necessary.

Now you have to believe me that the proper time tau of a twin along his curve between A and B is given by the "length" of the curve through spacetime.

$$\tau = \int_C d\tau$$

Here the "length" and therefore the proper time is calculated according to the strange 4-dim. relativistic Pythagoras t² - x².

As the two curves C and C' through spacetime are different for the two twins their proper times will differ.

$$\Delta\tau_{A\to B} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau$$

Edit:

Introducing the above mentioned coordinates (t,x) and the velocity v expressed as v = dx/dt we find for the proper time

$$\tau = \int_C d\tau = \int_0^T dt \sqrt{1-v^2}$$

where v=v(t) can be time-dependent. So for two twins starting at t=0 at A and meeting again at t=T at B their proper times and thefore the time dilation as difference of their proper times can be calculated using a difference of these two integrals. Note that both integrals are expressed in terms of a third coordinate time t which is neither identified with the proper time of the first nor of the second twin. In principle and for a physical definition this coordinate time t is not required; the proper times measured with co-moving clocks are sufficient.

This is the most general formula for defining differences of proper time in SRT. It can be used for arbitrary (timelike) curves C with arbitrary speed v.

Last edited: Jul 2, 2013
4. Jul 2, 2013

### Staff: Mentor

Why? The definition of inertial is pretty clear, as is the fact that the first postulate refers to the equivalence of inertial frames. Everything is up-front and clear, so in what way is it a "cheat"?

The key point of the twin paradox is asymmetry. A sloppy reading of the first postulate may lead a student to erroneously believe that all motion is equivalent, so the two twins are symmetric. This is the source of the paradox. Once it is pointed out that the first postulate only refers to inertial frames, then it becomes clear that the twins are asymmetric. In your "tweek"ed twin paradox the same resolution applies. They have different accelerations and therefore are asymmetric.

Nonsense. I have no idea how you could even begin to think that is a correct conclusion

Last edited: Jul 2, 2013
5. Jul 2, 2013

### Zeno Marx

I was thinking about notions of absolute velocity because I had been reading Lee Smolin's new book "Time Reborn" and he and his mates at the Perimeter institute seem to be proposing a notion of absolute rest which is an observer who will see the CMB the same temperature in all directions so that got me thinking about this

6. Jul 2, 2013

### WannabeNewton

The CMB frame is a privileged frame but it does not define a notion of absolute rest in the sense in which you are describing in post #1.

7. Jul 2, 2013

### Zeno Marx

I still think the proper time integrals unless I have done them wrong support my original naive conclusion though - ie: we have a ceratin "velocity budget" we can spend in either the time or space dimension so the faster we go in space the slower we go in time but this implies some refernce to how fast we are going my point of the tweek to this thought experiment was to remove the prefered observer so the time dilation only depends on your vel;ocity compare d to c over some distance and the observer on earth can;t claim to be inertial - this is kind of a subtle point but the very form of the proper time integral seems to imply knowing how fast you are going but how can you EVER know what your velocity is if all velocity is relative

8. Jul 2, 2013

### Staff: Mentor

If you really want to dig into exactly what the CMB does and does not tell you, you might want to ask around the Cosmology forum - doesn't have much to do with twin paradox, whether tweaked or not. Here's one thread from there.

9. Jul 2, 2013

### Zeno Marx

you have to put a figure in for 'v' but where do you get this from?

10. Jul 2, 2013

### WannabeNewton

Are you asking where the $v$ comes from? $\int d\tau = \int \sqrt{dt^{2} - dx^{2}} = \int dt\sqrt{1 - (\frac{\mathrm{d} x}{\mathrm{d} t})^{2}} = \int dt\sqrt{1 - v^{2}}$.

11. Jul 2, 2013

### Zeno Marx

i tseems we can only ever calculate velocity for others rather than ourselves but accelerations and gravitational feilds sem to slow and speed clocks in a way which is totally path dependant so if you get accelerated to a certain speed your clock locks at that rate and then satys there even if your frame is then inertial - infact i have a wacko suspicion that inertia is actually the resistance to change in the internal time metric the same way pressure and tension are resisatnce to the change in the internal spatial metric but hell that's just me playing with equations and drawing weird conclusions

12. Jul 2, 2013

### WannabeNewton

We can certainly calculate our own velocity: in our frames our velocity is zero. The rest of your post is nonsensical I'm afraid.

13. Jul 2, 2013

### Zeno Marx

I still don;t see how you haven;t put this into the equations by hand at some point - relative to what? we can it seems only draw conclusions about the universe relative to us but nothing about ourselves but the fact our clocks are alterered by gravitational feilds to me says they are also altered by acceleraqtions and velocity then means something in the same sence gravitational potential does?

14. Jul 2, 2013

### Zeno Marx

I mean change in velocity means something in the same sense as change of gravitational potential as far as how fast your clocks go

15. Jul 2, 2013

### WannabeNewton

16. Jul 2, 2013

### Staff: Mentor

You don't have to start with a known v. You can start with the known position of the object in space-time (its x and t coordinates) as a function of any convenient parameter, and then do the integration with respect to that parameter. It may be convenient to use the x and t values from a frame corresponding to an observer moving at a particular speed, but you can choose any speed you please; the x and t functions will take different forms but the integral will come out the same.

17. Jul 2, 2013

### pervect

Staff Emeritus
Newtonian and relativistic physics can both be written in terms of the principle of least action. I.e, there is some quantity , the action which is minimized (or more exactly locally minimized or extermized) by the natural motion of a body.

It turns out that the "action" of a point mass is just - m * proper time. This means that the equations of physics themselves say that a body that moves naturally (i.e. isn't subject to any external forces to make it acclerate) maximizes proper time.

There are several subtle points that I"ve glossed over, such as the difference between "extremal", "maximal" and "minimal", but these are the basics behind the "twin paradox". Learning a bit about Lagrangian physics and the principle of least action will be much, much, much more productive than speculations in the direction of some "absolute velocity".

Last edited: Jul 2, 2013
18. Jul 3, 2013

### Zeno Marx

I'm very much simpatico with least action physics richard feynman being my major scientific hero my point was really - ok maybe the thought experiment i posted didn;t do it properly but the idea wa to ask how you define velocity at all if you eliminate totally the concept of inertial observers because no observer over the course of his/her/its observership is ever completely inertial

19. Jul 3, 2013

### Staff: Mentor

Even if an observer is non-inertial you can still have an inertial reference frame. There is no need to use a frame where a particular observer is at rest, you can always use any inertial frame.

I.e., despite the traditional shorthand way of speaking, observers and reference frames are not the same thing.

20. Jul 3, 2013

### Zeno Marx

but my problem is how you define velocity WITHOUT inertial frames because we are all experiencing accelerations all the time so there is really not such a thing as an inertial frame except in deep space

Last edited by a moderator: Jul 3, 2013