# The twins paradox and explanation

1. Dec 22, 2007

### Stellar1

Hello,
I'm reading through my textbook and it claims the explanation for the twins paradox is that the space-traveler must experience acceleration during his journey... but could it not be said that, relative to him, the rest of the universe is accelerating in the opposite direction?

2. Dec 22, 2007

### Mentz114

I think that's pushing symmetry too far. The two situations require different amounts of energy for a start.

3. Dec 22, 2007

### JesseM

The twin will experience G-forces when he accelerates, while other parts of the universe that are moving inertially do not, so there is an objective way to decide who has accelerated in SR.

4. Dec 22, 2007

### Stellar1

One more question...

So from the perspective of the traveller, it looks as if time is going by slower for everything around him, correct? (Because in his frame of reference, it is everything else that is moving...)?

5. Dec 22, 2007

### JesseM

The traveller doesn't have a single inertial rest frame...his inertial rest frame before turning around is different from his inertial rest frame after turning around. In the frame where the traveller is at rest before turning around, the Earth's clock is indeed ticking slower than his before the turnaround, but after the turnaround his speed is even greater than the Earth's in this frame, and so his clock is ticking even slower. Likewise, in the frame where the traveller is at rest after turning around, the traveller is moving even faster than the Earth before the turnaround, so his clock is ticking slower until he turns around. The result is that each of these inertial frames predicts that the traveller's total elapsed time between leaving Earth and returning is less than the time elapsed on the Earth's clocks.

6. Dec 22, 2007

### Stellar1

Ok, forgetting about the paradox as it is written... lets simply make it into this: I have a twin brother. I decide to depart on an epic voyage in a straight line at 0.9c. So, in reference with me, time would seem to me to be going by slower for my brother, is that not so?

7. Dec 22, 2007

### JesseM

Yes, in your rest frame your brother would be aging more slowly, while in your brother's rest frame you'd be aging more slowly.

8. Dec 22, 2007

### Janus

Staff Emeritus

Yes, as long as the relative velocity between you and you brother is .9c, then your brother would age slower by your measurement.

9. Dec 22, 2007

### Stellar1

Ok, so if I decided to fly circles around him at 0.9c, given that the radius of my circle was small enough, I could observe him aging more slowely, correct?

10. Dec 22, 2007

### JesseM

No, in this case you wouldn't be moving inertially either (inertial motion = constant speed and direction), you'd be experiencing constant G-forces as you moved in a circle. The usual equations of SR like the time dilation equation only work in inertial frames. If you fly in circles around an inertial observer, then no matter what inertial frame one analyzes the problem in, you will age less than the observer at the center with each orbit (and if they are sending out light and you are watching their image, you will see them aging faster than you as well).

11. Dec 22, 2007

### Janus

Staff Emeritus
No, because in order to fly in a circle you have to accelerate. (acceleration is a change in velocity and velocity is measured in both speed and direction. Therefore changing direction while maintaining a constant speed is still accelerating. ) The rules dealing with acceleration are different than the rules dealing with inertial motion. You would in fact see your brother aging faster in this scenerio.

12. Dec 22, 2007

### Janus

Staff Emeritus
I've got to learn to type faster.

13. Dec 22, 2007

### Stellar1

Ok, I guess I'm just not yet at the level to deal with acceleration.

14. Dec 22, 2007

### Jorrie

You don't have to. I found that the easiest way to wrap my head around some of the issues of the 'twin paradox' is to consider time intervals as per the Lorentz transformations. It tells us that an inertial observer that is present at two events will always record less time between the events (i.e., proper time) than what any inertial observer not present at the same two events will record.

In the case of the twins, there are four events that define the inertial phases of the away-twin's voyage: departure, start turnaround, end turnaround and arrival back home. Only the away-twin is present at the two turnaround events and will hence record the lesser time interval between them and the departure and arrival events respectively. The accelerating period between the start and stop turnaround events can be insignificant if the trip is very long. It is however important to note that for the twins to reunite, at least one of them must accelerate at some point in time, but it can usually be ignored in the calculations.

15. Jan 20, 2008

### Xeinstein

Could you explain why I see my brother aging faster in this scenario?

Last edited: Jan 20, 2008
16. Jan 20, 2008

### tim_lou

imagine a rocket accelerating constantly upward, in my reference frame. One person at the top of the rocket sends a light signal to the bottom. Since the rocket is accelerating up, each consecutive light pulse needs to travel progressively less distance. Suppose that the signal at the top looks like tick... tick... tick, then the signal received at the bottom might look like tick.tick.tick.tick.

Similarly, when you are going in circle, you accelerations inward, and suppose your brother sends you his heart beats as signals... i.e. tick... tick... tick, you would see something like tick.tick.tick.tick. i.e. your brother is aging faster according to you.

Last edited: Jan 20, 2008
17. Jan 20, 2008

### Xeinstein

In your scenario, the direction of acceleration and that of velocity are the same.
But in this scenario, the distance between me and my brother does not change with time
Can we still use Doppler effect to explain it?

Last edited: Jan 20, 2008
18. Jan 20, 2008

### Xeinstein

In the CERN muon storage ring experiment, where the muons had a time of decay depending only on their velocity (in agreement with the time dilation formula) despite the fact that their acceleration was of 10^18g.

19. Jan 20, 2008

### JesseM

No one is saying that the rate of time dilation at any given moment depends on acceleration; in any inertial frame, a clock moving at velocity v at a given moment will be slowed down by a factor of $$\sqrt{1 - v^2/c^2}$$ at that moment. It is nevertheless true that if two clocks move apart and then reunite, the one that accelerated will always show less total time; the total time can be found using the integral $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2 /c^2} \, dt$$, where v(t) is the velocity as a function of time in the inertial coordinate system you're using, and this integral has the property that if you take the v(t) for two objects which start and stop and the same positions and times, the one with v(t) = constant will always yield a greater value than the one with a varying v(t). This can be thought of as a "geometrical" property of spacetime, similar to the fact that when you draw two paths between a pair of points on a piece of paper, the one with constant slope in whatever coordinate system you use (i.e. the path that's a straight line) will always have a shorter total length than the one whose slope is not constant (the path that is not totally straight)--"a straight line is the shortest distance between two points". Similarly, a straight worldline has the longest proper time between two points in the uncurved 4D spacetime of special relativity.

I elaborated on this analogy to plane geometry a bit in post #8 of this thread:

20. Jan 20, 2008

### Xeinstein

I think Janus did say that

Last edited: Jan 20, 2008