# Time Dilation Question

1. Aug 2, 2004

### Thomas2

Consider the following thought experiment: two clocks in inertial frames A and B are moving relatively to each other with speed v and both clocks are stopped and started through mechanical contacts at the end of the rod to which the clocks are mounted (see illustration).

What times do both clocks show after they have been stopped? A<B, A>B or A=B ? (Note: it shouldn't matter if A or B or both turn around to compare the times as the clocks are already stopped then).

2. Aug 2, 2004

### Janus

Staff Emeritus
Question: Is this diagram taken from the frame of A, B or another frame? This is important be if it is from A then From B, A will be length contracted such that the ends will not align and if it is from B then A will see B as length contracted so that the ends will not align. Also, where are the clocks located on these rods?, in the middle?

You also must realise that the clocks cannot start or stop the instant the contacts are triggered. The information that the contact has been activated cannot get from the contact to the clock at any speed greater than c.

3. Aug 3, 2004

### Thomas2

A length contraction of either rod should be irrelevant for my question as both clocks simultaneously start at the first contact and stop at the second.

The time it takes to relay the information about the contact to the clock merely produces a constant offset which is the same in both systems (if the clocks are in the middle of the rod) as the signal propagates in each reference frame independently.

So my question still stands: do both clocks show different or identical times when they are compared afterwards?

4. Aug 3, 2004

### mijoon

The word "simultaneously" is meaningless in this context. Plotting this experiment on a Minkowski diagram will be enlightening.

5. Aug 3, 2004

### Thomas2

It's only meaningless becaus for the experiment considered (see illustration) both clocks are started and stopped simultaneously by definition (when two point particles collide, they have identical space and time coordinates by definition).

This still doesn't answer my question though.

6. Aug 3, 2004

Staff Emeritus
The clocks at the two ends of the rod are spacelike separated, and there is no simulaneity between them. In all prblems with different points on a rod or different locations in a spaceship you have to be aware of this important relativistic ansatz.

7. Aug 3, 2004

### Staff: Mentor

Of course length contraction is relevant. As Janus points out, your diagram is ambiguous as it's not clear what viewpoint the diagram is from. My guess is that you mean for the rods to be identical--they have the same proper length.

So, I assume you intend identical clocks centered on identical rods.

Right.

If I understand your setup correctly, the clocks will show different times. This should be no surprise, as the arrangement is not symmetric: the trigger for starting & stopping the clocks occurs at the same place in the clock A frame, but at different places in the clock B frame.

8. Aug 3, 2004

### Thomas2

The diagram is obviously drawn from the viewpoint of reference frome A, but as velocities are relative this should be immaterial. You could as well have B resting and A moving or both moving (as long as the relative speed between A and B is v)

If the clocks are in the middle of the rods this shouldn't make any difference as it takes the same time for the trigger signal to reach the clock from either side of the rod.
As the situation is consequently symmetric, it would in my opinion therefore be a logical contradiction if the clocks show different times.

9. Aug 3, 2004

### Zanket

First simplify the diagram. Put clocks directly at the mechanical contacts. Then there is no time delay to stop or start the clocks. In this arrangement B has 2 clocks, but we can let them be synchronized since B’s frame is inertial. Next get rid of A’s rod, which is irrelevant. A can be just a clock with a mechanical contact.

The puzzle is analogous to the twin paradox, in which the answer depends upon how A and B accelerated to attain their velocity relative to one another.

Where A & B were initially at rest with respect to one another and acceleration is non-inertial: If they accelerated symmetrically to attain v relative to each other then A=B. If B remained inertial while A accelerated to attain v, then A<B. If A remained inertial while B accelerated to attain v, then A>B. The clock of whoever accelerated “the most” elapses less time.

Here are rough examples:

Let A & B be rockets that accelerate identically and directly toward each other from the Milky Way and Andromeda galaxies, respectively. Let both rockets attain v relative to each other at the moment of first contact. Then the situation is symmetrical, so A=B in this case.

Let the clocks on B be clocks at the Milky Way galaxy and Andromeda galaxy, respectively. Let A be a rocket launched from earth that accelerated in a giant loop to attain v as it passes the earth in a trip to the Andromeda galaxy. A rocket can in principle traverse between these galaxies in an arbitrarily short proper time, while clocks in the galaxies elapse at least 1 million years. So A<B in this case.

Let B be a rocket launched from rest relative to earth, accelerate in a giant loop to have attained v and have synchronized clocks as the bottom of rocket passes earth in a trip to the Andromeda galaxy. Let the length of the rocket, as measured by us on earth at this moment (when the rocket is length-contracted), be the distance between the Milky Way and Andromeda galaxies (so the rocket straddles the galaxies at this moment from our perspective, or from the perspective of someone in the Andromeda galaxy). Let A be a clock in the Andromeda galaxy. A rocket can in principle traverse between these galaxies in an arbitrarily short proper time, while clocks in the galaxies elapse at least 1 million years. So B<A in this case.

10. Aug 3, 2004

### Staff: Mentor

OK, so the diagram is drawn from the viewpoint of reference frame A. Not only is this relevant, it is critical. Your diagram shows that the A frame measures the B rod as being equal in length to the A rod. Which obviously means that the proper length of the B rod is $\gamma L$, where L is the proper length of the A rod. The two rods are not identical. And if you drew the diagram from B's frame, it would look very different.

Your setup depends on the rods having the precise relative velocity needed to make the contracted length of the B rod equal to the proper length of the A rod. (I liked my version better!)
The situation is still wildly asymmetric for reasons stated previously and because the proper lengths of the rods are different. The clocks still read different times. There is no logical contradiction.

11. Aug 5, 2004

### Thomas2

The whole point of the thought experiment as suggested by me (see illustration) is that no accelerations occur at all. Both observers move with constant speed v relatively to each other and a mutual mechanical contact starts and stops the clocks.

12. Aug 5, 2004

### Staff: Mentor

I agree that your thought experiment has nothing to do with the "twin paradox". Nonetheless, as I've stated, your setup is not symmetric and the two clocks read different times.

13. Aug 5, 2004

### Zanket

In my examples the observers are moving at constant speed v. They accelerated to get to v prior to the moment that they fit your illustration. Observers don’t get to v magically; they accelerate to it at some point in the history of the universe. How they accelerated relative to each other affects whose clock elapses more time even when they subsequently move at constant speed relative to each other.

14. Aug 5, 2004

### Staff: Mentor

previous acceleration is irrelevant

Since the clocks in Thomas2's thought experiment are in inertial frames when they are started and stopped, I don't see how their history of acceleration can affect the times that they read. (The experiment that you analyzed in your earlier post is very different from the one proposed by Thomas2 in this thread.)

15. Aug 5, 2004

### Zanket

In the examples in my post, the clocks are in inertial frames when they are started and stopped and during; in this way the experiments match that proposed by Thomas2. The examples show that the history of acceleration (prior to a clock starting) does affect the elapsed times on the clocks. Take a close look at the examples and see if you can find anything wrong with the conclusions.

16. Aug 5, 2004

### Staff: Mentor

Your examples all talk about time elapsed on the clocks during an acceleration. Who cares? The clocks aren't even on during that time. But even if they were, so what? We are only interested in the $\Delta t$ that each clock reads during the time that they are in inertial frames. Any offset due to their previous acceleration is irrelevant.

Just like with the twins. Sure, depending on their paths through spacetime, they will have different ages when they reunite. But once reunited, their clocks tick at the same rate once again.

17. Aug 5, 2004

### pervect

Staff Emeritus
You need to specify a bit more about how your contacts are working. If they are electrical contacts, you might be able to get the signal transmission up near light speed. If they are actually mechanical rods transmitting a displacement, the signal will travel down the rod at the speed of sound in the rod, which will be a snails pace compared to anything relativistic.

In no case will there be any instantaneous transmission of a signal.

18. Aug 5, 2004

### Staff: Mentor

As far as I can see, the exact mechanism for transmitting the signal from the contact point to the clock doesn't matter. It won't affect the answer to the thought experiment.

19. Aug 5, 2004

### Zanket

They do not. All time elapses in an inertial frame in the examples. For instance, the first example says “Let both rockets attain v relative to each other at the moment of first contact.” Once v is attained the frames are inertial. The clocks begin to elapse time at that moment.

Likewise, in the twin paradox, the twins’ clocks can elapse time differently while they are in inertial frames moving at constant velocity relative to each other. The twin who previously accelerated non-inertially has the slower clock when both are in inertial frames.

20. Aug 5, 2004

### Zanket

But the time it takes can be negligible. Above I suggested simplifying the experiment by putting the clocks directly at the mechanical contacts (switches). Then the transmission time between switch and clock can be infinitesimally small.