Expert: Expert - 11/6/2009

Question
QUESTION: Hi Expert!
I'm not an expert, but am hoping for a bit of a reality check as to where I have gone wrong in understanding Time Dilation.

Allow me to ask the question by giving a scenario.

There are two identical trains with an observer and a clock both ticking away and travelling away from each other at a constant velocity which is close to the speed of light. To make things easier, I will be one of the observers.

I begin with my train right next to the other, and set my clock to the same time as his, as we move off and get close to the speed of light, I will see the other clock slow down. When I reach a constant velocity, the other clock will appear to be at a constant speed which is slower than my own.

At any given point, I will observe the time on the other clock as being behind my own, and that gap will be increasing.

Both trains then stop and begin moving towards each other again, besides the dip in speed difference when we were accelerating, once we reach the same speed but in the opposite direction, I will be able to observe exactly the same difference in speed as I did earlier, and the other clock will still appear to be behind my own.

When we reach our initial point, we stop, and the other clock now appears to be moving at the same speed as my own, but appears to have a different time to my own. The person on the other train will see my clock as behind his, and I will see my clock as faster than his.

That’s a bit too trippy to be true, so where did I go wrong? Or should I just quit while I’m not too far behind and study arts instead?

Thanks heaps for taking the time to consider my silliness!

-   Alex

(oh, and may I ask, why can't we mention the gooey things with shells?)

ANSWER: What you described above is essentially correct. HOWEVER, you left something out: namely, acceleration. Instead of two clocks on two trains, each train moving away from each other, let's have three clocks, all of which start at rest on the surface of our Earth. Two of the clocks are put on the rockets, which ACCELERATE away from the third clock, but in opposite directions. The two clocks quickly reach an identical speed near that of light, and travel for an identical time (on their clocks). They both ACCELERATE (ie, change velocity) and travel back towards our Earth at this same near-light speed. When they each get there, they both DECELERATE to zero motion relative to the surface of the Earth. What will we see?

The two clocks will have the same time, and both of these times will be less than that of the clock that never accelerated. I emphasize that it is NOT the case that they "seem" to have less time or "appear" to have less time. There really HAS been less time transpired on these clocks that traveled than the one that did not -- and none of the three clocks can claim to have the "correct" time!

When you do the calculations rigorously, taking acceleration into account, you will get the above results. And we know this is correct every instance when we use a GPS system. How? Because GPS satellites have their precise timing signals adjusted to take this dilation into account -- if they didn't, after a few years GPS would be off by tens (if not hundreds) of meters.



Oh, and the reason I specifically exclude questions regarding eggs is because I (long ago) got tired of getting questions about how to do an egg-drop experiment. I still get them every few months, but that is down from the one or two a month I used to get.

---------- FOLLOW-UP ----------

QUESTION: Thanks for your answer; it has given me quite a bit to think about. I still am not quite there yet, though, if you will allow me to continue.

Apologies for the style I am using, I can’t really work out how to phrase it as a question, so instead I have put it down as an explanation. I’m sure you will read it and be able to pinpoint where I get it all wrong.

I shall need some labels for the clocks, so they are A (the ‘earth’ one which does not accelerate), B and C (on the rockets)

I might actually remove the earth from the equation so as we don’t have to worry about any gravitational effects on time.

At t0, all three clocks are in the same place and show the same time, and are moving at the same speed in their shared frame of reference.

At t1, B and C begin to accelerate at opposite directions and equal acceleration with respect to A. At this point dilation due to acceleration occurs and so B and C will be moving slower than A.

At t2, B and C are still accelerating, and have now gained speed. The acceleration will still be causing dilation so that B and C will be moving slower than A, and the speed will mean that B appears slower than C from the reference point of C (and C to B).

At t3, B and C stop accelerating.

The observer at clock A will notice that B and C show the same time, and are ticking at the same speed as each other, yet they are behind clock A, and moving slower than clock A.

The observer at clock B will notice that clock A is moving slower than clock B, and will be showing a different time. Taking both the speed dilation and acceleration dilation into account, it could be either ahead or behind depending on the values of the two. Clock C will appear to me moving even slower, and will be behind clock B.
At t4, B and C begin accelerating back towards each other (slowing down relative to A) and continue accelerating until their speed is the same as before the acceleration but is in the opposite direction.  The observations will be the same as in t2

At t5, B and C stop accelerating. The observations will be as in t3 but with greater differences in times.

At t6, B and C begin decelerating with respect to A, observations as in t2. The acceleration will still be causing dilation so that B and C will be moving slower than A, and the speed still will mean that B appears slower than C from the reference point of C (and C to B) The time on B, as it appears from clock C will be far behind clock C.

And at t7, B and C reach a speed of 0 with respect to A, stop accelerating and just so happen to be in exactly the same positions as in t0

An observer will notice that, as you say, B and C show the same time, but are both behind A, since less time has elapsed for them due to the acceleration, but what happened to the observations made at clock C? At what point did clock B make up the lost time? For the entire experiment, the observer at C saw clock B moving at a slower rate than clock C, yet now they show the same time.

Thank You for your help
-   Alex

(if the acceleration were to occur in 20 seconds and the speed to reach half of the speed of light... would an egg survive the forces. Sorry, I had to do it.)


Answer
Permit me to adjust the experiment EVER so slightly

Clock A stays on the Earth.

Clock B and Clock C accelerate away, but at t7, they do not return to Earth but instead to rendevous with each other. They each accelerate to a constant, but near-light, speed and come close enough to each other to observe the other's clocks close up.

The observor in B looks at Clock C and sees it is moving more slowly than Clock B. At the same time, the observor in C looks at Clock B and sees IT is moving more slowly than Clock C -- and both see the OTHER clock slowed down by EXACTLY the same amount!

I presume you are asking, "How can each clock be both behind and ahead of the other one? If one clock is behind a second clock, doesn't the second one HAVE to be in front of the first? Which clock actually *IS* in front?" The answer is a quite contrary to common sense: "NEITHER clock has the correct time, BOTH are in front of the other IN THEIR FRAME and behind the other IN THE OTHER CLOCK'S FRAME." Time, instead of being a constant throughout our Universe, changes depending on its motion relative to a frame. And there is NO absolute frame -- one is just as good as another. And, anytime two clocks are in the same frame, they will advance at the same rate. If they have accelerated and decelerated in exactly the same way relative to a frame in constant motion, then they will show the same difference in time elapsed with that third frame.

I'll say it again -- the clocks in the other frame do not "appear" or "seem" to be moving more slowly, they ARE moving more slowly.

If you are instead asking, "When do the two clocks, that have undegone acceleration, return to being the same?", the answer is that this occurs when they accelerate (ie, change their motion) back to the same frame of motion -- in this case, back to the frame where Clock A has always been. This exact calculation is quite involved; I've never even attempted it. But it's during the deceleration.


Also, an egg going from a speed of zero to half the speed of light -- ie, to 150,000,000 kilometers per second -- in twenty seconds would experience a force of about 750,000 g's. This would crush titanium, and also an egg.