# Thread: time dilation vs length contraction

1. ## time dilation vs length contraction

HI there- i was trying to explain relativity to some friends at a dinner party last night- that the speed of light in a vacuum is constant and therefore- if you have headlights on your spaceship... either time has to slow down, or length contract so that the speed of the photons emitted still equals c: given that speed = distance/ time.

It is dangerous to ask me (non-physics degree) any physics-related questions at dinner parties... but everybody else knew less than me!

So my question is:
1) is the statement above correct?
2) if true- how can you work out how much time dilates vs how much length contraction you get? couldn't you make the equations match with JUST length contraction without any time dilation? Or do they both happen in equal proportions?

thanks,
Plant

2. An excellent way to derive time dilation and length contraction is to use a "light clock," and then you can see that time dilation and length contraction are two sides of the same "constant c" coin. A light clock is two mirrors separated by a rigid rod of known length (in a frame moving with the clock), and a pulse of light bouncing back and forth between them, counting "ticks". Since the key postulate we can invoke here is the constancy of c, this would appear to be a perfectly good clock to an inertial observer moving with it (the "inertial" just means all our observers and clocks move with constant speed, and there's also no gravity around, so we are doing special relativity here).

Now imagine a second observer that sees the clock as being in motion perpendicular to the direction between the mirrors. The clock moves from point A to point B, and the light pulse follows a zigzag path. The two things that the two observers here must agree on are:
1) the pulse is moving at c at all times
2) the number of times the clock ticks between points A and B
It turns out they will also agree on the length of the rod between the mirrors, which might not surprise you, but hold that thought.
The thing the observers will not agree on is:
1) the time between each tick
2) another thing that is important but subtle.
This subtle thing the observers will not agree on requires more explanation. For the observer who thinks the clock is moving to gauge how much time "actually" passed by during its movement, they will need two clocks, one at point A and one at point B. These two clocks must be synchronized for the observer to use both of them to determine an elapsed time. However, the moving observer/clock will not agree that these two clocks are correctly synchronized. This is called the "relativity of simultaneity" and it is crucial-- without it, you will run into paradox after paradox. The take-home message is that if you are using two clocks instead of one, you are not really measuring time, because your synchronization of the clocks is simply choosing a "coordinate system" for time, and coordinate systems are not measurements-- they are arbitrary to the actual physics. So relativity of simultaneity is another way to say that observers in relative motion use different coordinate systems to talk about elapsed times that they cannot measure on a single clock.

OK, back to the light clock. Given the two things the two observers must agree on, you can use formulea as simple as the spirit of v=d/t and the Pythagorean formula for the zigzag path to calculate how many times the clock will tick from the point of view of both observers, and you can use the constraint that the number of ticks must be the same (it's clearly just a fact of the situation) to derive that the observers must perceive a different tick rate (time dilation). That's it, that's all you need-- no math more difficult than v=d/t and the Pythagorean theorem, high school stuff (recently we had a thread where someone was trying to sell the idea you had to know calculus to understand time dilation, but that was just baloney).

OK, so we now have time dilation, and we can resolve the apparent paradoxes it presents with the relativity of simultaneity. Now there's still one more wrinkle-- length contraction. To get that, simply turn the light clock so that the direction of the light pulse is the same as the direction of motion of the clock. So now the light follows a forward-back-forward-back path instead of a zigzag. But again, it's just simple formulae about how far the light goes and how long it takes to get there at c, and you can solve for the number of ticks. Setting that equal to the number of ticks seen by the person who thinks the light clock is stationary, and include the time dilation you got previously, and you find a new surprise-- the two observers must not agree on the length of a rod pointed along the direction of motion, so they don't agree on the span of the light clock here. This is length contraction-- the observer who sees the clock as moving also sees the rod as shortened, and that's the only way to get the right number of ticks, subject to the time dilation we got from pointing the clock the other way (or use two light clocks at the same time, oriented perpendicular to each other).

So both time dilation and length contraction emerge together, as consequences of the need for light to move at c in both these frames, and the need for the observers to agree on the number of times the clock "ticks" between A and B. So the spirit of your statement (1) was correct-- you need both to get consistency with the need for c to be constant, but the way it works out is a little more complicated than perhaps what you said.

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