From My Mind

One train car standing still.


Two train cars, one speeding past.

Once you accept that the speed of light is the same for observers in all frames of reference, you're stuck with an odd fact: time has to run at different rates in different frames of reference.

Here's why.

This article assumes you know the basic idea behind Special Relativity: that the speed of light is the same to observers in all all non-accelerating frames of reference. If that sounds mysterious, you can try my first article in this series.

The Setup

We're going leave earth in two identical spaceships. Our spaceships contain some important pieces of gear: A light source that will emit a flash of light, a mirror on the opposite wall to reflect the light, and a detector that registers and times the flashes. (See Figure 1.)

The detector is linked to the light, so it can mark the time the light flashes and the time that it receives the flash. Once the light flashes, the detector begins clocking time like a stopwatch and then stops when it receives the flash.

When the light flashes, the detector marks this as moment T1 (Time #1) and begins counting. In our design, when it detects the light flash, two ticks of its stopwatch have elapsed, so it marks this moment as T3. Because the path to the mirror is the same length as the path back, it must have taken the light an equal amount of time to travel both legs. Therefore, the halfway moment, T2, must be when the light hit the mirror.

When we are motionless and next to each other, our systems operate in sync. Both our lights flash at the same moment, and both our detectors receive the flash at the same moment.

Now It Gets Interesting

If your spaceship sits still and mine zips past you very fast, things change. (See Figure 2.)

My spaceship, on the right, enters at the bottom of the image and zips past you from bottom to top. Meanwhile, you in your spaceship, on the left, watch both our light/detector systems and notice something very weird.

From your point of view, both lights flash at T1. At this point, I'm down at the bottom of the diagram. Halfway through the process, when the light bounces off the mirror, I'm in the middle of the diagram, right next to you. And by the time my light flash reaches my detector, I've travelled to the top of the diagram.

Notice that, from your point of view, my speed makes the path my light must travel appear much longer than it does in your ship. In fact, as Figure 2 indicates, the light path appears to you to be three times as long! Each of the orange arrows represents one segment of the light path, for example, from the light to the mirror.

Now remember, the speed the light travels must remain constant, no matter what.

What that means is that, from your point of view, enough time must elapse for the light to travel three times as far in my ship as it does in yours. It takes two ticks for a complete path in your ship, but it takes four more ticks of your detector for it to make a complete path in my ship.

But remember that from my point of view, the light, mirror and detector are not moving, so the path is just two segments long. And sure enough, according to my detector, the light makes the journey in just two ticks.

So our detectors started together at T1, but when you seem my detector fire, yours says it's now T7 while mine says exactly what I would expect it to say, T3.

The only explanation is that time is running slower for me than for you from your point of view. Your clock shows T7 whereas mine shows T3!

Now It Gets Weird

But remember that motion is relative. I can just as easily claim that you are zipping past me, and I am standing still. From my point of view, the whole diagram is reversed, and it is your clock that is running slow.

This paradox lies at the heart of one of the weird things about spacetime: there's no such thing as simultaneous when you're considering two frames of reference.

From my frame of reference of view, my clock reaches T7 when I see yours reach T3. From your frame of reference, when my clock reaches T3, yours has counted to T7.

If this sounds incredibly weird, well, it is! Part of what's happening is that it's false to say our clocks both started at T1. It just looks that way from our different frames of reference.

(That things in different frames of reference can never be considered simultaneous is interesting enough to deserve it's own article and diagrams. Perhaps that will be part three in the series.)

The Twins Paradox

Suppose two twins, Al and Bob, build a very fast spaceship. Al stays on earth, while Bob takes off for a fast flight to a few local stars and returns. Because Bob is moving very fast, his clock appears to be running much slower from Al's point of view.

When Bob returns home to earth, he is returning to Al's frame of reference. But now Al is many years older than Bob! Bob's high-speed journey and return have allowed many years to pass on earth, while for Bob only a few years have passed.

Ah, you say, but you just told us it's all relative, so can't Bob claim he stood still, whereas Al and the universe were speeding around his stationary spaceship?

Actually, no. The difference is that Bob needed to accelerate to achieve that high speed. He also decelerated when he slowed back down and returned to earth. This changes the equation, and allows Al and Bob to determine which one of them took the journey.

Look at it another way. Bob speeds up and heads for a distant star. When he gets there, he slows down and lands. This returns him to a slower-moving frame of reference. During the journey, his clock ran slower compared to anyone in a slower-moving frame of reference. He's now many years younger than he would have been had he stayed on earth.

Now Al makes the same journey and joins Bob. The same time dilation effect applies to Al, and Al's clock runs slower while Bob lives out many years on his distant planet waiting for Al. Once Al does arrive, he will now be approximately the same age as Bob, since both have experienced approximately the same time dilation in their two journeys.

© 2008 Chris from MN

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published Sun Aug 3, 2008 1:31 PM CDT

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science, time, light, einstein, relativity, speed-of-light, special-relativity, twins-paradox, programmerdude