**Time Dilation**

We’ll start with time. Here’s a clock.-

Think of an hourglass, then replace the top and bottom with facing mirrors, and the sand with a rather large, glowing, and improbably slow photon bouncing between the mirrors once every time unit. We have one, and others have identical clocks in other reference frames.

It’s night time, and we’re sitting on the laboratory porch watching our clock. A darkened train passes by, another inertial reference frame, moving along at .866 c, a speed that has a Lorentz factor, γ, (gamma) of 2.0, i.e. half the clock rate, half the length, double the mass according to Einstein. Someone on the train has an identical clock, and instead of the vertical line our clock describes, we see the train photon tracing an up and down zig-zag path.

The train photon takes one time unit in the train frame, but the diagonal path we see it taking is longer than the vertical path our own photon takes. Since the rule is that everyone sees a photon traveling at the same speed, the time passage we see between reflections is longer than our own clock’s time unit. Time passes more slowly in the train frame. (Better and more detailed explanations are available on the web)

The odd thing is that the people on the train see our clock tracing out the same diagonal in the dark, so they think our clock is running more slowly than theirs. Their train runs off into the night, and we all are content to let the other go with their slow running clock, because we’ll never see them again, nor will they see us so it doesn’t matter how out of sync our clocks become. This is important in the paradox, and we’ll consider it again.

**Lorentz-FitzGerald Length Contraction**

So clocks run slowly in the other frame. Should that be enough to make everyone see light travelling the same speed, or do we need to apply the Lorentz-FitzGerald equation to length, too?

Let’s buy another clock, upscale this time, with two photons, the second running horizontally. Note that the photons coincide when they strike the corner reflector.

Now let’s put the clock aboard the train, still running to the right. In our lab frame, the horizontal photon at 1.0 c will be chasing the right mirror at .866 c. Without contraction, the vertical photon will have bounced several times before the horizontal photon’s ‘slow’ overtake of .134 c catches the right mirror and bounces. Since both horizontal and vertical photons must get back to the corner reflector at the same time in each frame, an uncontracted train frame doesn’t do it. We find that contracting the horizontal frame by the Lorentz factor, 2.0 in the case of .866 c, combined with the time dilation, gets the photons striking the corner mirror together. What about the far mirrors?

Let’s horizontally contract the train clock and step along the paths of the two photons. This diagram is to scale for .866 c, and overlapping snapshots have been removed for clarity, including the one at lab time 2.00, where the vertical photon bounces. The time is lab time; the dilated train time would be half that.

The horizontal photon strikes the right mirror and bounces at lab time 3.72, not the 2.00 time of the vertical photon. That’s a problem.

It also has to travel the entire distance back to the corner reflector in 0.28 units of lab time, but with the corner reflector moving rightward at .866 c and it moving leftward at 1.0 c, we have a closure rate of 1.866 c. The photon makes it right on time.

These snapshots were taken from the lab frame as the train passed. The continuous paths of the two photons are marked in red and green. At any point, the two paths are of equal length in the lab frame, regardless of how far along it appears they’ve travelled in the train frame.

Whoa, how do we allow a closure rate of 1.866 c when nothing can travel faster than light? That’s the closure rate in the lab frame, but the lab frame still sees the photon at 1.0 c. The train frame will also see the photon moving at 1.0 c, because we must use Einstein’s addition of velocities formula, where the velocities combined is then divided by the velocities multiplied plus 1.

That squares away the corner reflector bounces, but what about the discrepancy at the right mirror?

A digital clock at that bounce would read 1.0 on board the train, which would be 2.0 in lab frame time, the same 2.0 we found at the top mirror bounce. The return bounce at the corner reflector reads 4.0 lab time for both photons, which would be the expected 2.0 train time. Let’s take a closer look at the right mirror bounce.

The train time at the right mirror is different from the train time at the corner reflector, which we zeroed when the two photons departed. The only way to account for the discrepancy at the right mirror is for lab frame observers to see a spread of times along the length of the train, with earlier times toward the locomotive and later times cabooseward. On a long-enough train, the conductor in the caboose would have the on-board lottery results before the engineer in the locomotive even picked his numbers. (This is encompassed in the concept of relativity of simultaneity, which has many explanations on the web if you want more info.)

The spread is like time zones in the U.S., or more properly, like sundial time, a continuous change as you look from east to west. In our close up, the amount of change is .866 time units over a stretch of 1.0 distance units, not coincidentally v/c time units per Lorentz-contracted distance unit. Another piece of the puzzle we’ll use later on.

This non-intuitive spread of time aboard the train frame gives us a way of regarding the contraction we see. The definition of length is the distance measured between two point at the same time in the same frame. For a train 100 meters long in its own frame, in our lab frame the locomotive has not yet hit the point 100 meters ahead of where the caboose is now, and when it does, the caboose will have moved forward from that 100 meter trailing point.

The train observers will also see this spread of clock time in our own lab frame, and this is what allows both frames to regard the other frame as contracted.

Lorentz contraction, then, is as much a matter of time as of length, so getting our clocks set right is important. We’ll hit that next.

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