The Dog School of Mathematics Presents

6. The Light Clock

 The Light Clock

The light clock consists of a bar exactly 1/2 light second long with a perfectly reflecting mirror at each end. A light pulse is sent out from the "bottom" end, bounces off the top mirror and travels back to the bottom mirror. When it hits the bottom mirror the clock "ticks" and the light pulse reflects upward and the whole process repeats.

Since the light pulse travels twice the length of the stick between "ticks" and the length is 1/2 light second then we deduce that the ticks occur one second apart. At least in a frame in which the light clock is not moving.

Now we let the light clock move to the right with velocity v (this can be accomplished by having the light clock moving to the right or by having ourselves moving to the left to velocity v.) Now as the light clock speeds by the light pulse no longer goes straight up and down (in our frame) but zigzags up and down as it makes its way towards the right.

Let's watch the process in a "multiple exposure"

 The Moving Light Clock

Since the clock is moving the light pulse does not travel the distance of one light second between ticks. It travels in a slanted path and therefore goes farther than one light second. But we will observe the light pulse traveling at c [the second postulate!] since all observers measure light to have velocity c.

Since the light pulse travels farther in our frame than in the clock's frame but travels at the same speed in each frame, we are forced to conclude that the time between ticks is longer when measured in our frame than when measured in the light clock's frame. If a clock takes LONGER between ticks then it must be running SLOWER.

BUT WAIT

BUT WAIT (1): "But wait, isn't this just an artifact of light clocks? Wouldn't, maybe, some other clocks run "correctly" without slowing down??" Since the clock keeps correct time in its "proper" inertial frame any other clock that keeps correct time sitting next to it in the same frame will be synchronized with the light clock. [NOTE:  these clocks are together. The relativistic effect of synchronized clocks at a distance being unsynchronized when they are moving is not a point here.] Thus all these clocks will "tick" together. Thus ANY clock will go slower if it is measured from a frame in which it has velocity v. This will also be true for biological processes as well as psychological processes. Anyone moving along with the clock that we measure as running slow will rightly consider herself at rest [postulate 1!] and therefore notice no differences due to the motion
since in her frame there is no motion. Since any "clock" will be measured as running slower we can only conclude that "time" itself will be running slower in the frame of clock A.

BUT WAIT (2): "But wait, couldn't maybe the clock get shorter in the direction perpendicular to its travel (shorter up and down in the figure)?? Maybe then the light pulse would travel the same distance between ticks??" This could only be true if all objects got shorter in a direction at right angles to their motion. We learned that objects get shorter in the direction along their motion.  [See SR (4)] but there would be problems if they also contracted in the direction perpendicular to their motion. We can make this clear with a thought experiment.

Imagine two meter sticks. Stick A at rest and stick B moving. If we position things so that the 1/2 meter mark of stick A moves over the 1/2 meter mark of stick B then we can see if the ends match up as stick B passes by.

 Stick Moving Perpendicular to its Length

If there were sharp knives on the end of each meter stick then the knives in the shortened stick would cut the ends off the not-shortened stick. This would be observed by all observers and thus we could know who was moving and who was really stationary. This, however, contradicts the first postulate. By the first postulate we are completely justified in looking at the same incident from the point of view or the formerly moving stick (A). In A's frame the formerly stationary stick (B) is moving and if moving sticks are shorter in the direction perpendicular to their motion the now moving (but formerly stationary) stick (B) would cut off the ends of stick A.

Now either stick A has its ends cut off or not. Similarly with stick B. Thus the only consistent rule is that there is no contraction (or expansion for that matter) of things in a direction perpendicular
to their velocity.

By a similar symmetry argument we can also conclude that clocks which are synchronized in their proper frame will be synchronized when measured from a frame in which they are moving in a direction perpendicular to the line joining them.

 Two Clocks Moving Perpendicular to the Line Joining Them

Any argument we can imagine that says that clock A will be behind clock B could be applied to the same situation when we view it "standing on our heads." Thus by the symmetry of the situation we can conclude that if clocks A and B are synchronized in their proper frame then they are synchronized in the frame in which they are moving perpendicular to the line joining them.

[Note that this argument falls down if the clocks are moving in a direction parallel to the line joining them. In that case there is a "front" clock and a "rear" clock. The fact that one clock is in front of the other breaks the symmetry. The same idea works for the contraction of lengths along the direction a body travels. If we had knives on the ends of the meter sticks they would cut each other down the middle lengthwise. The end result would be sliced up meter sticks. This would be true in any frame.

Note also this argument falls down if there is something special about clock A over clock B. For instance if there were a gravitational field in which clock A was "uphill" from clock B, then there would by no violation of symmetry to say that one clock ran slower than the other. In fact, this is exactly what happens in the General Theory of Relativity which was developed by Einstein and published in 1916.

Thus we have THREE strange aspects of the behavior of clocks and meter sticks [and therefore space and time since their operational definitions are tied to the behavior of clocks and meter sticks]

THE REAL RULES

1) [relativity of simultaneity] Clocks which are synchronized in the
frame in which they are at rest are not synchronized when measured in
a frame in which they are moving in a direction parallel to the line
joining them. In that case the front clock will be behind the rear clock.

2) [length contraction] An object with length L when measured in the frame
in which it is at rest will have length less than L when measured in a
frame in which it is moving in a direction parallel to its length.

3) [time dilation] A clock which takes time t between ticks in the frame
in which it is at rest will take time t' between ticks when measured in a
frame in which it is moving.

We must take all three of these phenomena into account when analyzing things relativistically. The psychologically easiest one to ignore is RULE 1). We are so used to the concept of "now" being absolute that we find it difficult to imagine it not necessarily being so. Most of the "paradoxes" that are attributed to relativity by those who don't understand it are made by forgetting RULE 1).

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