relativity, clocks, and football

Recently (in january 1997), this query came up on sci.physics.relativity. (I've inserted my answers in yang444's query, below.)

yang444@pacbell.net
<32f111da.25817373@news.pacbell.net>
The standard scenario: In Inertial Reference Frame (IRF) S, there are observers, O1, O2, etc. all over the place, at rest in S, with synchronized clocks. O1 is at the origin of S. An observer, O', is at rest w.r.t. S', moving w.r.t. IRF S. He is at the origin of S'. As O' passes (coincides with) O1, they both take note that their clocks read t'=0, and t=0, respectively. [] When, later on, O' co-incides with O2, the two observers simultaneously read the two clocks.
The questions: Does this experiment make sense?

Yes.

Do the clocks agree or disagree?

The O' and O2 clocks disagree.

Do both observers read the same times on the clocks?

The O' clock reads less than the O2 clock. Both O' and O2 agree on this.

Let's generalize to show the symmetry. We have relatively moving S and S', each with a stream of equally placed clocks C1, C2, C3 and C'1 C'2 and C'3, and so on. The C clocks are synchronous in S, the C' clocks are synchronous in S'. The two streams of clocks are arranged to approach each other: first C1 and C'1 meet, then C1 meets C'2, then C'3 and son on, and vice versa. By coincidence, C1 and C'1 both read zero when they meet (for simplicity in description). Then, the C clocks will will see each C' clock they meet have a reading further and further ahead of their own. But note: each C' clock will also see each C clock they meet have a reading further and further ahead of their own. But note further: each C clock will see each C' clock it meets is ticking slower than itself. And each C' clock will see each C clock it meets ticking slower than itself.

If that's confusing, let's try to diagram first a similar situation, and then the passing clocks, so you can see graphically how these seemingly contradictory things actually fit together with no problems.

With visions of superbowls still dancing in our heads, consider this scenario. We have two football fields drawn on a single statium ground, with the goals of each field at an angle to the other. Consider this diagram

[overlapping-grids]

It shows this situation, drawn in blue and red. So let's call our overlapping football grids the "blue field" and the "red field". If you walk straight down the blue field, the red yardage markers you cross increase slower than the blue ones. At the same time, if you walk straight down the red field, the *blue* markers you cross increase slower.

For two such walkers then, the yardage markers from the other field they cross increase slower. But the opposite yardage needed to reach a given spot on your field increases faster. The sense of the comparison depends on whether you are tracking yardage lines you meet, or your yardage of somebody walking down the other field. And the effect is symmetric; it works the same way whichever field you pick as "this" vs "the other".

This is the same effect as the streams of clocks (except in normal instead of hyperbolic coordinates). Each yardage line is analogous to a given setting on a stream of clocks.

Now the same situation, but with the kinds of coordinates that SR uses. Consider this diagram

[clocks-meeting]

It shows the C and C' streams of clocks (three clocks shown) in a spacetime diagram. The paths of the clocks are drawn upwards on the diagram in thicker, arrowed lines. The "lines of simultaneity" (lines parallel to the x axis of S and S') are drawn in, and labeled with their clock setting.

On that diagram, we can see that when each C path crosses a C' path, the reading on the C clock of that pair is falling behind. For example, look at the C1 meets C'2 event. C1 reads about 0.5, and C'2 reads about 1. C1 meets C'3: C1=1, C'3=2. But if we follow C'1 instead, we see that at C'1 meets C2, C'1=0.5, C2=1, at C'1 meets C3, C'1=1, C3=2. (Well... the numbers aren't exact, because the diagram was drawn freestyle... but that's the essence.)

So each clock sees the clocks of the other system it encounters increase in reading faster, yet if you look at the S times via the S' lines of simultaneity, you get the usual result that each system sees the clocks of the other system ticking slower. Same events, different coordinate systems.

So. The clock properties that seem so strange are fully accounted for by this coordinate projection effect between two spacetime coordinate systems, just like drawing ordinary x/y coordinate systems at an angle.

For a bit more about spacetime diagrams and how to interpret them, see the comparison between time in SR and bricks; it gives more information on how to interpret spacetime diagrams.