Invariance of the y coordinate for a boost along the x axis

[homerT]

Homework Statement

I've been reading through Spacetime Physics by Taylor & Wheeler, but this argument about the invariance of the $y$ coordinate for inertial frames, one moving relative to the other on the $x$ axis, is tripping me up. I'll just write the text word for word:

Let two reference frames be two different latticeworks of meter sticks and clocks, one moving uniformly relative to the other, and in such a way that their $x$ axes coincide. Call one the of these frames the laboratory frame and the other -- moving in the positive $x$ direction relative to the laboratory frame -- the rocket frame. The rocket is unpowered and coasts along with constant velocity relative to the laboratory. Let the rocket and laboratory latticeworks be overlapping in the sense that there is a region of spacetime common to both frames. Test particles move through this common region of spacetime. From the motion of these test particles as recorded by his own clocks, an observer in each frame verifies that his frame is inertial.

A firecracker explodes. The explosion is recorded by the clock in the laboratory lattice nearest to the explosion. It is recorded also by the clock in the rocket lattice nearest to the explosion. How do the coordinates of the recording laboratory clock compare with the coordinates of the recording rocket clock? One result can be derived immediately from the principle of relativity: the recording laboratory and rocket clocks will have the same $y$ coordinates. To show this, let the recording clock carry a wet paint brush the makes marks on the laboratory lattice as it moves past. Figure 12 shows this for the special case, $y=1$ meter. The marks on the laboratory lattice serve to measure the laboratory $y$ coordinate of the $y=1$ rocket clock. These paint marks appear on the $y=1$ laboratory clocks rather than above or below them. For suppose that the paint marks appear on the lattice rods below the $y=1$ laboratory clocks: Then both observers will agree that the $y=1$ rocket clocks passed "inside" the $y=1$ laboratory clocks. Permanent paint marks would verify this for all to see. Similarly, if the paint marks appear on the lattice rods above the $y=1$ laboratory clocks, both observers will agree that the $y=1$ rocket clocks passed "outside" the $y=1$ laboratory clocks. In either case there would be a way to distinguish experimentally between the two frames.

Figure 12

Demonstration that $y$ coordinate of an event is the same in laboratory and rocket frames

I'm just not understanding this argument at all. There is a way to distinguish between whether the lab clocks were painted by the rocket or not if the lines painted at $y=1$ (rocket frame) weren't at $y=1$ (lab frame)? Is that the experiment to distinguish the two frames?

2. Homework Equations
The laws of physics are the same in any two inertial frames

3. The Attempt at a Solution
Just lost trying to understand how non-invariance of the $y$ coordinate makes the laws of physics different in between the inertial lab and rocket frames.

 

[tiny-tim]

welcome to pf!

hi homerT! welcome to pf!

it's a little confusing, but i think what they're saying is this …

all we've stipulated is that the y = 0 line (the x-axis) is the same in both systems

in other words, if one clock on y = 0 was using paint, then we've stipulated that the paint must fall on the clocks on y = 0 in the other frame

we haven't stipulated anything about clocks on y = 1

but if the paint from one clock on y = 1 fell inside the clocks on the other frame on y = 1, then that would be a way to distinguish between the frames, which can't happen, so the paint must fall exactly on the other frame's y = 1 clocks