Experiment: Gradient Colored Rod in Special Relativity

[yinfudan]

The length contraction in special relativity says that a rod moving along its axis will appear shorter by γ to a stationary observer. I think, however, not only the rod will appear shorter, but also each small segment of the rod will show its snapshot of different time as in the moving frame, in other words, the observer see part of rod in its current snapshot, but part of the rod in its past snapshot.

Am I correct? In order to make myself clearer, I come up with this experiment. Could you please tell me if it is valid or if there is any flaw. Thanks!

In a standard configuration, which frame O' is moving along the x-axis of frame O with speed v, clock is synchronized to 0 when origin O' coincides origin O. Also, let us use normalized scale. The unit of time t is second; the unit of x-axis x is light-second; and thus the velocity v = x/t is the ratio of speed with the light speed, 0<=v<1, γ=1 / sqrt(1-v²)

In frame O', there is a rod with length L', resting between x'=0 to x'=L'. The entire rod can change color uniformly with time. For example, the rod's color is blue at time t'=-L'*v but changes to red at time t'=0.

So for a stationary observer in frame O, at time t=0, he will see the rod's length to be L=L'/γ. That is not all. He will also see the rod's left end is red color, but the right end is blue color, and the color changes from red to blue gradually on the rod body.

 

[Fredrik]

You're right. This is easy to see if you draw a spacetime diagram.

 

[JesseM]

Yes, this is another consequence of the relativity of simultaneity. If you have two clocks attached to either end of the rod which are synchronized in the rod's rest frame, then assuming the rod is changing color uniformly in its own rest frame, snapshots of the two clocks showing the same time will always show that the color of the section of the rod next to them is the same too. Because of the relativity of simultaneity, in the frame where the rod is moving the two clocks show different times at any given moment, so at any given moment the section of the rod next to each clock must be a different color (since all frames must agree on local facts about the color of the section of the rod next to the clock when the clock is showing a specific time).

 

[yinfudan]

Thanks Fredrik and Jesse!

Another quick question: Due to length contract, the volume of the rod also decreases. Will the mass density of the rod increase for the observer?

 

[JesseM]

Thanks Fredrik and Jesse!

Another quick question: Due to length contract, the volume of the rod also decreases. Will the mass density of the rod increase for the observer?

If you define "mass density" as rest mass/volume then yes. It's possible there could be other ways of defining density in relativity (for instance, one might define it in terms of relativistic mass instead of rest mass, although this would still result in the density being higher in the frame where the object is moving), I'm not really sure.