Lengths perpendicular to relative motion are unchanged

[Mosis]

This has been bothering me for a long time. I can see that it's "obviously" true but I'm having trouble arguing it rigorously from what I know about special relativity (i.e. I "believe it" but when I try to (hypothetically) explain it to someone, I can't do so convincingly.)

I first found this problem in Hartle's book, in which he writes:

Imagine two meter sticks, one at rest and the other moving along an axis perpendicular to the first and perpendicular to its own length. There is an observer riding at the center of each meter stick.

(a) Argue that the symmetry about the x-axis implies that both observers will see the ends of the meter sticks cross simultaneously and that both observers will therefore agree if one meter stick is longer than the other.

(b) Argue that the lengths cannot be different without violating the principle of relativity.

So assuming that the observers will agree on which stick is shorter, then the lengths cannot be different. Suppose the observer "at rest" sees a length contraction of the other stick. Then both observers agree that the "moving" stick is contracted. However, the observer in motion is equally entitled to consider himself "at rest," in which case he should say (by the principle of relativity) that the other stick is contracted - contradicting the fact that both observers agree.

The part I'm having trouble with is why the meter sticks crossing simultaneously imply both observers must agree on which stick is shorter. I mean, in the case where the sticks are moving parallel to one another, the relativity of simultaneity in the measurements of lengths is precisely what gives length contraction with each observer claiming that the other stick is contracted, and in this case it's precisely the observers agreeing on which stick is "really" shorter that allows us to conclude that the lengths are not contracted. How does this difference come into the argument, though? What am I missing?

 

[Fredrik]

Endpoints of stick A: (0,0,0) and (1,0,0).
Endpoints of stick B: (0,0,vt) and (0,b,vt) (...where b is 1 m, but we're not supposed to assume that)

This is what you described, right?

This situation is perfectly symmetrical. The observers sitting on the sticks both see another stick, oriented in the up direction, moving to the left (if we imagine that both observers are facing the other stick at t=0 and that A considers "up" to be in the positive y direction while B considers "up" to be in the positive x direction).

 

[Entropia]

Thank you for sharing your thoughts on this topic. I can understand why this may be a confusing concept to wrap your head around. Let me try to provide a more detailed explanation that may help clarify things for you.

First, let's start with the concept of length contraction in special relativity. This phenomenon occurs when an object is in motion relative to an observer, causing it to appear shorter along its direction of motion. This is due to the fact that in special relativity, space and time are not absolute, but rather are relative to the observer's frame of reference. This means that an object's length can appear different to different observers depending on their relative motion.

Now, let's look at the scenario described in Hartle's book. We have two meter sticks, one at rest and one in motion perpendicular to the first. Both observers are at the center of their respective meter sticks. In this case, the lengths of the sticks are perpendicular to the relative motion between the two observers. This means that the length contraction phenomenon would not apply in this scenario, as it only occurs along the direction of motion.

Now, let's consider the concept of simultaneity. In special relativity, the concept of simultaneous events is also relative to the observer's frame of reference. This means that two events that may appear simultaneous to one observer, may not appear simultaneous to another observer in a different frame of reference. However, in the scenario described, both observers are at the center of their respective meter sticks, and the sticks are perpendicular to the relative motion between the two observers. This means that the events of the ends of the meter sticks crossing would appear simultaneous to both observers. This is due to the symmetry of the scenario about the x-axis, as noted by Hartle.

So, to summarize, in this scenario, the lengths of the meter sticks would not appear different to the two observers due to the perpendicular nature of the relative motion. And because the events of the ends of the meter sticks crossing would appear simultaneous to both observers, they would agree on which stick is shorter. This is why the lengths cannot be different without violating the principle of relativity.

I hope this explanation helps to clarify things for you. It is a complex concept, but with a deeper understanding of the principles of special relativity and how they apply in different scenarios, it will become clearer. Keep exploring and asking questions, as it will only deepen your understanding of this fascinating topic.