Question

I know that there is quite a lot of material on length contraction in special relativity (SR), but I am currently confused with the following case that I have not seen discussed before.

Imagine that Sara is inside a train that is parked at the station. And imagine that Sara has two rulers, one always aligned horizontally (along X) and one aligned vertically (along Y). Sara then notices that the window in her cabin is perfectly square with 1-meter sides according to the two rulers. Now imagine that there is also an observer outside the train, who also has an X and Y ruler, and who also measures the window to have 1-meter sides.

Sara goes to sleep, and in the meantime, the train accelerates quite a lot. When she wakes up, the train is now moving at constant speed at nearly the speed of light with respect to the station. I know that for the external observer, the train and all the stuff inside of it have been contracted along the X-axis (the axis of motion). So for the person outside, the train's windows are no longer perfectly square because the base is contracted while the height is the same (so they look like a generic rectangle). However, I am confused about what shape Sara sees for the window after she wakes up.

On the one hand, since Sara is moving together with the window, their relative speed is zero and no contraction should arise (this is the proper length of the base of the window). On the other hand, the Sara who is waking up, and the Sara before going to sleep are effectively using different frames of reference, so I could see an argument saying that they can't be seen exactly the very same X-length for the window. I understand if someone says that the two Saras both see that the X ruler indicates 1 meter for the base of the window because the ruler contracts together with the base, and so the X measurements will always indicate the same outcomes (if it indicates 1 meter at the station, it will indicate the same after Sara wakes up). But would not Sara (after waking up) be able to see that the X ruler looks shorter with respect to the Y ruler even though both indicate 1 meter? And so would not Sara see the window to be not perfectly square? In short, will the two Saras see their windows to be squares or not?
Options:
A) Sara would perceive the window as a perfect square with 1-meter sides, just as before the train's acceleration, as both rulers contract proportionally along the X-axis within her frame of reference.
B) Sara would perceive the window as a rectangle, noticing the X ruler appearing shorter with respect to the Y ruler after the train's acceleration, indicating a deviation from a perfect square.
C) Sara would observe the window as an elongated rectangle due to the increased speed of the train, causing greater length contraction along the X-axis, distorting the square shape.
D) Sara's perception of the window's shape will not change, as the relative speed between her and the window remains constant, implying no further length contraction along the X-axis.

Answer 1

Sara, being in the same inertial frame as the window and her rulers, will perceive the window to be a perfect square as she did before, as measurements within the same frame are not affected by length contraction.

Step-by-step explanation:

Length contraction is a fascinating consequence of Einstein's theory of special relativity. It tells us that objects moving at speeds close to the speed of light will appear contracted in the direction of motion to stationary observers. However, it must be clarified that this contraction only pertains to observers in different inertial frames of reference. Let's focus on Sara's situation in the moving train which is now traveling at a constant speed close to the speed of light. According to special relativity, Sara, who is at rest with respect to the train, would still perceive the window to be square with each side measuring 1 meter using her rulers. This is because she is in the same inertial frame as the train and the window, and therefore the rulers and the dimensions of the window contract equally.

To Sara, the lengths of the rulers along both the X and Y axes have proper lengths, so no length contraction is observed from her perspective. Consequently, the window retains its square shape. Meanwhile, an observer outside the train would indeed see the window as a rectangle, with the side along the direction of motion appearing contracted. The observed contraction from outside the train does not affect Sara's measurements, as she travels with the window and rulers at the same constant high speed.

So, option A is the correct choice: Sara would perceive the window as a perfect square with 1-meter sides, just as before the train's acceleration, as both rulers contract proportionally along the X-axis within her frame of reference.