I was watching this video about length contraction on youtube. It takes up an example where a rocket is moving towards a planet with velocity $$v$$ with respect to the earth. From that example, it derived the formula of length contraction. I found the same proof here in this question. The proof is as follows.

"Observers on Earth watch a spacecraft traveling at speed $$v$$ from Earth to, say, Neptune. The distance between the planets, as measured by the Earth observers, is $$L_0$$. The time required for the trip, measured from Earth, is" $$\Delta{t}=\frac{L_0}{v}$$

We now take on the view of an observer in the spaceship:

"The time between the departure of Earth and arrival of Neptune (observed from the spacecraft) is the"proper time," since the two events occur at the same point in space. Therefore the time interval is less for the spacecraft observers than for the Earth observers. That is, because of time dilation, the time for the trip as viewed by the spacecraft is" $$\Delta{t_0}=\Delta{t}\sqrt{1-\frac{v^2}{c^2}}$$ "Because the spacecraft observers measure the same speed but less time between these two events, they also measure the distance less. If we let $$L$$ be the distance between the planets as viewed by the spacecraft observers, then $$L = v\Delta{t_0}$$..."

From this, we get,

$$L = L_0\sqrt{1-\frac{v^2}{c^2}}$$ And,

"The length of an object is measured to be shorter when it is moving relative to the observer than when it is at rest."

Thus the person on the rocket will measure the distance between the planet and the earth shorter than the person on earth will measure. It all makes sense to me. But in the video, it also says that the person on earth will measure the length of the rocket shorter than it actually is, which can again be calculated by the same formula

$$L = L_0\sqrt{1-\frac{v^2}{c^2}}$$

But this time $$L_0$$ is the actual length of the rocket and $$L$$ is the shorter length of the rocket that a person from the earth will measure. Does length contraction apply for both the person in the rocket measuring the distance between planets and the person on earth measuring the length of the rocket? To resolve the twin paradox we concluded that time dilation only occurs for the twin on the rocket. Shouldn't this be the same case here, that the length contraction should only occur for either the person on the rocket on the person on earth?

• Does this answer your question? Length contraction Apr 17, 2022 at 10:36
• I get that the person on Earth will also measure the rocket shorter. But doesn't this create a paradox that since both of them are measuring contracted length, who of them is actually moving?
– Noor
Apr 17, 2022 at 14:40

Length contraction is entirely reciprocal between two inertial frames. If you are moving relative to me, and lengths in your frame seem to me to be shortened by 10%, say, then lengths in my frame will appear to you to be shortened by 10%.

Time dilation is reciprocal in the same way. Suppose four minutes pass on your watch while you coast between two events in my frame, and the time difference between those two events in my frame is five minutes. Given that, while four minutes pass on my watch I will coast between to events that are five minutes apart in your frame.

The twin paradox is another effect altogether, and is asymmetric.