# Length contraction, front middle or back

I still don't have a solid understanding of Length contraction. Imagine we have a ruler of length $L$ that starts at rest upon a ground with markings on it, then accelerates until nearly the speed of light.

For a stationary observer, the ruler accelerates at $a$ metres per second squared. Which of the points then, the red, blue, or green points is the one accelerating at $a$? The way I envision it, the faster it gets the more length contracted it is, so the accelerations at the red green and blue points must be different, depending on whether the ruler contracts to the front or to the back, or the middle. If the ruler contracts to the front, the red point is accelerating at $a$, and the green's acceleration must be greater to keep up. If the green point is accelerating at $a$, the red's acceleration must be less to contract. Or perhaps, there is no problem at all, it just depends on which point we take as the reference? Generally would we then take the blue point as the reference when we say a ruler is accelerating at $a$, and that the green point's acceleration is greater than $a$ and the red point's acceleration is less than $a$? This seems an ok solution. But then consider this problem instead, two rulers start side by side and accelerate to very high velocities.

Both rulers have rocket boosters, and therefore must to a stationary observer accelerate at the same rate. But when we say this, which acceleration do we mean? Does this mean that since the rocket boosters are at the back of the ruler, that the back of the ruler accelerates at $a$, meaning the front of both rulers must contract to the back, meaning that there will become a space between the rulers to a stationary observer? If this makes sense, than that means that wherever I place the boosters, there will always be a gap developed between the two rulers? I'm not too sure.

• Note that acceleration of a rigid raises all sorts of interesting issues. See this paper on the Arxiv a discussion. If you're only interested in steady speeds that simplifies things. If so can you adjust your question accordingly. Commented Dec 11, 2014 at 11:00
• If you want the ruler to accelerate in a way that ensures that the length will stay constant in the instantaneous comoving inertial rest frame of any point on the ruler, then the ruler must accelerate in a "Born rigid" way, which involves different points along the ruler having different proper accelerations (the back end's would be greater), and different instantaneous velocities at a given moment in any inertial frame. In this case different points on the ruler accelerate like different Rindler observers. Commented Jan 11, 2015 at 0:56
• Commented Apr 30, 2022 at 9:33

For a stationary observer, the ruler accelerates at $a$ metres per second squared.

One must be careful here.

If all points of the ruler have the same coordinate acceleration $a$, then the ruler length remains $L$ for the stationary observer.

If, on the other hand, the ruler is observed to contract, different points of the ruler have different coordinate acceleration.

Also, note that if $a$ is constant, the proper acceleration $\alpha$ of the ruler diverges as the speed, according to the stationary observer, approaches $c$.

Finally, note that if the points of the ruler maintain constant proper distance and constant proper acceleration, the points of the ruler are Rindler observers which means that the points 'to the rear' of the ruler must have greater proper acceleration than the points 'to the front' of the ruler.

All of this is to say that acceleration of extended bodies is complicated in SR and one must be careful to distinguish between coordinate and proper acceleration.

• If my first statement was correct, does that mean my second statement, that the two rulers will diverge no matter where the boosters are placed, is also correct?(unless the boosters are at red and orange points) Commented Dec 11, 2014 at 20:13
• @JoshuaLin, assuming the two rockets have the same proper acceleration and begin accelerating at the same time in the stationary frame, the (coordinate) distance between the rockets will be constant in that frame. However, according to either rocket, the distance between the rockets will increase. Commented Dec 11, 2014 at 20:37