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TLDR: if two unbounded objects are accelerated simultaneously, the ill-named length contraction states that in the proper frame, they move away from each other. But what about in real life where there is no such thing as unbounded objects: does a bus elongate in its proper frame after its acceleration to reach the highway?

Preliminaries:

A system $\Sigma$ consists of two initially unconnected subsystems, A and B.

  • State 1: A and B are stationary in the reference frame $R_0$ and are separated by a distance $L_0$.
  • State 2: A and B are set in motion simultaneously and identically in $R_0$, reaching the same constant velocity $v$ along their initial separation axis (AB). This state is defined such that A and B have been moving at this velocity for a time much longer than their acceleration phase (to avoid any debate about this stage).

Since A and B accelerated simultaneously in $R_0$, they remain at the same distance $L_0$ from each other in this frame. Thus, due to length contraction (or the relativity of simultaneity, whichever you prefer to invoke), in the proper frame of $\Sigma$, denoted $R_p$, the distance between A and B, $L_p$, is greater than $L_0$. This means that in $R_p$, A and B must have moved apart such that their separation increased from $L_0$ to $L_p$.

Actual Question:

Now, let's consider a similar situation, but this time A and B are connected or bounded to each other. As an example, take a spring, though the specific connecting mechanism does not matter.

  • In State 1, where the distance between A and B is $L_0$, the spring exerts no force, A and B do not move relative to each other, and the natural length of the spring is $L_0$.
  • We consider a case with dissipation, meaning that if A and B move relative to each other, they do not oscillate but instead reach an equilibrium position after damping effects.

The question is: What is the distance between A and B in each reference frame, $R_0$ and $R_p$, once $\Sigma$ has reached its cruising velocity $v$?

Dhragva
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2 Answers2

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The presence or absence of a connection does not make a difference to the kinematics. If $A$ and $B$ are accelerated at the same time and by the same amount in the inertial frame $R_0$, then the proper distance between them increases as you described.

The connection does not change that fact at all. What the connection does is it changes the forces that must be exerted on $A$ and $B$ to achieve the specified motion.

It changes the dynamics, not the kinematics. The increased proper distance is part of the kinematics.

Dale
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Yes, a bus moving at speed will be shorter than a bus at rest. It is all a matter of timing. The length of an object is the distance between the positions of its end points at a given moment in time. The positions of the ends of the bus at a given moment in time in a frame through which the bus is moving are the positions of its end points at two different times in its rest frame. Specifically, the front of the bus is considered earlier than the rear, which means that the rear has moved forward in the time gap and therefore the bus is shorter.

The effect applies not just to physical entities but to abstract distances between points in space.

Professor Sushing
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