Where is the logical hole in this reasoning? (special relativity)

Question

I cannot see the problem in the following reasoning, so I'd be glad if you could help me out.

The problem is defining a notion of "length" that is consistent with the constancy of the speed of light without using the notion of "ruler" or "measuring rod" but just light beams and clocks.

Given two stationary points $A$ and $B$ on the $x$-axis, we can locate the midpoint $M$ by putting a mirror in $A$ facing $B$, another in $B$ facing $A$ and sending a light signal towards both points from some point $C$ in between; if the reflected signal from $A$ arrives at the same time as the signal from $B$ then we are standing precisely on the midpoint of the segment $AB$. (we're supposedly working in an isotropic space-time).

I would like to define the length $L$ of $AB$ by setting $L:=\frac{1}{2}ct$, where $t_1$ is the time interval measured before at the midpoint, as described.

Now, let's suppose we have located the midpoint $M$ of $AB$ and that the segment is moving to the right with constant speed. This time, from an external observer's point of view, the light rays will bounce back from the mirror in $A$ (the leftmost point), then at a later time from $B$ and finally they will meet together at a time $t'$.

Geometry shows that $t'>t$; now I would like to say that the length $L'$ of the moving segment $AB$ is, as before, $L'=\frac{1}{2}ct'$, concluding that $L'>L$: by this idea the moving segment would appear, to a stationary observer, longer than the same segment "measured" by a comoving observer.

Maybe my definition of "length" of an object is non-orthodox, or maybe (surely!) I'm missing some important point here. Thank you for your help/observations!

Answer 1

If you define "length", without exception, as a constant multiple of the time between two events, it will behave no different than the time, really.

Instead, try this for a definition: the time taken for light to go from $M$ to $A$ (say), with a correction for the motion of $A$ during this time, so that we get the length at the "original" time, when light was released from $M$.

In this case, we get that the "length" of $MA$ is: $$L'_{MA} = ct'_{MA} - vt'_{MA} = (c-v)t'_{MA}$$

Now, since you have granted time dilation, we know that:

$$t'_{MA} = t_{MA}\sqrt{\frac{c+v}{c-v}}$$ and therefore, $$L'_{MA} = t_{MA}\sqrt{c^2 - v^2} = L_{MA}\sqrt{1-\frac{v^2}{c^2}}$$

Answer 2

This definition works but only in the frame of the object to be measured. More generally, if you add time as a 4th dimension then a distance is defined between two events and not two points in space. That's probably why you can't generalize your definition to a moving object.