# Why must the measurements of length be done over an instantaneous moment?

With respect to length contraction in special relativity, in order to measure the length of something (a meter stick say) in motion with uniform velocity to the right with respect to our stationary frame, it must be done in an instantaneous moment. My only guess is the ends of the stick must be measured (located on coordinates) at the same time. By "locate on coordinates," suppose we are standing on the ground facing a large sheet of graph paper in metric. Suppose the meter stick mentioned above were to fly horizontally to the right past us. We would need to measure both ends at the same time on the graph paper by marking them on the paper. If the right end is measured first and then the left end a few moments later, the length of the stick will measure shorter than it should be; this is due to error and not relativistic length contraction. Likewise if the left end is measured first and then the right end, it will measure longer than it should be and there is no such thing as length dilation.

So why must the length of something moving in another frame of reference be done in an instantaneous moment if my reasoning is wrong?

• I thought I answered my own question too but I worry I don't understand something. – Michael Lee Aug 12 '17 at 11:45

## 1 Answer

Well, you answered the question on your own. If you are in rest reference farme, it doesn't matter what time you measure the position of both edges. However, if the object is moving, of course you must measure both at the same time; if you don't, one end has "escaped" in the meantime and you'll measure a different lenght. It's okay, if you do know the velocity te object is moving with, you can also measure it in different times and then apply the correction, as the common sense says. There's nothing abour relativity here yet.

Special relativity enters here, not before. What Sp. Rel. tells you is the relation between those two measurements: $L_{rest}=\gamma L_{mov}$. Classically, both measurements (at rest, and moving but taken at the same time) should be the same, but in special Relativity they are different.

• You're right! When the stick is at rest, you can measure the ends in any order and over any time interval whereas if its in motion, the ends must be located simultaneously! Thank you so much! – Michael Lee Aug 12 '17 at 11:49