Length contraction clarification

[Trojan666ru]

this is the wrong interpretation of LT. It is measuring one
end of the rod at one place while the other end of rod at
another place.
This is wrong.
Because when you measure the end point, the front point
moves forward to different point.

 

[Trojan666ru]

The formula for spacetime interval is given by,
ds2 =- (cdt)2 + dl2
After a little bit of algebra we get
dt =|ds2|^1/2/c = dt[1
- (v/c)^2]^1/2.
The following
dt = |ds2|^1/2/c
is the expression for the proper time difference not
only for observers at rest in the coordinate system but
for all observers, however they move,

 

[Dale]

What tiny tim said is wrong
The distance between the rods won't contact

No, that part of what tiny tim said is correct (most of what he said is correct, only that one very small post wasn't).

You can write the worldline of the endpoint of any rod as $(t,x)=(t,p_i)$ where p_i is the position of the i'th endpoint in the rest frame. If you do a Lorentz transform on that then you can get the worldline in the primed frame, set t'=0 and see the distances. You will see that they all contract.

 

[Nugatory]

this is the wrong interpretation of LT. It is measuring one
end of the rod at one place while the other end of rod at
another place.
This is wrong.
Because when you measure the end point, the front point
moves forward to different point.

The at-rest observer can calculate the length of a rod by taking the time at which the front of the rod passed over the starting line of the track and subtracting that from the time at which the rear of the rod passed over the starting line. That gives us the time it took for the rod to pass that point, and we multiply that by the speed to get the length of the rod.

The same technique works to get the distance between the rods; we just use the times when the back of the leading rod and the front of the trailing rod pass the at-rest observer (or anything else that is at rest relative to him).

I really really strongly suggest that you try these calculations for yourself.

 

[Dale]

After a little bit of algebra we get
dt =|ds2|^1/2/c

You made a mistake with your algebra. This is not true in general, although it is true for dl=0.