But when Albert Einstein put his the special theory of relativity in September 1905.
Length contraction become a clear topic , because this contraction can be easily explained by using the special theory of relativity.
The second postulate in the special theory of relativity says that the speed of light in the vaccum is constant for all observers, regardless the motion of light source an observer, but you have to remember that any observable mass can’t travel with the speed of light.
but what would happened to the object which is moving with the speed of light. If a object is travelling with c then some physical dimensions of the object changes.
The physical dimensions which goes into change, is it’s length. It is observed that when does a object travel with a very high speed or with the speed of light, then the length of the object decreases.
Here a illustration is given to understand the length contraction phenomenon with different speed. The first pic have normal speed, and second pic have 80% of speed of light, and third pic is 99% of speed of light.
You can clearly see phenomenon of length contraction.
Length contraction is the phenomenon in which the the original length of the object decreases when they are traveling very fast.
This phenomenon is also known as Lorentz contraction or Lorentz Fitzgerald contraction. This contraction always occurs in the direction of propagation of objects.
This effect is negligible at daily basis speeds but, This effect become more significant when they are travelling with the speed of light.
If length contraction occurred due to the increasement in the object velocity in the direction of the motion. Then anyone can freely ask that why does length contraction phenomenon occurs? Why does object length decreases if it is travelling with very high speed? We all know that length contraction is true event not a illusion but many of us think that it is a work of illusion. If it is not a illusion,then how can we explain it.
Undoubtedly it’s explaination is somehow complicated if we try to explain it theoretically without any use of special theory of relativity, so here today we are going to explain it with the help special relativity. Let’s begin…
Let’s consider a two frame of reference S and S’ , having coordinates (xyz) and (x’y’z’)
S’ frame is moving with constant speed of velocity of v with respect to frame S along x axis. A metal rod of length L0 which is its proper length, put on the frame S’. If frame is moving with v velocity then this rod is also moving with v velocity along x axis.
Let’s marked the two ends point of rod is A and B. And if the distance of point of A and B with respect to S frame is x1 and x2. Then their proper length can be given as L0=x2-x1
And the distances of point A and B wrt to S’ frame is x1′ and x2′. And we know object is moving with v which is very high then it suffer with length contraction. And the contracted length is given as L. Which is given as L = x2′-x1′.
Now we have to use Lorentz Transformation equation to find the relation between x1 to x1′ and x2 to x2′.
The general equation of relation is given as below.
x = (x’+ vt’)/√[1-(v^2/c^2)]
Obtain the two relations by putting x1 and x2 in the above equation.
x1 = (x1’+ vt’)/√[1-(v^2/c^2)] and
x2 = (x2’+ vt’)/√[1-(v^2/c^2)]
Now subtract x1 from x2 to find the proper length of the rod.
L0 = x2-x1 =(x2’+ vt’)/√[1-(v^2/c^2)]-(x1’+ vt’)/√[1-(v^2/c^2)]
If we solve it then vt’ term get cancelled. And new equation will be.
L0 = ( x2′ – x1′)/√[1-(v^2/c^2)]
And we know that x2′-x1′ is equal to L which is contracted length. Then we get
L0 = L/√[1-(v^2/c^2)] if we find for L then it becomes L =L0×√[1-(v^2/c^2)] and here we clearly see that L is smaller than L0. This is value of contracted length if it is travelling with very high speed. If we put v = c in the above formula then we find that L become 0. Means objects having the speed of light then it’s length contracted to infinitly very small , which is somehow equal to zero, according to the equation.
So this is all about THE LENGTH CONTRACTION.
