A phenomenon in which the length of a moving object is measured to be shorter than its proper length is known as length contraction. In this article, you will read about the length contraction, its derivation and the theory of relativity.

What is Length Contraction?

Length contraction is basically the length measured in the rest frame of that object. It is also acknowledged as Lorentz–FitzGerald contraction or Lorentz contraction and is generally only perceptible at a considerable fraction of the speed of light. The length contraction phenomenon is only considered in the direction in which the object is travelling. Length contraction only becomes significant when the object approaches the speed of light relative to the observer. At everyday speeds, this effect is negligible for standard objects and can be overlooked for all regular purposes.

The shortening of the measured length, length of contraction (L), of an object moving relative to the observer’s frame is given by: L=L₀√(1−v²/c²)

We find its length L to be smaller than the proper length L_o, if we measure the length of any object moving relative to our frame, where L_o is the measured length of the object when it is stationary.

Derivation of Length Contraction

Suppose a cosmic ray colliding with a nucleus in the upper atmosphere of Earth produces a muon; then, to the Earth-bound observer, the velocity relative to the muon is given by

V = L_o/Δt

Since the object is moving relative to this observer, the time (Δt) is relative to the Earth-bound observer, and the velocity relative to the observer who is moving is given by

V = L/Δt_o

The moving observer observes the proper time Δt_o when he travels with the muon, and hence, the two velocities are identical; therefore,

L_o/Δt = L/Δt_o

As we know that Δt = γΔt_o, into the relationship substituting this equation gives:

L=L_oγ

Finally, we will get an equation relating the distances measured by different observers by substituting for γ

The shortening of the measured length, length of contraction (L) of an object moving relative to the observer’s frame is given by:

L=L_o√(1−v²/c²)

What is Proper Length?

When an observer measures the distance between two points and is at rest relative to both of the points, then such length is termed as proper length L_o.

Therefore, the proper length L_o will be measured by the Earth-bound observer because the points are relative to the Earth, that is, the point at which the muon is produced and the point at which it decays. The distance L observed through the muon is not the proper length because to the muon, the Earth, air, and clouds are in motion.

Special Theory of Relativity

It is a theorem that deals with the space-time structure, and this term was first presented by Einstein in the year 1905. Based on the following two hypotheses, Einstein explained this theory –

• Irrespective of the velocity of the observer, the laws of physics are equal for all.
• Irrespective of the motion of the source of light or the motion of the observer, the speed of light will always remain constant.

The foundation of time travel was laid by the Special Theory of Relativity According to Einstein, with the increase in velocity of the person, the rate of time ticking decreases. But this is tough to notice because, as compared to the increase in time, the decrease in the time is relatively very low. So, it can be presumed that you will be in a situation where time is still if you can run equal to the velocity of light. This phenomenon is termed as Time Dilation.

There are other amazing significances of this theory –

• Relativity of simultaneity: In a relative motion, two actions which are simultaneous for one person may not be simultaneous for another person.
• Shrinking of length: With respect to the observer, the objects appear shorter in the direction that they are moving.
• Mass-Energy Equivalence: E = mc², where E stands for energy, m for mass, and c for the velocity of light. This equation clarifies that the kinetic energy divided by the square of the speed of light is equal to the bigger relativistic weight of the object.

General Theory of Relativity

The General Relativity theory developed by Einstein states that being in acceleration or being at rest in the gravitational field, these two activities are physically identical. For example, a viewer can observe or see a ball falling on the rocket the same way as on the Earth. The reason behind this is the acceleration of the rocket, which is equal to 9.8 m/s². Special Relativity and Newton’s gravitational theory are related to this theory.

The consequences of the General Relativity theory are listed below:

• The passage of time is affected by gravity. Clocks, in general gravitational levels, run faster than the deeper gravitational wells. This is also known as Gravitational Time Dilation.
• Gravitational fields can be bent by light rays.
• The parts of the universe are moving away from Earth faster than the speed of light because the universe is expanding.

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