Imagine you are on a road trip with a friend. You say that you have 20km left to go, while your friend says that you have 30km left. You may disagree, but the disagreement can be resolved by measuring the distance ahead. At least, that is the case when you are travelling at everyday speeds.
However, when travelling at relativistic speeds, which are speeds close to the speed of light, the disagreement between two observers at different reference frames may be significant.
Figure 1. Distances may be described differently by various people, but at relativistic speeds, the distances are truly different.
What is length contraction?
We speak of length contraction when the length of an object travelling at a certain speed with respect to a frame of reference is measured to be shorter than its proper length. Proper length (L0) is the distance between two points observed by an observer who is at rest relative to both points.
Despite the fact that clocks measure different elapsed periods for the same procedure, relative speed, which is distance divided by elapsed time, is the same. This means that distance is also affected by the relative motion of the observer. These two affected quantities cancel each other out, leaving the speed to remain constant. For relative speed to be the same for two observers who see different times, they must also measure different distances.
Length contraction is the phenomenon that states that when an observer is moving at a speed close to the speed of light, distances obtained by various observers are not the same.
How do you calculate length contraction?
Having discussed what we mean by length contraction and proper length, let’s look at an example to explore how to calculate length contraction.
Let’s say a spaceship is moving at a velocity v that is close to the speed of light. An observer A on the earth and an observer B in the spaceship will observe different lengths for the distance covered by the spaceship.
Fig. 2 - Length contraction
We know that the velocity of the spaceship is the same for all observers. If we calculate the velocity v relative to the earth-bound observer A, we get:
\[v = \frac{L_0}{\Delta t}\]
Here, L0 is the proper length observed by the earth-bound observer A, while Δt is the time relative to the earth-bound observer A.
The velocity relative to the moving observer B is:
\[v=\frac{L}{\Delta t_0}\]
Here, Δt0 is the proper time observed by the moving observer B, while L is the distance observed by the moving observer B.
The two velocities are the same:
\[\frac{L_0}{\Delta t} = \frac{L}{\Delta t_0}\]
We know from time dilation that t = t0. Entering this into the previous equation, we get:
\[L = \frac{L_0}{\gamma}\]
We also know that:
\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]
Inserting y, we get the equation for length contraction as shown below:
\[L = L_0 \cdot \sqrt{1 - \frac{v^2}{c^2}}\]
Length contraction of an object’s length
One of the consequences of length contraction is that if an object is moving at a velocity near the speed of light, its length may be observed to be less than its proper length by an observer who is at rest relative to the motion. Let’s consider the following example.
Fig. 3 - Length contraction
Take a 10cm stick. Its length will no longer seem to be 10cm if it passes you at a speed near the speed of light.
The length of the stick while at rest is referred to as its proper length. When the stick is moving near the speed of light, the length measured will always be less than the proper length. When the stick’s speed is equal to the speed of light, the stick should, in theory, have no length.
What is an example of length contraction?
A great example of length contraction is when an object is travelling through space, as in the following example.
Let’s imagine an observer is travelling from the blue planet to the red one and travelling at the speed of y=30.00. The distance between the two planets is 4,000 light-years as measured by an earth-bound observer. What is the distance relative to the observer on the spaceship in measured kilometres?
Fig. 4 - Length contraction
If 4,000 light-years is the distance measured by the earth-bound observer, then this is the proper length L0. As we said, the relationship between proper length L0 and the length observed by the moving observer is:
\[L = \frac{L_0}{\gamma}\]
Adding the known variables L0 and y gives us:
\[L = \frac{4000 ly}{\gamma} = 0.1333 ly\]
1 light-year equals 9.46 ⋅ 1012 kilometres.
\[L = 0.1333 \cdot (9.46 \cdot 10^{12})\]
Hence, \(L = 1.26 \cdot 10^{12} [km]\).
Length Contraction - Key takeaways
- Length contraction is the phenomenon that occurs when the length of an item travelling at a certain speed is measured to be shorter than its proper length.
- Proper length (L0) is the distance between two points observed by an observer who is at rest relative to both points.
- If an object is moving at a velocity near the speed of light, its length will be observed to be less than its proper length by an observer who is at rest relative to the moving object.
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