Physics Lesson 18.5.3 - Quick Recap of Dilation and Contraction

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Welcome to our Physics lesson on Quick Recap of Dilation and Contraction, this is the third lesson of our suite of physics lessons covering the topic of Lorentz Transformations, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Quick Recap of Dilation and Contraction

Let's consider a bar at rest in S' which lies in the X'-axis. The coordinate of the left end of the bar is x'₁ and that of the right end is x'₂. The bar's length when measured in S' is

L' = x^'₂ - x^'₁

We want to calculate the length of the same bar when measured in the system S considered at rest. Let's suppose that we decide to measure the coordinate x₁ of the left end of the bar at the instant t₁ and in another instant t₂ we measure the coordinate of the right end of the bar. Obviously, we obtain different results for

L = x₂ - x₁

because the bar meanwhile is moving.

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Hence, it is more logical to make the measurement of the above coordinates in S in a fixed instant t, in order to avoid issues caused by the bar's displacement. The following equations derive from Lorentz transformations:

x₁' = x₁ - V ∙ t/√1 - V²/c² and x₂' = x₂ - V ∙ t/√1 - V²/c²

Subtracting the first coordinate in S' from the second one, we obtain

x₂' - x₁' = x₂ - V ∙ t/√1 - V²/c²-x₁ - V ∙ t/√1 - V²/c²
= x₂ - V ∙ t-x₁-V ∙ t/√1 - V²/c²
= x₂ - x₁/√1 - V²/c²

L' = L/√1 - V²/c²

Rearranging the last formula, we obtain

L = L' ∙ √1 - V²/c²

This is the well-known formula of length contraction in relativistic events.

Yet, let's suppose that two events occur successively at a fixed point of X' with coordinate x'₀. The instants of events' occurrence are t'₁ and t'₂ respectively. Applying the inverse Lorentz transformation, (i.e. exchanging the places and signs), we obtain

t₁ = t₁' + V/c² ∙ x₀'/√1 - V²/c² and t₂ = t₂' + V/c² ∙ x₀'/√1 - V²/c²

Again, subtracting the first equation from the second we obtain

t₂ - t₁ = t₂' + V/c² ∙ x₀'/√1 - V²/c² - t₁' + V/c² ∙ x₀'/√1 - V²/c²
= t₂' + V/c² ∙ x₀' - t₁'-V/c² ∙ x₀'/√1 - V²/c²
= t₂' - t₁'/√1 - V²/c²

Thus,

∆t = ∆t'/√1 - V²/c²

This too is the known equation of time dilation we have discussed in previous articles.

Example 2

A 20 m long rocket moves vertically upwards at 30 km/s. What is its length when measured from Earth surface?

Solution 2

Clues:

L' = 20 m
V = 30 km/s = 3 × 10⁴ m/s
c = 3 × 10⁸ m/s
L = ?

Using the equation

L = L' ∙ √1 - V²/c²

derived from Lorentz transformations, we obtain after substitutions

L = (20 m) ∙ √1 - 3 × 10⁴ m/s²/(3 × 10⁸ m/s)²
= (20 m) ∙ √1 - 10^-8
= (20 m) ∙ √10⁸ - 1/10⁸
= (20 m) ∙ 9999.99995/10000
= (20 m) ∙ 0.999999995
= 19.9999999 m

Thus, the contraction of length is only 0.0000001 m or 0.1 μm - practically undetectable.

You have reached the end of Physics lesson 18.5.3 Quick Recap of Dilation and Contraction. There are 4 lessons in this physics tutorial covering Lorentz Transformations, you can access all the lessons from this tutorial below.

More Lorentz Transformations Lessons and Learning Resources

Relativity Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
18.5	Lorentz Transformations
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
18.5.1	The Spacetime Lorentz Transformations
18.5.2	Galilean Transformations as Limit of Lorentz Transformations
18.5.3	Quick Recap of Dilation and Contraction
18.5.4	Lorentz Transformation of Velocity

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