Length Contraction Formula and Lorentz Factor

This section provides steps to derive the length contraction formula expressed with the Lorentz factor, based a thought experiment and the time dilation formula.

From the previous section, we collected two sets of observations from two reference frames:

L = c*T/2                     (6) - Carriage length observed by Amy
L' = (1-(v/c)**2)*c*T'/2     (21) - Carriage length observed by Bob

Now if we agree with the time dilation formula discussed earlier, we can derive the length contraction formula as below:

T' = T/sqrt(1-(v/c)**2)      (22) - Time dilation formula
gamma = 1/sqrt(1-(v/c)**2)   (23) - Set "gamma" as Lorentz factor
T' = gamma*T                 (24) - (23) in "gamma" format

L' = (1/gamma**2)*c*T'/2     (25) - (21) in "gamma" format
L' = (1/gamma**2)*c*(gamma*T)/2
                             (26) - Merge (24) into (25)
L' = (1/gamma)*c*T/2         (27) - Simplified (26)

L' = (1/gamma)*L             (28) - Merge (6) into (27)
   Length contraction formula

Congratulations, we have derived the length contraction formula! The formula tells us that the carriage is observed to be shorter by Bob on the ground than what observed by Amy moving with the carriage, because 1/gamma < 1.

The length contraction formula can also be expressed in mathematical format as:

Length Contraction Formula.jpg
Length Contraction Formula and Lorentz Factor

Last update: 2014.

Table of Contents

 About This Book

 Introduction of Space

 Introducion of Frame of Reference

 Introducion of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

Length Contraction in Special Relativity

 Length Contraction - Moving Object Is Shorter

 Demonstration of Length Contraction

Length Contraction Formula and Lorentz Factor

 Reciprocity of Length Contraction

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 References

 PDF Printing Version