Length Contraction Formula and Lorentz Factor

∟Length Contraction in Special Relativity

∟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                     (L.6) - Carriage length observed by Amy
L' = (1-(v/c)**2)*c*T'/2     (L.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)            (L.22) - Time dilation formula
gamma = 1/sqrt(1-(v/c)**2)         (L.23) - Set "gamma" as Lorentz factor
T' = gamma*T                       (L.24) - L.23 in "gamma" format

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

L' = (1/gamma)*L                   (L.28) - Merge L.6 into L.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.