Demonstration of Length Contraction

This section provides a thought experiment to demonstrate length contraction using a laser meter to observe and calculate the length of a carriage on a moving train from two different frames of references.

The most common way to demonstrate length contraction is to follow a thought experiment using a moving train with a carriage:

1. Have Amy performing the experiment on the moving train using the train as the frame of reference:

If Amy uses L as the length of the carriage and T as the total time for the light pulse to return to the meter, she will get the following formulas:

L = c*f                       (1) - Forward time-distance relation
L = c*b                       (2) - Backward time-distance relation
T = f + b                     (3) - Total time traveled in frame x
T = L/c + L/c                 (4) - Merge (1) and (2) into (3) 
T = 2*L/c                     (5) - Combine two terms

L = c*T/2                     (6) - Normalize on L
   Carriage length observed by Amy in frame x

2. Have Bob watching light pulse traveling forward in Amy's experiment on the ground using the ground as the frame of reference:

If Bob uses L' as the length of the carriage and F as the distance moved by the train within f' seconds, he will get the following formulas:

L' + F = c*f'                 (7) - Forward time-distance relation
F = v*f'                      (8) - Distance moved by the train
L' + v*f' = c*f'              (9) - Merge (8) into (7)

f' = L'/(c-v)                (10) - Normalize on f'           
   Forward travel time observed by Bob in frame x'

3. Have Bob watching light pulse traveling backward in Amy's experiment on the ground using the ground as the frame of reference:

If Bob uses B as the distance moved by the train within b' seconds, he will get the following formulas:

L' - B = c*b'                (11) - Backward time-distance relation
B = v*b'                     (12) - Distance moved by the train
L' - v*b' = c*b'             (13) - Merge (12) into (11)

b' = L'/(c+v)                (14) - Normalize on b'           
   Backward travel time observed by Bob in frame x'

4. Have Bob calculating the total travel time and carriage length:

If Bob uses T' as the total time for the light pulse to return to the meter, he will get the following formulas:

T' = f' + b'                 (15) - Total time traveled in frame x'
T' = L'/(c-v) + L'/(c+v)     (16) - Merge (10) and (14) into (15)

T' = L'*(1/(c-v) + 1/(c+v))  (17) - Taking L' out of two terms
T' = L'*((c+v)/(c**2-v**2) + (c-v)/(c**2-v**2))
                             (18) - Create same denominator
T' = 2L'*c/(c**2-v**2)       (19) - Combine two terms
T' = 2L'/(c*(1-v**2/c**2))   (20) - Normalize denominator

L' = (1-(v/c)**2)*c*T'/2     (21) - Normalize on L'
   Carriage length observed by Bob in frame x'
Length Contraction Experiment with Laser Meter
Length Contraction Experiment with Laser Meter

See next section on how to derive the relation between L and L'.

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