Formula for the Relativity of Simultaneity?

This section provides a thought experiment to derive the formula for the relativity of simultaneity.

To derive the formula for the relativity of simultaneity, let's use a thought experiment based on the example given in the previous section.

Part 1 - Amy on the Train: The first part of the thought experiment is to establish two simultaneous events in a moving reference frame. This part consists of the following:

If we let A and B be elapsed times of light pulses reaching the back wall and the front wall on Amy's clock, her observation can be expressed as:

A = 0.5*L/c                   (1) - time when light reaching the back
B = 0.5*L/c                   (2) - time when light reaching the front

B-A = 0                       (3) - reaching walls simultaneously 
   - Amy's observation in the moving frame

Part 2 - Bob on the Ground: The second part of the thought experiment is to observe the same events in a stationary reference frame. This part consists of the following:

If we let A' and B' be elapsed times of light pulses reaching the back wall and the front wall on Bob's clock, his observation can be expressed as:

A' = (0.5*L-E)/c              (4) - Time when light reaching the back
B' = (0.5*L+F)/c              (5) - Time when light reaching the front
E = A'*v                      (6) - distance moved by the back wall
F = B'*v                      (7) - distance moved by the front wall

A' = (0.5*L-A'*v)/c           (8) - merging (6) into (4)  
B' = (0.5*L+B'*v)/c           (9) - merging (7) into (5)  

A'*c + A'*v = 0.5*L          (10) - moving variables in (8)
B'*c - B'*v = 0.5*L          (11) - moving variables in (9)

A' = 0.5*L/(c+v)             (12) - moving variables in (10)
B' = 0.5*L/(c-v)             (13) - moving variables in (11)

B'-A' = 0.5*L/(c-v) - 0.5*L/(c+v)
                             (14) - difference between B' and A'

B'-A' = 0.5*L*(1/(c-v) - 1/(c+v))
B'-A' = 0.5*L*((c+v)/(c**2-v**2) - (c-v)/(c**2-v**2))
B'-A' = 0.5*L*((c+v) - (c-v))/(c**2-v**2)
B'-A' = 0.5*L*(2*v)/(c**2-v**2)

B'-A' = L*v/(c**2-v**2)      (15) - reaching walls at different times
   - Bob's observation in the stationary frame

Conclusion, two simultaneous events in Amy's frame observed as non-simultaneous in Bob's frame that is moving relatively.

Formula for Relativity of Simultaneity
Formula for Relativity of Simultaneity

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

The Relativity of Simultaneity

 What Is the Relativity of Simultaneity?

Formula for the Relativity of Simultaneity?

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 References

 PDF Printing Version