Physics Notes - Herong's Tutorial Notes - v3.24, by Herong Yang
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 (S.1) - time when light reaching the back B = 0.5*L/c (S.2) - time when light reaching the front B - A = 0 (S.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 (S.4) - Time when light reaching the back B' = (0.5*L+F)/c (S.5) - Time when light reaching the front E = A'*v (S.6) - distance moved by the back wall F = B'*v (S.7) - distance moved by the front wall A' = (0.5*L-A'*v)/c (S.8) - merging S.6 into S.4 B' = (0.5*L+B'*v)/c (S.9) - merging S.7 into S.5 A'*c + A'*v = 0.5*L (S.10) - moving variables in S.8 B'*c - B'*v = 0.5*L (S.11) - moving variables in S.9 A' = 0.5*L/(c+v) (S.12) - moving variables in S.10 B' = 0.5*L/(c-v) (S.13) - moving variables in S.11 B'-A' = 0.5*L/(c-v) - 0.5*L/(c+v) (S.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) (S.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.
Table of Contents
Introduction of Frame of Reference
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
Minkowski Spacetime and Diagrams
Introduction of Generalized Coordinates