Demonstration of Time Dilation

∟Demonstration of Time Dilation - Formula

This section continues the thought experiment to demonstrate time dilation. An isosceles triangle is used to derive the time dilation formula with Lorentz factor.

Part 3 - Time Dilation Formula: The third part of the thought experiment is to derive the time dilation formula using the Pythagorean theorem. This part consists of the following:

1. Establish an isosceles triangle with trajectories of the laser meter and light pulse in Bob's frame:

The trajectory of the laser meter forms the base of the isosceles triangle, D.
The trajectory of the light pulse forms two legs of the isosceles triangle, L' and L'.

2. Establish a relation between the leg length L and L' based on the Pythagorean theorem:

L'**2 = D**2 + L**2           (T.9) - Pythagorean theorem
2*D = v*T'                   (T.10) - maser meter moves at speed v
L'**2 = (v*T'/2)**2 + L**2   (T.11) - merging T.10 into T.9

3. Calculate time dilation factor:

2*L = c*T                     (T.1) - Amy's observation
2*L' = c*T'                   (T.3) - Bob's observation

L'**2 = (v*T'/2)**2 + (c*T/2)**2       (T.12) - merging T.1 into T.11
4*L'**2 = (v*T')**2 + (c*T)**2         (T.13) - multiplying by 4
4*(c*T'/2)**2 = (v*T')**2 + (c*T)**2   (T.14) - merging T.3 into T.13
(c*T')**2 = (v*T')**2 + (c*T)**2       (T.15) - simplifying left side
(c*T')**2 - (v*T')**2 = (c*T)**2       (T.16) - moving T' terms together
(c**2-v**2)*T'**2 = (c*T)**2           (T.17) - factoring T'

sqrt(c**2-v**2)*T' = c*T     (T.18) - taking square root
T' = (c/sqrt(c**2-v**2))*T   (T.19) - normalizing on T'
T' = (1/sqrt(1-(v/c)**2))*T  (T.20) - moving c into sqrt()
   # Time dilation formula

Voila! We derived the dilation formula. Bob's observed that Amy's bouncing light pulse clock is slower by a factor of (1/sqrt(1-(v/c)**2)), which is called Lorentz Factor (named after Hendrik Lorentz).