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:

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).

Isosceles Triangle to Calculate Time Dilation
Isosceles Triangle to Calculate Time Dilation

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

Time Dilation in Special Relativity

 Time Dilation - Moving Clock Is Slower

 Demonstration of Time Dilation - Amy on the Train

 Demonstration of Time Dilation - Bob on the Ground

Demonstration of Time Dilation - Formula

 What Is Lorentz Factor

 Reciprocity of Time Dilation

 Elapsed Time between Distant Events

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

 Introduction of Lagrangian

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait


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