**Herong's Notes on Physics** - Version 3.21, by Dr. Herong Yang

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

3. Calculate time dilation factor:

2*L = c*T (1) - Amy's observation 2*L' = c*T' (3) - Bob's observation L'**2 = (v*T'/2)**2 + (c*T/2)**2 (12) - merging (1) into (11) 4*L'**2 = (v*T')**2 + (c*T)**2 (13) - multiplying by 4 4*(c*T'/2)**2 = (v*T')**2 + (c*T)**2 (14) - merging (3) into (13) (c*T')**2 = (v*T')**2 + (c*T)**2 (15) - simplifying left side (c*T')**2 - (v*T')**2 = (c*T)**2 (16) - moving T' terms together (c**2-v**2)*T'**2 = (c*T)**2 (17) - factoring T' sqrt(c**2-v**2)*T' = c*T (18) - taking square root T' = (c/sqrt(c**2-v**2))*T (19) - normalizing on T' T' = (1/sqrt(1-(v/c)**2))*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).

*Last update: 2014.*

Table of Contents

Introducion of Frame of Reference

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

Elapsed Time between Distant Events

Length Contraction in Special Relativity

The Relativity of Simultaneity