# Chapter 22: The Special Theory of Relativity

**At Grade**Created by: CK-12

A GPS satellite travels at 14,000 kilometers per hour, and needs corrections for relativity to stay accurate.

Physics at the usual speeds found on Earth mostly follow our intuitive understanding of time and space. However, in the 20^{th} century, very accurate measurements of the speed of light found that these could not be true. This puzzle was solved by a new theory of time and space by Albert Einstein, called the Special Theory of Relativity. This chapter covers the paradox of the speed of light, time dilation, length contraction, simultaneity, and the General Theory of Relativity.

- 22.1.
## Galilean Relativity

- 22.2.
## Time Dilation and Length Contraction

- 22.3.
## Simultaneity

- 22.4.
## Mass-Energy Equivalence

- 22.5.
## General Theory of Relativity

### Chapter Summary

1. Postulates of Special Relativity

Postulate 1: The laws of physics have the same form in all inertial frames of reference.

Postulate 2: The speed of light in vacuum is constant in all inertial frames of reference.

2. Time dilation:

*Clocks moving relative to an observer who assumes himself “at rest,” run more slowly compared to the clocks in the “at rest” frame*. Such a slowing of time is called *time dilation*. The transformation equation between the two frames is given by

\begin{align*}t = \frac{t_p}{\sqrt{1-\frac{v^2}{c^2}}}\end{align*}

The time \begin{align*}t_p\end{align*} is the proper time (also called the lab or at rest frame).

3. Length contraction

Objects in motion with respect to a proper frame (or “rest frame”) of reference are contracted in the direction of their motion. The length of the (moving) object as measured from the proper frame is given by

\begin{align*}L = L_p\sqrt{1-\frac{v^2}{c^2}}\end{align*} where \begin{align*}L_p\end{align*} is the length measured in the proper frame.

4. Newton’s Equations with the proper transformations

\begin{align*}F &= \frac{ma}{\sqrt{1-\frac{v^2}{c^2}}}\\ P &= \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\end{align*}

5. The total energy \begin{align*}E\end{align*} is defined as \begin{align*}E = mc^2+KE\end{align*},

where \begin{align*}KE\end{align*} is the kinetic energy of an object and \begin{align*}m\end{align*} is its rest mass.

6. The correct relativistic addition of velocities equation which Einstein (and Lorentz) derived is

\begin{align*}v_r = \frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}\end{align*}

where \begin{align*}v_1\end{align*} is the velocity of an inertial frame \begin{align*}(S^\prime)\end{align*} measured in the proper frame \begin{align*}(S)\end{align*} and \begin{align*}v_2\end{align*} is the velocity of an object (or light wave) measured with respect \begin{align*}(S^\prime)\end{align*}. The relative velocity \begin{align*}v_r\end{align*} is with respect to frame \begin{align*}(S)\end{align*}.