Students will learn what is meant by time dilation in relativity theory and how to solve relative time problems.

### Key Equations

\begin{align*} \beta = \frac{v}{c}\end{align*}

An object moving with speed \begin{align*}v\end{align*} has a dimensionless speed \begin{align*} \beta \end{align*} calculated by dividing the speed \begin{align*}v\end{align*} by the speed of light (\begin{align*}\mathit{c} = 3\times10^8 \;\mathrm{m/s}\end{align*}). \begin{align*}0 \le \beta \le 1\end{align*}.

\begin{align*} \gamma = \frac{1}{\sqrt{1-\beta^2}}\end{align*}

The dimensionless Lorentz “gamma” factor \begin{align*}\gamma\end{align*} can be calculated from the speed, and tells you how much time dilation or length contraction there is. \begin{align*}1 \le \gamma \le \infty \end{align*}.

\begin{align*} T' = \gamma T \end{align*}

If a moving object experiences some event which takes a period of time \begin{align*}T\end{align*} (say, the amount of time between two heart beats), and the object is moving towards or away from you with Lorentz gamma factor \begin{align*}\gamma\end{align*}, the period of time \begin{align*}T'\end{align*} measured by you will appear longer.

- The speed of light will always be measured to be the same (about \begin{align*}3\times10^8 \;\mathrm{m/s}\end{align*})
*regardless*of your motion towards or away from the source of light. - In order for this bizarre fact to be true, we must reconsider what we mean by ‘space,’ ‘time,’ and related concepts, such as the concept of ‘simultaneous’ events. (Events which are seen as simultaneous by one observer might appear to occur at different times to an observer moving with a different velocity. Note that both observers see the same laws of physics, just a different sequence of events.)
- Clocks moving towards or away from you run more slowly, and objects moving towards or away from you shrink in length. These are known as Lorentz time dilation and length contraction; both are real, measured properties of the universe we live in.

Object | Speed (km/sec) | \begin{align*}\beta\end{align*} | Lorentz \begin{align*}\gamma\end{align*} Factor |
---|---|---|---|

Commercial Airplane | \begin{align*}0.25\end{align*} | \begin{align*}8\times10^{-7}\end{align*} | \begin{align*}1\end{align*} |

Space Shuttle | \begin{align*}7.8\end{align*} | \begin{align*}3\times10^{-5}\end{align*} | \begin{align*}1.00000000034\end{align*} |

UFO ☺ | \begin{align*}150,000\end{align*} | \begin{align*}0.5\end{align*} | \begin{align*}1.15\end{align*} |

Electron at the Stanford Linear Accelerator | \begin{align*}\sim300,000\end{align*} | \begin{align*}0.9999999995\end{align*} | \begin{align*}\sim100,000\end{align*} |

#### Example 1

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### Explore More

- Suppose you discover a speedy subatomic particle that exists for a nanosecond before disintegrating. This subatomic particle moves at a speed close to the speed of light. Do you think the lifetime of this particle would be
*longer*or*shorter*than if the particle were at rest? - The muon particle \begin{align*}(\mu-)\end{align*} has a half-life of \begin{align*}2.20\times10^{-6} \;\mathrm{s}\end{align*}. Most of these particles are produced in the atmosphere, a good \begin{align*}5-20 \;\mathrm{km}\end{align*} above Earth, yet we see them all the time in our detectors here on Earth. In this problem you will find out how it is possible that these particles make it all the way to Earth with such a short lifetime.
- Calculate how far muons could travel before half decayed, without using relativity and assuming a speed of \begin{align*}0.999\;\mathrm{c}\end{align*} (i.e. \begin{align*}99.9\end{align*}% of the speed of light)
- Now calculate \begin{align*}\gamma\end{align*}, for this muon.
- Calculate its 'relativistic' half-life.
- Now calculate the distance before half decayed using relativistic half-life and express it in kilometers. (This has been observed experimentally. This first experimental verification of time dilation was performed by Bruno Rossi at Mt. Evans, Colorado in 1939.)

- An alien spaceship moves past Earth at a speed of \begin{align*}.15 \;\mathrm{c}\end{align*} with respect to Earth. The alien clock ticks off \begin{align*}0.3\end{align*} seconds between two events on the spaceship. What will earthbound observers determine the time interval to be?
- Suppose your identical twin blasted into space in a space ship and is traveling at a speed of \begin{align*}0.100\;\mathrm{c}\end{align*}. Your twin performs an experiment which he clocks at \begin{align*}76\end{align*} minutes. You observe this experiment through a powerful telescope; what duration does the experiment have according to your clock? Now the opposite happens and you do the \begin{align*}76\end{align*} minute experiment. How long does the traveling twin think the experiment lasted?

#### Answers to Selected Problems

- longer
- \begin{align*}2.6 \times 10^8\;\mathrm{m/s}\end{align*}
- \begin{align*}1.34 \times 10^{-57}\;\mathrm{m}\end{align*}
- .