Students will learn how length contraction follows from time dilation and the meaning of the gamma factor.

### 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*}L' = \frac{L}{\gamma} \end{align*}

If you see an object of length\begin{align*}L\end{align*} moving towards you at a Lorentz gamma factor \begin{align*}\gamma\end{align*}, it will appear shortened (contracted) in the direction of motion to new length \begin{align*}L\end{align*} .

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*}1.00000000000\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*} |

### Watch this Explanation

### Explore More

- What would be the Lorentz gamma factor \begin{align*}\gamma\end{align*} for a space ship traveling at the speed of light c? If you were in this space ship, how wide would the universe look to you?
- How fast would you have to drive in your car in order to make the road \begin{align*}50\end{align*}% shorter through Lorentz contraction?
- In 1987 light reached our telescopes from a supernova that occurred in a near-by galaxy \begin{align*}160,000\end{align*} light years away. A huge burst of neutrinos preceded the light emission and reached Earth almost two hours ahead of the light. It was calculated that the neutrinos in that journey lost only \begin{align*}13\end{align*} minutes of their lead time over the light.
- What was the ratio of the speed of the neutrinos to that of light?
- Calculate how much space was Lorentz-contracted form the point of view of the neutrino.
- Suppose you could travel in a spaceship at that speed to that galaxy and back. It that were to occur the Earth would be \begin{align*}320,000\end{align*} years older. How much would you have aged?

#### Answers to Selected Problems

- \begin{align*}\gamma = \infty\end{align*}, the universe would be a dot
- \begin{align*}9.15 \times 10^7\;\mathrm{m/s}\end{align*}
- .