Length Measurement
 Hymnal Leaf, paper 18 x 11 cm. Constantinople circa 1775. Twenty five lines of Greek script and musical notations in black and red. Marginal medallions in red.

Length has been measured at least since ancient Egyptians stretched cords and knotted ropes to survey agricultural fields and construct pyramids. And until the middle of the twentieth century, the method used usually required some reference measuring rod generically noted by the letter $\sf{L}$. An ideal measuring rod is rigid, so its own length $\ell^{\sf{L}}$ is presumably a constant. For calibrated measuring rods $\ell^{\sf{L}}$ is known and conventionally expressed in abbreviated by (m). To measure the length $\ell^{\mathbf{AB}}$ between two space-time events $\mathbf{A}_{\it{i}}$ and $\mathbf{B}_{ \it{f}}$ count the least number $N$ of copies of $\sf{L}$ that can be fit between the two events. Then

$\left\| \, \bar{r} ^{\, \mathbf{B}}_{ \it{f}} - \bar{r} ^{\, \mathbf{A}}_{ \it{i}} \vphantom{\sum^{2}} \right\| = N \ell^{\sf{L}}$

The idea of a least number of copies is based on the historical practice of stretching a rope or surveyor's chain. More recently an optical method has been adopted. It requires a clock $\mathbf{\Theta}$ to measure an elapsed time $\Delta t$. To optically measure a length first measure the elapsed time in seconds for a photon to travel from $\mathbf{A}_{\it{i}}$ and $\mathbf{B}_{\it{f}}$. Then $\ell ^{\mathbf{AB}}$ is given in metres by $\Delta t$ multiplied by the integer number 299,792,458.

 Next step: velocity.
page revision: 32, last edited: 18 Jan 2013 06:59