special relativity

The equation explained

\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .

In wordsThe factor is one over the square root of one minus the square of the speed of the moving system divided by the velocity of light.OverviewThe factor

\beta

that multiplies every mixed combination in the transformed equations, the same factor § 3 introduced. It is 1 when the system is at rest and grows without limit as

v

approaches

V

.Full explanationThe paper repeats here the factor it defined on p. 900, because it appears in every transformed equation. It depends on

v

only through

\frac{v}{V}

, the speed as a fraction of the velocity of light, and only through its square, so reversing the direction of motion leaves it unchanged. At everyday speeds it is 1 to many decimal places; at

v = 0.6 \, V

it is exactly 1.25; for

v

equal to

V

it has no finite value, which is why the paper's systems always move more slowly than light.Every step

\frac{v}{V}

The speed of the moving system as a fraction of the velocity of light: 0.6 for a system moving at six tenths of it.

1 - \left( \frac{v}{V} \right)^2

Square the fraction and subtract it from one. For 0.6 this gives

1 - 0.36 = 0.64

. The result stays positive as long as

v

is less than

V

\beta = \frac{1}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}

Take the square root, 0.8 in the example, and then its reciprocal:

\beta = 1 / 0.8 = 1.25

. For

v = 0

every step gives 1.Historian's marginThe paper's

\beta

is the factor that modern texts write

\gamma

; modern texts use the letter beta for the ratio

v/c

instead. The site keeps the printed letter on the faces and says so in the notation (the paper's definition, p. 900; this display, p. 908).AlsoOverviewFull explanationEvery stepHistorian's margin

Read it in the paper