special relativity

The equation explained

\varphi(v) = \varphi(-v) ;

In wordsThe factor is the same for the speed v as for −v.OverviewBy symmetry, reversing the direction of the moving system cannot change the factor.Full explanationThe paper says this follows from reasons of symmetry, and its footnote gives an example. Suppose the only force in

K

is a magnetic component

N

. Then the relations give

k

an electric component

Y'

across the motion, and if the moving system moves the other way at the same speed, symmetry says

Y'

must keep its size and change its sign. The relations give the size in terms of the factor for

v

and for

-v

, and the two can match only if the factors are equal. The factor

\beta

is the same both ways, because it depends on

v

only through its square.Every step

X = Y = Z = L = M = 0, \quad N \neq 0

The footnote's example: in

K

the only force is a magnetic component

N

Y' = \varphi(v) \beta \left( -\frac{v}{V} N \right)

The relation for

Y'

from the previous displays, with this field put in: only the term in

N

survives.

Y' = \varphi(-v) \beta \frac{v}{V} N

The same field seen from a system moving the other way, with velocity

-v

: the sign of the term flips,

\beta

does not.

\varphi(v) = \varphi(-v)

Symmetry requires the second value to be the first with its sign changed and its size kept. That holds only if

\varphi(-v) = \varphi(v)

.AlsoOverviewFull explanationEvery step

Read it in the paper