special relativity

The equation explained

\begin{aligned} \frac{u_x - v}{1 - \frac{u_x v}{V^2}} &= u_\xi , \\ \frac{u_y}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\eta , &\qquad \varrho' = \frac{\partial X'}{\partial \xi} + \frac{\partial Y'}{\partial \eta} + \frac{\partial Z'}{\partial \zeta} = \beta \left( 1 - \frac{v u_x}{V^2} \right) \varrho . \\ \frac{u_z}{\beta \left( 1 - \frac{u_x v}{V^2} \right)} &= u_\zeta . \end{aligned}

In wordsThe velocity of the electricity in the moving system: along the motion, its velocity in the stationary system less the speed of the moving system, divided by one minus their product over the square of the velocity of light; across the motion, the stationary component divided by the factor and by the same denominator. Its density in the moving system, how much the moving system's electric force spreads out across its coordinates, is its density in the stationary system times the factor and one minus the same product.OverviewThe velocity and density of electricity as the moving system measures them. The velocity is the addition theorem of § 5; the density changes by a factor that depends on how fast the electricity moves.Full explanationThese are the quantities that make the transformed equations keep their form. The velocity

(u_\xi, u_\eta, u_\zeta)

is, as the paper points out, nothing but the velocity of the electric masses measured in

k

: it follows from the addition theorem of velocities of § 5, because a charge moving with velocity

u

K

moves with this velocity in

k

. The density

\varrho'

is again the divergence of

k

's electric force, and it equals

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

. Electricity at rest in

K

(

u_x = 0

) is denser by the factor

\beta

k

, because

k

finds its volume contracted. The paper remarks that it follows easily from these equations that a charged body whose charge is constant as seen from a system moving with it also has a constant charge in

K

(p. 917).Every step

d\xi = \beta (u_x - v) \, dt, \quad d\tau = \beta \left( 1 - \frac{u_x v}{V^2} \right) dt

A piece of electricity moves with velocity

u

K

: in a short time

dt

its

x

changes by

u_x \, dt

. § 3's

\xi = \beta (x - v t)

and

\tau = \beta \left( t - \frac{v}{V^2} x \right)

then change by these amounts.

u_\xi = \frac{d\xi}{d\tau} = \frac{u_x - v}{1 - \frac{u_x v}{V^2}}

The velocity along the motion in

k

: the change of

\xi

per unit of

k

's time;

\beta

cancels.

u_\eta = \frac{d\eta}{d\tau} = \frac{u_y}{\beta \left( 1 - \frac{u_x v}{V^2} \right)}

Across the motion

\eta = y

, so

d\eta = u_y \, dt

, and

\beta

stays in the denominator. The same for

u_\zeta

with

u_z

\varrho' = \beta \varrho + \beta \frac{v}{V^2} \frac{\partial X}{\partial t} - \beta \frac{v}{V} \left( \frac{\partial N}{\partial y} - \frac{\partial M}{\partial z} \right)

Now the density. Take the divergence of

k

's electric force, with

\frac{\partial}{\partial \xi} = \beta \left( \frac{\partial}{\partial x} + \frac{v}{V^2} \frac{\partial}{\partial t} \right)

and § 6's

X' = X

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

Z' = \beta \left( Z + \frac{v}{V} M \right)

, and collect

\frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} + \frac{\partial Z}{\partial z}

into

\varrho

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

The first field equation in

K

replaces the bracket by

\frac{1}{V} \left( u_x \varrho + \frac{\partial X}{\partial t} \right)

. The terms in

\frac{\partial X}{\partial t}

cancel, leaving

\beta \varrho - \beta \frac{v u_x}{V^2} \varrho

.AlsoOverviewFull explanationEvery step

Read it in the paper