Page:The principle of relativity (1920).djvu/87

 < | follows from the 4 values ω_{1}, ω_{2}, ω_{3}, ω_{4}; where (ω_{1}, ω_{2}, ω_{3}) are real, -iω_{4} real and positive and condition (19) is fulfilled.

The meaning of (ω_{1}, ω_{2}, ω_{3}, ω_{2}) here is, that they are the ratios of dx_{1}, dx_{2}, dx_{3}, dx_{4} to

(20) [sqrt](-(dx_{1}^2 + dx_{2}^2 + dx_{3}^2 + dx_{4}^2)) = dt[sqrt](1 - u^2),

The differentials donoting the displacements of matter occupying the spacetime point (x_{1}, x_{2}, x_{3}, x_{4}) to the adjacent space-time point.

After the Lorentz-transfornation is accomplished the vococity of matter in the new system of reference for the same space-time point (x´, y´, z´, t´) is the vector u´ with the ratios dx´/dt´, dy´/dt´, dz´/dt´, dl´/dt´, as components.

Now it is quite apparent that the system of values

x_{1} = ω_{1}, x_{2} = ω_{2}, x_{3} = ω_{3}, x_{4} = ω_{4}

is transformed into the values

x_{1}´ = ω_{1}´, x_{2}´ = ω_{2}´, x_{3}´ = ω_{3}´, x_{4}´ = ω_{4}´

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity u´ after the transformation as the first system of values has got for u before transformation.

If in particular the vector v of the special Lorentz-transformation be equal to the velecity vector u of matter at the space-time point (x_{1}, x_{2}, x_{3}, x_{4}) then it follows out of (10), (11), (12) that

ω_{1}´ = 0, ω_{2}´ = 0, ω_{3}´ = 0, ω_{4}´ = i

Under these circumstances therefore, the corresponding space-time point has the velocity v´ = 0 after the transformation, it is as if we transform to rest. We may call the invariant ρ[sqrt](1 - u^2) the rest-density of Electricity.