Page:The principle of relativity (1920).djvu/188

  Invariant of volume.

We see κ first the transformation law for the determinant g = | g_{μν} |. According to (11)

g´ = | [part]x_{μ}/[part]x_{σ´} [part]x_{ν}/[part]_{τ´} g_{μν} |

From this by applying the law of mutiplication twice, we obtain.

g´ = | [part]x_{μ}/[part]x_{σ´} | | [part]x_{ν}/[part]x_{τ´} | | g_{μν} | = | [part]x_{μ}/αx?]_{σ´} | g.

or [sqrt]g´ = | [part]x_{μ}/[part]x_{σ´} | [sqrt]g (A)

On the other hand the law of transformation of the volume element

dτ´ = [integral] dx_{1} dx_{2} dx_{3} dx_{4}

is according to the wellknown law of Jacobi.

dτ´ = | d´_{σ}/dx_{μ} | dτ. (B)

by multiplication of the two last equation (A) and (B) we get.

(18) = [sqrt]g dτ´ = [sqrt]g dτ.

Insted of [sqrt]g, we shall afterwards introduce [sqrt](-g) which has a real value on account of the hyperbolic character of the time-space continuum. The invariant [sqrt](-g)dτ, is equal in magnitude to the four-dimensional volume-element