_d_τ′ = ∫ _dx₁_ _dx₂_ _dx₃_ _dx₄_
is according to the wellknown law of Jacobi.
(B) $$ d\tau' = | \frac{dx'_{\sigma}}{dx_{\mu}} | d\tau $$
by multiplication of the two last equations (A) and (B) we get
(18) = √_g_ _d_τ′ = √_g_ _d_τ.
Instead of √_g_, we shall afterwards introduce √(-_g_) which has a real
value on account of the hyperbolic character of the time-space
continuum. The invariant √(-_g_)_d_τ, is equal in magnitude to the
four-dimensional volume-element measured with solid rods and clocks, in
accordance with the special relativity theory.
_Remarks on the character of the space-time continuum_—Our assumption
that in an infinitely small region the special relativity theory holds,
leads us to conclude that _ds²_ can always, according to (1) be
expressed in real magnitudes _d_X₁ ... _d_X_{_h_}. If we call _d_τ₀ the
“_natural_” volume element _d_X₁ _d_X₂ _d_X₃ _d_X₄ we have thus (18a)
_d_τ₀ = √(_g_)_i_τ.
Should √(-_g_) vanish at any point of the four-dimensional continuum it
would signify that to a finite co-ordinate volume at the place
corresponds an infinitely small “natural volume.” This can never be the
case; so that _g_ can never change its sign; we would, according to the
special relativity theory assume that _g_ has a finite negative value.
It is a hypothesis about the physical nature of the continuum
considered, and also a pre-established rule for the choice of
co-ordinates.
If however (-_g_) remains positive and finite, it is clear that the
choice of co-ordinates can be so made that this quantity becomes equal
to one. We would afterwards see that such a limitation of the choice of
co-ordinates would produce a significant simplification in expressions
for laws of nature.