Page:The principle of relativity (1920).djvu/189

 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 ds^2 can always, according to (1) be expressed in real magnitudes dX_{1} dX_{h}. If we call dτ_{0} the "natural" volume element dX_{1} dX_{2} dX_{3} dX_{4} we have thus (18a) dτ_{0}. = [sqrt](g)iτ.

Should [sqrt](-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.

In place of (18) it follows then simply that

dτ´ = d

from this it follows, remembering the law of Jacobi,

(19) | [part]x´_σ/dx_{μ} | = 1.