Page:LorentzGravitation1916.djvu/4

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coordinates will be of secondary importance; with a single exception (§ 13) it only serves for short calculations which we have to intercalate (for the proof of certain geometric propositions) and for establishing the final equations, which have to be used for the solution of special problems. In the discussion of the general principles coordinates play no part; and it is thus seen that the formulation of these principles can take place in the same way whatever be our choice of coordinates. So we are sure beforehand of the general covariancy of the equations that was postulated by Einstein.


§ 4. Einstein ascribes to a line-element PQ in the field-figure a length ds defined by the equation

\begin{array}{c}
ds^{2}=\sum(ab)g_{ab}dx_{a}dx_{b}\\
\\
\left(g_{ab}=g_{ba}\right)
\end{array} (1)

Here dx_{1},\dots dx_{4} are the changes of the coordinates when we pass from P to Q, while the coefficients g_{ab} depend in one way or another on the coordinates. The gravitation field is known when these 10 quantities are given as functions of x_{1},\dots x_{4} . Here it must be remarked that in all real cases the coordinates can be chosen in such a way that for one point arbitrarily chosen (1) becomes

ds^{2}=-dx_{1}^{2}-dx_{2}^{2}-dx_{3}^{2}+dx_{4}^{2}

This requires that the determinant g of the coefficients of (1) be always negative. The minor of this determinant corresponding to the coefficient g_{ab} will be denoted by G_{ab} .

Around each point P of the field-figure as a centre we may now construct an infinitesimal surface[1], which, when P is chosen as origin of coordinates, is determined by the equation

\sum(ab)g_{ab}x_{a}x_{b}=\epsilon^{2} (2)

where \epsilon is an infinitely small positive constant which we shall fix once for all. This surface, which we shall call the indicatrix, is a hyperboloid with one real axis and three imaginary ones. We shall also introduce the surface determined by the equation

\sum(ab)g_{ab}x_{a}x_{b}=-\epsilon^{2} (3)

which differs from (2) only by the sign of \epsilon^{2} . We shall call this the conjugate indicatrix. It is to be understood that the indicatrices and conjugate indicatrices take part in the changes to which the field-figure may be subjected. As these surfaces are infinitely small,

  1. A "surface" determined by one equation between the coordinates is a three-dimensional extension. It will cause no confusion if sometimes we apply the name of "plane" to certain two-dimensional extensions, if we speak e.g. of the "plane" determined by two line-elements.