"(44)"
Remarks upon the choice of co-ordinates.—It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that √(-g) = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. It is specially true for the tensor Bμν, which plays a fundamental rôle in the theory. By this simplification, Sμν vanishes of itself so that tensor Bμν reduces to Rμν.
I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special case.
C. THE THEORY OF THE GRAVITATION-FIELD
§13. Equation of motion of a material point in a gravitation-field. Expression for the field-components of gravitation.
A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimensional region, in which the co-ordinates K0 can be, and are, so chosen that gμν’s have special constant values of the expression (4).
Let us discuss this motion from the stand-point of any arbitrary co-ordinate-system K₁; it moves with reference to K₁ (as explained in §2) in a gravitational field. The laws of motion with reference to K₁ follow easily from the following consideration. With reference to K₀, the law of motion is a four-dimensional straight line and thus a geodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law of motion for the motion of the material-point with reference to K₁. If we put
"(45)"