Page:The principle of relativity (1920).djvu/233

 For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approximation the symmetrical solution of the equation:—

{ g_{ρσ} = -δ_{ρσ} - α(x_ρ x_σ)/r^3 (ρ and σ 1, 2, 3) { (70) { g_{ρ4} = g_{4ρ} = 0 (ρ 1, 2, 3) {    { g_{4 4} = 1 - α/r.

δ_{ρσ} is 1 or 0, according as ρ = σ or not and r is the quantity

+[sqrt](x_{1}^2 + x_{2}^2 + x_{3}^2).

On account of (68a) we have

(70a) α = κM/4π

where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately the field-equation outside the mass M.

Let us now investigate the influences which the field of mass M will have upon the metrical properties of the field. Between the lengths and times measured locally on the one hand, and the differences in co-ordinates dx_ν on the other, we have the relation

ds^2 = g_{μν} dx_μ dx_ν.

For a unit measuring rod, for example, placed parallel to the x axis, we have to put

ds^2 = -1, dx_2 = dx_3 = dx_4 = 0

then

-1 = g_{1 1}dx_{1}^2.