(68) ▽² _g₄₄_ = κρ.
The equations (67) and (68) together, are equivalent to Newton’s law of
gravitation.
For the gravitation-potential we get from (67) and (68) the exp.
(68a.) -κ/(8π) ∫ ρ_d_τ/_r_
whereas the Newtonian theory for the chosen unit of time gives
-K/_c²_ ∫ρ_d_τ/_r_,
where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating
them we get
(69) κ = 8πK/_c²_ = 1.87 x 10⁻²⁷.
§22. Behaviour of measuring rods and clocks in a statical
gravitation-field. Curvature of light-rays. Perihelion-motion of the
paths of the Planets.
In order to obtain Newton’s theory as a first approximation we had to
calculate only _g₄₄_, out of the 10 components _g__{μν} of the
gravitation-potential, for that is the only component which comes in the
first approximate equations of motion of a material point in a
gravitational field.
We see however, that the other components of _g__{μν} should also differ
from the values given in (4) as required by the condition _g_ = -1.
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})/_r³_ (ρ and σ 1, 2,
3)
{
(70) { _g__{ρ4} = _g__{4ρ} = 0 (ρ 1, 2, 3)
{
{ _g₄₄_ = 1 - α/_r_.
δ_{ρσ} is 1 or 0, according as ρ = σ or not and _r_ is the quantity
+√(_x₁²_ + _x₂²_ + _x₃²_).