Further it should be assumed that within the space-time region considered, gμν’s at infinite distances (using the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); i.e., we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.
We can assume that this approximation should lead to Newton’s theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a particle according to the equation (46). In the case of the special relativity theory, the components
dx₁/ds, dx₂/ds, dx₃/ds,
can take any values. This signifies that any velocity
v = √((dx₁/dx₄)² + (dx₂/dx₄)² + (dx₃/dx₄)²)
can appear which is less than the velocity of light in vacuum (v < 1). If we finally limit ourselves to the consideration of the case when v is small compared to the velocity of light, it signifies that the components
dx₁/ds, dx₂/ds, dx₃/ds,
can be treated as small quantities, whereas dx₄/ds is equal to 1, up to the second-order magnitudes (the second point of view for approximation).
Now we see that, according to the first view of approximation, the magnitudes γμντ’s are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.
By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—