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:—
d²xτ/dt² = Γ₄₄τ, where ds = dx₄ = dt,
or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,
It must be admitted, that this introduction of the energy-tensor of matter cannot be justified by means of the Relativity-Postulate alone; for we have in the foregoing analysis deduced it from the condition that the energy of the gravitation-field should exert gravitating action in the same way as every other kind of energy. The strongest ground for the choice of the above equation however lies in this, that they lead, as their consequences, to equations expressing the conservation of the components of total energy (the impulses and the energy) which exactly correspond to the equations (49) and (49a). This shall be shown afterwards.
§17. The laws of conservation in the general case.
The equations (52) can be easily so transformed that the second member on the right-hand side vanishes. We reduce (52) with reference to the indices μ and σ and subtract the equation so obtained after multiplication with ½ δμσ from (52).
We obtain,
(52a) ∂/∂xα(gσβ Γμβα - ½ δμσ gλβ Γλβα)
= -κ(tμσ + Tμσ)