(52) $$ \frac{\partial}{\partial x_{\alpha}} (g^{\sigmaeta}
\Gamma^{lpha}_{\mu\beta}) = - \kappa [(t^{\sigma}_{\mu} +
T^{\sigma}_{\mu}) - \frac{1}{2} \delta^{\sigma}_{\mu} (t + T)] $$ $$
\sqrt{-g} = 1 $$
where T = T_{μ}^μ (Laue’s Scalar). These are the general field-equations
of gravitation in the mixed form. In place of (47), we get by working
backwards the system
(53) $$ \frac{\partial \Gamma^{lpha}_{\mu u}}{\partial x_{\alpha}} +
\Gamma^{lpha}_{\mu\beta} \Gamma^{eta}_{\nu\alpha} = - \kappa
(T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T) $$
$$ \sqrt{-g} = 1 $$
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 δ_{μ}^σ from (52).
We obtain,
(52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
= -κ(_t__{μ}^σ + T_{μ}^σ)
we operate on it by ∂/∂_x__{σ}. Now,
∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
= -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
+ ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
The first and the third member of the round bracket lead to expressions
which cancel one another, as can be easily seen by interchanging the
summation-indices α, and σ, on the one hand, and β and λ, on the other.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
= ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
The second member of the expression on the left-hand side of (52a) leads
first to
- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
+ ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
The expression arising out of the last member within the round bracket
vanishes according to (29) on account of the choice of axes. The two
others can be taken together and give us on account of (31), the
expression
-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}