(51) { ∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) = -κ(_t_^σ_{μ} - ½ δ^σ_{μ}
_t_)
{ √(-_g_) = 1.
§16. General formulation of the field-equation of Gravitation.
The field-equations established in the preceding paragraph for spaces
free from matter is to be compared with the equation ▽²φ = 0 of the
Newtonian theory. We have now to find the equations which will
correspond to Poisson’s Equation ▽²φ = 4πκρ (ρ signifies the density of
matter).
The special relativity theory has led to the conception that the
inertial mass (Träge Masse) is no other than energy. It can also be
fully expressed mathematically by a symmetrical tensor of the second
rank, the energy-tensor. We have therefore to introduce in our
generalised theory energy-tensor τ^α_{σ} associated with matter, which
like the energy components _t_^α_{σ} of the gravitation-field (equations
49, and 50) have a mixed character but which however can be connected
with symmetrical covariant tensors. The equation (51) teaches us how to
introduce the energy-tensor (corresponding to the density of Poisson’s
equation) in the field equations of gravitation. If we consider a
complete system (for example the Solar-system) its total mass, as also
its total gravitating action, will depend on the total energy of the
system, ponderable as well as gravitational. This can be expressed, by
putting in (51), in place of energy-components _t__{μ}^σ of
gravitation-field alone the sum of the energy-components of matter and
gravitation, _i.e._,
_t__{μ}^σ + T_{μ}^σ.