(56) ∂/∂_x__{σ} (_t__{μ}^σ + T_{μ}^σ) = 0
From the field equations of gravitation, it also follows that the
conservation-laws of impulse and energy are satisfied. We see it most
simply following the same reasoning which lead to equations (49a); only
instead of the energy-components of the gravitational-field, we are to
introduce the total energy-components of matter and gravitational field.
§18. The Impulse-energy law for matter as a consequence of the
field-equations.
If we multiply (53) with ∂_g_^{μν}/∂_x__{σ}, we get in a way similar to
§15, remembering that
_g__{μν} ∂_g_^{μν}/∂_x__{σ} vanishes,
the equations ∂_t__{σ}^α/∂_x__{α} - ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
or remembering (56)
(57) ∂T_{σ}^α/∂_x__{α} + ½ ∂_g_^{μν}/∂_x__{σ} T_{μν} = 0
A comparison with (41b) shows that these equations for the above choice
of co-ordinates (√(-_g_) = 1) asserts nothing but the vanishing of the
divergence of the tensor of the energy-components of matter.
Physically the appearance of the second term on the left-hand side shows
that for matter alone the law of conservation of impulse and energy
cannot hold; or can only hold when _g_^{μν}’s are constants; _i.e._,
when the field of gravitation vanishes. The second member is an
expression for impulse and energy which the gravitation-field exerts per
time and per volume upon matter. This comes out clearer when instead of
(57) we write it in the form of (47).
(57a) ∂T_{σ}^α/∂_x__{α} = -Γ_{σβ}^α T_{α}^β.
The right-hand side expresses the interaction of the energy of the
gravitational-field on matter. The field-equations of gravitation
contain thus at the same time 4 conditions which are to be satisfied by
all material phenomena. We get the equations of the material phenomena
completely when the latter is characterised by four other differential
equations independent of one another.