From (55) and (52a) it follows that
(56) [part]/[part]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 [part]g^{μν}/[part]x_{σ}, we get in a way similar to §15, remembering that
g_{μν} [part]g^{μν}/[part]x_{σ} vanishes,
the equations [part]t_{σ}^α/[part]x_{α} - 1/2 [part]g^{μν}/[part]x_{σ} T_{μν} = 0
or remembering (56)
(57) [part]T_{σ}^α/[part]x_{α} + 1/2 [part]g^{μν}/[part]x_{σ} T_{μν} = 0
A comparison with (41b) shows that these equations for the above choice of co-ordinates ([sqrt](-g) = 1) asserts nothing but the vanishing of the divergence of the tensor of the energy-components of matter.
