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 ∂gμν/∂xσ, we get in a way similar to §15, remembering that
gμν ∂gμν/∂xσ vanishes,
the equations ∂tσα/∂xα - ½ ∂gμν/∂xσ Tμν = 0
or remembering (56)
(57) ∂Tσα/∂xα + ½ ∂gμν/∂xσ Tμν = 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 gμν’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) ∂Tσα/∂xα = -Γσβα Tαβ.