∂_g__{σ}^{μν}/∂_x__{α} = ∂_g__{α}^{μν}/∂_x__{σ}
and consequently
_g__{σ}^{μν} ∂/∂_x__{α} (∂H/∂_g__{α}^{μν}) = ∂/∂_x__{α}
(_g__{σ}^{μν} ∂H/∂_g__{α}^{μν})
- ∂H/∂_g__{α}^{μν} ∂_g__{α}^{μν}/∂_x__{σ}
we obtain the equation
∂/∂_x__{α} (_g__{σ}^{μν} ∂H/∂_g__{α}^{μν}) - ∂H/∂_x__{σ} = 0
or
{ ∂_t__{σ}^α/∂_x__{α} = 0
(49) { -2κ_t__{σ}^{α} = _g__{σ}^{μν} ∂H/∂_g__{α}^{μν} - δ_{σ}^{α} H.
Owing to the relations (48), the equations (47) and (34),
(50) κ_t__{σ}^{α} = ½ δ_{σ}^{α} _g_^{μν} Γ_{μβ}^{α} Γ_{να}^{β}
- _g_^{μν} Γ_{μβ}^{α} Γ_{νσ}^{β}.
It is to be noticed that _t__{σ}^{α} is not a tensor, so that the
equation (49) holds only for systems for which √-_g_ = 1. This equation
expresses the laws of conservation of impulse and energy in a
gravitation-field. In fact, the integration of this equation over a
three-dimensional volume V leads to the four equations
(49a) _d_/_dx₄_ {∫_t__{σ}^4 _d_V} = ∫(_t__{σ}^1 α₁
+ _t__{σ}² α₂ + _t__{σ}³ α₃)_d_S
where α₁, α₂, α₂ are the direction-cosines of the inward-drawn normal to
the surface-element _d_S in the Euclidean Sense. We recognise in this
the usual expression for the laws of conservation. We denote the
magnitudes _t_^α_{σ} as the energy-components of the gravitation-field.
I will now put the equation (47) in a third form which will be very
serviceable for a quick realisation of our object. By multiplying the
field-equations (47) with _g_^{νσ}, these are obtained in the mixed
forms. If we remember that
_g_^{νσ} ∂Γ^α_{μν}/∂_x__{α} = ∂/∂_x__{α} (_g_^{νσ} Γ^α_{μν}) -
∂_g_^{νσ}/∂_x__{α} Γ^α_{μν},
which owing to (34) is equal to
∂/∂_x__{α} (._g_^{νσ} Γ^α_{μν}) - _g_^{νβ} Γ^σ_{αβ} Γγ^α_{μν}
- _g_^{σβ} Γ^ν_{βα} Γ^α_{μν},
or slightly altering the notation, equal to
∂/∂_x__{α} (_g_^{σβ} Γ^α_{μβ}) - _g_^{mn} Γ^σ_{mβ} Γ^β_{_n_μ}
- _g_^{νσ} Γ^α_{μβ} Γ^β_{να}.
The third member of this expression cancels with the second member of
the field-equations (47). In place of the second term of this
expression, we can, on account of the relations (50), put
κ (_t_^σ_{μ} - ½ δ^σ_{μ} _t_), where _t_ = _t_^α_{α}