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For $h_{ab}=\delta g_{ab}$, the Ricci scalar varies asThe last two terms form a covariant divergence and hence contribute only a boundary term to a compactly supported variation of the Einstein-Hilbert action.

Put $h_{ab}=\delta g_{ab}$, $h=g^{ab}{h}_{ab}$, and $h^{ab}=g^{ac}{g}^{bd}{h}_{cd}$. Throughout this part use the Riemann curvature tensor convention printed in the paper. Varying $g^{ac}{g}_{cb}={\delta }^a{}_b$ gives the inverse-metric variation. The determinant identity $\delta \log |\det g|=g^{ab}{h}_{ab}$ gives the volume form variation. Together these areIn Lorentzian coordinates the latter reads $\delta \sqrt{-g}=\frac{1}{2}\sqrt{-g}h$.

For the connection, vary metric compatibility ${\nabla }_a{g}_{bc}=0$:Add the equations with derivatives $a,b$, subtract that with derivative $c$, and use symmetry of the lower indices of the Levi-Civita connection. This givesThe difference of two connections is a tensor, so this formula is covariant.

Varying the coordinate curvature formula yields the Palatini identityThe contractions of the connection variation areFinally, $\delta R=(\delta g^{ab})R_{ab}+g^{ab}\delta R_{ab}$ gives the metric variation of scalar curvature:The double divergence may equivalently be written ${\nabla }_a{\nabla }_b{h}^{ab}$: relabeling the contracted dummy indices and using symmetry of $h^{ab}$ gives the same expression. The derivative terms are a covariant divergence, useful when varying a curvature-dependent action.

For the torsion-free Levi-Civita connection, cyclically summing the displayed coordinate expression and using ${\Gamma }^a{}_{bc}={\Gamma }^a{}_{cb}$ gives the algebraic first Bianchi identityApply the Jacobi identity to three covariant-derivative commutators, or differentiate the coordinate formula in normal coordinates. The third-derivative terms and the derivatives of quadratic connection terms cancel cyclically, giving the differential second Bianchi identityContracting the first and third curvature indices and then contracting once more yieldsor the contracted Bianchi identity

For the metric variation $h_{ab}=\delta g_{ab}$,while the two double-divergence terms in the supplied metric variation of scalar curvature become a boundary term after integration by parts for tensor fields. The Einstein-Hilbert action therefore contributes $G_{ab}+\Lambda g_{ab}$. Varying the Proca action with respect to the metric gives the Proca stress-energy tensor, and the full Einstein-Proca theory equation is

Since $\delta F_{ab}=2{\nabla }_{[a}\delta A_{b]}$, variation with respect to $A_b$ and one integration by parts give the covariant Proca equationTaking its covariant divergence givesbecause the resulting contractions pair the symmetric Ricci tensor with the antisymmetric electromagnetic field tensor. Since $m\ne 0$, the Lorenz constraint in Proca theory follows:Moreover, $F=dA$ and $d^2=0$, so the Electromagnetic Bianchi identity is

Finally, use that identity to differentiate the matter tensor. The result groups naturally asBoth terms vanish by the Proca equation and its Lorenz constraint in Proca theory. Thus ${\nabla }^a{T}_{ab}=0$, exactly as required by the contracted Bianchi identity applied to the Einstein field equations. The vector equation is therefore consistent with the gravitational equation.

The trace of the electromagnetic stress-energy tensor in four spacetime dimensions isFor its covariant divergence, the source-free Maxwell equations eliminate the derivative of the first factor. Contracting the Bianchi identity ${\nabla }_{[a}{F}_{bc]}=0$ with $F^{ac}$ giveswhich cancels the derivative of the trace term. HenceFor $J^a=T^{ab}{V}_b$, stress-energy conservation and symmetry of $T^{ab}$ now imply

Local energy-momentum conservation is the vanishing covariant divergence of the energy-momentum tensor: