First variation, Hilbert tensor and conservation of the scalar-field model

(Field variation of the density.) Write $\mathrm{\Phi }(u):=\frac{1}{2}{g}^{ab}{\mathrm{\partial }}_a{\phi }_u{\mathrm{\partial }}_b{\phi }_u+V({\phi }_u)$, so that $\mathrm{\Delta }S({\phi }_u,g)=-\frac{K}{c}{\int }_M[\mathrm{\Phi }(u)-\mathrm{\Phi }(0)]{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Differentiating the integrand at $u=0$, ${\mathrm{\partial }}_u\mathrm{\Phi }=(g^{ab}{\mathrm{\partial }}_a\phi ){\mathrm{\partial }}_b\delta \phi +V^{\mathrm{\prime }}(\phi )\delta \phi$, and by [L4] the derivative of the integral is the integral of the derivative.

(Integration by parts.) Consider the smooth compactly supported vector field $J^b:=\delta \phi g^{ab}{\mathrm{\partial }}_a\phi$; by [L1], ${\mathrm{\nabla }}_b{J}^b=({\mathrm{\partial }}_b\delta \phi )g^{ab}{\mathrm{\partial }}_a\phi +\delta \phi {\mathrm{\square }}_g\phi$, and ${\int }_M{\mathrm{\nabla }}_b{J}^b{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$ by [L2]. Hence ${\int }_M(g^{ab}{\mathrm{\partial }}_a\phi ){\mathrm{\partial }}_b\delta \phi {\mathrm{v}\mathrm{o}\mathrm{l}}_g=-{\int }_M\delta \phi {\mathrm{\square }}_g\phi {\mathrm{v}\mathrm{o}\mathrm{l}}_g$, and step 1.1 gives $\frac{d}{du}\mathrm{\Delta }S=\frac{K}{c}{\int }_M({\mathrm{\square }}_g\phi -V^{\mathrm{\prime }}(\phi ))\delta \phi {\mathrm{v}\mathrm{o}\mathrm{l}}_g$, which is the variation formula in (a). Every compactly supported smooth scalar $f$ is an admissible first variation, by ${\phi }_u=\phi +uf$. Let $E={\mathrm{\square }}_g\phi -V\mathrm{\prime }(\phi )$. If all such variations are stationary, positivity of $K\mathrm{/}c$ gives $\int Ef{\mathrm{v}\mathrm{o}\mathrm{l}}_g=0$. On any oriented coordinate chart and for every compactly supported symmetric tensor $q$, take $f=g_{ab}{q}^{ab}$, extended by zero. Then $\int (E{g}_{ab})q^{ab}w{d}^{4}x=0$, so [L5] gives $Eg=0$. Nondegeneracy of $g$ forces $E=0$ at every point. Conversely $E=0$ makes every variation derivative vanish by the displayed formula. This proves all of (a).

(Metric variation.) With $\mathrm{\Psi }(s):=\frac{1}{2}{g}_s^{ab}{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi +V(\phi )$, the difference is $\mathrm{\Delta }S(s)=-\frac{K}{c}{\int }_M[\mathrm{\Psi }(s){\mathrm{v}\mathrm{o}\mathrm{l}}_{g_s}-\mathrm{\Psi }(0){\mathrm{v}\mathrm{o}\mathrm{l}}_g]$ along the metric family, differentiate at $s=0$: ${\mathrm{\partial }}_s\mathrm{\Psi }=\frac{1}{2}{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi q^{ab}$, and by [L2] ${\mathrm{\partial }}_s{\mathrm{v}\mathrm{o}\mathrm{l}}_g=-\frac{1}{2}{g}_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Therefore ${\mathrm{\partial }}_s\left(\mathrm{\Psi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g\right){\mid }_0=\frac{1}{2}[{\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi -g_{ab}\mathrm{\Psi }]q^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g=\frac{1}{2K}{T}_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ by the definition of $T$ in The scalar-field matter model of the classical gravitational framework clause (c). Differentiating under the integral by [L4] and multiplying by $-K\mathrm{/}c$ gives (b). More precisely fix the base pair $(g,\phi )$ and define the real-valued functional on all smooth Lorentzian/scalar pairs differing from it in a compact set by the action difference of the Definition. Its domain admits every compact metric direction by the stability lemma, and this calculation applies to every jointly smooth compact metric family at the base pair, so it satisfies the exact Hilbert definition; uniqueness then follows from Uniqueness of the Hilbert stress-energy tensor.

(Conservation identity.) By [L1], ${\mathrm{\nabla }}^a{T}_{ab}=K[{\mathrm{\nabla }}^a({\mathrm{\partial }}_a\phi {\mathrm{\partial }}_b\phi )-{\mathrm{\nabla }}_b(\frac{1}{2}{\mathrm{\partial }}^c\phi {\mathrm{\partial }}_c\phi +V(\phi )\left)\right]$. The first term is $K[({\mathrm{\nabla }}^a{\mathrm{\partial }}_a\phi ){\mathrm{\partial }}_b\phi +{\mathrm{\partial }}_a\phi {\mathrm{\nabla }}^a{\mathrm{\partial }}_b\phi ]=K[{\mathrm{\square }}_g\phi {\mathrm{\partial }}_b\phi +{\mathrm{\partial }}_a\phi {\mathrm{\nabla }}_b{\mathrm{\partial }}^a\phi ]$ by torsion freedom, and the second is $K[{\mathrm{\partial }}_a\phi {\mathrm{\nabla }}_b{\mathrm{\partial }}^a\phi +V^{\mathrm{\prime }}(\phi ){\mathrm{\partial }}_b\phi ]$. Subtracting gives ${\mathrm{\nabla }}^a{T}_{ab}=K({\mathrm{\square }}_g\phi -V^{\mathrm{\prime }}(\phi )){\mathrm{\partial }}_b\phi$, which is the identity in (c). The forward conservation consequence follows by substitution. At a point with $d\phi \mathrm{\ne }0$ choose a tangent vector $X$ with $d\phi (X)\mathrm{\ne }0$; if conservation holds, evaluating the identity on $X$ and using $K>0$ gives ${\mathrm{\square }}_g\phi -V\mathrm{\prime }(\phi )=0$. If such points are dense, smoothness gives the same equation everywhere. For the necessity of this caveat take $\phi =a$ constant and a smooth potential with $V\mathrm{\prime }(a)\mathrm{\ne }0$: then $T=-KV(a)g$ is conserved by metric compatibility, while the scalar equation fails. This proves all of (c).