First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional

(Variation of the inverse metric.) Differentiate the identity $g(s{)}^{ac}g(s{)}_{cb}={\delta }^a{}_b$ at $s=0$ using [L5]: $q^{ac}{g}_{cb}+g^{ac}{h}_{cb}=0$. Multiplying by $g^{-1}$ twice gives $q^{ab}=-g^{ac}{g}^{bd}{h}_{cd}$, the first assertion of (a).

(Variation of the Christoffel symbols.) Fix $p$, use the point-normal chart of [L2] at $p$, and let $g(s)$ be the given family; at $p$, the Christoffel formula of [L1] applied to $g(s)$ gives $\mathrm{\Gamma }(s{)}^a{}_{bc}(p)=\frac{1}{2}g(s{)}^{ad}(p)({\mathrm{\partial }}_{b}g(s{)}_{dc}+{\mathrm{\partial }}_{c}g(s{)}_{bd}-{\mathrm{\partial }}_{d}g(s{)}_{bc})(p)$. Differentiating at $s=0$ and using $g(0)=g$, the derivatives of the metric at $p$ may be replaced by covariant derivatives because ${\mathrm{\Gamma }}^a{}_{bc}(p)=0$; this gives $\delta {\mathrm{\Gamma }}^a{}_{bc}(p)=\frac{1}{2}{g}^{ad}({\mathrm{\nabla }}_b{h}_{cd}+{\mathrm{\nabla }}_c{h}_{bd}-{\mathrm{\nabla }}_d{h}_{bc})(p)$, the displayed formula. Both sides are tensor fields: the left side is a difference of Christoffel-symbol families, whose transformation law differs only by a metric-independent term that cancels, and the right side is built from tensors. Since the identity holds at every point in some point-normal chart, it holds on $U$; this proves (b).

(Realizability.) Let $h_0$ be smooth symmetric with compact support in $U$. By [L6] applied to the Lorentzian metric $g$ and the field $h_0$, there is ${\lambda }_0>0$ such that $g+\mu h_0$ is a smooth Lorentzian metric of signature $(-,+,+,+)$ for every $\mathrm{\mid }\mu \mathrm{\mid }<{\lambda }_0$. For $\mathrm{\mid }\lambda \mathrm{\mid }<{\lambda }_0$ and $\mathrm{\mid }s\mathrm{\mid }<1$ put $g(s):=g+s\lambda h_0$, so that $\mu =s\lambda$ ranges over $(-{\lambda }_0,{\lambda }_0)$; then $g(s)$ is Lorentzian for every $s\in (-1,1)$, equals $g$ off $K=\mathrm{supp}{h}_0$, and its components are smooth in $(s,x)$. For an arbitrary prescribed $q$, take $h_{ab}=-g_{ac}{g}_{bd}{q}^{cd}$ and restrict $g+sh$ to the stability interval; step 1.1 gives exactly that inverse-metric derivative. This proves the remaining assertions of (a).

(Palatini identity.) In the point-normal chart at $p$, the chart expression of [L3] reads $R(s{)}^a{}_{bcd}={\mathrm{\partial }}_c\mathrm{\Gamma }(s{)}^a{}_{db}-{\mathrm{\partial }}_d\mathrm{\Gamma }(s{)}^a{}_{cb}+{\sum }_e(\mathrm{\Gamma }(s{)}^a{}_{ce}\mathrm{\Gamma }(s{)}^e{}_{db}-\mathrm{\Gamma }(s{)}^a{}_{de}\mathrm{\Gamma }(s{)}^e{}_{cb})$. At $p$ the quadratic terms have a factor $\mathrm{\Gamma }(0{)}^e{}_{xy}(p)=0$, so differentiating at $s=0$ gives $\delta R^a{}_{bcd}(p)={\mathrm{\partial }}_c\delta {\mathrm{\Gamma }}^a{}_{db}(p)-{\mathrm{\partial }}_d\delta {\mathrm{\Gamma }}^a{}_{cb}(p)={\mathrm{\nabla }}_c\delta {\mathrm{\Gamma }}^a{}_{db}(p)-{\mathrm{\nabla }}_d\delta {\mathrm{\Gamma }}^a{}_{cb}(p)$, where the last equality again uses $\mathrm{\Gamma }(p)=0$. Both sides are tensors, so the identity holds on $U$. Contracting $a=c$ with [L3], $\delta {\mathrm{Ric}}_{bd}=\delta R^a{}_{bad}={\mathrm{\nabla }}_a\delta {\mathrm{\Gamma }}^a{}_{db}-{\mathrm{\nabla }}_d\delta {\mathrm{\Gamma }}^a{}_{ab}$, which is (c).

(Variation of the density times scalar curvature.) From [L4], ${\mathrm{\partial }}_{s}w(s){\mathrm{\mid }}_0=-\frac{1}{2}w{g}_{ab}{q}^{ab}$, and from step 1.1 and (c), ${\mathrm{\partial }}_{s}S(s){\mathrm{\mid }}_0={\mathrm{\partial }}_s(g(s{)}^{bd}\mathrm{Ric}(s{)}_{bd}){\mathrm{\mid }}_0=q^{bd}{\mathrm{Ric}}_{bd}+g^{bd}\delta {\mathrm{Ric}}_{bd}$. Therefore ${\mathrm{\partial }}_s(w(s)S(s)){\mid }_0=w[{\mathrm{Ric}}_{bd}{q}^{bd}-\frac{1}{2}S{g}_{bd}{q}^{bd}+g^{bd}({\mathrm{\nabla }}_a\delta {\mathrm{\Gamma }}^a{}_{db}-{\mathrm{\nabla }}_d\delta {\mathrm{\Gamma }}^a{}_{ab}\left)\right]\mathrm{.}$ The first two terms are $w{G}_{bd}{q}^{bd}$ by [L3]. For the last two terms, metric compatibility ($\mathrm{\nabla }{g}^{-1}=0$, [L1]) gives $g^{bd}{\mathrm{\nabla }}_a\delta {\mathrm{\Gamma }}^a{}_{db}={\mathrm{\nabla }}_a(g^{bd}\delta {\mathrm{\Gamma }}^a{}_{db})$ and $g^{bd}{\mathrm{\nabla }}_d\delta {\mathrm{\Gamma }}^a{}_{ab}={\mathrm{\nabla }}_d(g^{bd}\delta {\mathrm{\Gamma }}^a{}_{ab})={\mathrm{\nabla }}_a(g^{ab}\delta {\mathrm{\Gamma }}^d{}_{db})$ after relabelling the second index; their difference is ${\mathrm{\nabla }}_a{V}^a$ with $V^a=g^{bd}\delta {\mathrm{\Gamma }}^a{}_{db}-g^{ab}\delta {\mathrm{\Gamma }}^d{}_{db}$, so ${\mathrm{\partial }}_s(wS){\mathrm{\mid }}_0=w{G}_{ab}{q}^{ab}+w{\mathrm{\nabla }}_a{V}^a$. Since ${\mathrm{\partial }}_a(w{V}^a)=w{\mathrm{\nabla }}_a{V}^a$ by the density-divergence formula [L7], this is (d).

(Chart integral.) The action-difference integrand $[(S(s)-2\mathrm{\Lambda })w(s)-(S(0)-2\mathrm{\Lambda })w(0)]$ is smooth in $(s,x)$ on $I\times x(U)$ and supported in the fixed compact set $K$; all its $s$-derivatives are supported in $K$. By [L8], $F$ is differentiable at $0$ with $F^{\mathrm{\prime }}(0)={\int }_U{\mathrm{\partial }}_s((S(s)-2\mathrm{\Lambda })w(s)){\mathrm{\mid }}_0{d}^{4}x$. By [L4] and (d), ${\mathrm{\partial }}_s((S(s)-2\mathrm{\Lambda })w(s)){\mathrm{\mid }}_0=w{G}_{ab}{q}^{ab}+{\mathrm{\partial }}_a(w{V}^a)+\mathrm{\Lambda }{g}_{ab}{q}^{ab}w=w(G_{ab}+\mathrm{\Lambda }{g}_{ab})q^{ab}+{\mathrm{\partial }}_a(w{V}^a)$, the last step by [L7]. The function ${\mathrm{\partial }}_a(w{V}^a)$ has compact support in $K\subseteq U$, so its integral vanishes by [L8]; hence $F^{\mathrm{\prime }}(0)={\int }_U(G_{ab}+\mathrm{\Lambda }{g}_{ab})q^{ab}w{d}^{4}x$, which is (e).

(Particle functional.) Clause (f) is exactly [L9]: the lemma's hypotheses are the stated ones, and its conclusions are the formulas (1)-(3) quoted in (f). Multiplying the length functional by the positive constant $mc$ gives $S_p(H(s,\cdot ))=-mcL(s)$, so ${\mathrm{\partial }}_s{S}_p(H(s,\cdot )){\mathrm{\mid }}_0=-mc{L}^{\mathrm{\prime }}(0)$; since $mc>0$, the vanishing of $L^{\mathrm{\prime }}(0)$ for every admissible variation is equivalent to the vanishing of ${\mathrm{\partial }}_s{S}_p{\mathrm{\mid }}_0$, and [L9] identifies this with $D_t{\gamma }^{\mathrm{\prime }}=0$.