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Compactly Supported Metric Variation of a Weighted Scalar-Curvature Action

Statement

Let $n \geq 2$ , let $U \subseteq R^{n}$ be open, and let $g (s) = (g_{M N} (s, x))$ , for $s$ in an open interval about $0$ , be a smooth family of Lorentzian metrics of signature $(- + \dots +)$ on $U$ . Assume $g (0) = g$ , and that $g (s) = g$ outside one compact set $K \subset U$ for every $s$ . Let $f \in C^{\infty} (U, R)$ be independent of $s$ , put $q^{M N} : = \partial_{s} g (s)^{M N} ∣_{s = 0}$ , and use the Levi-Civita Christoffel formula and curvature convention

R^{P}_{Q M N} = \partial_{M} Γ^{P}_{N Q} - \partial_{N} Γ^{P}_{M Q} + Γ^{P}_{M R} Γ^{R}_{N Q} - Γ^{P}_{N R} Γ^{R}_{M Q}, R_{Q N} = R^{P}_{Q P N}, R = g^{Q N} R_{Q N} .

Write $w = \sqrt{∣ \det (g_{M N}) ∣}$ , $\nabla$ for the Levi-Civita connection, and $□ f = g^{M N} \nabla_{M} \nabla_{N} f$ . The inverse-metric variation changes the density by $\dot{w} = - \frac{1}{2} w g_{M N} q^{M N}$ . The chart action integral over $U$ of the difference of the two densities is differentiable at $s = 0$ , and its first variation is

{\frac{d}{d s} ∣}_{0} \int_{U} (w (s) f R (s) - w f R) d^{n} x = \int_{U} w [f (R_{M N} - \frac{1}{2} g_{M N} R) + g_{M N} □ f - \nabla_{M} \nabla_{N} f] q^{M N} d^{n} x .

All integrands in this difference and its derivative have compact support in $K$ . Equivalently, for compactly supported inverse-metric variation $q$ , the bulk coefficient of $q^{M N}$ in the weighted scalar-curvature action is the tensor displayed above. No field equations or physical assumptions are used.

For any smooth compactly supported vector-density components $J^{M}$ on $U$ , the same chart integral convention gives $\int_{U} \partial_{M} J^{M} d^{n} x = 0$ .

Facts & Assumptions

Given: The smooth Lorentzian metric family $g (s)$ , compact support $K$ , fixed smooth scalar $f$ , and the chart curvature and integral conventions of the Statement.

Connection conventions: Indices range from $1$ to $n$ , repeated upper/lower indices are summed, and $(g^{M N})$ is the inverse of $(g_{M N})$ . The Levi-Civita Christoffel formula means $Γ^{P}_{M N} : = \frac{1}{2} g^{P Q} (\partial_{M} g_{N Q} + \partial_{N} g_{M Q} - \partial_{Q} g_{M N})$ , applied to each $g (s)$ . For a scalar $a$ , vector $X$ , and covector $α$ , define $\nabla_{M} a : = \partial_{M} a$ , $\nabla_{M} X^{P} : = \partial_{M} X^{P} + Γ^{P}_{M Q} X^{Q}$ , and $\nabla_{M} α_{N} : = \partial_{M} α_{N} - Γ^{Q}_{M N} α_{Q}$ . Extend to tensors by the product rule and contraction, and set $\nabla^{P} : = g^{P M} \nabla_{M}$ . In particular, $\nabla_{M} \nabla_{N} f = \partial_{M} \partial_{N} f - Γ^{P}_{M N} \partial_{P} f$ .

[F1]

For a differentiable invertible matrix curve $A (s)$ , $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ (The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ ).

[F2]

Every continuous real function on a closed nondegenerate rectangle in $R^{m}$ , $m \geq 1$ , is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable).

[F3]

If $f$ is Riemann integrable on a product of two nondegenerate closed rectangles, then its iterated section integrals in either coordinate block equal the multiple integral whenever the sections are integrable (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[F4]

If $G^{'} = u$ on $[a, b]$ and $u$ is integrable, then $\int_{a}^{b} u = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F5]

A continuous map from a compact metric space to a metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F6]

The Riemann integral on a rectangle is linear and order preserving, and $∣ \int_{Q} u ∣ \leq \int_{Q} ∣ u ∣$ (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ).

[F7]

If $K$ is compact in the subspace $U \subseteq R^{n}$ , every ambient open cover of $K$ in $R^{n}$ has a finite subcover; compact subsets of Euclidean space are bounded, and closed coordinate boxes are compact (A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct; differentiate the inverse metric, determinant, connection, and curvature, then integrate compactly supported coordinate divergences by Fubini and the fundamental theorem of calculus

1.1givenF1algebra

Put $h_{M N} : = \partial_{s} g (s)_{M N} ∣_{0}$ . Differentiating $g (s)^{M A} g (s)_{A N} = δ^{M}_{N}$ gives $q^{M A} g_{A N} + g^{M A} h_{A N} = 0$ , hence $h_{M N} = - g_{M A} g_{N B} q^{A B}$ . The determinant formula [F1] and $\det (g) < 0$ give $\dot{w} = \frac{1}{2} w g^{M N} h_{M N} = - \frac{1}{2} w g_{M N} q^{M N}$ ; the first equality differentiates $w = (- \det g)^{1 / 2}$ , and the second contracts the inverse-variation identity.

2.1givenstep 1.1algebra

The explicit Christoffel formula gives $Γ^{P}_{M N} = Γ^{P}_{N M}$ and $Γ^{Q}_{M N} g_{Q P} + Γ^{Q}_{M P} g_{N Q} = \partial_{M} g_{N P}$ , so $\nabla g = 0$ . Put $C^{P}_{M N} : = \partial_{s} Γ (s)^{P}_{M N} ∣_{0}$ . Differentiating metric compatibility gives $\nabla_{M} h_{N Q} = C^{R}_{M N} g_{R Q} + C^{R}_{M Q} g_{N R}$ . Adding this identity with $M, N$ interchanged and subtracting its version with derivative index $Q$ , the symmetry $C^{P}_{M N} = C^{P}_{N M}$ yields $C^{P}_{M N} = \frac{1}{2} g^{P Q} (\nabla_{M} h_{N Q} + \nabla_{N} h_{M Q} - \nabla_{Q} h_{M N})$ . This is $δ Γ$ ; its coordinate change law is tensorial because the inhomogeneous connection-transformation term is independent of $s$ . Directly differentiating the coordinate curvature formula gives $\partial_{P} δ Γ^{P}_{N M} - \partial_{N} δ Γ^{P}_{P M}$ plus the four variations of its quadratic terms. Expanding $\nabla_{P} δ Γ^{P}_{N M} - \nabla_{N} δ Γ^{P}_{P M}$ gives the same four terms; the remaining terms cancel as $- Γ^{R}_{P N} δ Γ^{P}_{R M} + Γ^{R}_{N P} δ Γ^{P}_{R M} = 0$ by symmetry of the lower connection indices. Thus $δ R_{M N} = \nabla_{P} δ Γ^{P}_{N M} - \nabla_{N} δ Γ^{P}_{P M}$ . With $V^{P} : = g^{M N} δ Γ^{P}_{M N} - g^{P M} δ Γ^{N}_{N M}$ , differentiating $R = g^{M N} R_{M N}$ yields $δ R = R_{M N} q^{M N} + \nabla_{P} V^{P}$ .

3.1step 1.1step 2.1algebra

Substitute $h_{M N} = - g_{M A} g_{N B} q^{A B}$ from step 1.1 into the connection variation from step 2.1 and contract its indices. Writing $q : = g_{M N} q^{M N}$ , direct contraction gives $g^{M N} δ Γ^{P}_{M N} = - \nabla_{M} q^{M P} + \frac{1}{2} \nabla^{P} q$ and $g^{P M} δ Γ^{N}_{N M} = - \frac{1}{2} \nabla^{P} q$ , hence $V^{P} = - \nabla_{M} q^{M P} + \nabla^{P} q$ .

3.2step 1.1step 2.1algebra

By steps 1.1–2.1 and the product rule, $δ (w f R) = w f (R_{M N} - \frac{1}{2} g_{M N} R) q^{M N} + w f \nabla_{P} V^{P}$ . This is the pointwise variation of the density before integration by parts.

3.3givenstep 2.1F1algebra

In the explicit Christoffel formula, symmetry of $g^{M Q}$ cancels the $\partial_{M} g_{P Q}$ and $- \partial_{Q} g_{M P}$ terms after summation, giving $Γ^{M}_{M P} = \frac{1}{2} g^{M Q} \partial_{P} g_{M Q}$ . Apply [F1] along the $x^{P}$ coordinate line and differentiate $w = \sqrt{- \det g}$ to obtain $w^{- 1} \partial_{P} w = \frac{1}{2} g^{M Q} \partial_{P} g_{M Q}$ . Thus $w \nabla_{P} V^{P} = w \partial_{P} V^{P} + w Γ^{P}_{P Q} V^{Q} = \partial_{P} (w V^{P})$ for the vector $V$ defined in step 2.1.

4.1step 3.1step 3.3F2F3F4F7algebra

Since $K$ is compact in the subspace $U \subseteq R^{n}$ , any ambient open cover of $K$ restricts to an open cover in $U$ and has a finite subcover; thus $K$ is compact as a subset of $R^{n}$ by [F7]. Heine--Borel [F7] makes $K$ closed and bounded. If $K \neq \emptyset$ , choose $p \in R^{n}$ and $ρ > 0$ with $K \subset B (p, ρ)$ , and set $Q : = \prod_{j = 1}^{n} [p_{j} - (ρ + 1), p_{j} + (ρ + 1)]$ ; then $K \subset int Q$ . If $K = \emptyset$ , take any nondegenerate closed coordinate box $Q$ . In either case [F7] makes $Q$ compact. The family of all balls $B (p, r)$ with $p \in K$ and $\overline{B} (p, 2 r) \subset U$ covers $K$ ; compactness gives a finite subcover, whose union has closure in $U$ . The field $V$ has compact support because $q$ and its first derivatives do, by step 3.1. Extend $w f V^{P}$ by zero to $Q$ ; it vanishes outside $K$ and near $\partial U$ , so the extension is continuously differentiable. By the chart-integral convention, its integral over $Q$ is the chart integral over $U$ . Each coordinate section is continuously differentiable and compactly supported, so [F2] makes it integrable; [F3] reduces its integral to iterated one-dimensional integrals, and [F4] makes each coordinate integral of $\partial_{P} (w f V^{P})$ zero at the endpoints. The identical argument proves the Statement's divergence-integral identity for any compactly supported smooth vector-density $J^{P}$ . Using step 3.3 and the product rule for $f \partial_{P} (w V^{P})$ gives $\int_{U} w f \nabla_{P} V^{P} d^{n} x = - \int_{U} w (\nabla_{P} f) V^{P} d^{n} x$ .

5.1step 3.1step 4.1F2F3F4algebra

Substitute $V^{P} = - \nabla_{M} q^{M P} + \nabla^{P} q$ from step 3.1 into the integration-by-parts identity in step 4.1. Integrating its two terms by parts by the same compact-support argument gives $- \int_{U} w (\nabla_{P} f) V^{P} d^{n} x = \int_{U} w [- (\nabla_{M} \nabla_{N} f) q^{M N} + g_{M N} (□ f) q^{M N}] d^{n} x$ ; the Hessian is symmetric because $f$ is a scalar and the connection is torsion-free.

6.1givenstep 3.2step 4.1step 5.1F2F4F5F6F7algebra∎

Let $Q$ be the closed box from step 4.1 and choose a nondegenerate closed parameter interval $J$ about $0$ contained in the open parameter domain of $g (s)$ . On $J \times Q$ , define $a (s, x) = w (s) f R (s) - w f R$ by its formula for $x \in U$ and by zero for $x \notin U$ . The density difference and its $s$ -derivative are supported in $K \subset U$ and vanish near $\partial U$ by step 4.1, so both extend smoothly by zero. By the chart-integral convention, the action difference over $U$ equals $\int_{Q} a (s, x) d^{n} x$ . The product $J \times Q$ is a closed coordinate box and is compact by [F7]. Hence $\partial_{s} a$ is uniformly continuous there by [F5]. For each fixed $x$ , $t \mapsto \partial_{t} a (t, x)$ is continuous, hence integrable on every closed subinterval of $J$ by [F2]. Since $a (0, x) = 0$ , for $s > 0$ the fundamental theorem [F4] on $[0, s]$ gives $a (s, x) / s = s^{- 1} \int_{0}^{s} \partial_{t} a (t, x) d t$ ; for $s < 0$ , [F4] on $[s, 0]$ gives $a (s, x) / s = ∣ s ∣^{- 1} \int_{s}^{0} \partial_{t} a (t, x) d t$ . Thus in either case the quotient is an average of $\partial_{t} a (t, x)$ over the segment joining $0$ to $s$ . Uniform continuity implies that for every $ε > 0$ , all sufficiently small $∣ s ∣$ satisfy $\sup_{x \in Q} ∣ a (s, x) / s - \partial_{s} a (0, x) ∣ < ε$ . By [F6], the absolute difference of the rectangle integrals is at most $ε$ times its volume. Thus the difference quotients of the action integrals converge to the integral of $\partial_{s} a (0, \cdot)$ . Combining that derivative with steps 3.2 and 5.1 proves the stated differentiability and first-variation formula.

Compactly Supported Metric Variation of a Weighted Scalar-Curvature Action

Statement

Facts & Assumptions

Proof

Depends on

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