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Weighted Einstein Hilbert Variation in Ten Dimensions

Statement

Let $U \subseteq R^{10}$ be an open oriented coordinate domain in a smooth ten-manifold. For $s$ in an open interval about $0$ , let $g (s)$ be a family of smooth nondegenerate Lorentzian metrics of inertia $(9, 1, 0)$ , jointly smooth in $(s, x)$ , with $g (0) = g$ and $g (s) = g$ outside one fixed compact set $K \subset U$ . Put $q^{M N} : = {\partial_{s} g (s)^{M N} ∣}_{s = 0}$ . For each $s$ use the connection with coefficients $Γ^{P}_{M N} (s) = \frac{1}{2} g (s)^{P Q} (\partial_{M} g (s)_{N Q} + \partial_{N} g (s)_{M Q} - \partial_{Q} g (s)_{M N})$ : for vector fields, $(\nabla_{X} Y)^{P} = X^{M} (\partial_{M} Y^{P} + Γ^{P}_{M N} Y^{N})$ . On scalars set $\nabla_{X} f = X (f)$ and extend to covariant tensors by duality and the product rule. This is the torsion-free metric-compatible (Levi-Civita) connection, as verified below. All unmarked quantities are evaluated at $s = 0$ . Fix the curvature convention $R^{P}_{Q M N} = \partial_{M} Γ_{N Q}^{P} - \partial_{N} Γ_{M Q}^{P} + Γ_{M R}^{P} Γ_{N Q}^{R} - Γ_{N R}^{P} Γ_{M Q}^{R}$ , $R_{Q N} = R^{P}_{Q P N}$ , and $R = g^{Q N} R_{Q N}$ . Write $w (s) = \sqrt{∣ \det (g (s)_{M N}) ∣}$ , $w = w (0)$ , $G_{M N} = R_{M N} - \frac{1}{2} g_{M N} R$ , and $□ f = g^{M N} \nabla_{M} \nabla_{N} f$ . Let $f \in C^{\infty} (U)$ be independent of $s$ , and define the bulk first variation by

$δ S_{f} [g] (q) : = {\frac{d}{d s} ∣}_{s = 0} \int_{U} (w (s) f R (s) - w f R) d^{10} x .$

The density difference is supported in $K$ , so this integral is finite even when either unvaried action integral is not. Every smooth compactly supported symmetric $q^{M N}$ is admissible: the inverse-metric path $g (s)^{M N} = g^{M N} + s q^{M N}$ remains Lorentzian for sufficiently small $∣ s ∣$ , since the support of $q$ is compact and the Lorentzian matrices form an open set. The bulk first variation is

$δ \int_{U} w f R d^{10} x = \int_{U} w [f G_{M N} + g_{M N} □ f - \nabla_{M} \nabla_{N} f] q^{M N} d^{10} x .$

This is a local bulk identity for compactly supported variations; no boundary condition on the unvaried region is needed.

Statement references

Smooth manifolds and their smooth charts

Oriented smooth manifolds and oriented charts

A smooth tensor field

Affine connection on a smooth manifold

Christoffel symbols of an affine connection

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

Facts & Assumptions

Given: The ten-dimensional chart, metric, connection, curvature convention, scalar $f$ , and compactly supported smooth symmetric variation $q^{M N}$ in the Statement.

[F1]

The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ : for an invertible differentiable matrix curve, $δ \det g = \det (g) tr (g^{- 1} δ g)$ ; differentiating $g^{M P} g_{P N} = δ_{N}^{M}$ gives $δ g_{M N} = - g_{M P} g_{N Q} q^{P Q}$ .

[F2]

Affine connection on a smooth manifold supplies the affine-connection linearity and Leibniz rules; Christoffel symbols of an affine connection supplies the convention $\nabla_{\partial_{M}} \partial_{N} = Γ^{P}_{M N} \partial_{P}$ . Neither definition supplies metric compatibility or zero torsion. The coefficients used here are defined in the Statement, and their required properties are proved in step 1.1.

[L1]

If $V^{M}$ is a smooth compactly supported vector field on $U$ , then $\int_{U} \partial_{M} (w V^{M}) d^{10} x = 0$ : extend $w V^{M}$ by zero to a closed coordinate box containing its support. The extension and its derivatives are smooth, hence integrable by Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable. Apply Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections to integrate each coordinate derivative along its coordinate lines; The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ gives zero because the extension vanishes near both endpoints. From [F1] and the Christoffel formula in the Statement, $Γ^{M}_{M P} = w^{- 1} \partial_{P} w$ , so $w \nabla_{M} V^{M} = \partial_{M} (w V^{M})$ .

Proof

1.1givenF1F2algebra

The inverse metric is jointly smooth: each entry is a cofactor divided by the nonzero determinant. Thus the displayed coefficients are smooth and the formula for $\nabla_{X} Y$ directly satisfies the affine-connection rules in [F2]. Their symmetry in $M, N$ gives zero torsion, and substitution gives $\partial_{M} g_{N Q} = Γ^{P}_{M N} g_{P Q} + Γ^{P}_{M Q} g_{N P}$ , hence $\nabla g = 0$ , for every $s$ . Conversely, cyclically permuting this identity and using symmetry recovers the displayed coefficients, proving uniqueness on $U$ . Put $h_{M N} = δ g_{M N} = - g_{M P} g_{N Q} q^{P Q}$ by [F1]; then $δ w = - \frac{1}{2} w g_{M N} q^{M N}$ . Write $C^{P}_{M N} = δ Γ^{P}_{M N}$ . Differentiating metric compatibility gives $\nabla_{M} h_{N Q} = C^{P}_{M N} g_{P Q} + C^{P}_{M Q} g_{N P}$ ; combining its three cyclic versions and using $C^{P}_{M N} = C^{P}_{N M}$ yields $C^{P}_{M N} = \frac{1}{2} g^{P Q} (\nabla_{M} h_{N Q} + \nabla_{N} h_{M Q} - \nabla_{Q} h_{M N})$ .

2.1F2step 1.1algebra

Differentiating the stated coordinate formula for $R^{P}_{Q M N}$ and grouping the connection terms into covariant derivatives gives $δ R_{Q N} = \nabla_{P} δ Γ_{N Q}^{P} - \nabla_{N} δ Γ_{P Q}^{P}$ . Contracting with $g^{Q N}$ and inserting step 1.1 yields $δ R = R_{M N} q^{M N} - \nabla_{M} \nabla_{N} q^{M N} + □ (g_{M N} q^{M N})$ . The first identity follows by cancellation of the quadratic-connection terms in the curvature derivative; the second uses $\nabla g = 0$ .

3.1givenL1step 1.1step 2.1algebra∎

Joint smoothness and the common compact support permit differentiation under the integral: on a closed coordinate box containing $K$ , the difference quotients of the zero-extended density difference converge uniformly to its $s$ -derivative by the one-variable fundamental theorem of calculus and uniform continuity of that derivative. Varying $w f R$ , the density and algebraic curvature terms are $w f (R_{M N} - \frac{1}{2} g_{M N} R) q^{M N}$ . The two remaining terms from step 2.1 integrate by parts twice, with no boundary contribution because $q$ has compact support: $\int w f [- \nabla_{M} \nabla_{N} q^{M N} + □ (g_{M N} q^{M N})] = \int w [g_{M N} □ f - \nabla_{M} \nabla_{N} f] q^{M N}$ . By [L1], compactly supported coordinate divergences integrate to zero; the Christoffel trace identity in [L1] converts the covariant divergence to a coordinate divergence. The scalar Hessian is symmetric because the connection is torsion-free, so the twice-integrated coefficient has the displayed index order. Adding these coefficients proves the formula.

Weighted Einstein Hilbert Variation in Ten Dimensions

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources