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Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $\overset{ˉ}{g}$ be a smooth background metric with Levi-Civita connection $\overset{ˉ}{\nabla}$ , and let $H^{a} = g^{b c} (Γ^{a}_{b c} - {\overset{ˉ}{Γ}}^{a}_{b c})$ be the harmonic-gauge vector of The harmonic gauge vector, the reduced Einstein equation and the Einstein residual. Then:

(a) [principal part] Let $\hat{g}$ be a smooth Lorentzian metric on an open set $U \subseteq M$ and let $h$ be a smooth symmetric covariant $2$ -tensor on $U$ . The derivative at $\hat{g}$ in the direction $h$ of the operator $g \mapsto {Ric}_{a b} [g] - {\overset{ˉ}{\nabla}}_{(a} H_{b)} [g]$ equals $- \frac{1}{2} {\hat{g}}^{c d} {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} h_{a b} + L_{a b} (h, \overset{ˉ}{\nabla} h; \hat{g}, \overset{ˉ}{g}),$ where $L$ is a smooth expression linear in $h$ and its first $\overset{ˉ}{\nabla}$ -derivatives with coefficients built from $\hat{g}, {\hat{g}}^{- 1}$ , its first and second $\overset{ˉ}{\nabla}$ -derivatives, $\overset{ˉ}{g}$ and the background curvature. Thus the coefficients depend on the full second jet of the linearization metric. In particular the principal symbol is $- \frac{1}{2} {\hat{g}}^{c d} ξ_{c} ξ_{d}$ times the identity on the metric slot at every covector $ξ$ .

(b) [first-order reduction] Fix constants $κ > 0$ , $Λ \in R$ and a prescribed smooth symmetric covariant $2$ -tensor field $T$ on the chart domain. Fix a chart $(x^{0}, x^{1}, x^{2}, x^{3})$ with $x^{0} = t$ . Introduce the metric variables $u : = (g_{a b}, p_{a b} : = {\overset{ˉ}{\nabla}}_{0} g_{a b}, q_{a b i} : = {\overset{ˉ}{\nabla}}_{i} g_{a b})$ (symmetry in $a, b$ understood) and the relation variables ${\overset{ˉ}{\nabla}}_{0} g_{a b} - p_{a b} = 0$ and ${\overset{ˉ}{\nabla}}_{0} q_{a b i} - {\overset{ˉ}{\nabla}}_{i} p_{a b} = C_{a b i}$ , where $C_{a b i} = - {\overset{ˉ}{R}}^{c}_{a 0 i} g_{c b} - {\overset{ˉ}{R}}^{c}_{b 0 i} g_{a c}$ . Second covariant derivatives include their derivative-slot connection. These known background-curvature terms are zeroth order in the first-order state and vanish for the flat coordinate background. Then the reduced Einstein equation can be solved for the second time derivatives in the form ${\overset{ˉ}{\nabla}}_{0} p_{a b} = - (g^{00})^{- 1} [2 g^{0 i} {\overset{ˉ}{\nabla}}_{i} p_{a b} + g^{i j} {\overset{ˉ}{\nabla}}_{i} q_{a b j} + R_{a b} (x, u, T; κ, Λ, \overset{ˉ}{g})],$ where the source is smooth in the first-order metric state and the prescribed matter data, with no derivatives of $T$ , and includes $2 κ (T_{a b} - \frac{1}{2} (g^{c d} T_{c d}) g_{a b}) + 2 Λ g_{a b}$ . The resulting first-order system for $u$ is symmetric hyperbolic on every region where $g^{00} < 0$ and the spatial block $(g^{i j})$ is positive definite: the coefficient of ${\overset{ˉ}{\nabla}}_{0} u$ is the block diagonal $diag (I, - g^{00} I, g^{i j} I)$ (on the three slots $g, p, q$ ), which is positive definite there, and the coefficients of the spatial derivatives are symmetric. More generally, in a coupled reduction one may substitute a specified smooth algebraic stress map $T_{a b} = T_{a b} (x, u, z)$ , symmetric in $a, b$ , where $z$ is a specified additional first-order matter state ranging in an open Euclidean set. This keeps the same metric principal matrices; symmetric hyperbolicity of the combined metric-matter system requires a separate check of the matter equations.

Facts & Assumptions

Given: the manifold, metrics and gauge vector of the statement, and in (a) the metric $\hat{g}$ and direction $h$ ; in (b) the fixed matter tensor or stated algebraic stress map and constants $κ, Λ$ .

[L1]

Difference of connections: for $h = g - \overset{ˉ}{g}$ , $Δ^{a}_{b c} : = Γ^{a}_{b c} - {\overset{ˉ}{Γ}}^{a}_{b c} = \frac{1}{2} g^{a d} ({\overset{ˉ}{\nabla}}_{b} h_{d c} + {\overset{ˉ}{\nabla}}_{c} h_{b d} - {\overset{ˉ}{\nabla}}_{d} h_{b c})$ (Christoffel symbols of an affine connection, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L2]

Substituting $Γ = \overset{ˉ}{Γ} + Δ$ into the coordinate curvature formula and contracting gives ${Ric}_{a b} [g] = {Ric}_{a b} [\overset{ˉ}{g}] + {\overset{ˉ}{\nabla}}_{c} Δ^{c}_{a b} - {\overset{ˉ}{\nabla}}_{b} Δ^{c}_{a c} + Δ^{c}_{c d} Δ^{d}_{a b} - Δ^{c}_{b d} Δ^{d}_{a c} .$ This identity uses the curvature convention of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula and The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors; the background covariant derivatives collect the terms linear in $\overset{ˉ}{Γ}$ .

[L3]

Lowering the gauge vector in [L1] gives the exact identity $H_{b} = g^{c d} ({\overset{ˉ}{\nabla}}_{c} g_{b d} - \frac{1}{2} {\overset{ˉ}{\nabla}}_{b} g_{c d})$ . Differentiating $g^{c e} g_{e d} = δ_{d}^{c}$ in a metric direction $h$ gives $δ g^{c d} = - g^{c e} h_{e f} g^{f d}$ ; the spatial derivative version is ${\overset{ˉ}{\nabla}}_{k} g^{c d} = - g^{c e} ({\overset{ˉ}{\nabla}}_{k} g_{e f}) g^{f d}$ .

[L4]

The commutator on a covariant two-tensor is $[{\overset{ˉ}{\nabla}}_{0}, {\overset{ˉ}{\nabla}}_{i}] g_{a b} = - {\overset{ˉ}{R}}^{c}_{a 0 i} g_{c b} - {\overset{ˉ}{R}}^{c}_{b 0 i} g_{a c}$ . It follows by applying the vector commutator in The commutator of covariant derivatives on a vector field and its contracted component identity (a) to each slot of the scalar pairing, using the tensor derivative rule in Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields; commutators in any two directions have the same slot formula.

[L5]

The metric and its inverse give symmetric bilinear forms; $g^{00} < 0$ on a chart adapted to a spacelike slice with normal time coordinate, and $(g^{i j})$ is then positive definite (The inverse metric, musical isomorphisms and index raising, The harmonic gauge vector, the reduced Einstein equation and the Einstein residual).

[L6]

The exact reduced equation is ${Ric}_{a b} - \nabla_{(a} H_{b)} = κ (T_{a b} - \frac{1}{2} (g^{c d} T_{c d}) g_{a b}) + Λ g_{a b}$ (The harmonic gauge vector, the reduced Einstein equation and the Einstein residual clause (b)); its right side is algebraic in the metric and stress tensor.

Proof

technique · direct

1.1L1algebra

(The difference formula.) Write $h = g - \overset{ˉ}{g}$ and $Δ^{a}_{b c} = Γ^{a}_{b c} - {\overset{ˉ}{Γ}}^{a}_{b c}$ . Subtracting the two Christoffel formulas and using ${\overset{ˉ}{\nabla}}_{k} h_{a b} = \partial_{k} h_{a b} - {\overset{ˉ}{Γ}}^{c}_{k a} h_{c b} - {\overset{ˉ}{Γ}}^{c}_{k b} h_{a c}$ gives [L1] directly: the algebraic identity $Γ^{a}_{b c} = {\overset{ˉ}{Γ}}^{a}_{b c} + \frac{1}{2} g^{a d} ({\overset{ˉ}{\nabla}}_{b} h_{d c} + {\overset{ˉ}{\nabla}}_{c} h_{b d} - {\overset{ˉ}{\nabla}}_{d} h_{b c})$ is equivalent to the Christoffel formula for $g$ after insertion of $\partial_{k} h_{a b}$ .

2.1step 1.1L1L2L3L4algebra

(Exact quasilinear principal part.) Insert [L1] into [L2]. Terms in which a derivative hits $g^{- 1}$ and the $Δ Δ$ terms depend smoothly only on $g, g^{- 1}, \overset{ˉ}{\nabla} g$ . The second-derivative terms are $\frac{1}{2} g^{c d} ({\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{a} g_{d b} + {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{b} g_{d a} - {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} g_{a b} - {\overset{ˉ}{\nabla}}_{b} {\overset{ˉ}{\nabla}}_{a} g_{c d})$ . By [L3] the second-derivative terms of ${\overset{ˉ}{\nabla}}_{(a} H_{b)}$ are $\frac{1}{2} g^{c d} ({\overset{ˉ}{\nabla}}_{a} {\overset{ˉ}{\nabla}}_{c} g_{b d} + {\overset{ˉ}{\nabla}}_{b} {\overset{ˉ}{\nabla}}_{c} g_{a d} - {\overset{ˉ}{\nabla}}_{a} {\overset{ˉ}{\nabla}}_{b} g_{c d})$ . Subtraction cancels the first, second and fourth Ricci terms after commuting derivatives. By [L4] the commutators contain only background curvature times $g$ . Consequently ${Ric}_{a b} [g] - {\overset{ˉ}{\nabla}}_{(a} H_{b)} [g] = - \frac{1}{2} g^{c d} {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} g_{a b} + Q_{a b} (g, g^{- 1}, \overset{ˉ}{\nabla} g; \overset{ˉ}{g}, \overset{ˉ}{R})$ , with $Q$ smooth and containing no second derivatives of $g$ . The actual reduced operator using $\nabla_{(a} H_{b)}$ differs by $Δ^{c}_{a b} H_{c}$ , another smooth first-jet term.

3.1step 2.1L3algebra

(Linearization and its lower-order coefficients.) Differentiate step 2.1 at $\hat{g}$ . Variation of the wave term is $- \frac{1}{2} {\hat{g}}^{c d} {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} h_{a b} + \frac{1}{2} {\hat{g}}^{c e} h_{e f} {\hat{g}}^{f d} {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} {\hat{g}}_{a b}$ . Variation of $Q$ is linear in $h, \overset{ˉ}{\nabla} h$ , with coefficients depending smoothly on $\hat{g}, {\hat{g}}^{- 1}, \overset{ˉ}{\nabla} \hat{g}$ and the background. This proves precisely (a), including the second-jet dependence. Replacing ${\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} h$ by $ξ_{c} ξ_{d} h$ defines the stated principal-symbol convention.

3.2step 2.1L4L5L6algebra

(First-order reduction.) By step 2.1 the exact reduced metric equation has the form $g^{c d} {\overset{ˉ}{\nabla}}_{c} {\overset{ˉ}{\nabla}}_{d} g_{a b} + 2 S_{a b} - 2 {\tilde{Q}}_{a b} = 0$ , where $S_{a b} = κ (T_{a b} - \frac{1}{2} (g^{c d} T_{c d}) g_{a b}) + Λ g_{a b}$ by [L6] and $\tilde{Q} = Q + Δ^{c}_{a b} H_{c}$ is the actual first-jet remainder from step 2.1. With $p = {\overset{ˉ}{\nabla}}_{0} g$ , $q_{i} = {\overset{ˉ}{\nabla}}_{i} g$ , its mixed derivatives are ${\overset{ˉ}{\nabla}}_{0} q_{i} = {\overset{ˉ}{\nabla}}_{i} p + C_{i}$ by [L4], and its spatial derivatives are ${\overset{ˉ}{\nabla}}_{i} q_{j}$ , including the prescribed derivative-slot connection terms. Where $g^{00} < 0$ , solving for ${\overset{ˉ}{\nabla}}_{0} p$ gives the displayed equation, with background commutator and connection terms included in the smooth geometric remainder $R_{a b}^{g e o m} (x, u; \overset{ˉ}{g})$ and $R_{a b} = R_{a b}^{g e o m} + 2 κ (T_{a b} - \frac{1}{2} (g^{c d} T_{c d}) g_{a b}) + 2 Λ g_{a b}$ . Append ${\overset{ˉ}{\nabla}}_{0} g = p$ and the three commutator relations. This is a first-order system on the metric jet variables with no second metric derivatives or derivatives of $T$ left in its sources. A smooth algebraic stress substitution $T (x, u, z)$ has the same property for the enlarged state, without asserting any property of its as yet unspecified matter equations.

4.1step 3.2L5algebra∎

(Explicit symmetric matrices.) Move the equation in step 3.2 to the form $- g^{00} \partial_{t} p - 2 g^{0 k} \partial_{k} p - g^{i j} \partial_{i} q_{j} =$ lower-order terms, and multiply the relation $\partial_{t} q_{i} - \partial_{i} p =$ lower-order terms by $g^{j i}$ . In the state ordering $(g, p, q_{i})$ , suppressing the identity on symmetric metric components, the time matrix is $A^{0} = diag (1, - g^{00}, (g^{i j}))$ . For each $k$ , the only nonzero entries of $A^{k}$ are $(A^{k})_{p p} = - 2 g^{0 k}$ , $(A^{k})_{p q_{j}} = - g^{k j}$ and $(A^{k})_{q_{i} p} = - g^{i k}$ . Thus every $A^{k}$ is symmetric, even when $g^{0 k} \neq 0$ , and $A^{0}$ is positive definite under exactly the two stated block inequalities. Passing from background-covariant to component derivatives changes only smooth zeroth-order state terms. No initially orthogonal chart hypothesis is needed.

Remarks

Role. Clause (a) is the statement that the harmonic-gauge reduction has the same principal part as the wave operator and hence is a hyperbolic (not elliptic) system; clause (b) is the form in which the local well-posedness theorem Local well-posedness and finite-speed independence of the extension for symmetric hyperbolic systems on a coordinate neighbourhood is applied in Local existence for the reduced Einstein equation from Cauchy data on a spacelike slice.
Scope. The reduction is local and uses any chart satisfying the two stated block inequalities; no global chart or global hyperbolicity is asserted.

Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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