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The reduced dust evolution equations: Weyl system, trace preservation, transport equations and symmetric hyperbolicity

Statement

Work in the reduction conventions of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns with $q : = U (r) + r χ = 0$ (the dust continuity equation in the comoving gauge) and let $E, B$ be the electric-magnetic variables of Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction. Then:

(a) [Weyl evolution] the reduced Weyl equations are $0 = P_{i j} = L_{U} E_{i j} + D_{k} B_{l (i} ϵ_{j)}^{k l} - 3 χ_{(i}^{l} E_{j) l} - 2 χ^{l}_{(i} E_{j) l} + h_{i j} χ^{k l} E_{k l} + 2 χ E_{i j} + \frac{1}{2} κ r (χ_{(i j)} - \frac{1}{3} χ h_{i j}),$ $0 = Q_{i j} = L_{U} B_{i j} - D_{k} E_{l (i} ϵ_{j)}^{k l} - χ^{l}_{(i} B_{j) l} - 2 χ_{(i}^{l} B_{j) l} + χ B_{i j} + χ^{k l} B^{p q} ϵ_{p k (i} ϵ_{q l j)},$ where $L_{U}$ is the Lie derivative along $U$ and the bracket on the indices $i, j$ has weight $\frac{1}{2}$ ; they are obtained by projecting the Bianchi identity of the prescribed curvature onto the symmetric tracefree spatial parts, the density-derivative terms cancelling by $q = 0$ ;

(b) [trace preservation] extend the displayed symmetric equations of (a) algebraically to all symmetric spatial $E, B$ . Contracting this extension with $h^{i j}$ gives $U (t_{E}) + 2 χ t_{E} = 0$ and $U (t_{B}) = 0$ , where $t_{T} = h^{i j} T_{i j}$ . Thus $h^{i j} E_{i j}$ and $h^{i j} B_{i j}$ satisfy homogeneous transport equations along $U$ ; consequently zero initial traces remain zero, and the evolution may be restricted to the five-dimensional tracefree space;

(c) [transport equations] with the parallel-comoving gauge one has $\partial_{s} e_{a}^{μ} = - Γ_{a}^{j}_{0} e_{j}^{μ}, \partial_{s} Γ_{a}^{i}_{j} = R^{p r e i}_{j 0 a} - Γ_{k}^{i}_{j} Γ_{a}^{k}_{0}, \partial_{s} r = - r χ,$ the first two expressing zero time torsion and zero time-curvature residual, the third the continuity equation; for tentative data the frame bracket is used in the curvature definition so that these hold identically;

(d) [symmetric hyperbolicity] using the algebraic connection variables $χ_{i j}$ of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns, equations (a) and (c) form a first-order quasilinear system for $(e_{a}^{μ}, Γ_{a}^{i}_{j}, E_{i j}, B_{i j}, r)$ . In the parallel-comoving frame, the equations for $(e, Γ, r)$ in (c) have only the transport derivative $U = \partial_{s}$ as their principal derivative. The $E, B$ block has time symbol $z I$ and spatial curl symbol $J (ξ) = (\begin{array}{cc} 0 & C (ξ) \\ - C (ξ) & 0 \end{array}), (C (ξ) B)_{i j} = ϵ_{k l (i} ξ^{k} B_{j)}^{l},$ with the positive Frobenius product on symmetric spatial tensors. By The symmetric-tensor curl symbol is skew-adjoint with norm at most the wave number, giving positivity of the dust principal block, $J (ξ)$ is symmetric and $∥ J (ξ) ∥ \leq ∣ ξ ∣$ . Thus for a future-positive timelike conormal $ζ$ , whose components satisfy $z : = ζ (U) > ∣ ζ_{⊥} ∣$ , the full principal symbol is block diagonal with transport block $z I$ and Weyl block $z I + J (ζ_{⊥})$ , both positive definite. Hence the entire reduced dust system is symmetric hyperbolic on every spacelike initial surface, including a tilted one.

Facts & Assumptions

Given: the reduction data, the gauge conditions, and the reconstruction of the Weyl tensor from $(E, B)$ .

[L1]

The metric-compatible parallel frame has $h_{i j} = - δ_{i j}$ , $ϵ_{i j k} = ϵ_{0 i j k}$ , $Γ_{0}^{i}_{j} = 0$ , and $Γ_{a}^{b}_{0} = Γ_{a}^{0}_{b} = - χ_{a b}$ . The projected derivative $D$ retains the arbitrary spatial connection coefficients; the independent-frame curvature uses the actual frame bracket. The curvature-Bianchi residual to be projected is $F_{j k l} = \nabla^{i} C_{i j k l} - \frac{1}{2} (\nabla_{k} S_{l j} - \nabla_{l} S_{k j})$ , and $q_{j} = \nabla_{i} (r U^{i} U_{j}) = q U_{j}$ in this acceleration-free gauge (Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns, Tentative coframe, Cartan torsion and curvature, and the curvature residual of the dust reduction).

[L2]

Exact reconstruction: $C_{0 i 0 j} = E_{i j}$ , $C_{0 i j k} = B_{i p} ϵ_{p j k}$ and $C_{a i p j} = - δ_{i p} E_{j a} + δ_{i j} E_{p a} + δ_{a p} E_{j i} - δ_{a j} E_{p i}$ , with $l_{00} = l_{a b} δ^{a b} / 3 = - 1$ . The Schouten components are those of Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction; constant $Λ$ differentiates to zero because $\nabla γ = 0$ (Reconstruction of a Weyl tensor from its electric and magnetic parts: symmetries, trace-freeness and recovery, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L3]

Curl symbol: the operator $C (ξ)$ on symmetric spatial tensors is skew-adjoint for the positive Frobenius product and has norm at most $∣ ξ ∣$ (The symmetric-tensor curl symbol is skew-adjoint with norm at most the wave number, giving positivity of the dust principal block). $h^{i j} L_{U} h_{i j}$ -type contractions satisfy $L_{U} h_{i j} = 2 χ_{(i j)}$ in the acceleration-free comoving gauge. [L3]

[L4]

In (c), $R^{p r e}$ and $χ$ are algebraic functions of the current reduction variables; the displayed right sides for $e_{a}^{μ}$ , $Γ_{a}^{i}_{j}$ and $r$ contain no spatial derivatives. In the parallel frame, the principal time derivative in $L_{U} E$ and $L_{U} B$ is $\partial_{s}$ ; connection and frame terms are lower order (Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns, Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction).

Proof

technique · direct

1.1L1L2algebra

(Electric projection.) Put $X_{i j} = χ_{i j}$ , $θ = χ = - tr X$ , and ${\dot{T}}_{i j} = (\nabla_{U} T)_{i j} = \partial_{s} T_{i j}$ for parallel-frame components. Spatial repeated indices in this step denote explicit Euclidean sums. The four lower-index connection actions in $\nabla^{a} C_{a i 0 j}$ give ${\dot{E}}_{i j} + ϵ_{i a p} D_{a} B_{j p} - tr (X) E_{i j} + X_{a i} E_{a j} - X_{a p} C_{a i p j}$ ; inserting [L2] makes this ${\dot{E}}_{i j} + ϵ_{i a p} D_{a} B_{j p} + 2 X_{a i} E_{a j} + X_{j p} E_{p i} - 2 tr (X) E_{i j} - δ_{i j} X_{a p} E_{a p}$ . Also $\nabla_{0} S_{j i} = κ \dot{r} δ_{i j} / 3$ and $\nabla_{j} S_{0 i} = κ r X_{j i}$ , so the symmetric Schouten contribution is $κ r X_{(i j)} / 2 - κ \dot{r} δ_{i j} / 6$ . Its Euclidean trace is $- κ q / 2$ ; imposing $q = 0$ or removing the trace yields $κ r (X_{(i j)} + θ δ_{i j} / 3) / 2$ . Symmetrize the Weyl terms and use $(L_{U} T)_{i j} = {\dot{T}}_{i j} - X_{i k} T_{k j} - X_{j k} T_{i k}$ , $χ_{i}^{l} = - X_{i l}$ and $χ^{l}_{i} = - X_{l i}$ to obtain exactly the displayed $P_{i j}$ . The derivative term is the displayed positive symmetric curl of $B$ .

1.2L1algebra

(Transport equations (c).) The first equation is the vanishing of the time torsion in the parallel-comoving gauge: with $Γ_{0}^{i}_{j} = 0$ and $e_{0} = U$ the zero-torsion identity is exactly $\partial_{s} e_{a}^{μ} = - Γ_{a}^{j}_{0} e_{j}^{μ}$ , the displayed transport equation. The second is the vanishing of the time-curvature residual $Δ^{i}_{j 0 a} = 0$ written in the same gauge; for tentative data the actual curvature is defined with the frame bracket, so the independent-frame curvature identity reads $R^{a c t i}_{j 0 a} = \partial_{s} Γ_{a}^{i}_{j} + Γ_{a}^{k}_{0} Γ_{k}^{i}_{j}$ , giving the displayed equation when the time-curvature residual vanishes. The third is $q = 0$ , with no division by $r$ : $U (r) + r χ = 0$ reads $\partial_{s} r = - r χ$ .

2.1L1L2step 1.1algebra

(Magnetic projection.) Contract the last pair with $ϵ_{i}^{k l} / 2$ . The time derivative is ${\dot{B}}_{j i}$ . The spatial derivative part is $ϵ_{i j l} D_{a} E_{l a} - ϵ_{i k l} D_{k} E_{l j}$ , whose first term disappears on symmetrization. The four connection-slot contributions before contraction are $- tr (X) B_{j p} ϵ_{p k l} + X_{a j} B_{a p} ϵ_{p k l} - X_{a k} B_{l p} ϵ_{p a j} + X_{a l} B_{k p} ϵ_{p a j}$ . The first two contract to $θ B_{i j} + X_{a j} B_{a i}$ . The last two contract to $- ϵ_{i k l} X_{a k} B_{l p} ϵ_{p a j}$ ; interchanging dummy indices and using the symmetry of $B$ makes its symmetric part $+ X_{k l} B_{p q} ϵ_{p k (i} ϵ_{q l j)}$ . Hence the symmetric Weyl projection is ${\dot{B}}_{i j} - curl (E)_{i j} + sym (X^{T} B)_{i j} + θ B_{i j} + X_{k l} B_{p q} ϵ_{p k (i} ϵ_{q l j)}$ . The Schouten contribution is $- κ ϵ_{i}^{k j} D_{k} r / 6$ , antisymmetric in $i, j$ , so its symmetric projection vanishes. Converting $\dot{B}$ to $L_{U} B$ by step 1.1 gives exactly the displayed $Q$ with positive final double-epsilon term. No symmetry of $X$ , hypersurface orthogonality or restriction on tilt was used.

3.1step 1.1step 2.1L3algebra

(Trace preservation (b).) On the algebraic extension to symmetric spatial tensors, let $t_{T} = h^{i j} T_{i j}$ . Since $L_{U} h_{i j} = 2 χ_{(i j)}$ , one has $h^{i j} L_{U} T_{i j} = U (t_{T}) + 2 χ^{i j} T_{i j}$ . The symmetric curl has identically zero trace: $ϵ^{i k l} D_{k} T_{l i} = 0$ by symmetry in $l, i$ . The electric matter term has zero trace. The electric coupling contractions combine to $- 2 χ^{i j} E_{i j} + 2 χ t_{E}$ , giving $h^{i j} P_{i j} = U (t_{E}) + 2 χ t_{E}$ . For the magnetic double-epsilon term the identity $\sum_{i} ϵ_{p k i} ϵ_{q l i} = δ_{p q} δ_{k l} - δ_{p l} δ_{k q}$ gives $h^{i j} χ^{k l} B^{p q} ϵ_{p k (i} ϵ_{q l j)} = χ^{k l} B_{k l} - χ t_{B}$ . All magnetic coupling contractions therefore cancel the $2 χ^{i j} B_{i j}$ Lie contribution and yield $h^{i j} Q_{i j} = U (t_{B})$ . These homogeneous scalar ODEs preserve zero initial traces along every flow line, including where $r = 0$ ; symmetry is preserved by the displayed symmetrizations. Thus the tracefree restriction of the extended system is invariant and is precisely the projected Weyl system derived in steps 1.1–1.2.

4.1step 1.2L3L4algebra∎

(Full principal symbol and positivity (d).) Let $Y = (e_{a}^{μ}, Γ_{a}^{i}_{j}, r)$ denote the transport variables, written in independent components, and use the tracefree symmetric components of $(E, B)$ from step 3.1. By [L4], the principal symbol on $Y$ is $z I$ . In the $P$ equation, raising the two spatial indices of $ϵ_{j k l}$ with $h^{i j} = - δ^{i j}$ contributes a positive sign, so $ϵ_{j}^{k l} = ϵ_{j k l}$ . Thus the coefficient of a spatial covector $ξ$ in $D_{k} B_{l (i} ϵ_{j)}^{k l}$ is $ϵ_{k l (i} ξ^{k} B_{j) l} = C (ξ) B$ , where the last equality uses the Euclidean raising convention in The symmetric-tensor curl symbol is skew-adjoint with norm at most the wave number, giving positivity of the dust principal block. The corresponding term in $Q$ is its negative acting on $E$ . Hence the Weyl spatial symbol is exactly $J (ξ) = (\begin{matrix} 0 & C (ξ) \\ - C (ξ) & 0 \end{matrix})$ . By [L3] and The symmetric-tensor curl symbol is skew-adjoint with norm at most the wave number, giving positivity of the dust principal block, $C (ξ)$ is skew-adjoint with norm at most $∣ ξ ∣$ , so $J (ξ)$ is symmetric with norm at most $∣ ξ ∣$ . For a future-positive timelike conormal $ζ$ , $z = ζ (U) > ∣ ζ_{⊥} ∣$ ; hence the full symbol is $diag (z I_{Y}, z I_{E, B} + J (ζ_{⊥}))$ , whose quadratic form is bounded below by $\min {z, z - ∣ ζ_{⊥} ∣}$ times the squared norm. It is therefore positive definite on every spacelike initial surface, irrespective of tilt.

Remarks

Role. Clause (d) is the hyperbolicity input for the local dust existence theorem; the residual and subsidiary equations that make the reduced system equivalent to the full Einstein-dust system are analysed in Subsidiary system of the dust reduction: Cartan identities, closed homogeneous residual system and vanishing of the residuals.

Depends on

Dependency tree · 2 levels

Sources

H. Friedrich and A. D. Rendall, The Cauchy Problem for the Einstein Equations, Living Reviews in Relativity 3 (2000), article 1 (arXiv:gr-qc/0002074), Section 4.2 (The Einstein-Euler System): the hyperbolic reduction of the Einstein-Euler system, the positivity conditions on specific enthalpy and speed of sound (4.38), the frame variables and their reduced equations, and the pressure-free (dust) subsidiary system of Sections 4.2.4-4.2.5, equations (4.26)-(4.48) and (4.56)-(4.63)