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Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data

Statement

Assume Countable Choice and Dependent Choice. Let $Σ$ be a smooth oriented three-manifold without boundary, with smooth Riemannian metric $h$ , smooth symmetric covariant tensor $K$ , and smooth spatial vectors $e, b$ . Fix $κ > 0$ , $μ_{0} > 0$ , $Λ \in R$ , use signature $(-, +, +, +)$ and $K (X, Y) = g (\nabla_{X} n, Y)$ , and suppose the four Einstein–Maxwell constraints of Gauss-Codazzi and Maxwell initial constraints hold. In SI convention use length coordinate $x^{0} = c t$ , dimensionless $g, h$ , $K$ in $m^{- 1}$ , and $e = E / c$ , $b = B$ in tesla. Reconstruct the complete initial two-form on $R N \oplus T Σ$ by $F (N, e_{i}) = - e_{i}, F (e_{i}, e_{j}) = ϵ_{i j k} b_{k},$ where $N$ is the future unit normal symbol and $(e_{i})$ is a positively oriented $h$ -orthonormal frame. Then the data have a smooth time-oriented local source-free Einstein–Maxwell development on an open variable-width neighborhood of the entire initial surface. After restriction to a smaller collar, the interior of its intrinsic domain of dependence is a globally hyperbolic neighborhood with $Σ$ Cauchy. Any two developments of this same model and complete data are related near the entire initial surface by a data-fixing, time-orientation-preserving isometry carrying $F$ to $F$ . There is no uniform positive time width for noncompact $Σ$ and no global-in-time conclusion.

Facts & Assumptions

Given: Countable Choice, Dependent Choice, the smooth oriented data and constants in the Statement, and the specified tetrad, field and extrinsic-curvature conventions.

[L1]

Draft model Electrovacuum tensor Cauchy system fixes $G + Λ g = κ T$ , $T_{a b} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ , $d F = d ⋆_{g} F = 0$ , and the curvature and orientation/sign conventions. Its item file is listed in batch 6; its status is draft and its precheck is pending; the application remains conditional.

[L2]

Draft constraint result Gauss-Codazzi and Maxwell initial constraints states, with this $K$ and field convention, $D_{i} e^{i} = D_{i} b^{i} = 0, R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2} = 2 Λ + \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2}), D^{j} K_{j i} - D_{i} {tr}_{h} K = - \frac{κ}{μ_{0}} (e \times_{h} b)_{i} .$ Its item file is present but listed in batch 6 and has verification precheck pending, so the interface remains conditional.

[L3]

Draft stress identity Maxwell stress algebra and covariant conservation gives $\nabla^{a} T_{a b} = - F_{b c} j^{c}$ for smooth fields satisfying $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ . The exact source-free Hodge conversion is $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}), ε_{a b c d} : = ({v o l}_{g})_{a b c d},$ with indices raised by $g^{- 1}$ . It follows by expanding $d ⋆_{g} F = 3 \nabla_{[a} (⋆_{g} F)_{c d]}$ , using metric-compatibility $\nabla_{a} ε_{b c d e} = 0$ , $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ and $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ ; the coordinate divergence uses $Γ^{a}_{a c} = \partial_{c} \log \sqrt{∣ \det g ∣}$ and the vanishing contraction $Γ^{b}_{a c} F^{a c}$ . Hence the source-free equations in [L1] meet both hypotheses with $j = 0$ before the unreduced Einstein equation is used. The file exists but is listed in batch 6 and its precheck is pending; this input remains conditional.

[L4]

Draft harmonic initialization Harmonic initial data relative to a fixed smooth background specifies all initial metric first derivatives, including auxiliary reduction variables, so $H = 0$ and the initial reduced state is compatible with the chosen smooth background. Its file exists but is listed in batch 6 and its precheck is pending; this use remains conditional.

[F1]

Positive symmetric Maxwell field-strength reduction gives the symmetric Maxwell principal block and positive time matrix; Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction gives the metric wave block; Coupled reduced Einstein–Maxwell local evolution combines them, verifies smooth coefficients and no derivative coupling through the algebraic stress, and applies the conditional local solver.

[F2]

Local high-Sobolev and smooth electrovacuum includes the local smooth existence clause on arbitrary smooth initial slices. It proves the derivative-variable compatibility, local Maxwell constraint propagation and Einstein harmonic recovery, conditional on the CP01 carriers named above.

[F3]

Maxwell Gauss constraints propagate by form transport proves local recovery of $d F = d ⋆_{g} F = 0$ from the normal-slot equations $ι_{n} d F = ι_{n} d ⋆_{g} F = 0$ and zero initial Gauss divergences for $C^{1}$ metric, field and transverse future unit normal; the smooth local developments here satisfy these hypotheses. Einstein harmonic gauge propagation with Maxwell on shell proves local $H = 0$ and recovery of the unreduced Einstein equation from the normal constraints and conserved stress.

[F4]

Local geometric development for a fixed covariant Einstein-matter model with complete constrained data applies to a fixed covariant matter model with a symmetric-hyperbolic reduction, complete smooth matter initialization, propagated auxiliary compatibility, on-shell stress conservation, and the Einstein constraints. Its actual published carrier has precheck pending; its statement and full local proof were inspected.

[F5]

Causal and Cauchy-surface inputs for smooth electrovacuum developments collects the exact local causal-neighborhood, globally hyperbolic Cauchy-neighborhood, and choice interfaces used in the local-development conclusion.

Proof

1.1givenL1L2algebra

On the abstract initial bundle $R N \oplus T Σ$ , the displayed formulas assign the three normal-spatial components and the three spatial-spatial components of an antisymmetric two-form. These are all six independent components of a two-form in four dimensions, so they determine a unique complete initial $F$ without choosing a potential or assuming global exactness. They are smooth because $h, e, b$ are smooth and the reconstruction is tensorial under oriented orthonormal-frame changes.

1.2givenL4F1F2F3F4algebra

In each relatively compact initial coordinate ball choose product coordinates $(x^{0}, x^{i})$ with $x^{0} = 0$ on the slice and $\partial_{0} = n$ there. Choose a smooth positive spatial background $γ$ and put $Q^{i} : = h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k})$ . Relative to $\overset{ˉ}{g} = - (d x^{0})^{2} + γ_{i j} (x) d x^{i} d x^{j}$ , prescribe $g_{i j} = h_{i j}$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $\partial_{0} g_{i j} = 2 K_{i j}$ , $\partial_{0} g_{00} = - 2 {tr}_{h} K$ , and $\partial_{0} g_{0 i} = h_{i j} Q^{j}$ . The same Christoffel computation as the periodic construction [L4], now carried out on this local chart, gives $H^{0} = - {tr}_{h} K + {tr}_{h} K = 0$ and $H^{i} = - Q^{i} + h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k}) = 0$ . This local jet is smooth and compatible with the first-order reduction. The metric block has time matrix $diag (I, - g^{00} I, g^{i j} I)$ by [F1]'s reduced Einstein principal-part supplier, and the Maxwell block has time matrix $α^{- 1} I_{6}$ by [F1]. On the open state neighborhood where $g^{00} < 0$ , $(g^{i j}) > 0$ and $α > 0$ , their block sum is positive definite; every spatial matrix is symmetric. Since $T (g, F)$ is algebraic in the state, it contributes only to lower-order terms and creates no off-diagonal derivative blocks. The prescribed local high-Sobolev construction [F2] applies after the compact chart localization in its proof and yields a reduced solution. Its derivative relations propagate; [F3] restores both Maxwell equations and the full Einstein equation on a smaller local domain. Every smooth physical solution in the same harmonic chart extracts to this same coefficient system because the component equations are precisely the harmonic-reduced Einstein equation and field-strength Maxwell evolution. This verifies the local symmetric-hyperbolic, recovery and compatibility hypotheses of [F4].

2.1step 1.1L1F4algebra

The equations $d F = 0$ and $d ⋆_{g} F = 0$ are covariant: pullback commutes with exterior differentiation, and for an orientation-preserving diffeomorphism $ϕ$ , $ϕ^{*} (⋆_{g} F) = ⋆_{ϕ^{*} g} (ϕ^{*} F)$ . The algebraic stress $T (g, F)$ is likewise natural under pullback. Thus the matter field, equations, stress and initial reconstruction define one fixed covariant model of the kind required by [F4]; no coordinate-dependent potential gauge is introduced.

2.2step 1.1step 1.2L1L2L3F3F4

Step 1.2, using [F3], has already recovered $d F = d ⋆_{g} F = 0$ on the local development; [L1] fixes the corresponding source-free model and stress tensor. The exact Hodge conversion in [L3] gives $\nabla_{a} F^{a b} = 0$ , so the stress identity applies with $j = 0$ and yields $\nabla^{a} T_{a b} = 0$ , independently of the unreduced Einstein equation. The constraints in the Statement are exactly [L2]'s four normal-projection constraints with the same sign, orientation, $κ, μ_{0}, Λ$ and $K$ conventions. This verifies the conservation and Einstein-constraint hypotheses of [F4].

3.1step 1.1step 2.1step 2.2step 1.2L2L3L4F4F5∎

We have now checked every specific input of [F4]: a fixed covariant model by step 2.1, complete matter data and smooth initialization by step 1.1 and the local harmonic jet in step 1.2, a smooth symmetric-hyperbolic reduction and reconstruction by step 1.2, on-shell stress conservation by step 2.2, and the model's Einstein constraints by [L2]. The choice assumptions are those in the Statement. Applying [F4] gives the smooth local development, variable-width collar, intrinsic globally hyperbolic Cauchy neighborhood and data-fixing isometry matching $F$ . The causal qualifications in [F5] confirm that no compactness or uniform collar width is used.

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources