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The Einstein residual through the gauge vector, and the subsidiary wave equation satisfied by the gauge vector

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , let $\overset{ˉ}{g}$ be a smooth background metric, let $H^{a} = g^{b c} (Γ^{a}_{b c} - {\overset{ˉ}{Γ}}^{a}_{b c})$ be the harmonic-gauge vector and let $E_{a b} = G_{a b} + Λ g_{a b} - κ T_{a b}$ be the Einstein residual of a smooth symmetric tensor $T$ (The harmonic gauge vector, the reduced Einstein equation and the Einstein residual). Then:

(a) On every open set where $(g, T)$ solves the reduced Einstein equation, the identity $E_{a b} = \frac{1}{2} (\nabla_{a} H_{b} + \nabla_{b} H_{a} - g_{a b} \nabla_{c} H^{c})$ holds on that set; and

(b) if $(g, T)$ solves the reduced Einstein equation on an open set $U$ and $\nabla^{a} T_{a b} = 0$ on $U$ , then the gauge vector satisfies the homogeneous linear wave equation $□_{g} H_{b} + {Ric}_{b a} H^{a} = 0 on U,$ where $□_{g} H_{b} = \nabla^{a} \nabla_{a} H_{b}$ .

Facts & Assumptions

Given: the manifold and connection, the metrics $g, \overset{ˉ}{g}$ , the gauge vector $H$ , the constants $Λ, κ$ , the tensor $T$ and the residual $E$ .

[L1]

Trace of the reduced equation: if $(g, T)$ solves ${Ric}_{a b} - \nabla_{(a} H_{b)} = κ (T_{a b} - \frac{1}{2} T g_{a b}) + Λ g_{a b}$ , then $S - \nabla_{c} H^{c} = - κ T + 4 Λ$ with $S = g^{a b} {Ric}_{a b}$ and $T = g^{a b} T_{a b}$ (The harmonic gauge vector, the reduced Einstein equation and the Einstein residual, The inverse metric, musical isomorphisms and index raising).

[L2]

Contracted second Bianchi identity: $\nabla^{a} G_{a b} = 0$ (The second Bianchi identity and the vanishing divergence of the Einstein tensor, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L3]

Commutator identity: $\nabla^{a} \nabla_{b} H_{a} - \nabla_{b} \nabla^{a} H_{a} = R_{b a} H^{a}$ , with $R_{b a}$ the Ricci tensor in the framework's sign convention (The commutator of covariant derivatives on a vector field and its contracted component identity).

[L4]

Metric compatibility: $\nabla^{a} g_{a b} = 0$ , and $\nabla$ is torsion-free and compatible with $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

Proof

technique · direct

1.1L1algebra

(The residual identity (a).) Substituting the reduced equation into the definition of $E$ and using the trace identity [L1]: $E_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b} + Λ g_{a b} - κ T_{a b} = \nabla_{(a} H_{b)} + κ (T_{a b} - \frac{1}{2} T g_{a b}) + Λ g_{a b} - \frac{1}{2} S g_{a b} + Λ g_{a b} - κ T_{a b} = \nabla_{(a} H_{b)} - \frac{1}{2} (\nabla^{c} H_{c}) g_{a b}$ , because $S = \nabla^{c} H_{c} - κ T + 4 Λ$ and the $κ T$ , $Λ$ terms cancel in pairs ( $- \frac{1}{2} κ T g_{a b} - \frac{1}{2} (- κ T) g_{a b} = 0$ and $Λ g_{a b} + Λ g_{a b} - \frac{1}{2} \cdot 4 Λ g_{a b} = 0$ ). Since $\nabla_{(a} H_{b)} - \frac{1}{2} (\nabla^{c} H_{c}) g_{a b} = \frac{1}{2} (\nabla_{a} H_{b} + \nabla_{b} H_{a} - g_{a b} \nabla^{c} H_{c})$ , this is (a).

1.2L2L4algebra

(Vanishing of the same divergence.) By definition $E = G + Λ g - κ T$ and [L2], [L4] give $\nabla^{a} E_{a b} = 0 + Λ \nabla^{a} g_{a b} - κ \nabla^{a} T_{a b} = - κ \nabla^{a} T_{a b} = 0$ on $U$ by the on-shell conservation hypothesis.

2.1step 1.1L3L4algebra

(Divergence of the residual.) Take the divergence of (a): $\nabla^{a} E_{a b} = \frac{1}{2} (\nabla^{a} \nabla_{a} H_{b} + \nabla^{a} \nabla_{b} H_{a} - \nabla_{b} \nabla^{c} H_{c}) = \frac{1}{2} (□_{g} H_{b} + R_{b a} H^{a})$ , where the last equality uses the commutator identity [L3] in the form $\nabla^{a} \nabla_{b} H_{a} - \nabla_{b} \nabla_{c} H^{c} = R_{b a} H^{a}$ (with [L4] to raise the index freely).

3.1step 2.1step 1.2algebra∎

(Conclusion (b).) Comparing steps 2.1 and 1.2 gives $□_{g} H_{b} + R_{b a} H^{a} = 0$ on $U$ .

Remarks

Role. Clause (b) is the subsidiary equation whose zero Cauchy data force the harmonic gauge to persist; the Cauchy data are computed in Propagation of the harmonic gauge and of the Einstein constraints: a constrained reduced solution is an Einstein solution.

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 2.4 (The Evolution Equations): the splitting of the field equations into evolution and constraint parts, gauge source functions and the harmonically reduced field equations (2.39), and the subsidiary system obtained from their divergence