Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Action Hessian and gauge normalization

Identical frozen background, operator, and rods › Action Hessian and gauge normalization · lines 426–454

XX-C@1#eq:hyperbolic

The second-derivative metric part of $H$ is

\frac{M ^{2}}{4} T □_{g ˉ} + \frac{M ^{2}}{2} D^{†} D, D h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h .

The second term is cancelled by $r F^{†} F$ . Mixed entries have at most one derivative. Consequently

L = □_{g ˉ} id_{V} + smooth lower-order terms .

It is formally self-adjoint for $(z, w)_{A} = (z, A w)_{0}$ . This verifies the normally hyperbolic hypotheses, but the pairing is not positive definite.

The compact-spacetime residual condition $Q ξ = 0$ has only the zero compactly supported solution by Cauchy uniqueness. This does not eliminate nonzero homogeneous solutions with general Cauchy data.

Displayed formulas in this passage

XX-C@1#eq:hyperbolic · line 440
$L = □_{g ˉ} id_{V} + smooth lower-order terms .$
unlabeled · line 429
$\frac{M ^{2}}{4} T □_{g ˉ} + \frac{M ^{2}}{2} D^{†} D, D h = \nabla^{μ} h_{μν} - \frac{1}{2} \nabla_{ν} tr_{g ˉ} h .$

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section: Action Hessian and gauge normalization