Manuscript XV-F-patch@1 · XV-F-patch.tex · lemma · 6d6ef16d

Lemma: The mixed Hessian and its factor

Patch J: the target, blocks, stress, and map clauses › One stress polarization fixed by the displayed vertex · lines 1684–1718 · proof included or linked

XV-F-patch@1#lem:Hessian, XV-F-patch@1#eq:Hessian

Back to manuscript

The stress in Definition def:polarization is the unique symmetric bilinear polarization of $T_{0}$ . For the displayed vertex (eq:vertex),

V_{g}^{''}_{γ ϕ} [h, ψ] = - \int g h_{μν} T^{μν} (ϕ, ψ) d^{4} y .

Proof. Expanding the quadratic expression gives

T_{0} [ϕ + s ψ] = T_{0} [ϕ] + 2 s T (ϕ, ψ) + s^{2} T_{0} [ψ] .

The coefficient of $s$ is therefore $2 T (ϕ, ψ)$ . The metric derivative of $- \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ]$ in direction $h$ is $- \frac{1}{2} \int g h_{μν} T_{0}^{μν} [ϕ]$ . Differentiating this in the scalar direction $ψ$ cancels the factor $1/2$ , yielding (eq:Hessian).

Uniqueness follows from the polarization identity for a quadratic map over a field of characteristic zero:

T (ψ_{1}, ψ_{2}) = \frac{1}{2} (T_{0} [ψ_{1} + ψ_{2}] - T_{0} [ψ_{1}] - T_{0} [ψ_{2}]) .

Displayed formulas in this passage

XV-F-patch@1#eq:Hessian · line 1689
$V_{g}^{''}_{γ ϕ} [h, ψ] = - \int g h_{μν} T^{μν} (ϕ, ψ) d^{4} y .$
unlabeled · line 1699
$T_{0} [ϕ + s ψ] = T_{0} [ϕ] + 2 s T (ϕ, ψ) + s^{2} T_{0} [ψ] .$
unlabeled · line 1713
$T (ψ_{1}, ψ_{2}) = \frac{1}{2} (T_{0} [ψ_{1} + ψ_{2}] - T_{0} [ψ_{1}] - T_{0} [ψ_{2}]) .$

Lemma: The mixed Hessian and its factor

Displayed formulas in this passage

Nearby in this section