Manuscript XI-E@1 · XI-E.tex · theorem · 46e55514

Theorem: Cubic source and tree variation

Route 3: the cubic Einstein tree and its computed variation › Variation from tensor-density covariance · lines 1522–1574 · proof included or linked

XI-E@1#thm:cubicvar, XI-E@1#eq:dJ, XI-E@1#eq:du, XI-E@1#eq:dC, XI-E@1#eq:duvac

Back to manuscript

Let $δ_{0} γ = K ξ$ , with $ξ \in g_{U}$ , and hold the free scalar fixed. Then

δ_{0} J_{3} [γ] δ_{0} u_{3} δ_{0} C_{3} = L_{ξ}^{dens} (A γ) - A (L_{ξ} γ), = - G_{gr}^{R} (L_{ξ}^{dens} (A γ) - A (L_{ξ} γ)), = A_{1} [δ_{0} u_{3}; ϕ] .

If $A γ = 0$ and $r = L_{ξ} γ$ , the response simplifies to

δ_{0} u_{3} = G_{gr}^{R} A r = r - 2 R E_{0}^{R} D r .

Proof. Naturality of the invariant Einstein Euler tensor density under the infinitesimal flow of $κ ξ$ is exact. Its expansion is $A γ + κ J_{3} [γ] + \dots$ . The metric variation is $K ξ + κ L_{ξ} γ$ . At order $κ$ , covariance gives

A (L_{ξ} γ) + δ_{0} J_{3} [γ] = L_{ξ}^{dens} (A γ),

proving (eq:dJ). Applying the fixed linear response operator proves (eq:du). Linearity of $A_{1}$ proves (eq:dC).

If $A γ = 0$ , only $- A r$ remains in (eq:dJ). Since $r$ is compact inside $U$ , its data vanish initially. Use

A + 2 D^{*} D = - □ P, G_{gr}^{R} D^{*} = R E_{0}^{R}

and the left inverse identity on compact tests to obtain

G_{gr}^{R} A r = r - 2 G_{gr}^{R} D^{*} D r = r - 2 R E_{0}^{R} D r .

Displayed formulas in this passage

XI-E@1#eq:dJ · line 1526
$δ_{0} J_{3} [γ] δ_{0} u_{3} δ_{0} C_{3} = L_{ξ}^{dens} (A γ) - A (L_{ξ} γ), = - G_{gr}^{R} (L_{ξ}^{dens} (A γ) - A (L_{ξ} γ)), = A_{1} [δ_{0} u_{3}; ϕ] .$
XI-E@1#eq:du · line 1526
$δ_{0} J_{3} [γ] δ_{0} u_{3} δ_{0} C_{3} = L_{ξ}^{dens} (A γ) - A (L_{ξ} γ), = - G_{gr}^{R} (L_{ξ}^{dens} (A γ) - A (L_{ξ} γ)), = A_{1} [δ_{0} u_{3}; ϕ] .$
XI-E@1#eq:dC · line 1526
$δ_{0} J_{3} [γ] δ_{0} u_{3} δ_{0} C_{3} = L_{ξ}^{dens} (A γ) - A (L_{ξ} γ), = - G_{gr}^{R} (L_{ξ}^{dens} (A γ) - A (L_{ξ} γ)), = A_{1} [δ_{0} u_{3}; ϕ] .$
XI-E@1#eq:duvac · line 1541
$δ_{0} u_{3} = G_{gr}^{R} A r = r - 2 R E_{0}^{R} D r .$
unlabeled · line 1555
$A (L_{ξ} γ) + δ_{0} J_{3} [γ] = L_{ξ}^{dens} (A γ),$
unlabeled · line 1566
$A + 2 D^{*} D = - □ P, G_{gr}^{R} D^{*} = R E_{0}^{R}$
unlabeled · line 1571
$G_{gr}^{R} A r = r - 2 G_{gr}^{R} D^{*} D r = r - 2 R E_{0}^{R} D r .$

Theorem: Cubic source and tree variation

Displayed formulas in this passage

Nearby in this section