Manuscript XII-B@1 · XII-B.tex · proposition · 229bc869

Proposition: Off-shell variational integrability

An explicit Einstein–Hilbert layer discrepancy › Simply weighting the source is not variational · lines 723–773 · proof included or linked

XII-B@1#prop:integrability, XII-B@1#eq:Hessianantisymmetry

Back to manuscript

For nonconstant $σ_{ε}$ , the full source $σ_{ε} J_{3}$ cannot be the Euler derivative of a twice differentiable action on an off-shell neighborhood containing the unrestricted pure-trace configurations and compact metric variations.

Proof. If such an action existed, restrict it to $γ = ψ η$ . Its first derivative would be

F_{σ}_{ε} [ψ] = - 3 σ_{ε} (\partial ψ)^{2} - 6 σ_{ε} ψ □ ψ .

Symmetry of mixed second variations requires

\int h F_{σ}_{ε}^{'} [ψ] k d^{4} x = \int k F_{σ}_{ε}^{'} [ψ] h d^{4} x

for every pair of compact variations $h, k$ .

Choose $ψ = 1$ on a neighborhood of their supports. There

F_{σ}_{ε}^{'} [1] k = - 6 σ_{ε} □ k .

Compact integration by parts gives

\int h F_{σ}_{ε}^{'} [1] k d^{4} x - \int k F_{σ}_{ε}^{'} [1] h d^{4} x = 6 \int (\partial_{μ} σ_{ε}) (h \partial^{μ} k - k \partial^{μ} h) d^{4} x .

Take a nonzero real $h$ supported strictly inside the layer and put $k = t h$ . Both variations are compact, and

h \partial_{t} (t h) - t h \partial_{t} h = h^{2} .

Since the ramp depends only on time, (eq:Hessianantisymmetry) equals

6 \int σ_{ε}^{'} (t) h^{2} d^{4} x > 0.

This contradicts symmetry of the Hessian.

Displayed formulas in this passage

XII-B@1#eq:Hessianantisymmetry · line 752
$\int h F_{σ}_{ε}^{'} [1] k d^{4} x - \int k F_{σ}_{ε}^{'} [1] h d^{4} x = 6 \int (\partial_{μ} σ_{ε}) (h \partial^{μ} k - k \partial^{μ} h) d^{4} x .$
unlabeled · line 734
$F_{σ}_{ε} [ψ] = - 3 σ_{ε} (\partial ψ)^{2} - 6 σ_{ε} ψ □ ψ .$
unlabeled · line 739
$\int h F_{σ}_{ε}^{'} [ψ] k d^{4} x = \int k F_{σ}_{ε}^{'} [ψ] h d^{4} x$
unlabeled · line 748
$F_{σ}_{ε}^{'} [1] k = - 6 σ_{ε} □ k .$
unlabeled · line 764
$h \partial_{t} (t h) - t h \partial_{t} h = h^{2} .$
unlabeled · line 769
$6 \int σ_{ε}^{'} (t) h^{2} d^{4} x > 0.$

Proposition: Off-shell variational integrability

Displayed formulas in this passage

Nearby in this section