Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Euler sign

Sign, density, and switch audit › Euler sign · lines 2039–2063

With signature $(-, +, +, +)$ and the scalar action (eq:switchedaction), variation gives

\frac{δ S _{g} ^{sc}}{δ ϕ} = D_{g} [g] ϕ .

Since

D_{g} [g] = P_{m} + κ V_{g} + κ^{2} W_{g} + \dots,

the first-order equation is

P_{m} ϕ_{1, g} = - V_{g} ϕ_{0},

hence

ϕ_{1, g} = - E_{m} V_{g} ϕ_{0} .

If an alternative convention defines the perturbation operator with the opposite sign, the response formula must change sign as well. The two conventions cannot be mixed.

Displayed formulas in this passage

unlabeled · line 2043
$\frac{δ S _{g} ^{sc}}{δ ϕ} = D_{g} [g] ϕ .$
unlabeled · line 2049
$D_{g} [g] = P_{m} + κ V_{g} + κ^{2} W_{g} + \dots,$
unlabeled · line 2053
$P_{m} ϕ_{1, g} = - V_{g} ϕ_{0},$
unlabeled · line 2057
$ϕ_{1, g} = - E_{m} V_{g} ϕ_{0} .$