Manuscript X-E-ST@1 · X-E-ST.tex · proposition · 44614221

Proposition: Quadratic Einstein–Hilbert coefficient

E3: the derived free graviton operator › Quadratic form and sign · lines 1016–1056 · proof included or linked

X-E-ST@1#prop:quadratic, X-E-ST@1#eq:FP

Up to a total derivative,

S_{FP} [l] = \frac{1}{4} \int l_{μν} G_{lin}^{μν} [l] d^{4} x, \frac{δ S _{FP}}{δ l _{μν}} = \frac{1}{2} G_{lin}^{μν} [l] .

This is the quadratic coefficient of (eq:action), not a separately imported free action. For transverse-traceless spatial configurations,

S_{FP} = \frac{1}{8} \int (\partial_{t} l_{ij} \partial_{t} l_{ij} - \partial_{a} l_{ij} \partial_{a} l_{ij}) d^{4} x .

Thus the physical transverse-traceless kinetic term has positive sign.

Proof. For compact $l$ , set $I (s) = \int J_{η + s l} R [η + s l] d^{4} x$ . Lemma lem:palatini gives

I^{'} (0) = 0, I^{''} (0) = - \int l_{μν} G_{lin}^{μν} [l] d^{4} x .

Multiplication by $- 1/ (2 κ^{2})$ , with $s = κ$ , gives (eq:FP). Symmetry of the second variation gives its Euler derivative. For trace-free divergence-free $l$ , $G_{lin} [l] = - \frac{1}{2} □ l$ . One integration by parts gives the displayed kinetic form. Compact variation gives the same local density result without a decay assumption.

Displayed formulas in this passage

X-E-ST@1#eq:FP · line 1019
$S_{FP} [l] = \frac{1}{4} \int l_{μν} G_{lin}^{μν} [l] d^{4} x, \frac{δ S _{FP}}{δ l _{μν}} = \frac{1}{2} G_{lin}^{μν} [l] .$
unlabeled · line 1030
$S_{FP} = \frac{1}{8} \int (\partial_{t} l_{ij} \partial_{t} l_{ij} - \partial_{a} l_{ij} \partial_{a} l_{ij}) d^{4} x .$
unlabeled · line 1044
$I^{'} (0) = 0, I^{''} (0) = - \int l_{μν} G_{lin}^{μν} [l] d^{4} x .$

Nearby in this section

section: Quadratic form and sign