Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Curvature and action

One action, one gauge-fixing term, and two sign checks › Curvature and action · lines 434–483

XI-A@1#eq:ricci, XI-A@1#eq:EH, XI-A@1#eq:scalar-action, XI-A@1#eq:gf

Fix the curvature convention

R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ},

with the Levi-Civita Christoffel symbols and $G_{μν} = R_{μν} - \frac{1}{2} q_{μν} R$ . Use

S_{EH} [q] = - \frac{2}{κ ^{2}} \int_{M_{T}} - q R [q] d^{4} x .

There is no cosmological constant and no gravitational switch.

The unchanged X-A scalar density is

L_{q} [ϕ] L_{η} [ϕ] S_{g}^{sc} [q, ϕ] = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}), = \frac{1}{2} (\partial_{μ} ϕ \partial^{μ} ϕ - m^{2} ϕ^{2}), = \int_{M_{T}} ((1 - g) L_{η} [ϕ] + g L_{q} [ϕ]) d^{4} x .

Actions are interpreted through compact interior variations and compact differences. Numerical finiteness on every smooth configuration is not assumed. No metric boundary variation is taken.

Define

γ ˉ_{μν} = γ_{μν} - \frac{1}{2} η_{μν} τ, D^{ν} (γ) = \partial_{μ} γ ˉ^{μν} .

The single frozen gauge-fixing term is

S_{gf} [γ] = \int_{M_{T}} D_{ν} (γ) D^{ν} (γ) d^{4} x .

Its coefficient has been fixed; no gauge-parameter family is used.

Displayed formulas in this passage

XI-A@1#eq:ricci · line 437
$R_{μν} = \partial_{ρ} Γ^{ρ}_{μν} - \partial_{ν} Γ^{ρ}_{μ ρ} + Γ^{ρ}_{ρ λ} Γ^{λ}_{μν} - Γ^{ρ}_{ν λ} Γ^{λ}_{μ ρ},$
XI-A@1#eq:EH · line 448
$S_{EH} [q] = - \frac{2}{κ ^{2}} \int_{M_{T}} - q R [q] d^{4} x .$
XI-A@1#eq:scalar-action · line 456
$L_{q} [ϕ] L_{η} [ϕ] S_{g}^{sc} [q, ϕ] = \frac{1}{2} - q (q^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} ϕ^{2}), = \frac{1}{2} (\partial_{μ} ϕ \partial^{μ} ϕ - m^{2} ϕ^{2}), = \int_{M_{T}} ((1 - g) L_{η} [ϕ] + g L_{q} [ϕ]) d^{4} x .$
XI-A@1#eq:gf · line 478
$S_{gf} [γ] = \int_{M_{T}} D_{ν} (γ) D^{ν} (γ) d^{4} x .$
unlabeled · line 471
$γ ˉ_{μν} = γ_{μν} - \frac{1}{2} η_{μν} τ, D^{ν} (γ) = \partial_{μ} γ ˉ^{μν} .$