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

The relational scalar

Unswitched harmonic coordinates › The relational scalar · lines 839–889

VII-A@1#eq:Phi, VII-A@1#eq:Of, VII-A@1#eq:Phiexp, VII-A@1#eq:A1, VII-A@1#eq:A2

For a fixed label-space test density represented by $f \in C_{c}^{\infty} (U)$ , define

Φ_{g} [g; ϕ_{0}] (y) O_{f} [g] (g, ϕ_{0}) = ϕ_{g} [g; ϕ_{0}] (X_{g}^{- 1} (y)), = \int f (y) Φ_{g} [g; ϕ_{0}] (y) d^{4} y .

The smearing is performed in coordinate-label space. Therefore no spacetime volume density is to be inserted into (eq:Of). An equivalent spacetime expression would contain the Jacobian of $X_{g}$ .

Expanding,

Φ_{g} = ϕ_{0} + κ A_{1}^{(g)} + κ^{2} A_{2}^{(g)} + O (κ^{3}),

where

A_{1}^{(g)} A_{2}^{(g)} = ϕ_{1, g} - w^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2, g} - w^{μ} \partial_{μ} ϕ_{1, g} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

The notation $A_{1}^{(g)}$ emphasizes that the first-order expression itself is unchanged; its admitted gauge domain is changed.

Displayed formulas in this passage

VII-A@1#eq:Phi · line 843
$Φ_{g} [g; ϕ_{0}] (y) O_{f} [g] (g, ϕ_{0}) = ϕ_{g} [g; ϕ_{0}] (X_{g}^{- 1} (y)), = \int f (y) Φ_{g} [g; ϕ_{0}] (y) d^{4} y .$
VII-A@1#eq:Of · line 843
$Φ_{g} [g; ϕ_{0}] (y) O_{f} [g] (g, ϕ_{0}) = ϕ_{g} [g; ϕ_{0}] (X_{g}^{- 1} (y)), = \int f (y) Φ_{g} [g; ϕ_{0}] (y) d^{4} y .$
VII-A@1#eq:Phiexp · line 861
$Φ_{g} = ϕ_{0} + κ A_{1}^{(g)} + κ^{2} A_{2}^{(g)} + O (κ^{3}),$
VII-A@1#eq:A1 · line 870
$A_{1}^{(g)} A_{2}^{(g)} = ϕ_{1, g} - w^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2, g} - w^{μ} \partial_{μ} ϕ_{1, g} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
VII-A@1#eq:A2 · line 870
$A_{1}^{(g)} A_{2}^{(g)} = ϕ_{1, g} - w^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2, g} - w^{μ} \partial_{μ} ϕ_{1, g} + (w^{ν} \partial_{ν} w^{μ} - v^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$