Manuscript XII-C@1 · XII-C.tex · section · dead1004

A finite unintegrated density prescription

C1: the density representative and actual cubic source › A finite unintegrated density prescription · lines 552–619

XII-C@1#eq:finite-curvature, XII-C@1#eq:ell3, XII-C@1#eq:J3

Write $τ = tr_{η} γ$ and set

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν},

b_{0} = 1, b_{1} = \frac{1}{2} τ, b_{2} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν} .

For $n = 1, 2, 3$ , define

Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} γ_{ν λ} + \partial_{ν} γ_{μ λ} - \partial_{λ} γ_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .

The declared representative is

ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) .

It is the coefficient of $κ$ in the unintegrated signed Einstein–Hilbert density. Equivalently, it is $- 2$ times the coefficient of $κ^{3}$ in $- q R [q]$ .

Every term is local, cubic, and of jet order at most two. The formulas specify the density itself, not only its Euler equivalence class.

For clarity, the original source can also be evaluated finitely. Put

V_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, V_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1} .

Then

J_{3}^{μν} = 2 (V_{2}^{μν} + b_{1} V_{1}^{μν} - γ^{μα} V_{1, α}^{ν} - γ^{ν α} V_{1, α}^{μ}) .

Compact variation of (eq:ell3) gives $EL_{γ} (ℓ_{3}) = J_{3}$ with the XI-A sign.

Displayed formulas in this passage

XII-C@1#eq:finite-curvature · line 565
$Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ λ} (\partial_{μ} γ_{ν λ} + \partial_{ν} γ_{μ λ} - \partial_{λ} γ_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .$
XII-C@1#eq:ell3 · line 587
$ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) .$
XII-C@1#eq:J3 · line 607
$J_{3}^{μν} = 2 (V_{2}^{μν} + b_{1} V_{1}^{μν} - γ^{μα} V_{1, α}^{ν} - γ^{ν α} V_{1, α}^{μ}) .$
unlabeled · line 555
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{λ} γ^{λ ν},$
unlabeled · line 560
$b_{0} = 1, b_{1} = \frac{1}{2} τ, b_{2} = \frac{1}{8} τ^{2} - \frac{1}{4} γ_{μν} γ^{μν} .$
unlabeled · line 601
$V_{1, μν} = R_{1, μν} - \frac{1}{2} η_{μν} r_{1}, V_{2, μν} = R_{2, μν} - \frac{1}{2} η_{μν} r_{2} - \frac{1}{2} γ_{μν} r_{1} .$