Manuscript VII-B@1 · VII-B.tex · section · be632835

Finite formulas for the gravitational vertices

An explicit action and vertex ledger › Finite formulas for the gravitational vertices · lines 1598–1685

VII-B@1#eq:gamma-coeff, VII-B@1#eq:grav1, VII-B@1#eq:grav2

The following formulas specify the gravitational vertices without suppressing integrations by parts.

Define

T_{λ μν} (h) = \partial_{μ} h_{λ ν} + \partial_{ν} h_{λ μ} - \partial_{λ} h_{μν} .

Write

Γ^{ρ}_{μν} = κ γ_{1}^{ρ}_{μν} + κ^{2} γ_{2}^{ρ}_{μν} + κ^{3} γ_{3}^{ρ}_{μν} + O (κ^{4}),

where

γ_{r}^{ρ}_{μν} = \frac{1}{2} (- 1)^{r - 1} (h^{r - 1})^{ρ}_{σ} η^{σ λ} T_{λ μν} (h), r = 1, 2, 3,

and $h^{0} = id$ .

For two Christoffel-type arrays $P, Q$ , set

Q_{μν} (P, Q) = P^{ρ}_{μ σ} Q^{σ}_{ν ρ} - P^{ρ}_{μν} Q^{σ}_{ρ σ} .

Then

L_{grav, 1} = - 2 [L_{grav, 2} = - 2 [A_{1}^{μν} Q_{μν} (γ_{1}, γ_{1}) + η^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1}))], A_{2}^{μν} Q_{μν} (γ_{1}, γ_{1}) + A_{1}^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1})) + η^{μν} (Q_{μν} (γ_{1}, γ_{3}) + Q_{μν} (γ_{2}, γ_{2}) + Q_{μν} (γ_{3}, γ_{1}))] .

Finally,

V_{j}^{grav} = \int g_{grav} L_{grav, j} d^{4} x, j = 1, 2.

These expressions are finite local polynomials with smooth compact switches. Their local coefficient kernels satisfy the microcausal cone condition: the covectors in a diagonal conormal sum to zero, and hence cannot all lie in the same nonspacelike cone unless they all vanish. Derivatives do not enlarge that wavefront bound.

Displayed formulas in this passage

VII-B@1#eq:gamma-coeff · line 1621
$γ_{r}^{ρ}_{μν} = \frac{1}{2} (- 1)^{r - 1} (h^{r - 1})^{ρ}_{σ} η^{σ λ} T_{λ μν} (h), r = 1, 2, 3,$
VII-B@1#eq:grav1 · line 1640
$L_{grav, 1} = - 2 [L_{grav, 2} = - 2 [A_{1}^{μν} Q_{μν} (γ_{1}, γ_{1}) + η^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1}))], A_{2}^{μν} Q_{μν} (γ_{1}, γ_{1}) + A_{1}^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1})) + η^{μν} (Q_{μν} (γ_{1}, γ_{3}) + Q_{μν} (γ_{2}, γ_{2}) + Q_{μν} (γ_{3}, γ_{1}))] .$
VII-B@1#eq:grav2 · line 1640
$L_{grav, 1} = - 2 [L_{grav, 2} = - 2 [A_{1}^{μν} Q_{μν} (γ_{1}, γ_{1}) + η^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1}))], A_{2}^{μν} Q_{μν} (γ_{1}, γ_{1}) + A_{1}^{μν} (Q_{μν} (γ_{1}, γ_{2}) + Q_{μν} (γ_{2}, γ_{1})) + η^{μν} (Q_{μν} (γ_{1}, γ_{3}) + Q_{μν} (γ_{2}, γ_{2}) + Q_{μν} (γ_{3}, γ_{1}))] .$
unlabeled · line 1604
$T_{λ μν} (h) = \partial_{μ} h_{λ ν} + \partial_{ν} h_{λ μ} - \partial_{λ} h_{μν} .$
unlabeled · line 1612
$Γ^{ρ}_{μν} = κ γ_{1}^{ρ}_{μν} + κ^{2} γ_{2}^{ρ}_{μν} + κ^{3} γ_{3}^{ρ}_{μν} + O (κ^{4}),$
unlabeled · line 1633
$Q_{μν} (P, Q) = P^{ρ}_{μ σ} Q^{σ}_{ν ρ} - P^{ρ}_{μν} Q^{σ}_{ρ σ} .$
unlabeled · line 1672
$V_{j}^{grav} = \int g_{grav} L_{grav, j} d^{4} x, j = 1, 2.$