Manuscript X-E@1 · X-E.tex · section · 293fae6a

Matter variation and the vertex identity

E1: the gravitational action and the metric Euler equation › Matter variation and the vertex identity · lines 664–702

X-E@1#eq:stress, X-E@1#eq:matter-variation, X-E@1#eq:T1, X-E@1#eq:T-divergence

With

T^{μν} [q, ϕ] = q^{μα} q^{ν β} \partial_{α} ϕ \partial_{β} ϕ - q^{μν} (\frac{1}{2} q^{α β} \partial_{α} ϕ \partial_{β} ϕ - \frac{1}{2} m^{2} ϕ^{2}), T^{μν} [ϕ] := T^{μν} [η, ϕ],

one has $\partial L_{q} / \partial q_{μν} = - \frac{1}{2} - q T^{μν} [q, ϕ]$ (no $\partial q$ enters $L_{q}$ ), hence

\frac{δ S ^{sc} _{g}}{δ q _{μν}} = - \frac{1}{2} g - q T^{μν} [q, ϕ], - q T^{μν} [q, ϕ] = T^{μν} [ϕ] + κ T^{μν}_{(1)} [h, ϕ] + O (κ^{2}),

T^{μν}_{(1)} [h, ϕ] = a^{μα} \partial_{α} ϕ \partial^{ν} ϕ + a^{ν β} \partial^{μ} ϕ \partial_{β} ϕ - b \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} a^{μν} \partial_{α} ϕ \partial^{α} ϕ - \frac{1}{2} η^{μν} a^{α β} \partial_{α} ϕ \partial_{β} ϕ + \frac{1}{2} b η^{μν} \partial_{α} ϕ \partial^{α} ϕ + \frac{1}{2} m^{2} a^{μν} ϕ^{2} .

The flat stress tensor obeys

\partial_{μ} T^{μν} [ϕ] = (P_{m} ϕ) \partial^{ν} ϕ .

Displayed formulas in this passage

X-E@1#eq:stress · line 667
$T^{μν} [q, ϕ] = q^{μα} q^{ν β} \partial_{α} ϕ \partial_{β} ϕ - q^{μν} (\frac{1}{2} q^{α β} \partial_{α} ϕ \partial_{β} ϕ - \frac{1}{2} m^{2} ϕ^{2}), T^{μν} [ϕ] := T^{μν} [η, ϕ],$
X-E@1#eq:matter-variation · line 678
$\frac{δ S ^{sc} _{g}}{δ q _{μν}} = - \frac{1}{2} g - q T^{μν} [q, ϕ], - q T^{μν} [q, ϕ] = T^{μν} [ϕ] + κ T^{μν}_{(1)} [h, ϕ] + O (κ^{2}),$
X-E@1#eq:T1 · line 685
$T^{μν}_{(1)} [h, ϕ] = a^{μα} \partial_{α} ϕ \partial^{ν} ϕ + a^{ν β} \partial^{μ} ϕ \partial_{β} ϕ - b \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} a^{μν} \partial_{α} ϕ \partial^{α} ϕ - \frac{1}{2} η^{μν} a^{α β} \partial_{α} ϕ \partial_{β} ϕ + \frac{1}{2} b η^{μν} \partial_{α} ϕ \partial^{α} ϕ + \frac{1}{2} m^{2} a^{μν} ϕ^{2} .$
X-E@1#eq:T-divergence · line 698
$\partial_{μ} T^{μν} [ϕ] = (P_{m} ϕ) \partial^{ν} ϕ .$