Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

Matter variation and the complete Euler equation

E1: action, Palatini variation, and switched Noether identity › Matter variation and the complete Euler equation · lines 613–632

X-E-ST@1#eq:vertex, X-E-ST@1#eq:stress-div

The first matter vertex is therefore

V_{1, g} [l, ϕ] = - \frac{1}{2} \int g l_{μν} T_{0}^{μν} [ϕ] d^{4} x, \frac{δ V _{1, g}}{δ l _{μν}} = - \frac{1}{2} g T_{0}^{μν}, \frac{δ V _{1, g}}{δ ϕ} = - L_{1} [l] ϕ .

For later use, direct differentiation gives

\partial_{μ} T_{0}^{μν} [ϕ] = (P_{m} ϕ) \partial^{ν} ϕ .

Displayed formulas in this passage

X-E-ST@1#eq:vertex · line 616
$V_{1, g} [l, ϕ] = - \frac{1}{2} \int g l_{μν} T_{0}^{μν} [ϕ] d^{4} x, \frac{δ V _{1, g}}{δ l _{μν}} = - \frac{1}{2} g T_{0}^{μν}, \frac{δ V _{1, g}}{δ ϕ} = - L_{1} [l] ϕ .$
X-E-ST@1#eq:stress-div · line 627
$\partial_{μ} T_{0}^{μν} [ϕ] = (P_{m} ϕ) \partial^{ν} ϕ .$