Manuscript XI-A@1 · XI-A.tex · lemma · b8211ecc

Lemma: Stress sign and inherited vertex

Metric Euler derivative through coupling order two › Stress tensor and the first vertex · lines 599–626 · proof included or linked

XI-A@1#lem:vertex, XI-A@1#eq:vertex

For a covariant metric variation,

δ_{q} L_{q} = - \frac{1}{2} - q T_{q}^{μν} δ q_{μν} .

In particular,

V_{1, g} = \frac{1}{2} \int g (a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2}) d^{4} x = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} d^{4} x .

Proof. Use $δ - q = \frac{1}{2} - q q^{μν} δ q_{μν}$ and $δ q^{α β} = - q^{α μ} q^{β ν} δ q_{μν}$ . For the second assertion,

a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2} = - γ^{μν} p_{μ} p_{ν} + τ ℓ_{0} = - γ_{μν} T_{0}^{μν} .

Displayed formulas in this passage

XI-A@1#eq:vertex · line 607
$V_{1, g} = \frac{1}{2} \int g (a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2}) d^{4} x = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} d^{4} x .$
unlabeled · line 602
$δ_{q} L_{q} = - \frac{1}{2} - q T_{q}^{μν} δ q_{μν} .$
unlabeled · line 621
$a_{1}^{μν} p_{μ} p_{ν} - m^{2} b_{1} ϕ^{2} = - γ^{μν} p_{μ} p_{ν} + τ ℓ_{0} = - γ_{μν} T_{0}^{μν} .$