Manuscript VIII-B@3 · VIII-B-v3.tex · proposition · 3aa6baa1

Proposition: Stress identities do not imply Einstein dynamics

Non-identifications and separate successor obligations › What the constructor does not derive · lines 2378–2407 · proof included or linked

VIII-B@3#prop:no-einstein

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Symmetry, conservation, and zero trace of a tensor on a declared metric do not imply

G_{μν} (g) = 8 π G_{N} t_{μν} .

Proof. This independent counterexample is in $3 + 1$ dimensions. Take $η_{4} = diag (- 1, 1, 1, 1)$ and the massless scalar $ϕ = a T$ , with $a \neq = 0$ . Its improved flat-space stress is

t_{μν} = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{4, μν} (\partial ϕ)^{2} + \frac{1}{6} (η_{4, μν} □ - \partial_{μ} \partial_{ν}) ϕ^{2} .

Since $□ ϕ^{2} = - 2 a^{2}$ , direct substitution gives

t_{00} = \frac{a ^{2}}{2}, t_{ii} = \frac{a ^{2}}{6}, t_{0 i} = t_{ij} = 0 (i \neq = j) .

This constant tensor is symmetric, conserved, and traceless. However $G_{μν} (η_{4}) = 0$ and $t \neq = 0$ . For $G_{N} > 0$ , the proposed equation fails.

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$G_{μν} (g) = 8 π G_{N} t_{μν} .$
unlabeled · line 2392
$t_{μν} = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} η_{4, μν} (\partial ϕ)^{2} + \frac{1}{6} (η_{4, μν} □ - \partial_{μ} \partial_{ν}) ϕ^{2} .$
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$t_{00} = \frac{a ^{2}}{2}, t_{ii} = \frac{a ^{2}}{6}, t_{0 i} = t_{ij} = 0 (i \neq = j) .$

Proposition: Stress identities do not imply Einstein dynamics

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