Manuscript Oai282-A@1 · Oai282-A.tex · proposition · 5b6134ec

Proposition: A conserved traceless tensor need not source its declared flat metric

Stage 2: conditional improvement and its affine charges › Why this is not an Einstein equation · lines 761–787 · proof included or linked

Oai282-A@1#prop:einstein

The conditions of symmetry, conservation, and zero trace do not imply

G_{μν} (g) = 8 π G_{N} t_{μν}

for the metric on which the tensor was calculated.

Proof. Consider the classical massless scalar $ϕ = a x^{0}$ , $a \neq = 0$ , on $g = η$ . Its canonical stress tensor, improved by

\frac{1}{6} (η_{μν} □ - \partial_{μ} \partial_{ν}) ϕ^{2},

has constant components

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

It is symmetric, conserved, and traceless. However $G_{μν} (η) = 0$ , whereas $t \neq = 0$ . For $G_{N} > 0$ , the displayed Einstein equation fails.

Displayed formulas in this passage

unlabeled · line 766
$G_{μν} (g) = 8 π G_{N} t_{μν}$
unlabeled · line 776
$\frac{1}{6} (η_{μν} □ - \partial_{μ} \partial_{ν}) ϕ^{2},$
unlabeled · line 780
$t_{00} = \frac{a ^{2}}{2}, t_{ii} = \frac{a ^{2}}{6}, t_{0 i} = t_{ij} = 0 (i \neq = j) .$