Manuscript Oai365-A@1 · Oai365-A.tex · proposition · a3840e3d

Proposition: Same cones, different Einstein tensor

Regularity, metric jets, and dynamics separation › Even a pointwise cone field does not determine curvature · lines 1584–1612 · proof included or linked

Oai365-A@1#prop:conformal

Back to manuscript

At a point where $φ = 0$ and $d φ = 0$ ,

G_{μν} (G) = 2 (η_{μν} □ φ - \partial_{μ} \partial_{ν} φ) .

There are smooth compactly supported choices of $φ$ for which this is nonzero, although all cones agree with the flat ones.

Proof. Substitute $e^{2 φ} η$ into the Christoffel formula. At the selected point the quadratic first-derivative terms vanish, and contraction gives

R_{μν} = - 2 \partial_{μ} \partial_{ν} φ - η_{μν} □ φ, R = - 6 □ φ .

Subtracting half the scalar curvature times the metric proves the formula.

Choose $φ = a (y^{1})^{2} /2$ near the point, multiplied by a smooth cutoff equal to one there. Then $□ φ = a$ , and $G_{00} = - 2 a \neq = 0$ if $a \neq = 0$ .

Displayed formulas in this passage

unlabeled · line 1587
$G_{μν} (G) = 2 (η_{μν} □ φ - \partial_{μ} \partial_{ν} φ) .$
unlabeled · line 1601
$R_{μν} = - 2 \partial_{μ} \partial_{ν} φ - η_{μν} □ φ, R = - 6 □ φ .$

Proposition: Same cones, different Einstein tensor

Displayed formulas in this passage

Nearby in this section