Manuscript Em02-E@1 · Em02-E-v1.tex · proposition · f3795050

Proposition: Induced metric through second order

Clock slices and harmonic spatial labels › Clock-adapted description · lines 458–494 · proof included or linked

Em02-E@1#prop:induced, Em02-E@1#eq:q1, Em02-E@1#eq:q2

The coefficients are

q_{1, ij} q_{2, ij} = h_{ij} + u_{1} \partial_{t} (a^{2}) δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2}) δ_{ij} + u_{1}^{2} \partial_{t}^{2} (a^{2}) δ_{ij} + 2 (h_{0 i} \partial_{j} u_{1} + h_{0 j} \partial_{i} u_{1}) - 2 \partial_{i} u_{1} \partial_{j} u_{1} .

Proof. The spatial tangent vectors are

\partial_{i} Ψ_{λ} = \partial_{i} + (λ \partial_{i} u_{1} + λ^{2} \partial_{i} u_{2} /2) \partial_{t} .

Taylor-expand the spatial-spatial metric component at $t = s$ . The mixed components first contribute at degree two, as $λ h_{0 i} λ \partial_{j} u_{1}$ . The background time-time component is $- 1$ , so its degree-two contribution is $- λ^{2} \partial_{i} u_{1} \partial_{j} u_{1}$ . With the factorial convention of (eq:field-jet), these are exactly (eq:q1)–(eq:q2).

Displayed formulas in this passage

Em02-E@1#eq:q1 · line 461
$q_{1, ij} q_{2, ij} = h_{ij} + u_{1} \partial_{t} (a^{2}) δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2}) δ_{ij} + u_{1}^{2} \partial_{t}^{2} (a^{2}) δ_{ij} + 2 (h_{0 i} \partial_{j} u_{1} + h_{0 j} \partial_{i} u_{1}) - 2 \partial_{i} u_{1} \partial_{j} u_{1} .$
Em02-E@1#eq:q2 · line 461
$q_{1, ij} q_{2, ij} = h_{ij} + u_{1} \partial_{t} (a^{2}) δ_{ij}, = k_{ij} + 2 u_{1} h ˙_{ij} + u_{2} \partial_{t} (a^{2}) δ_{ij} + u_{1}^{2} \partial_{t}^{2} (a^{2}) δ_{ij} + 2 (h_{0 i} \partial_{j} u_{1} + h_{0 j} \partial_{i} u_{1}) - 2 \partial_{i} u_{1} \partial_{j} u_{1} .$
unlabeled · line 480
$\partial_{i} Ψ_{λ} = \partial_{i} + (λ \partial_{i} u_{1} + λ^{2} \partial_{i} u_{2} /2) \partial_{t} .$

Proposition: Induced metric through second order

Displayed formulas in this passage

Nearby in this section