Manuscript XX-D@1 · XX-D.tex · section · c83a526d

The native two-rod inverse-distance term

Why the missing infrared hypotheses cannot be inferred › The native two-rod inverse-distance term · lines 1628–1711

XX-D@1#eq:matrix, XX-D@1#eq:RRtail

The two-rod causal kernel is more weakly decaying than the compact-source one-rod response. Its leading term can be checked from the native normalization alone.

In orthonormal spatial components $h_{ij} = a^{- 2} h_{ij}$ , the map $C$ is time independent. Translation invariance makes $E_{B}$ a kernel $e (t, t^{'}; x - y)$ . For fixed times it has compact spatial support. Its spatial moment satisfies

m (t^{'}, t^{'}) = 0, \partial_{t} m (t, t^{'})_{t = t^{'}} = - a (t^{'})^{- 3} A^{- 1} .

Define

K_{ij} (r) = \frac{δ _{ij} - n _{i} n _{j}}{8 π r}, r = ∣ r ∣, n = r / r .

Its Fourier multiplier is $k_{i} k_{j} / ∣ k ∣^{4}$ . The double-rod causal kernel is the convolution of $K_{ij}$ with the compact spatial distribution $C^{ia} e (C^{j b})^{†}$ .

The native normalization gives

k_{i} k_{j} C^{ia} A^{- 1} (C^{j b})^{†} = \frac{2}{M ^{2}} ∣ k ∣^{2} δ_{ab} + \frac{p + 1}{2 M ^{2}} k_{a} k_{b} .

For verification, the metric coefficient of $k_{i} C^{ia}$ is

B^{a}_{k l} = \frac{1}{2} k_{a} δ_{k l} - k_{(k} δ_{l) a} .

It has trace $k_{a} /2$ and contraction

B^{a} : B^{b} = \frac{1}{2} ∣ k ∣^{2} δ_{ab} + \frac{1}{4} k_{a} k_{b} .

Four-dimensional trace reversal subtracts $k_{a} k_{b} /8$ , and its normalization is $4/ M^{2}$ . The scalar contribution is $p^{2} k_{a} k_{b} / A^{2}$ . This proves (eq:matrix).

Consequently, for $t = t_{0} + δ$ and small nonzero $δ$ ,

E_{R R}^{ab} (t_{0} + δ, t_{0}; r) = - \frac{δ}{16 π M ^{2} a ( t _{0} ) ^{3} r} ((p + 9) δ_{ab} - (p + 1) n_{a} n_{b}) + O (δ^{2} r^{- 1}) + O_{δ} (r^{- 2}) .

The calculation uses

F^{- 1} (∣ k ∣^{- 2}) = \frac{1}{4 π r}, F^{- 1} (k_{a} k_{b} ∣ k ∣^{- 4}) = \frac{δ _{ab} - n _{a} n _{b}}{8 π r}

and the compact-distribution moment expansion of the convolution.

For $n$ along axis $a$ , the $aa$ coefficient in parentheses is eight. Hence the inverse-distance component is nonzero.

Displayed formulas in this passage

XX-D@1#eq:matrix · line 1662
$k_{i} k_{j} C^{ia} A^{- 1} (C^{j b})^{†} = \frac{2}{M ^{2}} ∣ k ∣^{2} δ_{ab} + \frac{p + 1}{2 M ^{2}} k_{a} k_{b} .$
XX-D@1#eq:RRtail · line 1689
$E_{R R}^{ab} (t_{0} + δ, t_{0}; r) = - \frac{δ}{16 π M ^{2} a ( t _{0} ) ^{3} r} ((p + 9) δ_{ab} - (p + 1) n_{a} n_{b}) + O (δ^{2} r^{- 1}) + O_{δ} (r^{- 2}) .$
unlabeled · line 1643
$m (t^{'}, t^{'}) = 0, \partial_{t} m (t, t^{'})_{t = t^{'}} = - a (t^{'})^{- 3} A^{- 1} .$
unlabeled · line 1650
$K_{ij} (r) = \frac{δ _{ij} - n _{i} n _{j}}{8 π r}, r = ∣ r ∣, n = r / r .$
unlabeled · line 1671
$B^{a}_{k l} = \frac{1}{2} k_{a} δ_{k l} - k_{(k} δ_{l) a} .$
unlabeled · line 1676
$B^{a} : B^{b} = \frac{1}{2} ∣ k ∣^{2} δ_{ab} + \frac{1}{4} k_{a} k_{b} .$
unlabeled · line 1698
$F^{- 1} (∣ k ∣^{- 2}) = \frac{1}{4 π r}, F^{- 1} (k_{a} k_{b} ∣ k ∣^{- 4}) = \frac{δ _{ab} - n _{a} n _{b}}{8 π r}$

The native two-rod inverse-distance term

Displayed formulas in this passage

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