Manuscript XX-C@1 · XX-C.tex · theorem · 7a8cab37

Theorem: Native two-rod inverse-distance tail

An intrinsic inverse-distance obstruction for two rods › The nonzero leading coefficient · lines 1548–1603 · proof included or linked

XX-C@1#thm:tworod, XX-C@1#eq:twoasymptotic

Back to manuscript

Let $t^{'} = t_{0}$ and $t = t_{0} + δ$ , with $δ$ sufficiently small and nonzero. For large $r = ∣ z ∣$ ,

E_{R R}^{ab} (t_{0} + δ, t_{0}; z) = - \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}) .

In particular, the native two-rod commutator is not $O (r^{- 2})$ for all nearby unequal times.

Proof. Equation (eq:jump) gives

m (t_{0} + δ, t_{0}) = - δ a (t_{0})^{- 3} A^{- 1} + O (δ^{2}) .

Insert this moment into the large-distance expansion of (eq:ERR).

By Lemma lem:sourcecoefficient, the leading Fourier multiplier is

- \frac{δ}{a ( t _{0} ) ^{3}} (\frac{2}{M ^{2}} \frac{δ _{ab}}{∣ k ∣ ^{2}} + \frac{p + 1}{2 M ^{2}} \frac{k _{a} k _{b}}{∣ k ∣ ^{4}}) .

Using

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}

gives (eq:twoasymptotic).

For example, choose $n$ parallel to coordinate axis $a$ and take component $aa$ . The coefficient in square brackets is then $(p + 9) - (p + 1) = 8$ . For small nonzero $δ$ , the $O (δ^{2})$ term cannot cancel that leading coefficient. Hence the inverse-distance tail is nonzero.

Displayed formulas in this passage

XX-C@1#eq:twoasymptotic · line 1553
$E_{R R}^{ab} (t_{0} + δ, t_{0}; z) = - \frac{δ}{16 π M ^{2} a ( t _{0} ) ^{3} r} [(p + 9) δ_{ab} - (p + 1) n_{a} n_{b}] + O (δ^{2} r^{- 1}) + O_{δ} (r^{- 2}) .$
unlabeled · line 1567
$m (t_{0} + δ, t_{0}) = - δ a (t_{0})^{- 3} A^{- 1} + O (δ^{2}) .$
unlabeled · line 1577
$- \frac{δ}{a ( t _{0} ) ^{3}} (\frac{2}{M ^{2}} \frac{δ _{ab}}{∣ k ∣ ^{2}} + \frac{p + 1}{2 M ^{2}} \frac{k _{a} k _{b}}{∣ k ∣ ^{4}}) .$
unlabeled · line 1586
$F^{- 1} (∣ k ∣^{- 2}) = \frac{1}{4 π r}, F^{- 1} (k_{a} k_{b} ∣ k ∣^{- 4}) = \frac{δ _{ab} - n _{a} n _{b}}{8 π r}$

Theorem: Native two-rod inverse-distance tail

Displayed formulas in this passage

Nearby in this section