Manuscript I-B-VP@3 · I-B-VP-v3.tex · theorem · 93c04eb4

Theorem: Long-range tail

Scale-dependent diagonal nonnegative extensions · lines 2319–2364 · proof included or linked

I-B-VP@3#thm:large

As $r \to \infty$ at fixed $λ > 0$ ,

δ F = \frac{ν}{4 π ^{2} α ( 1 + 2 ν ) λ r ^{2 + 2 ν}} + O (r^{- 2 - 4 ν}) .

The leading stress coefficients are

δ T^{r}_{r} δ T^{t}_{t} = δ T^{z}_{z} δ T^{φ}_{φ} = \frac{ν [( 1 + 2 ν ) - 2 ξ ( 1 + ν ) ( 3 + 2 ν )]}{2 π ^{2} α λ ( 1 + 2 ν ) ( 3 + 2 ν ) r ^{4 + 2 ν}} + O (r^{- 4 - 4 ν}), = \frac{ν ( 1 + ν ) [ 4 ξ ( 1 + ν ) ( 3 + 2 ν ) - ( 2 ν ^{2} + 3 ν + 2 )]}{2 π ^{2} α λ ( 1 + 2 ν ) ( 3 + 2 ν ) r ^{4 + 2 ν}} + O (r^{- 4 - 4 ν}), = - (3 + 2 ν) [δ T^{r}_{r}]_{leading} + O (r^{- 4 - 4 ν}) .

Where a leading coefficient is nonzero, its relative correction is $O ((M r)^{- 2 ν})$ . No relative assertion is made at a zero coefficient.

Proof. For $ε \to \infty$ ,

\frac{x ^{2 ν}}{x ^{2 ν} + ε ^{2 ν}} = ε^{- 2 ν} x^{2 ν} + O (ε^{- 4 ν} x^{4 ν}) .

All resulting $K$ -Bessel moments, including the derivative moments, converge. Evaluate them with Appendix app:mellin and the defining relation between $M$ and $λ$ . This gives $δ F$ and $T^{r}_{r}$ . Conservation at power $r^{- 4 - 2 ν}$ gives the angular coefficient; the trace identity gives the time coefficient. The same moment estimate supplies the absolute remainder.

Displayed formulas in this passage

unlabeled · line 2328
$δ T^{r}_{r} δ T^{t}_{t} = δ T^{z}_{z} δ T^{φ}_{φ} = \frac{ν [( 1 + 2 ν ) - 2 ξ ( 1 + ν ) ( 3 + 2 ν )]}{2 π ^{2} α λ ( 1 + 2 ν ) ( 3 + 2 ν ) r ^{4 + 2 ν}} + O (r^{- 4 - 4 ν}), = \frac{ν ( 1 + ν ) [ 4 ξ ( 1 + ν ) ( 3 + 2 ν ) - ( 2 ν ^{2} + 3 ν + 2 )]}{2 π ^{2} α λ ( 1 + 2 ν ) ( 3 + 2 ν ) r ^{4 + 2 ν}} + O (r^{- 4 - 4 ν}), = - (3 + 2 ν) [δ T^{r}_{r}]_{leading} + O (r^{- 4 - 4 ν}) .$
unlabeled · line 2322
$δ F = \frac{ν}{4 π ^{2} α ( 1 + 2 ν ) λ r ^{2 + 2 ν}} + O (r^{- 2 - 4 ν}) .$
unlabeled · line 2351
$\frac{x ^{2 ν}}{x ^{2 ν} + ε ^{2 ν}} = ε^{- 2 ν} x^{2 ν} + O (ε^{- 4 ν} x^{4 ν}) .$

Theorem: Long-range tail

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