| … | … | @@ -6851,7 +6851,7 @@ Metric |
| 6851 | 6851 | |
| 6852 | 6852 | \begin_layout Standard |
| 6853 | 6853 | Poincare ball metric with curvature |
| 6854 | | \begin_inset Formula $K=-1/A^{2}$ |
| 6854 | \begin_inset Formula $\kappa=-1/A^{2}$ |
| 6855 | 6855 | \end_inset |
| 6856 | 6856 | |
| 6857 | 6857 | , |
| … | … | @@ -7466,12 +7466,12 @@ and the flat–space version would have been |
| 7466 | 7466 | \end_inset |
| 7467 | 7467 | |
| 7468 | 7468 | To regain flat space |
| 7469 | | \begin_inset Formula $K=0$ |
| 7469 | \begin_inset Formula $\kappa=0$ |
| 7470 | 7470 | \end_inset |
| 7471 | 7471 | |
| 7472 | 7472 | in the limit, |
| 7473 | 7473 | set |
| 7474 | | \begin_inset Formula $K\to0$ |
| 7474 | \begin_inset Formula $\kappa\to0$ |
| 7475 | 7475 | \end_inset |
| 7476 | 7476 | |
| 7477 | 7477 | which means |
| … | … | @@ -7523,7 +7523,36 @@ with |
| 7523 | 7523 | maybe, |
| 7524 | 7524 | I guess. |
| 7525 | 7525 | Who knows. |
| 7526 | | Assuming it is, |
| 7526 | Well, |
| 7527 | not entirely; |
| 7528 | in fact, |
| 7529 | Claude assumed that the |
| 7530 | \begin_inset Quotes eld |
| 7531 | \end_inset |
| 7532 | |
| 7533 | grand spin quantum number |
| 7534 | \begin_inset Quotes erd |
| 7535 | \end_inset |
| 7536 | |
| 7537 | |
| 7538 | \begin_inset Formula $K=0$ |
| 7539 | \end_inset |
| 7540 | |
| 7541 | which is OK if all one wants is the ground state, |
| 7542 | but clearly fails for the vacuum states. |
| 7543 | I didn't catch this early enough, |
| 7544 | so the below proceeds with |
| 7545 | \begin_inset Formula $K=0$ |
| 7546 | \end_inset |
| 7547 | |
| 7548 | . |
| 7549 | That's OK, |
| 7550 | for now. |
| 7551 | |
| 7552 | \end_layout |
| 7553 | |
| 7554 | \begin_layout Standard |
| 7555 | Assuming Claude is otherwise correct, |
| 7527 | 7556 | the radial equations become |
| 7528 | 7557 | \begin_inset Formula |
| 7529 | 7558 | \begin{align*} |
| … | … | @@ -7596,10 +7625,101 @@ Pöschl-Teller potential |
| 7596 | 7625 | \end_inset |
| 7597 | 7626 | |
| 7598 | 7627 | . |
| 7628 | Never heard of that before. |
| 7599 | 7629 | So it knows about this. |
| 7600 | 7630 | Someone must have figured this out. |
| 7601 | 7631 | \end_layout |
| 7602 | 7632 | |
| 7633 | \begin_layout Standard |
| 7634 | Next, |
| 7635 | we can solve the above. |
| 7636 | Clause guesses that its hypergeometric, |
| 7637 | and arrives at the exact solution |
| 7638 | \begin_inset Formula |
| 7639 | \[ |
| 7640 | u=\frac{\sqrt{\xi^{2}-1}}{\xi}\;{}_{2}F_{1}\left(a,b;2;1-\xi^{2}\right) |
| 7641 | \] |
| 7642 | |
| 7643 | \end_inset |
| 7644 | |
| 7645 | with |
| 7646 | \begin_inset Formula $\xi=\cosh\rho$ |
| 7647 | \end_inset |
| 7648 | |
| 7649 | and |
| 7650 | \begin_inset Formula $a=\left(3+\sqrt{1+4\epsilon^{2}}\right)/4$ |
| 7651 | \end_inset |
| 7652 | |
| 7653 | and |
| 7654 | \begin_inset Formula $b=\left(3-\sqrt{1+4\epsilon^{2}}\right)/4$ |
| 7655 | \end_inset |
| 7656 | |
| 7657 | OK, |
| 7658 | that's trctable. |
| 7659 | In chart coords its |
| 7660 | \begin_inset Formula |
| 7661 | \[ |
| 7662 | u=r\;{}_{2}F_{1}\left(a,b;2;-\frac{r^{2}}{1-r^{2}}\right) |
| 7663 | \] |
| 7664 | |
| 7665 | \end_inset |
| 7666 | |
| 7667 | It seems plausible that this might reduce to sphere bessel functions in the |
| 7668 | \begin_inset Formula $A\to\infty$ |
| 7669 | \end_inset |
| 7670 | |
| 7671 | flat space limit. |
| 7672 | It should reduce to |
| 7673 | \begin_inset Formula $j_{0}$ |
| 7674 | \end_inset |
| 7675 | |
| 7676 | with |
| 7677 | \begin_inset Formula $v$ |
| 7678 | \end_inset |
| 7679 | |
| 7680 | going over to |
| 7681 | \begin_inset Formula $j_{1}$ |
| 7682 | \end_inset |
| 7683 | |
| 7684 | . |
| 7685 | I'm not doing this now; |
| 7686 | first we have to get the |
| 7687 | \begin_inset Formula $K\ne0$ |
| 7688 | \end_inset |
| 7689 | |
| 7690 | . |
| 7691 | \end_layout |
| 7692 | |
| 7693 | \begin_layout Standard |
| 7694 | Lets pause to look at the boundary conditions. |
| 7695 | Claude claims that |
| 7696 | \begin_inset Formula |
| 7697 | \[ |
| 7698 | \left.\frac{v}{u}\right|_{r=R}=-\tan\frac{\Theta}{2} |
| 7699 | \] |
| 7700 | |
| 7701 | \end_inset |
| 7702 | |
| 7703 | which cranked through is supposed to be |
| 7704 | \begin_inset Formula |
| 7705 | \[ |
| 7706 | u^{\prime}\sin\frac{\Theta}{2}=\left(\epsilon\cos\frac{\Theta}{2}-\frac{2+\tanh^{2}\rho_{R}}{\tanh\rho_{R}}\sin\frac{\Theta}{2}\right)u |
| 7707 | \] |
| 7708 | |
| 7709 | \end_inset |
| 7710 | |
| 7711 | which Claude claims is a |
| 7712 | \begin_inset Quotes erd |
| 7713 | \end_inset |
| 7714 | |
| 7715 | Robin boundary condition |
| 7716 | \begin_inset Quotes erd |
| 7717 | \end_inset |
| 7718 | |
| 7719 | . |
| 7720 | Never heard of that but OK. |
| 7721 | \end_layout |
| 7722 | |
| 7603 | 7723 | \begin_layout Paragraph* |
| 7604 | 7724 | Punt |
| 7605 | 7725 | \end_layout |