Derivation of the Normal Modes

  We would like to describe the spatial and angular dependence of the normal modes $\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m}(\vec{x},\hat{n})$ in a coordinate-free way by constructing them out of covariant derivatives of ${\bf Q}^{(m)}$ contracted with some orthonormal basis $(\hat{n},\hat{m}_1,\hat{m}_2)$.The lowest $j={\rm max}(\vert m\vert,\vert s\vert)$ modes can be written as [3,4],  

$$\begin{array} {rcl}\, {}_{0}^{\vphantom{m}} G_{j}^{m} &=& n^... ...t{m}_1 \pm i \hat{m}_2)^{i_2} Q_{i_1 i_2}^{(m)} \,,\end{array}$$

and satisfy (Appendix B),  

$$\, {}_{s}^{\vphantom{m}} G_{\ell}^{m}(\vec{x},\hat{n}) = (-... ...om{m}} Y_{\ell}^{m}(\hat{n})] \exp[i\delta(\vec{x},\vec{k})]\,,$$ (55)

with $\ell=j$. We demand that the higher $\ell$-modes also do so, to maintain the division of spin and orbital angular momentum defined in flat space [1].

We begin the construction by choosing some arbitrary point $\vec{x}_0$,and using a spherical coordinate system around it, $\vec{x}-\vec{x}_0=\sqrt{-K}\, \chi(-\hat{n})$.Now $\hat{n}$ defines both the intrinsic angular coordinate system and the angular coordinates for the spatial location $\vec{x}(\chi,\hat{n})$.This reduction in the dimension of the space is sufficient since the end goal is to derive how the intrinsic and orbital angular dependence in the same direction $\hat{n}$ adds. In physical terms, only those photons directed toward the observer can contribute to the local angular dependence there. First expand the lowest mode in spin-spherical harmonics  

$$\, {}_{s}^{\vphantom{m}} {G}_{j}^{m}(\chi,\hat{n};\nu) = \... ...\nu) \, \, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m}({\hat{n}})\, ,$$ (56)

where recall that the dimensionless wavenumber is $\nu=q/\sqrt{-K}$.We obtain the explicit expressions for $\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}$and their recursion relations in Appendix B by simple comparison between equations (C1) and (C3). At the origin they satisfy  

$$\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}(0,\nu) = {1 \over 2\ell+1}\delta_{\ell,j} \, ,$$ (57)

which both fixes the normalization of the modes and manifestly obeys Eq. (C2). As $\chi \rightarrow 0$ only the local angular dependence remains, as expressed in the Kronecker delta of Eq. (C4). Because the spatial variation of the normal mode Q^(m) across a shell at fixed radius $\chi$ must be added to the local dependence, even a mode of fixed j has a sum over all $\ell$ in its angular dependence which contributes at any other point.

This generation of higher $\ell$ structure as $\chi$ increases suggests that we can use the radial structure of $\, {}_{s}^{\vphantom{m}} {G}_{j}^{m}$ to generate the higher $\ell$ modes. From the radial recursion relation for $\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}$Eq. (B9), let us make the ansatz  

$${1 \over \sqrt{-K}} n^i (\, {}_{s}^{\vphantom{m}} {G}_{\ell}... ... s \over \ell(\ell+1)} \, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m}.$$ (58)

That this series generates modes with the desired properties can be shown by returning to the spherical coordinate system. By explicit substitution of the radial form for $\, {}_{s}^{\vphantom{m}} {G}_{j}^{m}$ of Eq. (C3) and by noting that in this coordinate system  

$${1 \over \sqrt{-K}} n^i (\, {}_{s}^{\vphantom{m}} {G}_{\ell}... ... -{d \over d\chi} (\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m})\,,$$ (59)

we obtain  

$$\, {}_{s}^{\vphantom{m}} G_{\ell}^{m}(0,\hat{n}) = (-i)^\el... ...2\ell+1}} [\, {}_{s}^{\vphantom{m}} Y_{\ell}^{m}(\hat{n})] \,,$$ (60)

(up to a phase factor) as desired. Since we have shown this for an arbitrary point, it is clear that Eq. (C2) holds in general. Note that this construction requires  

$$\int {d\Omega \over 4\pi}\, \big\vert [\, {}_{s}^{\vphantom{... ...over 2\ell_1+1} \delta_{\ell_1,\ell_2}\, \delta_{m_1 ,m_2} \,,$$ (61)

for all $\vec{x}$, as in the flat case of Eq. (24), and defines our normalization convention.