Newman-Penrose (1966)

In their 1966 paper, Newman and Penrose [4], Newman and Penrose first introduced the spin-weighted spherical harmonics, ${}_sY_{ℓ m}$. They use the standard (physicists') convention for spherical coordinates and introduce the stereographic coordinate [Eq. (3.8)]

\[ζ = e^{iϕ} \cot\frac{θ}{2}.\]

Spin weight

They are a little ambiguous about the relationship of the complex basis vector $m^\mu$ to the coordinates:

The vectors $\Re(m^\mu)$ and $\Im(m^\mu)$ may be regarded as orthogonal tangent vectors (of length $2^{-1/2}$) at each point of the surface. [...] If spherical polar coordinates are used, a natural choice for $m^\mu$ is to make $\Re(m^\mu)$ and $\Im(m^\mu)$ tangential, respectively, to the curves $ϕ = \mathrm{const}$ and $θ = \mathrm{const}.$

The ambiguity is in the sign implied by "tangential", but the natural choice is to assume they mean that the components are positive multiples of $\partial_θ$ and $\partial_ϕ$ respectively, in which case we have

\[m^\mu = \frac{1}{\sqrt{2}} \left[ \partial_θ + i \csc θ \partial_ϕ \right],\]

which is precisely our $𝐦$. They define the spin weight in terms of behavior of a quantity under rotation of $m^\mu$ in its own plane as

\[(m^\mu)' = e^{iψ} m^\mu = \frac{1}{\sqrt{2}} \left[ \left(\cos ψ\partial_θ - \sin ψ\csc θ \partial_ϕ\right) + i \left(\cos ψ\csc θ \partial_ϕ + \sin ψ\partial_θ\right) \right],\]

and a quantity $\eta$ has spin weight $s$ if it transforms as [Eq. (3.3)]

\[\eta' = e^{i s ψ} \eta.\]

Raising the spherical coordinates $(θ, ϕ)$ to Euler angles $(ϕ, θ, -ψ)$, we see that the rotor $𝐑_{ϕ, θ, -ψ}$ rotates the $𝐳$ basis vector to the point $(θ, ϕ)$, and it rotates $(𝐱 + i 𝐲) / \sqrt{2}$ onto $(m^\mu)'$. Supposing that these quantities are functions of Euler angles, we can write

\[\eta(ϕ, θ, -ψ) = e^{i s ψ} \eta(ϕ, θ, 0), \qquad \text{or} \qquad \eta(ϕ, θ, γ) = e^{-i s γ} \eta(ϕ, θ, 0).\]

Thus, the operator with eigenvalue $s$ is $i \partial_γ$, which is exactly our $R_z$. This is the definition of spin weight adopted by this package, so there is nothing to test numerically here; the rest of this page tests the operators and functions built on it.

The operators $\eth$ and $\bar{\eth}$

Newman and Penrose define the spin-raising operator $\eth$ acting on a function of spin weight $s$ in their Eq. (3.8) as

\[\eth \eta = -\left(\sin θ\right)^s \left\{ \frac{\partial}{\partial θ} + \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left\{\left(\sin θ\right)^{-s} \eta\right\},\]

along with its complex conjugate,

\[\bar{\eth} \eta = -\left(\sin θ\right)^{-s} \left\{ \frac{\partial}{\partial θ} - \frac{i}{\sin θ} \frac{\partial}{\partial ϕ} \right\} \left\{\left(\sin θ\right)^{s} \eta\right\},\]

which lowers the spin weight. On the Summary page — and in detail on the calculation page — we show that these are precisely $\eth = R_+ = R_x + i R_y$ and $\bar{\eth} = -R_- = -(R_x - i R_y)$ in terms of our right angular-momentum operators, so that

\[\eth\, {}_sY_{ℓ,m} = \sqrt{(ℓ-s)(ℓ+s+1)}\, {}_{s+1}Y_{ℓ,m}, \qquad \bar{\eth}\, {}_sY_{ℓ,m} = -\sqrt{(ℓ+s)(ℓ-s+1)}\, {}_{s-1}Y_{ℓ,m}.\]

The minus sign in the second relation is real, and is the reason $\bar{\eth} \neq R_-$. Below, we implement Newman and Penrose's differential operators literally (using automatic differentiation) and verify these relations on the spin-weighted spherical harmonics of this package.

Spin-weighted spherical harmonics

Newman and Penrose then compute the spin-weighted spherical harmonics, up to normalization, as [Eq. (3.21)]

\[{}_sY_{ℓ, m} \propto \frac{1}{\left[(ℓ-s)! (ℓ+s)!\right]^{1/2}} \left(1 + ζ \bar{ζ}\right)^{-ℓ} \sum_p ζ^p (-\bar{ζ})^{p+s-m} \binom{ℓ-s}{p} \binom{ℓ+s}{p+s-m},\]

where the sum is over all integers $p$ such that the binomial coefficients are nonzero. Note that, because $ζ$ includes a factor $e^{iϕ}$, this expression is proportional to $e^{i(m-s)ϕ}$ rather than $e^{imϕ}$: it is referred to the tangent basis defined by the stereographic coordinate, which is rotated relative to the $(\boldsymbol{θ}, \boldsymbol{ϕ})$ basis by the angle $ϕ$. Since Newman and Penrose only give this expression up to proportionality, the test below establishes the exact relationship to our ${}_sY_{ℓ,m}$.

Implementing formulas

We begin by writing code that implements the formulas from Newman and Penrose. We encapsulate the formulas in a module so that we can test them against the SphericalFunctions package.

module NewmanPenrose

We'll use some predefined utilities to make the code look more like the equations, and ForwardDiff to evaluate the derivatives in the differential operators exactly.

import ..ConventionsUtilities: 𝒾, ❗import ForwardDiff

The stereographic coordinate is

ζ(θ, ϕ) = exp(𝒾 * ϕ) * cot(θ / 2)

The spin-raising operator $\eth$ of Eq. (3.8) acts on a function η(θ, ϕ) of spin weight s, returning a new function of $(θ, ϕ)$. We evaluate the partial derivatives with forward-mode automatic differentiation.

function ð(η, s)    function ðη(θ, ϕ)        η̃(θ, ϕ) = sin(θ)^(-s) * η(θ, ϕ)        ∂θ = ForwardDiff.derivative(θ′ -> η̃(θ′, ϕ), θ)        ∂ϕ = ForwardDiff.derivative(ϕ′ -> η̃(θ, ϕ′), ϕ)        -sin(θ)^s * (∂θ + 𝒾 / sin(θ) * ∂ϕ)    endend

The spin-lowering operator $\bar{\eth}$ is the complex conjugate of $\eth$, meaning that it is obtained by conjugating every explicit $i$ — including the one hidden in $s$, whose sign flips under conjugation because the conjugate of a spin-weight-$s$ quantity has spin weight $-s$:

function ð̄(η, s)    function ð̄η(θ, ϕ)        η̃(θ, ϕ) = sin(θ)^(s) * η(θ, ϕ)        ∂θ = ForwardDiff.derivative(θ′ -> η̃(θ′, ϕ), θ)        ∂ϕ = ForwardDiff.derivative(ϕ′ -> η̃(θ, ϕ′), ϕ)        -sin(θ)^(-s) * (∂θ - 𝒾 / sin(θ) * ∂ϕ)    endend

Newman and Penrose's unnormalized expression for the spin-weighted spherical harmonics in terms of $ζ$ is a sum over all $p$ for which both binomial coefficients are nonzero, which means $\max(0, m-s) \leq p \leq \min(ℓ-s, ℓ+m)$:

function ₛYₗₘ_ζ(s, ℓ, m, θ, ϕ)    z = ζ(θ, ϕ)    (1 / √((ℓ-s)❗ * (ℓ+s)❗)) * (1 + z * conj(z))^(-ℓ) *    sum(        z^p * (-conj(z))^(p+s-m) * binomial(big(ℓ-s), p) * binomial(big(ℓ+s), p+s-m)        for p ∈ max(0, m-s):min(ℓ-s, ℓ+m);        init=zero(Complex{typeof(θ)})    )end

Finally, to apply the differential operators we need a version of our spin-weighted spherical harmonics that can be differentiated. The functions provided by SphericalFunctions are evaluated with recursions in floating-point arithmetic, which is not suitable for automatic differentiation, so we transcribe the explicit formula from the Summary page:

\[{}_{s}Y_{ℓ,m}(θ, ϕ) = (-1)^s\sqrt{\frac{2ℓ+1}{4π}}\, e^{imϕ} \sum_{k = k_1}^{k_2} \frac{(-1)^k[(ℓ+m)!(ℓ-m)!(ℓ-s)!(ℓ+s)!]^{1/2}} {(ℓ+m-k)!\,(ℓ+s-k)!\,k!\,(k-s-m)!} \left(\cos\frac{θ}{2}\right)^{2ℓ+m+s-2k} \left(\sin\frac{θ}{2}\right)^{2k-s-m},\]

with $k_1 = \max(0, m+s)$ and $k_2 = \min(ℓ+m, ℓ+s)$. We check below that this agrees with the package, before using it in the differential tests.

We write the phase $e^{imϕ}$ as cis(m * ϕ) rather than exp(𝒾 * m * ϕ) because Base.exp(::Complex) short-circuits to Complex(exp(real(z)), imag(z)) whenever iszero(imag(z)). That branch is correct for numbers, but it is only accurate to first order in the imaginary part, so under automatic differentiation at $ϕ = 0$ it silently discards every derivative beyond the first — which would break the second-order $\bar{\eth}\eth$ test below. cis computes Complex(cos, sin) unconditionally, and so differentiates correctly to all orders.

function ₛYₗₘ(s, ℓ, m, θ, ϕ)    T = float(typeof(θ))    (-1)^s * √((2ℓ+1) / (4T(π))) * cis(m * ϕ) *    sum(        (-1)^k * T(√((ℓ+m)❗ * (ℓ-m)❗ * (ℓ-s)❗ * (ℓ+s)❗) / ((ℓ+m-k)❗ * (ℓ+s-k)❗ * (k)❗ * (k-s-m)❗)) *        cos(θ/2)^(2ℓ+m+s-2k) * sin(θ/2)^(2k-s-m)        for k ∈ max(0, m+s):min(ℓ+m, ℓ+s);        init=zero(Complex{T})    )end

end  # module NewmanPenrose

Tests

We can now test the functions against the equivalent functions from the SphericalFunctions package. We will need to test approximate floating-point equality, so we set absolute and relative tolerances (respectively) in terms of the machine epsilon:

ϵₐ = 100eps()ϵᵣ = 1000eps()

We only test up to

ℓₘₐₓ = 4

and

sₘₐₓ = 2

because the formulas are slow, and this will be sufficient to sort out any sign or normalization differences, which are the most likely source of error. For the same reason we use a modest grid of points:

θϕs = θϕrange(rng, Float64, 7)

The differential operators involve $1/\sin θ$, so for them we avoid the poles. Even so, the factor $\sin^{-s} θ$ inside $\eth$ amplifies rounding errors near the poles by a factor of order $1/\sin^{|s|} θ$, so we evaluate the differential tests in BigFloat arithmetic (the automatic differentiation is exact, so this leaves only the analytic comparison), while still requiring agreement to Float64 precision:

θϕs_big = [(big(θ), big(ϕ)) for (θ, ϕ) ∈ θϕrange(rng, Float64, 5; avoid_poles=1e-3)]

We will also need the imaginary unit from the utilities module.

import .ConventionsUtilities: 𝒾

First, we check that our transcription of the explicit formula agrees with the package:

for (θ, ϕ) ∈ θϕs    for (ℓ, Yˡ) ∈ SphericalFunctions.sYlmCalculator(θ, ϕ, ℓₘₐₓ, -sₘₐₓ:sₘₐₓ)        for s ∈ -min(ℓ, sₘₐₓ):min(ℓ, sₘₐₓ), m ∈ -ℓ:ℓ            @test NewmanPenrose.ₛYₗₘ(s, ℓ, m, θ, ϕ) ≈ Yˡ[s, m] atol=ϵₐ rtol=ϵᵣ        end    endend

Next, we apply Newman and Penrose's $\eth$ and $\bar{\eth}$ to our spin-weighted spherical harmonics, and verify the raising and lowering relations — including the minus sign in the lowering relation. When the raised or lowered spin weight would exceed $ℓ$ in magnitude, the result must vanish.

for (θ, ϕ) ∈ θϕs_big    for (s, ℓ, m) ∈ sℓmrange(ℓₘₐₓ, sₘₐₓ)        Y(θ, ϕ) = NewmanPenrose.ₛYₗₘ(s, ℓ, m, θ, ϕ)        ðY = NewmanPenrose.ð(Y, s)(θ, ϕ)        ð̄Y = NewmanPenrose.ð̄(Y, s)(θ, ϕ)        if abs(s+1) ≤ ℓ            @test ðY ≈ √((ℓ-s) * (ℓ+s+1)) * NewmanPenrose.ₛYₗₘ(s+1, ℓ, m, θ, ϕ) atol=ϵₐ rtol=ϵᵣ        else            @test ðY ≈ 0 atol=ϵₐ        end        if abs(s-1) ≤ ℓ            @test ð̄Y ≈ -√((ℓ+s) * (ℓ-s+1)) * NewmanPenrose.ₛYₗₘ(s-1, ℓ, m, θ, ϕ) atol=ϵₐ rtol=ϵᵣ        else            @test ð̄Y ≈ 0 atol=ϵₐ        end    endend

A consequence of these two relations is that $\bar{\eth}\eth$ acts on ${}_sY_{ℓ,m}$ as multiplication by $-(ℓ-s)(ℓ+s+1)$, which is the eigenvalue relation Newman and Penrose use to characterize the spin-weighted spherical harmonics. We check it directly:

for (θ, ϕ) ∈ θϕs_big    for (s, ℓ, m) ∈ sℓmrange(ℓₘₐₓ, sₘₐₓ)        Y(θ, ϕ) = NewmanPenrose.ₛYₗₘ(s, ℓ, m, θ, ϕ)        ð̄ðY = NewmanPenrose.ð̄(NewmanPenrose.ð(Y, s), s+1)(θ, ϕ)        @test ð̄ðY ≈ -(ℓ-s) * (ℓ+s+1) * Y(θ, ϕ) atol=ϵₐ rtol=ϵᵣ    endend

Finally, we compare Newman and Penrose's $ζ$-expression to the package. Since they only give it up to proportionality, we determine the constant of proportionality — which turns out to be $(-1)^ℓ \sqrt{4π / [(2ℓ+1)(ℓ+m)!(ℓ-m)!]}$ — and assert it exactly. (Newman and Penrose encapsulate this as $a_{l,m}$, but only say its "exact values are inessential for our purposes".) There is also factor of $e^{-isϕ}$ to account for the rotation from the stereographic tangent basis to the $(\boldsymbol{θ}, \boldsymbol{ϕ})$ basis, as discussed above. Because $ζ$ itself diverges at the north pole, we again use the pole-avoiding BigFloat grid.

for (θ, ϕ) ∈ θϕs_big    for (ℓ, Yˡ) ∈ SphericalFunctions.sYlmCalculator(θ, ϕ, ℓₘₐₓ, -sₘₐₓ:sₘₐₓ)        for s ∈ -min(ℓ, sₘₐₓ):min(ℓ, sₘₐₓ), m ∈ -ℓ:ℓ            c = (-1)^ℓ * exp(-𝒾 * s * ϕ) * √(4big(π) / ((2ℓ+1) * factorial(ℓ+m) * factorial(ℓ-m)))            @test NewmanPenrose.ₛYₗₘ_ζ(s, ℓ, m, θ, ϕ) ≈ c * Yˡ[s, m] atol=ϵₐ rtol=ϵᵣ        end    endend

These successful tests show that Newman and Penrose's $\eth$ and $\bar{\eth}$ act on this package's spin-weighted spherical harmonics exactly as the conventions pages claim, and that their stereographic expression for ${}_sY_{ℓ,m}$ is proportional to ours with the constant given above.

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