Thorne (1980)

Thorne [51] is the standard reference for multipole expansions of gravitational radiation, and its conventions for the spherical harmonics are inherited by much of the gravitational-wave literature. Thorne uses the standard (physicists') spherical coordinates $(θ, ϕ)$, and writes the indices of the spherical harmonics as superscripts: $Y^{ℓ m}$.

Thorne's Eq. (2.7) gives the scalar spherical harmonics for $m \geq 0$ as

\[Y^{ℓ m}(θ, ϕ) = C^{ℓ m}\, \left(e^{iϕ} \sin θ\right)^m \sum_{j=0}^{\lfloor (ℓ-m)/2 \rfloor} a^{ℓ m j}\, (\cos θ)^{ℓ-m-2j},\]

with the coefficients given in Eq. (2.8),

\[C^{ℓ m} = (-1)^m \left[ \frac{(2ℓ+1)\,(ℓ-m)!}{4π\,(ℓ+m)!} \right]^{1/2}, \qquad a^{ℓ m j} = \frac{(-1)^j}{2^ℓ\, j!\, (ℓ-j)!}\, \frac{(2ℓ-2j)!}{(ℓ-m-2j)!},\]

and the harmonics with negative $m$ are defined through the conjugation relation of Eq. (2.9b),

\[Y^{ℓ, -m} = (-1)^m\, \overline{Y^{ℓ m}}.\]

The factor $(-1)^m$ in $C^{ℓ m}$ is the Condon–Shortley phase, so we expect agreement with our spherical harmonics. Thorne's paper also defines the "pure-spin" vector and tensor harmonics built from these scalar harmonics, which are outside the scope of this package.

Implementing formulas

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

module Thorne

We'll use some predefined utilities to make the code look more like the equations.

import ..ConventionsUtilities: 𝒾, ❗

The coefficients of Eq. (2.8). We capture the floating-point type T to ensure that we don't lose precision when converting π and the factorials to floating-point numbers.

function C(ℓ, m, ::Type{T}) where {T}    (-1)^m * √T(((2ℓ+1) * (ℓ-m)❗) / (4big(π) * (ℓ+m)❗))endfunction a(ℓ, m, j, ::Type{T}) where {T}    T((-1)^j / (2^ℓ * (j)❗ * (ℓ-j)❗) * ((2ℓ-2j)❗ / (ℓ-m-2j)❗))end

Equation (2.7) then gives the spherical harmonics for $m \geq 0$, and Eq. (2.9b) extends them to negative $m$:

function Y(ℓ, m, θ::T, ϕ::T) where {T<:Real}    if m < 0        return (-1)^m * conj(Y(ℓ, -m, θ, ϕ))    end    sinθ, cosθ = sincos(θ)    C(ℓ, m, T) * (exp(𝒾 * ϕ) * sinθ)^m * sum(        a(ℓ, m, j, T) * cosθ^(ℓ-m-2j)        for j ∈ 0:(ℓ-m)÷2;        init=zero(T)    )end

end  # module Thorne

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 = 5

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.

First, we check that Thorne's conjugation relation (2.9b) is consistent with the explicit formula — that is, that the $(-1)^m$ in $C^{ℓ m}$ and the $e^{imϕ}$ factor really do reproduce the relation. Thorne.Y defines negative $m$ through that relation, so comparing the two would compare the relation with itself. Instead, we apply Eq. (2.7) directly to negative $m$, which Thorne does not do but which is valid: the sum is Rodrigues' formula for the associated Legendre function, which holds for $-ℓ ≤ m ≤ ℓ$, and every factorial in it has a non-negative argument. For negative $m$ the sum cancels down to a multiple of $\sin^{2|m|} θ$, so it is evaluated in BigFloat, where that cancellation costs nothing that matters, even beside the poles.

function Y₂₇(ℓ, m, θ::T, ϕ::T) where {T<:Real}    sinθ, cosθ = sincos(θ)    Thorne.C(ℓ, m, T) * (exp(ConventionsUtilities.𝒾 * ϕ) * sinθ)^m * sum(        Thorne.a(ℓ, m, j, T) * cosθ^(ℓ-m-2j)        for j ∈ 0:(ℓ-m)÷2;        init=zero(T)    )endfor (θ, ϕ) ∈ θϕrange(rng, Float64, 15; avoid_poles=1e-3)    for ℓ ∈ 0:ℓₘₐₓ        for m ∈ 1:ℓ            @test Y₂₇(ℓ, -m, big(θ), big(ϕ)) ≈ (-1)^m * conj(Thorne.Y(ℓ, m, θ, ϕ)) atol=ϵₐ rtol=ϵᵣ        end    endend

Now we compare to the SphericalFunctions package:

for (θ, ϕ) ∈ θϕrange(rng)    for (ℓ, Yˡ) ∈ SphericalFunctions.YlmCalculator(θ, ϕ, ℓₘₐₓ), m ∈ -ℓ:ℓ        @test Thorne.Y(ℓ, m, θ, ϕ) ≈ Yˡ[m] atol=ϵₐ rtol=ϵᵣ    endend

This successful test shows that the spherical harmonics defined by Thorne agree with the spherical harmonics defined by the SphericalFunctions package.

This page was generated using Literate.jl.