Aberration of Gravitational Waves
Setup
Consider two inertial observers, A and B, both of whom pass through the spacetime origin on their respective worldlines. B moves at constant velocity $v⃗$ relative to A, with speed $β = |v⃗| < 1$. Associated to A is a set of mode weights — a spin-weighted function on the sphere, expressed as a function of A's proper time — describing either a field that A emits (to future null infinity $ℐ^+$) or a field that A receives (from past null infinity $ℐ^-$). The goal of this section is to find B's description of the same physical field.
We introduce a sign variable $ε ∈ \{+1,-1\}$ to track the two cases:
\[ε = \begin{cases} +1 & \text{emitted field ($\mathscr{I}^+$)} \\ -1 & \text{received field ($\mathscr{I}^-$)} \end{cases}\]
This single variable turns out to control the sign of every frame-dependent quantity in the transformation. In the code, it is exposed as the emitted keyword argument of Scri.aberration.
Waves and null wavevectors
A monochromatic plane wave in flat Minkowski spacetime has the form
\[Φ(x) = A \exp(i\, k_μ x^μ),\]
where the phase $φ(x) = k_μ x^μ$ is a Lorentz scalar. Its gradient $k_μ = ∂_μ φ$ is the covariant wavevector. Inserting this into the wave equation $□Φ = 0$ gives
\[k^μ k_μ = 0,\]
the null condition, with metric signature ${-}{+}{+}{+}$.
Frame decomposition and the ε convention
Given observer A with four-velocity $t^μ = (1,0,0,0)$ (at rest in A's frame), we define
\[ω = -k_μ t^μ = k^t,\]
the frequency as measured by A. The null condition then forces the spatial wavevector $\vec{k} = (k^x, k^y, k^z)$ to satisfy $|\vec{k}| = ω$. We write
\[k^μ = ω(1,\, ε\hat{n}),\]
where $\hat{n}$ is a unit 3-vector and $ε = \pm 1$:
- $ε = +1$ (outgoing, $ℐ^+$): $\hat{n}$ points away from A — it is the propagation direction of the wave. (Think of gravitational waves emitted by A; each frequency component travels outward in direction $\hat{n}$.)
- $ε = -1$ (incoming, $ℐ^-$): $\hat{n}$ points toward A — it is the direction from which the wave arrives. (Think of a plane wave whose source is far away in the direction $\hat{n}$.)
This is identical to the null section $σ_ε: \hat{n} \mapsto (1, ε\hat{n})$ introduced in the BMS group page in its discussion of the celestial sphere. The same rotor language connects naturally to the Spacetime Algebra developed in Quaternionic.jl, where null vectors arise from combinations of boost and rotation generators in the even subalgebra of $\mathrm{Cl}(3,1)$.
Lorentz transformation of the wavevector
B's four-velocity in A's frame is $(t_B)^μ = (γ, γ\vec{v})$, where $γ = 1/\sqrt{1-β^2}$. B measures the frequency
\[ω' = -k_μ (t_B)^μ.\]
Expanding with $k^μ = ω(1, ε\hat{n})$ and writing $\cos Θ = \hat{n} \cdot \hat{v}$ (where $\hat{v} = \vec{v}/β$ is the unit boost direction):
\[ω' = γω(1 - εβ\cos Θ).\]
The angle $Θ$ is not an independent definition: it is the inner product of the spatial wavevector with the boost direction, $\cos Θ = (k^i \hat{v}_i)/ω$, read off directly from the contraction $k_μ (t_B)^μ$. When $ε = +1$, this is $\cos Θ = \hat{n}\cdot\hat{v}$ with $\hat{n}$ the propagation direction; when $ε = -1$ it is $\cos Θ = \hat{n}\cdot\hat{v}$ with $\hat{n}$ the direction toward the source.
The conformal factor
The ratio
\[K = \frac{ω}{ω'} = \frac{1}{γ(1 - ε\vec{v}\cdot\hat{n})}\]
is exactly the conformal factor introduced in the BMS page: for $ε = +1$ it reduces to $K = 1/[γ(1-\vec{v}\cdot\hat{n})]$ while for $ε = -1$ it becomes $K = 1/[γ(1+\vec{v}\cdot\hat{n})]$.
The aberration formula
In addition to changing the frequency, B's boost changes the direction from which the wave appears to come. For a boost along $\hat{z}$, the new direction $Θ'$ (measured from $\hat{z}$) satisfies
\[\cos Θ' = \frac{\cos Θ - εβ}{1 - εβ\cos Θ}.\]
[⚠️ Verify: MTW §22.5 (or a nearby exercise) for the standard form of this formula; the $ε = +1$ case is the standard result.] [⚠️ Verify: Schutz §2.7–2.8 derives $ω' = γω(1-β\cos Θ)$ for $ε = +1$; check whether the aberration formula itself also appears there.]
This is derived by applying the Lorentz boost directly to the spatial wavevector components: with $k^μ = ω(1, ε\sinΘ, 0, ε\cos Θ)$ (choosing the boost in the $x$-$z$ plane for simplicity), the boosted spatial $z$-component is
\[k'^z = γ(k^z - β k^t) = γω(ε\cos Θ - β),\]
and dividing by $ω' = γω(1 - εβ\cos Θ)$ gives the formula above.
Half-angle form
The half-angle substitution $\cos Θ = (1-t^2)/(1+t^2)$, $t = \tan(Θ/2)$, turns the aberration formula into
\[\tan\!\frac{Θ'}{2} = e^{-εφ}\,\tan\!\frac{Θ}{2}, \qquad φ = \operatorname{atanh}(β).\]
This compact form, involving the rapidity $φ$, is the one used in the rotor implementation.
Geometric sign
The direction of the shift depends on $ε$:
$ε = +1$ (emitted): $Θ' > Θ$ for $β > 0$. In B's frame the wave appears at a larger angle from the boost axis than in A's frame. Equivalently, in A's frame the radiation is beamed toward the boost axis. This is relativistic beaming familiar from astrophysical jets [4].
$ε = -1$ (received): $Θ' < Θ$ for $β > 0$. Moving observers see incoming sources shifted toward their direction of motion — the classical stellar-aberration effect. This is the convention used in Penrose-Rindler Vol. 1 around Eq. (1.3.5) [8], which works on the past celestial sphere. [⚠️ Verify: confirm Eq. (1.3.5) is for incoming null vectors, giving opposite sign to the $ε = +1$ case.]
General boost: the rotor formulation
For a boost in an arbitrary direction $\vec{v}$, the aberration is a rotation of $\hat{n}$ in the plane spanned by $\hat{n}$ and $\vec{v}$. Let $Θ'$ denote the boosted-frame angle between $\hat{n}'$ and the boost axis (this is the angle B observes), and let $Θ$ denote the corresponding rest-frame angle (the angle A uses). They are related by the half-angle formula
\[\tan\!\frac{Θ}{2} = e^{-εφ}\,\tan\!\frac{Θ'}{2}.\]
The rotation that maps $\hat{n}'$ to the correct rest-frame direction is generated by the aberration rotor
\[B' = \exp\!\left(\frac{\hat{n}' \times \vec{v}}{|\hat{n}' \times \vec{v}|}\,\frac{Θ'-Θ}{2}\right).\]
For $ε = +1$, $Θ' > Θ$ so the exponent is positive; for $ε = -1$, $Θ' < Θ$ and the exponent is negative, reversing the rotation.
Crucially, $B'$ does not merely map the direction $\hat{n}' \to \hat{n}$: it also rotates the tangent frame at each point on the sphere, producing the spin-weight phase factor that enters the mode transformation of spin-weighted functions. This is why we work with full rotors rather than unit 3-vectors.
The function Scri.aberration implements this for the product $R' = B' R_{\mathrm{pix}}$, where $R_{\mathrm{pix}}$ is the pixel rotor encoding both the direction and the tangent frame orientation at a grid point. The derivation of $B'$ is given in Appendix C of [5], Eqs. (C6)–(C8).
Geometric observations and tests
Each of the following observations is a direct consequence of the formula above. They are precise enough to serve as tests, and each maps to a @testitem in test/test_aberration.jl.
$β = 0$: no boost, no aberration
At $β = 0$, $φ = 0$ and $Θ = Θ'$ for both values of $ε$. The aberration rotor is the identity, so $R' = R_{\mathrm{pix}}$ regardless of the boost direction. This holds for all input rotors and all choices of $\hat{v}$.
Tests: "aberration: identity at β=0" and "aberration: emitted=false — identity at β=0".
Pole invariance: no tangent rotation along the boost axis
When $\hat{n}' \parallel \pm\hat{v}$, the cross product $\hat{n}' \times \vec{v} = 0$ and the rotation axis in $B'$ vanishes. Consequently $B' = 1$ and the rotor is unchanged.
This covers both the north pole ($\hat{n}' = +\hat{v}$) and the south pole ($\hat{n}' = -\hat{v}$), and holds for both $ε = +1$ and $ε = -1$ since the cross product is independent of the sign convention.
Tests: "aberration: pole invariance" and "aberration: emitted=false — pole invariance".
Equatorial formula: $\cos Θ = εβ$
Take the boost along $\hat{z}$ and consider an equatorial direction $\hat{n}'$ ($Θ' = π/2$). Setting $\cos Θ' = 0$ in the aberration formula gives
\[\cos Θ = εβ.\]
For $ε = +1$: the rest-frame direction is in the northern hemisphere, $\cos Θ = +β > 0$. For $ε = -1$: the rest-frame direction is in the southern hemisphere, $\cos Θ = -β < 0$.
Tests: "aberration: geometric sign — equatorial pixel maps closer to boost axis (future ℐ⁺)" and "aberration: emitted=false — equatorial pixel maps farther from boost axis (past ℐ⁻)".
Azimuthal symmetry
For a boost along $\hat{z}$, any rotation $R_z$ about $\hat{z}$ commutes with the aberration correction:
\[R'(R_z R_{\mathrm{pix}},\, \vec{v}) = R_z\, R'(R_{\mathrm{pix}},\, \vec{v}).\]
This follows because $R_z$ acts on $\hat{n}'$ by rotating it about the boost axis, which leaves the angle $Θ'$ — and therefore the magnitude of the aberration — unchanged, while rotating the axis $\hat{n}' \times \vec{v}$ by the same $R_z$.
Test: "aberration: azimuthal symmetry — z-rotation commutes with z-boost".
Round-trip: $ε = +1$ and $ε = -1$ are inverses
Applying the $ε = +1$ aberration and then the $ε = -1$ aberration with the same $\vec{v}$ recovers the identity:
\[R'(R'(R, \vec{v};\, \text{emitted}=\mathtt{true}),\, \vec{v};\, \text{emitted}=\mathtt{false}) = R.\]
This is immediate from the half-angle formula: $e^{-εφ}$ with $ε = +1$ followed by $e^{-εφ}$ with $ε = -1$ gives $e^{-φ} e^{+φ} = 1$.
Tests: "aberration: round-trip with inverse gives identity" (using the old boosted_rotor helper) and "aberration: emitted=false round-trip gives identity" (using both keyword values).
Wigner rotation: composition of non-collinear boosts
Two successive boosts in different directions do not compose to a pure boost; their composition in the Spin group is a boost followed by a Wigner rotation. Specifically, if the composite Lorentz transformation decomposes as $(v⃗_{\mathrm{eff}}, R_W)$ (in the sense of Quaternionic.jl's vR decomposition), then
\[R'\!\left(R'(R, \vec{v}_1),\, \vec{v}_2\right) = R'\!\left(R_W R,\, \vec{v}_{\mathrm{eff}}\right).\]
The Wigner rotation $R_W$ is a purely spatial rotation that has no classical analogue; it appears because the Lorentz group is not simply the direct product of rotations and boosts.
Test: "aberration: two-boost composition matches vR decomposition (Wigner rotation)".