GitHub

The BMS Group

Bondi-Sachs coordinates, metric, and gauge

The null cone and the celestial sphere

  • Future null cone and celestial sphere

    • $𝒩⁺$ with metric asymptotically Minkowski $ηᵦᵧ$

    • celestial sphere $𝕊²$ is the space of future null rays $𝒩⁺/ℝ⁺$

      • choose some $t$ vector
      • each null vector decomposes as a term proportional to $t$ plus a spatial vector
      • each null vector then gives rise to a unique spatial unit vector $n̂$ by normalizing the spatial part
      • consider $𝕊²$ to be represented by these spatial unit vectors
      • also use coordinates $xᴬ$, which may be $(θ, φ)$

    • section $σ:𝕊²→𝒩⁺$ via $σ:n̂↦ℓᵝ=(1,n̂)$. That is, $σᵝ(xᴬ) = (1, n̂(xᴬ))$

    • (note that $ℓᵝ = -ηᵝᵞ(dt-dr)ᵧ \sim -ηᵝᵞ(du)ᵧ$)

    • induced metric on $𝕊²$ is the pullback

      \[dΩ² = σ^* η = ηᵦᵧ \frac{∂σᵝ}{∂xᴬ} \frac{∂σᵞ}{∂xᴮ} dxᴬ dxᴮ\]

      or

      \[g_{AB} = ηᵦᵧ \frac{∂σᵝ}{∂xᴬ} \frac{∂σᵞ}{∂xᴮ}.\]

Lorentz transformations $ℒ$ and the conformal factor $K$

  • Lorentz transformations preserve $𝒩⁺$ but not the section with $ℓ⁰=1$.

  • Define $K$ in terms of time component ${Λ⁰}_μ ℓ^μ = γ(1-v⃗⋅n̂) = 1/K$.

  • Lorentz transformation $Λ$ induces a transformation of the section as $σ' = K (Λ ∘ σ)$ — where we have to rescale by $K$ to get back to the preferred section with $ℓ^{0'}=1$.

  • Differentiating ${σ'}ᵝ = K {Λᵝ}ᵧσᵞ$ by $xᴬ$ gives us two terms, the first differentiating $K$ and the second differentiating $σ$:

    \[\frac{∂{σ'}ᵝ}{∂xᴬ} = \frac{∂K}{∂xᴬ} {Λᵝ}ᵧσᵞ + K {Λᵝ}ᵧ \frac{∂σᵞ}{∂xᴬ}.\]

    That derivative is then contracted twice with the Minkowski metric to give us the new metric in the primed frame:

    \[g'_{AB} = ηᵦᵧ \frac{∂{σ'}ᵝ}{∂xᴬ} \frac{∂{σ'}ᵞ}{∂xᴮ}.\]

    We will expand this expression, but we need a few preliminary results first. First, note that in all resulting terms, we have the $Λ$s contracted with the metric, which they preserve:

    \[ηᵦᵧ {Λᵝ}ᵤ {Λᵞ}ᵥ = ηᵤᵥ.\]

    Next, we will need the fact that the section is null, which means that

    \[ηᵦᵧ σᵝ σᵞ = 0.\]

    Finally, we can differentiate that expression to find that

    \[ηᵦᵧ σᵝ \frac{∂σᵞ}{∂xᴬ} = 0.\]

    We can now use the expression for the derivative of $σ'$ to expand the expression for $g'_{AB}$, and use these results to simplify, then find

    \[g'_{AB} = K² ηᵦᵧ \frac{∂{σ}ᵝ}{∂xᴬ} \frac{∂{σ}ᵞ}{∂xᴮ} = K² g_{AB}.\]

    That is, ${dΩ'}² = K² dΩ²$. This is the key result: Lorentz transformations transform the unit sphere metric on the celestial sphere by a conformal factor $K²$.

  • Combine that with the fact that Bondi gauge requires the angular part of the metric to be asymptotically $r² dΩ²$, and if the transformation is an isometry we must have the asymptotic relation $r² dΩ² \sim {r'}² {dΩ'}²$, and we find that $r \sim K r'$, which is Sachs's definition of $K$.

To do

Here's a more mechanical and unenlightening derivation, though it may be more familiar, so it could be useful to have both.

  • Conformal factor under boost:
    • aberration formula $\cos θ' = (\cos θ - β) / (1 - β \cos θ)$
    • differentiate to find $\sin θ'\, dθ' = K² \sin θ\, dθ$
    • use $\sin² θ' = 1-\cos² θ'$ to find $\sin θ' = K \sin θ$
    • also have $dθ' = K dθ$
    • combine to show that a boost along $z$ transforms the unit sphere metric as ${dΩ'}² = K² dΩ²$.
    • rotations preserve the unit sphere metric, so arbitrary Lorentz transformations transform the unit sphere metric as ${dΩ'}² = K² dΩ²$, with $K$ as we defined it.

Supertranslations $𝒮$

BMS

Denote the (proper orthochronous) Lorentz group by $ℒ$ and the supertranslations by $𝒮$. Then the BMS group is the semidirect product of these two groups:

\[\text{BMS} = 𝒮 ⋊ ℒ.\]

In particular, note that $𝒮$ is a normal subgroup of $\text{BMS}$, but $ℒ$ is not.

That is, for $Λ ∈ ℒ$ and $α ∈ 𝒮$, the element $Λ α Λ⁻¹$ is still an element of $𝒮$, but $α Λ α⁻¹$ is not an element of $ℒ$. This fact will be useful later.

We can write an arbitrary element of $\text{BMS}$ uniquely [12, page 173] as $(Λ, α)$ for some elements $Λ ∈ ℒ$ and $α ∈ 𝒮$. We interpret this as the composition of the supertranslation $α$ followed by the Lorentz transformation $Λ$.[1] The group operation is then

\[(Λ₂, α₂) (Λ₁, α₁) = (Λ₂ Λ₁, α₁ + Λ₁⁻¹ α₂),\]

where $Λ₁⁻¹ α₂$ is the function that first rotates the argument of $α₂$ by $Λ₁$ (not its inverse), and then evaluates $α₂$ at that rotated argument.

The supertranslations form an abelian subgroup of $\text{BMS}$ because

\[(\text{id}, α₂) (\text{id}, α₁) = (\text{id}, α₁ + α₂) = (\text{id}, α₂ + α₁) = (\text{id}, α₁) (\text{id}, α₂).\]

We have essentially constructed the BMS group, but Sachs [13] actually derived it from the asymptotic metric conditions, and analyzed the group structure after the fact [14].

Sachs [13] was the first to describe the BMS group. (He referred to it as the "Generalized Bondi-Metzner group", or GBM group; later authors renamed it the Bondi-Metzner-Sachs group to honor his contribution.) His slightly later paper [14] was more specifically about the BMS group itself, and proved some important properties, including:

  1. The supertranslations form an abelian normal subgroup $N$ of the generalized Bondi-Metzner group; the factor group is isomorphic to the orthochronous homogeneous Lorentz group.
  2. The translations form a normal four-dimensional subgroup of the proper Bondi Metzner group.
  3. If $N'$ is a four dimensional normal subgroup of the proper GBM group then $N'$ is contained in the supertranslation group $N$.
  4. The only normal four dimensional subgroup of the GBM group is the translation group.

Decomposition of BMS

Essentially by our definition, we have already decomposed $\text{BMS}$ into the supertranslations and the Lorentz transformations. However, it can also be useful to further decompose these parts. Once we have chosen a frame $𝐭, 𝐱, 𝐲, 𝐳$ to work with, we can conventionally decompose any element of the BMS group as shown in the diagram below.

BMS Supertranslations S α Lorentz L Λ Proper supertranslations Spacetime translations Rotations R Boosts v⃗ Time translations δt Space translations δx⃗ Spherical coordinates (θ, φ) Spin angle ψ 1-D boosts η Null rotations ζ

The supertranslations decompose naturally into "proper" supertranslations and ordinary spacetime translations — the latter of which further decompose into time translations $δt$ and space translations $δx⃗$. In general, the supertranslations are functions on the sphere, which we decompose into spherical harmonics. If the supertranslation is a pure spacetime translation, then the only nonzero spherical harmonic modes are the $ℓ=0$ mode for time translations and the $ℓ=1$ modes for space translations. Equivalently, we can think of such translations in terms of their physical parameters $δt$ and $δx⃗$.

The Lorentz transformations decompose naturally into rotations $R$ and boosts $v⃗$. It is already convenient to represent the rotation as a quaternion, which can be constructed in numerous ways, including the generator of the rotation, the axis and angle of the rotation, the Euler angles, or the spherical coordinates and spin angle (which is essentially a different version of Euler angles). Finally, a general boost may be decomposed into a 1-D boost of rapidity $η$ in the $(θ, φ)$ direction, and a null rotation of parameter $ζ$ about that direction. This is not often an intuitive decomposition, but it is important in discussions of boost weight.

  • 1We can equivalently write an arbitrary element of $\text{BMS}$ in the opposite order: as some $Λ'$ followed by some $α'$. We have $α' Λ' = Λ' \left(Λ'⁻¹ α' Λ'\right)$, and the fact that $𝒮$ is a normal subgroup of $\text{BMS}$ guarantees that $Λ'⁻¹ α' Λ'$ is still an element of $𝒮$, so either order is permissible.