The issue of gauge invariance, the understanding of which is not obvious or easy, must now be addressed. The claim is that the work described here is in fact gauge (or BMS) invariant.
First of all we have, , or its real part, . On , for each choice of spacetime interior and solution of the Einstein–Maxwell equations, we have its UCF, either in its complex version, , or its real version, Eq. (6.21). The geometric picture of the UCF is a one-parameter family of slicings (complex or real) of or . This is a geometric construct that has a different appearance or description in different Bondi coordinate systems. It is this difference that we must investigate. We concentrate on the complex version.
Under the action of the supertranslation, Eq. (2.63), we have:
The action of the homogeneous Lorentz transformations, Eq. (2.64),
Before discussing the relevant effects of the Lorentz transformations on our considerations we first digress and describe an important technical issue concerning representation of the homogeneous Lorentz group.
The representation theory of the Lorentz group, developed and described by Gelfand, Graev and Vilenkin  used homogeneous functions of two complex variables (homogeneous of degrees, and ) as the representation space. Here we summarize these ideas via an equivalent method [33, 26] using spin-weighted functions on the sphere as the representation spaces. In the notation of Gelfand, Graev and Vilenkin, representations are labeled by the two numbers or by , with . The ‘’ is the same ‘’ as in the spin weighted functions and ‘’ is the conformal weight  (sometimes called ‘boost weight’). The different representations are written as . The special case of irreducible unitary representations, which occur when are not integers, plays no role for us and will not be discussed. We consider only the case when are integers so that the take integer or half integer values. If and are both positive integers or both negative integers, we have, respectively, the positive or negative integer representations. The representation space, for each , are the functions on the sphere, , that can be expanded in spin-weighted spherical harmonics, , so that
Under the action of the Lorentz group (7.4) – (7.5), they transform asIt is this fact which we heavily utilize. More specifically we have two related situations: (1) when the are both positive integers (or ) there will be finite dimensional invariant subspaces, , which are spanned by the basis vectors , with given in the range, . All the finite dimensional representations can be obtained in this manner. And (2) when the are both negative integers (i.e., we have a negative integer representation) there will be an infinite dimensional invariant subspace, , described elsewhere . One, however, can obtain a finite dimensional representation for each negative integer case by the following construction: One forms the factor space, . This space is isomorphic to one of the finite dimensional spaces associated with the positive integers. The explicit form of the isomorphism, which is not needed here, is given in Held et al. [33, 27].
Of major interest for us is not so much the invariant subspaces but instead their interactions with their compliments (the full vector space modulo the invariant subspace). Under the action of the Lorentz transformations applied to a general vector in the representation space, the components of the invariant subspaces remain in the invariant subspace but in addition components of the complement move into the invariant subspace. On the other hand, the components of the invariant subspaces do not move into the complement subspace: the transformed components of the compliment involve only the original compliment components. The transformation thus has a nontrivial Jordan form.
Rather than give the full description of these invariant subspaces we confine ourselves to the few cases of relevance to us.
We write the GCF as
After the Lorentz transformation, the geometric slicings have not changed but their description in terms of has changed to that of . This leads to
Using the transformation properties of the invariant subspace and its compliment we see that the coordinate transformation must have the form:
Treating the and as functions of , we have
It then follows that transforms asare the coefficients of harmonic expansions of the and respectively. We have shown that a Lorentz transformation induces a specific coordinate transformation (and associated vector transformation) on these coefficients.
This is the justification for calling the harmonics of the mass aspect a Lorentzian four-vector. (Technically, the Bondi four-momentum is a co-factor but we have allowed ourselves a slight notational irregularity.)
What finite tensor transformation this corresponds to is a slightly more complicated question than that of the previous examples of Lorentzian vectors. In fact, it corresponds to the Lorentz transformations applied to (complex) self-dual antisymmetric two-index tensors . We clarify this with an example from Maxwell theory: from a given E and B, the Maxwell tensor, , and then its self-dual version can be constructed:
A Lorentz transformation applied to the tensor, , is equivalent  to the same transformation applied to
This allows us to assign Lorentzian invariant physical meaning to our identifications of the complex mass dipole moment and angular momentum vector, .
Consider pairs of conformally weighted functions (), , with weights respectively, . They are considered to be in dual spaces. Our claim is that the integrals of the form
We first point out that under the fractional linear transformation, , given by Eq. (7.5), the area element on the sphere
There are several immediate simple applications of Eq. (7.19). By choosing an arbitrary function, say and , we immediately have a Lorentzian scalar,
If this is made more specific by choosing , we have the remarkable result (proved in Appendix D) that this scalar yields the -space metric norm of the “velocity” , via
A simple variant of this arises by taking the derivative of Eq. (7.24) with respect to , and multiplying by an arbitrary vector, leading to
Many other versions can easily be found.
Living Rev. Relativity 15, (2012), 1
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