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 [27] 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 [60] (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 as
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:
in other words it moves the higher harmonic coefficients down to the
![]() |
Treating the and
as functions of
, we have
![]() |
It then follows that transforms as
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 [42]. 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 [43] 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.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
![]() This work is licensed under a Creative Commons License. E-mail us: |