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, Equation (291
). 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, Equation (67), we have:
The action of the homogeneous Lorentz transformations, Equation (68),
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 [16] used homogeneous functions of two complex variables (homogeneous of degrees,
and
) as the representation space. Here we summarize these ideas via an equivalent method [21
, 15]
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 [44] (sometimes called ‘boost weight’). The different representations are written as
. The special case of irreducible unitary representations, which occur when
are not
integers, play 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, Equation (335), 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 non-trival Jordan form.
Rather than give the full description of these invariant subspaces we confine ourselves to the few cases of relevance to us.
I. Though our interest is primarily in the negative integer representations, we first address the positive
integer case of the and
, [
], representation. The harmonics,
form the invariant subspace. The cut function,
, for each fix values of
, lies in this
space.
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
II. A second important example concerns the mass aspect, (where we have introduced the for
simplicity of treatment of numerical factors)
This is the justification for calling the harmonics of the mass aspect a Lorentzian
four-vector. Technically, the Bondi four-momentum is a co-vector but we have allowed ourselves a slight
notational irregularity.
III. The Weyl tensor component, , has
and
,
. The associated
finite dimensional factor space is isomorphic to the finite part of the
representation. We have that
The question of what finite tensor transformation this corresponds to is slightly more complicated 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 [29]. 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 [30] to the same transformation
applied to
These observations allow us to assign Lorentzian invariant physical meaning to our identifications of the
Bondi momentum, and the complex mass dipole moment and angular momentum vector,
.
IV. Our last example is a general discussion of how to construct Lorentzian invariants from the
representation spaces. Though we will confine our remarks to just the cases of , it is easy to extend
them to non-vanishing
by having the two functions have respectively spin-weight
and
.
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, , Equation (337
), the area
element on the sphere
This leads immediately to
the claimed result. There are several immediate simple applications of Equation (352). By choosing an arbitrary
function, say
, we immediately have a Lorentzian scalar,
If this is made more specific by chosing , we have the remarkable result (proved in
Appendix D) that this scalar yields the
-space metric via
A simple variant of this arises by taking the derivative of (357) with respect to
and multiplying by
an arbitrary vector,
leading to
Many other versions can easily be found.
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