4.2
-space and the good-cut equation
Equation (177), written in earlier literature as
is a well-known and well-studied partial differential equation, often referred to as the “good-cut
equation” [19, 20
]. For sufficiently regular
(which is assumed here) it has been proven [20
]
that the solutions are determined by points in a complex four-dimensional space,
, referred to as
-space, i.e., solutions are given as
Later in this section, by choosing an arbitrary complex analytic world line in
-space,
,
we describe how to construct the shear-free angle field,
. First, however, we discuss properties
and the origin of Equation (180).
Roughly or intuitively one can see how the four complex parameters enter the solution from the
following argument. We can write Equation (179) as the integral equation
with
where
is the kernel of the
operator (the solution to the homogeneous good-cut equation)
and
is the Green’s function for the
operator [23]. By iterating this equation, with
the kernel being the zeroth iterate, i.e.,
one easily sees how the four
enter the solution. Basically, the
come from the solution to the
homogeneous equation.
It should be noted again that the
is composed of the
harmonics,
Furthermore, the integral term does not contribute to these lowest harmonics. This means that solutions
can be written
with
containing spherical harmonics
and higher.
We note that using this form of the solution implies that we have set stringent coordinate conditions on
the
-space by requiring that the first four spherical harmonic coefficients be the four
-space
coordinates. Arbitrary coordinates would just mean that these four coefficients were arbitrary functions of
other coordinates. How these special coordinates change under the BMS group is discussed
later.
Remark: It is of considerable interest that on
-space there is a natural quadratic complex
metric – demonstrated in Appendix D – that is given by the surprising relationship [34, 20]
Update
Remarkably this turns out to be a
Ricci-flat metric with a nonvanishing anti-self-dual Weyl tensor, i.e., it is intrinsically a complex
vacuum metric. For vanishing Bondi shear,
-space reduces to complex Minkowski space (i.e.,
).
4.2.1 Solutions to the shear-free equation
Returning to the issue of the solutions to the shear-free condition, i.e., Equation (174),
, we see
that they are easily constructed from the solutions to the good-cut equation,
. By
choosing an arbitrary complex world line in the
-space, i.e.,
we write the GCF as
or, from Equation (185),
This leads immediately, via Equations (178) and (179), to the parametric description of the shear-free
stereographic angle field
, as well as the Bondi shear
:
We denote the inverse to Equation (191) by
The asymptotic twist of the asymptotically shear-free NGC is exactly as in the flat-space case,
As in the flat-space case, the derived quantity
plays a large role in applications. (In the case of the Robinson–Trautman metrics [55
, 28
]
is the basic
variable for the construction of the metric.)
Using the gauge freedom,
, in a slightly different way than in the Minkowski-space
case, we impose the simple condition
A Brief Summary: The description and analysis of the asymptotically shear-free NGCs in
asymptotically-flat spacetimes is remarkably similar to that of the flat-space regular shear-free NGCs. We
have seen that all regular shear-free NGCs in Minkowski space and asymptotically-flat spaces are generated
by solutions to the good-cut equation, with each solution determined by the choice of an arbitrary complex
analytic world line in complex Minkowski space or
-space. The basic governing variables are the
complex GCF,
, and the stereographic angle field on
,
, restricted to real
. In every sense, the flat-space case can be considered as a special case of the asymptotically-flat
case.
In Sections 5 and 6, we will show that in every asymptotically flat spacetime a special
complex-world line (along with its associated NGC and GCF) can be singled out using physical
considerations. This special GCF is referred to as the (gravitational) UCF, and is denoted by