### 5.2 The Robinson–Trautman metrics

The algebraically-special type II Robinson–Trautman (RT) metrics are expressed in conventional RT
coordinates, (), now real, by [71]
with
The unknowns are the Weyl component (closely related to the Bondi mass), which is a function only
of (real) and the variable, , both of which are variables in the RT equation (see below).
There remains the freedom
which often is chosen so that = constant. However, we make a different choice. In the spherical
harmonic expansion of ,
the is chosen by normalizing the four-vector, , to one, i.e., . The final field equation, the
RT equation, is
These spacetimes, via the Goldberg–Sachs theorem, possess a degenerate shear-free PND field, , that is
surface-forming, (i.e., twist free). Using the tetrad constructed from we have that the Weyl components
are of the form
Furthermore, the metric contains a ‘real timelike world line, ,’ with normalized velocity vector
. All of these properties allow us to identify the RT metrics as being analogous to the real
Liénard–Wiechert solutions of the Maxwell equations.
Assuming for the moment that we have integrated the RT equation and know , then, by
the integral

the UCF for the RT metrics has been found. The freedom of adding to the integral is just
the supertranslation freedom in the choice of a Bondi coordinate system. From
a variety of information can be obtained: the Bondi shear, , is given parametrically by
as well as the angle field by
In turn, from this information the RT metric (in the neighborhood of ) can, in principle, be
re-expressed in terms of the Bondi coordinate system, though in practice one must revert to
approximations. These approximate calculations lead, via the Bondi mass aspect evolution equation, to
both Bondi mass loss and to equations of motion for the world line, . An alternate
approximation for the mass loss and equations of motion is to insert the spherical harmonic expansion of
into the RT equation and look at the lowest harmonic terms. We omit further details
aside from mentioning that we come back to these calculations in a more general context in
Section 6.