### 5.2 The Robinson–Trautman metrics

The algebraically-special type II Robinson–Trautman (RT) metrics are expressed in conventional RT
coordinates, (), now real, by [55]
Update, 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 satisfy 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 PNV 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. 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.