where
*L*,
*H*
and
*K*
are gauge dependent functions of
which would vanish in a Bondi frame [139,
92]. The calculation of the Bondi mass requires regularization of
this expression by numerical techniques so that the coefficient
can be picked off. The task is now similar to the experimental
determination of the mass of an object by using non-inertial
instruments in a far zone which contains
*O*
(1/
*r*) radiation fields. But it can be done!

It was accomplished in Stewart's code by reexpressing the formula for the Bondi mass in terms of the well-behaved fields of the conformal formalism [135]. In the Pittsburgh code, it was accomplished by re-expressing the Bondi mass in terms of renormalized metric variables which regularize all calculations at and make them second order accurate in grid size [69]. The calculation of the Bondi news function (which provides the waveforms of both polarization modes) is an easier numerical task than the Bondi mass. It has also been implemented in both of these codes, thus allowing the important check of the Bondi mass loss formula.

An alternative approach to computing the Bondi mass is to
adopt a gauge which corresponds more closely to an inertial or
Bondi frame at
and simplifies the asymptotic limit. Such a choice is the null
quasi-spherical gauge in which the angular part of the metric is
proportional to the unit sphere metric and as a result the gauge
term
*K*
vanishes in Eq. (18). This gauge was adopted by the Canberra group in developing a
3-D characteristic evolution code [21] (see Sec.
3.5
for further discussion). It allowed accurate computation of the
Bondi mass as a limit as
of the Hawking mass [17].

Mainstream astrophysics is couched in Newtonian concepts, some of which have no well defined extension to general relativity. In order to provide a sound basis for relativistic astrophysics, it is crucial to develop general relativistic concepts which have well defined and useful Newtonian limits. Mass and radiation flux are fundamental in this regard. The results of characteristic codes show that the energy of a radiating system can be evaluated rigorously and accurately according to the rules for asymptotically flat spacetimes, while avoiding the deficiencies that plagued the ``pre-numerical'' era of relativity: (i) the use of coordinate dependent concepts such as gravitational energy-momentum pseudotensors; (ii) a rather loose notion of asymptotic flatness, particularly for radiative spacetimes; (iii) the appearance of divergent integrals; and (iv) the use of approximation formalisms, such as weak field or slow motion expansions, whose errors have not been rigorously estimated.

Characteristic codes have extended the role of the Bondi mass from that of a geometrical construct in the theory of isolated systems to that of a highly accurate computational tool. The Bondi mass loss formula provides an important global check on the preservation of the Bianchi identities. The mass loss rates themselves have important astrophysical significance. The numerical results demonstrate that computational approaches, rigorously based upon the geometrical definition of mass in general relativity, can be used to calculate radiation losses in highly nonlinear processes where perturbation calculations would not be meaningful.

Numerical calculation of the Bondi mass has been used to
explore both the Newtonian and the strong field limits of general
relativity [69]. For a quasi-Newtonian system of radiating dust, the numerical
calculation joins smoothly on to a post-Newtonian expansion of
the energy in powers of 1/
*c*, beginning with the Newtonian mass and mechanical energy as the
leading terms. This comparison with perturbation theory has been
carried out to
, at which stage the computed Bondi mass peels away from the
post-Newtonian expansion. It remains strictly positive, in
contrast to the truncated post-Newtonian behavior which leads to
negative values.

A subtle feature of the Bondi mass stems from its role as one component of the total energy-momentum 4-vector, whose calculation requires identification of the translation subgroup of the Bondi-Metzner-Sachs group [121]. This introduces boost freedom into the problem. Identifying the translation subgroup is tantamount to knowing the conformal transformation to a conformal Bondi frame [139] in which the time slices of have unit sphere geometry. Both Stewart's code and the Pittsburgh code adapt the coordinates to simplify the description of the interior sources. This results in an arbitrary foliation of . The determination of the conformal factor which relates the 2-metric of a slice of to the unit sphere metric is an elliptic problem equivalent to solving the second order partial differential equation for the conformal transformation of Gaussian curvature. In the axisymmetric case, the PDE reduces to an ODE with respect to the angle , and is straightforward to solve [69]. The integration constants determine the boost freedom along the axis of symmetry.

The non-axisymmetric case is more complicated. Stewart [135] proposes an approach based upon the dyad decomposition

The desired conformal transformation is obtained by first
relating
conformally to the flat metric of the complex plane. Denoting
the complex coordinate of the plane by
, this relationship can be expressed as
. The conformal factor
*f*
can then be determined from the integrability condition

This is equivalent to the classic Beltrami equation for finding isothermal coordinates. It would appear to be a more effective scheme than tackling the second order PDE directly, but numerical implementation has not yet been carried out.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
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