It was accomplished in Stewart’s code by re-expressing the formula for the Bondi mass in terms of the
well-behaved fields of the conformal formalism [281]. 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 made them second-order accurate in grid size [143
]. 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 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
vanishes in Equation (29
). This gauge was adopted by Bartnik and Norton
at Canberra in their development of a 3D characteristic evolution code [36
] (see Section 4 for further
discussion). It allowed accurate computation of the Bondi mass as a limit as
of the Hawking
mass [33
].
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 [143]. 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
, 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 [257]. This introduces boost freedom into the problem. Identifying the translation subgroup is
tantamount to knowing the conformal transformation to an inertial Bondi frame [288] 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 a non-standard
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
, which is
straightforward to solve [143]. The integration constants determine the boost freedom along the axis of
symmetry.
The non-axisymmetric case is more complicated. Stewart [281] proposes an approach based upon the dyad decomposition
The desired conformal transformation is obtained by first relating
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
![]() This work is licensed under a Creative Commons License. E-mail us: |