3.5 Three-D Characteristic Evolution3 Characteristic Evolution Codes3.3 The Bondi Problem

3.4 The Bondi Mass

Numerical calculations of asymptotic quantities such as the Bondi mass must overcome severe technical difficulties arising from the necessity to pick off nonleading terms in an asymptotic expansion about infinity. For example, in an asymptotically inertial frame (called a Bondi frame at tex2html_wrap_inline1511), the mass aspect tex2html_wrap_inline1807 must be picked off from the asymptotic expansion of Bondi's metric quantity V, (see Eq. (16Popup Equation)) of the form tex2html_wrap_inline1811 . This is similar to the experimental task of determining the mass of an object by measuring its far field. What makes the job much more difficult is that, in most computational setups, (the global null problem or the Cauchy-characteristic matching problem), the gauge choice does not correspond to a Bondi frame at tex2html_wrap_inline1511 . One must deal with an arbitrary foliation of tex2html_wrap_inline1511 into retarded time slices which are determined by the details of the interior geometry. As a result, V has the more complicated asymptotic behavior (given in the axisymmetric case)


where L, H and K are gauge dependent functions of tex2html_wrap_inline1825 which would vanish in a Bondi frame [17Jump To The Next Citation Point In The Article, 49]. The calculation of the Bondi mass requires regularization of this expression by numerical techniques so that the coefficient tex2html_wrap_inline1827 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 [53Jump To The Next Citation Point In The Article]. In the Pittsburgh code, it is accomplished by re-expressing the Bondi mass in terms of renormalized metric variables which regularize all calculations at tex2html_wrap_inline1511 and make them second order accurate [57Jump To The Next Citation Point In The Article]. 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.

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 most 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 space-times, 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 space-times; (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.

The above codes have extended the role of the Bondi mass from that of a geometrical construct in the theory of gravitational radiation 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 [57Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline1835, 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.

Another subtle numerical problem associated with 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 [58]. This introduces boost freedom into the problem. Identifying the translation subgroup is tantamount to knowing the conformal transformation to a conformal Bondi frame [17] in which the time slices of tex2html_wrap_inline1515 have unit sphere geometry. An important feature providing flexibility to both Stewart's code and the Pittsburgh code is that the coordinates are adopted to simplify the description of the interior sources. This results in an arbitrary foliation of tex2html_wrap_inline1515 . The determination of the conformal factor which relates the 2-metric tex2html_wrap_inline1841 of a slice of tex2html_wrap_inline1515 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 tex2html_wrap_inline1583, and is straightforward to solve [57]. The integration constants determine the boost freedom along the axis of symmetry.

The non-axisymmetric case is more complicated. Stewart [53] proposes a promising approach based upon the dyad decomposition


The desired conformal transformation is obtained by first relating tex2html_wrap_inline1841 conformally to the flat metric of the complex plane. Denoting the complex coordinate of the plane by tex2html_wrap_inline1849, this relationship can be expressed as tex2html_wrap_inline1851 . 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.

3.5 Three-D Characteristic Evolution3 Characteristic Evolution Codes3.3 The Bondi Problem

image Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de