3.4 The Bondi mass

Numerical calculations of asymptotic quantities such as the Bondi mass must pick off non-leading terms in an asymptotic expansion about infinity. This is similar to the experimental task of determining the mass of an object by measuring its far field. For example, in an asymptotically-inertial Bondi frame at + ℐ (in which the metric takes an asymptotically-Minkowski form in null spherical coordinates)), the mass aspect β„³ (u, πœƒ,Ο•) is picked off from the asymptotic expansion of Bondi’s metric quantity V (see Equation (27View Equation)) of the form V = r − 2β„³ + π’ͺ (1βˆ•r). In gauges, which incorporate some of the properties of an asymptotically-inertial frame, such as the null quasi-spherical gauge [36Jump To The Next Citation Point] in which the angular metric is conformal to the unit-sphere metric, this can be a straightforward computational problem. However, the job can be more difficult if the gauge does not correspond to a standard Bondi frame at ℐ+. One must then deal with an arbitrary coordinatization of ℐ+, which is determined by the details of the interior geometry. As a result, V has a more complicated asymptotic behavior, given in the axisymmetric case by
r2(L sinπœƒ ) V − r = ---------,πœƒ+ re2(H −K) × [ sinπœƒ ] ( −2(H −K)) 2(H,πœƒ-sin-πœƒ),πœƒ- 2 2 1 − e + sinπœƒ + K, πœƒπœƒ + 3K,πœƒ cotπœƒ + 4(H,πœƒ) − 4H,πœƒK,πœƒ − 2(K,πœƒ) 2H −1 − 2e β„³ + π’ͺ (r ), (29 )
where L, H, and K are gauge dependent functions of (u,πœƒ), which would vanish in an inertial Bondi frame [288Jump To The Next Citation Point, 176Jump To The Next Citation Point]. The calculation of the Bondi mass requires regularization of this expression by numerical techniques so that the coefficient β„³ 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 π’ͺ (1βˆ•r) radiation fields. But it has been done!

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 [281Jump To The Next Citation Point]. In the Pittsburgh code, it was accomplished by re-expressing the Bondi mass in terms of renormalized metric variables, which regularize all calculations at ℐ+ and made them second-order accurate in grid size [143Jump To The Next Citation Point]. 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 + ℐ 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 Equation (29View Equation). This gauge was adopted by Bartnik and Norton at Canberra in their development of a 3D characteristic evolution code [36Jump To The Next Citation Point] (see Section 4 for further discussion). It allowed accurate computation of the Bondi mass as a limit as r → ∞ of the Hawking mass [33Jump To The Next Citation Point].

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 [143Jump To The Next Citation Point]. 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 1βˆ•c, beginning with the Newtonian mass and mechanical energy as the leading terms. This comparison with perturbation theory has been carried out to π’ͺ (1βˆ•c7), 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 [288Jump To The Next Citation Point] in which the time slices of ℐ+ 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 ℐ+. The determination of the conformal factor, which relates the 2-metric hAB of a slice of ℐ+ 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 πœƒ, 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

h dxA dxB = m dxA m¯ dxB. (30 ) AB A B
The desired conformal transformation is obtained by first relating hAB conformally to the flat metric of the complex plane. Denoting the complex coordinate of the plane by ζ, this relationship can be expressed as dζ = efm dxA A. The conformal factor f can then be determined from the integrability condition
m [A∂B ]f = ∂[AmB ]. (31 )
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.

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