The formalism for expressing Einstein’s equations as an evolution system based on characteristic, or null-cone, coordinates is based on work originally due to Bondi (1960) and Bondi et al. (1962) for axisymmetry, and extended to the general case by Sachs (1962). The formalism is covered in the review by Winicour (2005), to which the reader is referred for an in-depth discussion of its development and the associated literature.
Most work on characteristic evolution uses, or is an adpatation of, a finite difference code that was originally developed at the University of Pittsburgh and has become known as the PITT null code. The early work that eventually led to the PITT code was for the case of axisymmetry (Isaacson et al. 1983; Bishop et al. 1990; Gómez et al. 1994), and a general vacuum code was developed in the mid-1990s (Bishop et al. 1996b, 1997b; Lehner 1998, 1999, 2001). Subsequently, the code was extended to the non-vacuum case (Bishop et al. 1999b, 2005), and code adaptations in terms of variables, coordinates and order of accuracy have been investigated (Gómez 2001; Gómez and Frittelli 2003; Reisswig et al. 2007, 2013a). Spectral, rather than finite difference, implementations have also been developed, for both the axially symmetric case (de Oliveira and Rodrigues 2009) and in general (Handmer and Szilágyi 2015). One potential difficulty, although in practice it has not been important in characteristic extraction, is the development of caustics during the evolution, and algorithms to handle the problem have been proposed (Stewart and Friedrich 1982; Corkill and Stewart 1983). There are also approaches that use outgoing null cones but for which the coordinates are not Bondi–Sachs (Bartnik 1997; Bartnik and Norton 2000).
Shortly after the publication of the Bondi and Bondi–Sachs metrics and formalism, the idea of conformal compactification was introduced. This led to the well-known asymptotic description of spacetime, and the definitions of asymptotic flatness, past, future and spacelike infinity (\(I^+,I^-,I^0\)), and of past and future null infinity (\({\mathcal {J}}^-,\mathcal {J}^+\)) (Penrose 1963); see also Penrose (1964, 1965b) and Tamburino and Winicour (1966); and the reviews by Adamo et al. (2012) and Frauendiener (2004). The key result is that gravitational radiation can be defined unambiguously in an asymptotically flat spacetime only at null infinity. The waves may be expressed in terms of the Bondi news \({{\mathcal {N}}}\) (see Eq. (271) below), the Newman–Penrose quantity \(\psi _4\), or the wave strain \((h_+,h_\times )\).
After a characteristic code has been run using a compactified radial coordinate as in Eq. (259), the metric is known at \(\mathcal {J}^+\), and so it would seem to be straightforward to calculate the emitted gravitational radiation. Unfortunately, this is not in general the case because of gauge, or coordinate freedom, issues. The formulas do take a very simple form when expressed in terms of coordinates that satisfy the Bondi gauge condition in which the asymptotic flatness property is obviously satisfied, and for which conditions set at \(\mathcal {J}^+\) are propagated inwards along radial null geodesics. However, in a numerical simulation that is not the case: coordinate conditions are fixed on an extraction worldtube (in the case of characteristic extraction), or perhaps on a worldline (Siebel et al. 2003) or ingoing null hypersurface, and then propagated outwards to \(\mathcal {J}^+\). The result is that the geometry at and near \(\mathcal {J}^+\) may appear very different to one that is foliated by spherical 2-surfaces of constant curvature. Of course, the Bondi gauge and the general gauge are related by a coordinate transformation, and formulas for \({{\mathcal {N}}}\) and \(\psi _4\) are obtained by constructing the transformation.
An explicit formula in the general gauge for the news was obtained in Bishop et al. (1997b) (“Appendix 2”); and a calculation of \(\psi _4\) was reported in Babiuc et al. (2009), but the formula produced was so lengthy that it was not published. These formulas have been used in the production of most waveforms calculated by characteristic codes. An alternative approach, in which the coordinate transformation is explicit, rather than partially implicit, was suggested (Bishop and Deshingkar 2003) but has not been further used or developed. Recently, a formula for the wave strain \((h_+,h_\times )\), which is the quantity used in the construction of templates for gravitational-wave data analysis, was derived (Bishop and Reisswig 2014). An important special case is that of the linearized approximation, in which deviations from the Bondi gauge are small. The resulting formulas for \({{\mathcal {N}}}\), \(\psi _4\) and \((h_+,h_\times )\), are much simpler and so much easier to interpret than in the general case. Further these formulas are widely used because the linearized approximation often applies to the results of a waveform computation in a realistic scenario.
We set the context for this section by summarizing the Einstein equations in characteristic coordinates, and outlining the characteristic evolution procedure. The focus of this section is formulas for gravitational waves, and we next present the formulas in the simplest case, when the coordinates satisfy the Bondi gauge conditions. Much of the remainder of the section will be devoted to formulas for gravitational waves in the general gauge, and will include a discussion of conformal compactification. This section makes extensive use of spin-weighted spherical harmonics and the eth formalism, which topics are discussed in “Appendix 2”.
The Einstein equations in Bondi–Sachs coordinates
We start with coordinates based upon a family of outgoing null hypersurfaces. Let u label these hypersurfaces, \(\phi ^{^{_A}}\)
\((A=2,3)\) be angular coordinates labelling the null rays, and r be a surface area coordinate. In the resulting \(x^\alpha =(u,r,\phi ^{^{_A}})\) coordinates, the metric takes the Bondi–Sachs form
$$\begin{aligned} ds^2= & {} -\left( e^{2\beta }(1 + W_c r) -r^2h_{_{AB}}U^{^{_A}}U^{^{_{_{B}}}}\right) \, du^2 \nonumber \\&- 2e^{2\beta } \, du \, dr -2r^2 h_{_{AB}}U^{^{_{_{B}}}} \, du \, d\phi ^{^{_A}} + r^2h_{_{AB}} \, d\phi ^{^{_A}} \, d\phi ^{^{_{_{B}}}}, \end{aligned}$$
(250)
where \(h^{^{_{AB}}}h_{BC}=\delta ^{^{_A}}_C\) and \(\det (h_{_{AB}})=\det (q_{_{AB}})\), with \(q_{_{AB}}\) a metric representing a unit 2-sphere; \(W_c\) is a normalized variable used in the code, related to the usual Bondi–Sachs variable V by \(V=r+W_c r^2\). It should be noted here that different references use various notations for what is here denoted as \(W_c\), and in particular (Bishop et al. 1997b) uses W with \(W :=r^2W_c\). As discussed in Sect. 1, we represent \(q_{_{AB}}\) by means of a complex dyad \(q_{_{A}}\), then \(h_{_{AB}}\) can be represented by its dyad component \(J:=h_{_{AB}}q^{^{_A}}q^{^{_{_{B}}}}/2\). We also introduce the fields \(K :=\sqrt{1+J \bar{J}}\) and \(U :=U^{^{_A}}q_{_{A}}\). The spin-weight s of a quantity is defined and discussed in section “Spin-weighted fields” in “Appendix 2”; for the quantities used in the Bondi–Sachs metric
$$\begin{aligned} s(W_c)= & {} s(\beta )=0,\qquad s(J)=2,\qquad s(\bar{J})=-2, \nonumber \\ s(K)= & {} 0, \qquad s(U)=1,\qquad s(\bar{U})=-1. \end{aligned}$$
(251)
We would like to emphasize two matters: (a) The metric Eq. (250) applies quite generally, and does not rely on the spacetime having any particular properties. (b) There are many different metrics of the form Eq. (250) that describe a given spacetime, and changing from one to another is known as a gauge transformation (about which more will be said later).
The form of the Einstein equations for the general Bondi–Sachs metric has been known for some time, but it was only in 1997 (Bishop et al. 1997b) that they were used for a numerical evolution. [see also Gómez and Frittelli 2003 for an alternative semi-first-order form that avoids second angular derivatives (\(\eth ^2, \bar{\eth }^2, \bar{\eth }\eth \))]. The equations are rather lengthy, and only the hypersurface and evolution equations are given in that paper, in an “Appendix”.Footnote 12 See also section “Computer algebra” in “Appendix 3”. Here, in order to make the discussion of the Einstein equations precise but without being overwhelmed by detail, we give the equations in vacuum in the linearized case, that is when any second-order term in the quantities \(J, \beta , U, W_c\) can be ignored. The Einstein equations are categorized into three classes, hypersurface, evolution, and constraint. The hypersurface equations are
$$\begin{aligned} R_{11}:&\quad \frac{4}{r}\partial _r \beta =0, \end{aligned}$$
(252)
$$\begin{aligned} q^{^{_A}} R_{1A}:&\quad \frac{1}{2r} \left( 4 \eth \beta - 2 r \eth \partial _r\beta + r \bar{\eth } \partial _r J +r^3 \partial ^2_r U +4 r^2 \partial _r U \right) = 0, \end{aligned}$$
(253)
$$\begin{aligned} h^{^{_{AB}}} R_{_{AB}}:&\quad (4-2\eth \bar{\eth }) \beta +\frac{1}{2}(\bar{\eth }^2 J + \eth ^2\bar{J}) +\frac{1}{2r^2}\partial _r(r^4\eth \bar{U}+r^4\bar{\eth }U)\nonumber \\&\quad -2 r^2 \partial _r W_{c} -4 r W_c =0. \end{aligned}$$
(254)
The evolution equation is
$$\begin{aligned} q^{^{_A}} q^{^{_{_{B}}}} R_{_{AB}}: \;\; -2\eth ^2\beta + \partial _r(r^2 \eth U) - 2r \partial _r J - r^2 \partial ^2_r J +2 r \partial _r\partial _u (rJ)= 0. \end{aligned}$$
(255)
The constraint equations are Reisswig et al. (2007)
$$\begin{aligned} R_{00}:&\quad \frac{1}{2r^2} \bigg ( r^3 \partial ^2_r W_{c}+4r^2\partial _r W_{c}+2rW_c+r\eth \bar{\eth } W_c +2 \eth \bar{\eth } \beta \nonumber \\&\quad -4 r \partial _u\beta - r^2 \partial _u(\eth \bar{U} + \bar{\eth }U)+2 r^2 \partial _u W_{c} \bigg ) = 0, \end{aligned}$$
(256)
$$\begin{aligned} R_{01}:&\quad \frac{1}{4r^2} \bigg (2r^3 \partial ^2_r W_{c}+8r^2\partial _r W_{c}+4rW_c+4 \eth \bar{\eth }\beta -\partial _r(r^2\eth \bar{U}+r^2\bar{\eth }U)\bigg )=0, \end{aligned}$$
(257)
$$\begin{aligned} q^{^{_A}} R_{0A}:&\quad \frac{1}{4} \bigg ( 2r \eth \partial _r W_{c}+2 \eth W_c+ 2 r(4 \partial _r U + r \partial _r^2 U)+4 U +(\eth \bar{\eth }U-\eth ^2\bar{U})\nonumber \\&\quad +2 \bar{\eth }\partial _u J-2 r^2 \partial _r\partial _u U-4 \eth \partial _u\beta \bigg )=0. \end{aligned}$$
(258)
An evolution problem is normally formulated in the region of spacetime between a timelike or null worldtube \(\varGamma \) and future null infinity (\({\mathcal {J}}^+\)), with (free) initial data J given on \(u=0\), and with boundary data for \(\beta ,U,\partial _r U,W_c,J\) satisfying the constraints given on \(\varGamma \) (Fig. 9). (In characteristic extraction, the data satisfies the Einstein equations inside \(\varGamma \), and so the issue of ensuring that the boundary data must satisfy the characteristic constraint equations does not arise). The hypersurface equations are solved to find \(\beta ,U,W_c\), and then the evolution equation gives \(\partial _u J\) and thence J on the “next” null cone. See Kreiss and Winicour (2011) and Babiuc et al. (2014) for a discussion of the well-posedness of the problem.
We extend the computational grid to \(\mathcal {J}^+\) by compactifying the radial coordinate r by means of a transformation \(r \rightarrow x=f(r)\) where \(\lim _{r \rightarrow \infty } f(r)\) is finite. In characteristic coordinates, the Einstein equations remain regular at \(\mathcal {J}^+\) under such a transformation. In practice, in numerical work the compactification is usually
$$\begin{aligned} r \rightarrow x=\frac{r}{r+r_\varGamma }. \end{aligned}$$
(259)
However, for the purpose of extracting gravitational waves, it is more convenient to express quantities as power series about \(\mathcal {J}^+\), and so we compactify using
$$\begin{aligned} r\rightarrow \rho =1/r. \end{aligned}$$
(260)
(Common practice has been to use the notation \(\ell \) for 1 / r, but since we will have expressions involving the compactified radial coordinate and spherical harmonics such a notation would be confusing). Starting from the Bondi–Sachs metric Eq. (250), we make the coordinate transformation (260) to obtain
$$\begin{aligned} ds^2= & {} \rho ^{-2}\left( -\left( e^{2\beta }(\rho ^2+\rho W_c) -h_{_{AB}}U^{^{_A}}U^{^{_{_{B}}}}\right) \, du^2 +2e^{2\beta }\, du \, d\rho -2 h_{_{AB}}U^{^{_{_{B}}}} \, du \, d\phi ^{^{_A}}\right. \nonumber \\&\left. + h_{_{AB}} \, d\phi ^{^{_A}} \, d\phi ^{^{_{_{B}}}}\right) . \end{aligned}$$
(261)
In contravariant form,
$$\begin{aligned} {g}^{11}=e^{-2\beta }\rho ^3(\rho + W_c), \quad {g}^{1A}=\rho ^2e^{-2\beta } U^{^{_A}}, {g}^{10}=\rho ^2e^{-2\beta }, \quad {g}^{^{_{AB}}}=\rho ^2h^{^{_{AB}}}, \quad {g}^{0A}={g}^{00}=0. \end{aligned}$$
(262)
Later, we will need to use the asymptotic Einstein equations, that is the Einstein equations keeping only the leading order terms when the limit \(r\rightarrow \infty \), or equivalently \(\rho \rightarrow 0\), is taken. We write the metric variables as \(J=J_{(0)}+J_{(1)}\rho \), and similarly for \(\beta , U\) and \(W_c\). Each Einstein equation is expressed as a series in \(\rho \) and only leading order terms are considered. There is considerable redundancy, and instead of 10 independent relations we find (see section “Computer algebra” in “Appendix 3”)
$$\begin{aligned}&\displaystyle R_{11}=0 \rightarrow \beta _{(1)}=0, \end{aligned}$$
(263)
$$\begin{aligned}&\displaystyle q^{^{_A}} R_{1{\scriptscriptstyle A}}=0 \rightarrow -2\eth \beta _{(0)} +e^{-2\beta _{(0)}} K_{(0)} U_{(1)}+e^{-2\beta _{(0)}} J_{(0)} \bar{U}_{(1)}=0, \end{aligned}$$
(264)
$$\begin{aligned}&\displaystyle h^{^{_{AB}}}R_{_{AB}}=0\rightarrow 2W_{c(0)} - \eth \bar{U}_{(0)} - \bar{\eth }U_{(0)}=0, \end{aligned}$$
(265)
$$\begin{aligned}&\displaystyle q^{^{_A}} q^{^{_{_{B}}}} R_{_{AB}}=0 \rightarrow 2K_{(0)}\eth U_{(0)} + 2\partial _u J_{(0)} +\bar{U}_{(0)}\eth J_{(0)}+U_{(0)}\bar{\eth } J_{(0)}\nonumber \\&\displaystyle +J_{(0)}\eth \bar{U}_{(0)}-J_{(0)}\bar{\eth } U_{(0)}=0. \end{aligned}$$
(266)
The above are for the fully nonlinear case, with the linearized approximation obtained by setting \(K_{(0)}=e^{-2\beta _{(0)}}=1\), and ignoring products of J and U terms.
The Bondi gauge
In the Bondi gauge, the form of the Bondi–Sachs metric is manifestly asymptotically flat since it tends to Minkowskian form as \(r\rightarrow \infty \). In order to see what conditions are thus imposed, the first step is to write the Minkowskian metric in compactified Bondi–Sachs coordinates. Starting from the Minkowski metric in spherical coordinates \((t,r,\phi ^{^{_A}})\), we make the coordinate transformation \((t,r)\rightarrow (u,\rho )\) where
$$\begin{aligned} u=t-r,\;\;\rho =\frac{1}{r} \end{aligned}$$
(267)
to obtain
$$\begin{aligned} ds^2=\rho ^{-2}\left( -\rho ^2 du^2+2 du\, d\rho +q_{_{AB}}d\phi ^{^{_A}}\,d\phi ^{^{_{_{B}}}} \right) . \end{aligned}$$
(268)
We use the notation \(\;\tilde{ }\;\) to denote quantities in the Bondi gauge. The metric of Eqs. (261) and (262) still applies, with the additional properties as \({\tilde{\rho }}\rightarrow 0\),
$$\begin{aligned} \tilde{J}&=0,\quad \tilde{K}=1,\quad \tilde{\beta }=0,\quad \tilde{U}=0,\quad \tilde{W}_c=0, \nonumber \\ \partial _{\tilde{\rho }}\tilde{K}&=0,\;\partial _{\tilde{\rho }}\tilde{\beta }=0,\; \partial _{\tilde{\rho }}\tilde{U}=0,\;\partial _{\tilde{\rho }}\tilde{W}_{c}=0. \end{aligned}$$
(269)
The undifferentiated conditions can be regarded as defining the Bondi gauge, being motivated by the geometric condition that the metric Eq. (261) should take the form Eq. (268) in the limit as \(\rho \rightarrow 0\). The conditions on \(\partial _{\tilde{\rho }}\tilde{\beta }\), \(\partial _{\tilde{\rho }}\tilde{U}\) and \(\partial _{\tilde{\rho }}\tilde{K}\) follow from the asymptotic Eqs. (263), (264), and (423) respectively; and the condition on \(\partial _{\tilde{\rho }}\tilde{W}_{c}\) is obtained from the asymptotic Einstein equation \(h^{^{_{AB}}}R_{_{AB}}=0\) to second order in \(\rho \) and applying the Bondi gauge conditions already obtained. The null tetrad in the Bondi gauge will be denoted by \(\tilde{\ell }^\alpha ,\tilde{n}_{_{[NP]}}^\alpha , \tilde{m}^\alpha \), with components to leading order in \(\tilde{\rho }\) [obtained by applying the coordinate transformation (267) to Eq. (96)]
$$\begin{aligned} \tilde{\ell }^\alpha =\left( 0,- \frac{\tilde{\rho }^2}{\sqrt{2}},0,0\right) ,\qquad \tilde{n}_{_{[NP]}}^\alpha =\left( \sqrt{2},\frac{\tilde{\rho }^2}{\sqrt{2}},0,0\right) ,\qquad \tilde{m}^\alpha =\left( 0,0,\frac{\tilde{\rho } q^{^{_A}}}{\sqrt{2}}\right) . \end{aligned}$$
(270)
The gravitational news was defined by Bondi et al. (1962) and is
$$\begin{aligned} {{\mathcal {N}}}=\frac{1}{2} \partial _{\tilde{u}} \partial _{\tilde{\rho }}\tilde{J}, \end{aligned}$$
(271)
evaluated in the limit \(\tilde{\rho }\rightarrow 0\), and is related to the strain in the TT gauge by
$$\begin{aligned} {{\mathcal {N}}}=\frac{1}{2} \partial _{\tilde{u}}\lim _{\tilde{r}\rightarrow \infty } \tilde{r}\left( h_+ +i h_\times \right) = \frac{1}{2}\partial _{\tilde{u}}H, \end{aligned}$$
(272)
where the rescaled strain H is
$$\begin{aligned} H:=\lim _{\tilde{r}\rightarrow \infty } \tilde{r}\left( h_+ +i h_\times \right) =\partial _{\tilde{\rho }} \tilde{J}, \end{aligned}$$
(273)
which result is a straightforward consequence of the relation \(\tilde{J}=h_++ih_\times \) discussed in section “Spin-weighted representation of deviations from spherical symmetry” in “Appendix 2”. When using the Newman–Penrose formalism to describe gravitational waves, it is convenient to introduce
$$\begin{aligned} \psi ^0_4= \lim _{\tilde{r}\rightarrow \infty } \tilde{r}\psi _4 \;\;\left( =\lim _{\tilde{\rho }\rightarrow 0} \frac{\psi _4}{\tilde{\rho }} \right) , \end{aligned}$$
(274)
since it will be important, when considering conformal compactification (Sect. 6.5), to have a quantity that is defined at \(\tilde{\rho }=0\). In the Bondi gauge, as shown in section “Computer algebra” in “Appendix 3”, \(\psi ^0_4\) simplifies to
$$\begin{aligned} \psi ^0_4= \partial ^2_{\tilde{u}} \partial _{\tilde{\rho }}\bar{\tilde{J}}, \end{aligned}$$
(275)
evaluated in the limit \(\tilde{\rho }\rightarrow 0\). Thus \(\psi ^0_4\), \({{\mathcal {N}}}\) and H are related by
$$\begin{aligned} \psi ^0_4=2 \partial _{\tilde{u}} {\bar{\mathcal {N}}}= \partial ^2_{\tilde{u}} \bar{H}. \end{aligned}$$
(276)
General gauge
We construct quantities in the general gauge by means of a coordinate transformation to the Bondi gauge, although this transformation is largely implicit because it does not appear in many of the final formulas. The transformation is written as a series expansion in \(\rho \) with coefficients arbitrary functions of the other coordinates. Thus it is a general transformation, and the requirements that \(g^{\alpha \beta }\) must be of Bondi–Sachs form, and that \(\tilde{g}^{\alpha \beta }\) must be in the Bondi gauge, impose conditions on the transformation coefficients. The transformation is (Fig. 10)
$$\begin{aligned} u \rightarrow {\tilde{u}}=u+u_0+\rho A^u, \qquad \rho \rightarrow {\tilde{\rho }}=\rho \omega +\rho ^2 A^\rho , \qquad \phi ^{^{_A}} \rightarrow {\tilde{\phi }}^{^{_A}}=\phi ^{^{_A}}+\phi ^{^{_A}}_0+\rho A^{^{_A}}, \end{aligned}$$
(277)
where the transformation coefficients \(u_0,A^u,\omega ,A^\rho ,\phi ^{^{_A}}_0,A^{^{_A}}\) are all functions of u and \(\phi ^{^{_A}}\) only. Conditions on the coefficients are found by applying the tensor transformation law
$$\begin{aligned} {{{\tilde{g}}}}^{\alpha \beta }=\frac{\partial {\tilde{x}}^\alpha }{\partial x^\mu } \frac{\partial {\tilde{x}}^\beta }{\partial x^\nu } { g}^{\mu \nu }, \qquad \hbox {and}\qquad { g}_{\alpha \beta }=\frac{\partial {\tilde{x}}^\mu }{\partial x^\alpha } \frac{\partial {\tilde{x}}^\nu }{\partial x^\beta }{{\tilde{g}}}_{\mu \nu }, \end{aligned}$$
(278)
for specific cases of \(\alpha ,\beta \), using the form of the metric in Eqs. (261) and (262) and also applying the conditions in Eq. (269) to \({{\tilde{g}}}^{\alpha \beta }\) and \({{\tilde{g}}}_{\mu \nu }\) (Bishop et al. 1997b; Bishop and Deshingkar 2003; Bishop and Reisswig 2014). The procedure is shown in some detail for one case, with the other cases being handled in a similar way. The actual calculations are performed by computer algebra as discussed in section “Computer algebra” in “Appendix 3”.
From Eqs. (262) and (269), \({{\tilde{g}}}^{01}=\tilde{\rho }^2+ \mathcal {O}(\tilde{\rho }^4)\). Then using the contravariant transformation in Eq. (278) with \(\alpha =0,\beta =1\), we have
$$\begin{aligned} \tilde{\rho }^2+\mathcal {O}(\tilde{\rho }^4)=\rho ^2\omega ^2+\mathcal {O}({\rho }^4)= \frac{\partial {\tilde{u}}}{\partial x^\mu } \frac{\partial {\tilde{\rho }}}{\partial x^\nu } { g}^{\mu \nu }. \end{aligned}$$
(279)
Evaluating the right hand side to \(\mathcal {O}({\rho }^2)\), the resulting equation simplifies to give
$$\begin{aligned} (\partial _u+U^{^{_{_{B}}}}\partial _{_{B}})u_{0}=\omega e^{2\beta }-1. \end{aligned}$$
(280)
The remaining conditions follow in a similar way
$$\begin{aligned} 0+\mathcal {O} (\tilde{\rho }^4)= & {} {{{\tilde{g}}}}^{A1}, \qquad \hbox {so that to }\mathcal {O} ({\rho }^2),\qquad (\partial _u+U^{^{_{_{B}}}}\partial _{_{B}})\phi ^{^{_A}}_0=-U^{^{_A}}, \end{aligned}$$
(281)
$$\begin{aligned} 0+\mathcal {O} (\tilde{\rho }^4)= & {} {{{\tilde{g}}}}^{11}, \qquad \hbox {so that to }\mathcal {O} ({\rho }^3),\qquad (\partial _u+U^{^{_{_{B}}}}\partial _{_{B}})\omega =-\omega W_c/2, \end{aligned}$$
(282)
$$\begin{aligned} 0= & {} {{\tilde{g}}}^{00},\qquad \hbox {so that to }\mathcal {O}({\rho }^2),\quad \; 2\omega A^u=\frac{J\bar{\eth }^2u_0+\bar{J}\eth ^2 u_0}{2}-K\eth u_0 \bar{\eth }u_0.\nonumber \\ \end{aligned}$$
(283)
In the next equations, \(X_0=q_{_{A}} \phi _0^{^{_A}}, A=q_{_{A}} A^{^{_A}}\); the introduction of these quantities is a convenience to reduce the number of terms in the formulas, since \(\phi _0^{^{_A}}, A^{^{_A}}\) do not transform as 2-vectors. As a result, the quantity \(\zeta =q+ip\) also appears, and the formulas are specific to stereographic coordinates. We find
$$\begin{aligned} 0=\tilde{q}_{_{A}} {{\tilde{g}}}^{0A}, \end{aligned}$$
(284)
so that to \(\mathcal {O}({\rho }^2)\)
$$\begin{aligned} 0=&\, 2A\omega +2A_u X_0 U \bar{\zeta } e^{-2\beta } +K\eth u_0 (2 +\bar{\eth }X_0+2X_0\bar{\zeta })\nonumber \\&+K\bar{\eth }u_0\eth X_0 \bar{\eth }u_0 (2+\bar{\eth }X_0+2 X_0\bar{\zeta })-\bar{J}\eth u_0 \eth X_0, \end{aligned}$$
(285)
$$\begin{aligned} \det (q_{_{AB}})=&\det (g_{_{AB}})\rho ^4, \end{aligned}$$
(286)
so that at \(\rho =0\)
$$\begin{aligned} \omega =\frac{1+q^2+p^2}{1+\tilde{q}^2+\tilde{p}^2} \sqrt{1+\partial _q q_{0}+\partial _p p_{0}+\partial _q q_{0}\partial _p p_{0} -\partial _p q_{0}\partial _q p_{0}}, \end{aligned}$$
(287)
and
$$\begin{aligned} J=\frac{q^{^{_A}} q^{^{_{_{B}}}} g_{_{AB}}}{2} \rho ^2, \end{aligned}$$
(288)
so that at \(\rho =0\)
$$\begin{aligned} \qquad J= \frac{(1+q^2+p^2)^2}{2(1+\tilde{q}^2+\tilde{p}^2)^2\omega ^2} \eth X_0 (2+\eth \bar{X}_0+2\bar{X} \zeta ). \end{aligned}$$
(289)
Explicit expressions for the null tetrad vectors \(n_{_{[NP]}}^\alpha \) and \(m^\alpha \) (but not \(\ell ^\alpha \)) will be needed. \(n_{_{[NP]}}^\alpha \) is found by applying the coordinate transformation Eq. (278) to \(\tilde{n}_{_{[NP]}\alpha }\) (Eq. (270)) and then raising the index, giving to leading order in \(\rho \)
$$\begin{aligned} {n}_{_{[NP]}}^\alpha =\left( \frac{e^{-2\beta }\sqrt{2}}{\omega }, \rho \frac{e^{-2\beta }(2\partial _u\omega +\bar{U}\eth \omega +U\bar{\eth }\omega +2W_c\omega )}{\sqrt{2}\omega ^2}, \frac{U^{^{_A}}e^{-2\beta }\sqrt{2}}{\omega }\right) . \end{aligned}$$
(290)
The calculation of an expression for \(m^\alpha \) is indirect. Let \(F^{^{_A}}\) be a dyad of the angular part of the general gauge metric, so it must satisfy Eq. (384), then Bishop et al. (1997b); Babiuc et al. (2009),
$$\begin{aligned} F^{^{_A}}=\left( \frac{q^{^{_A}}\sqrt{K+1}}{2}-\frac{\bar{q}^{^{_A}} J}{2\sqrt{(K+1)}}\right) , \end{aligned}$$
(291)
with \(F^{^{_A}}\) undetermined up to an arbitrary phase factor \(e^{-i\delta (u,x^{^{_A}})}\). We then define \(m_{_{[G]}}^\alpha \)
$$\begin{aligned} m_{_{[G]}}^\alpha = e^{-i\delta }\rho (0,0,F^{^{_A}}). \end{aligned}$$
(292)
The suffix \({}_{_{[G]}}\) is used to distinguish the above form from that defined in Eq. (270) since \(m_{_{[G]}}^\alpha \ne m^\alpha \). However, it will be shown later (see Sects. 6.5.1, 6.5.2 and section “Computer algebra” in “Appendix 3”) that the value of gravitational-wave descriptors is unaffected by the use of \(m_{_{[G]}}^\alpha \) rather than \(m^\alpha \) in its evaluation; thus it is permissible, for our purposes, to approximate \(m^\alpha \) by \(m_{_{[G]}}^\alpha \). We now transform \(m_{_{[G]}}^\alpha \) in Eq. (292) to the Bondi gauge,
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha =\frac{\partial \tilde{x}^\alpha }{\partial x^\beta } m_{_{[G]}}^\beta =\left( \partial _{_{B}}u_0 e^{-i\delta }\frac{\tilde{\rho }}{\omega } F^{^{_{_{B}}}}, \partial _{_{B}}\omega e^{-i\delta }\frac{\tilde{\rho ^2}}{\omega ^2} F^{^{_{_{B}}}}, \partial _{_{B}} \phi ^{^{_A}}_0 e^{-i\delta }\frac{\tilde{\rho }}{\omega } F^{^{_{_{B}}}}\right) . \end{aligned}$$
(293)
The component \(\tilde{m}^1_{_{[G]}}\) is of the same order as \(\tilde{\ell }^1\), and so \(\tilde{m}_{_{[G]}}^\alpha \) and \(\tilde{m}^\alpha \) are not equivalent. It can be checked (see section “Computer algebra” in “Appendix 3”) (Bishop and Reisswig 2014) that the angular part of \(\tilde{m}_{_{[G]}}^\alpha \) is equivalent to \(\tilde{m}^\alpha \), since \(\tilde{m}_{_{[G]}}^\alpha \tilde{m}_\alpha =0\) and
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha \bar{\tilde{m}}_\alpha =\nu , \end{aligned}$$
(294)
where \(|\nu |=1\).
Since we actually require \(\nu =1\), Eq. (294) can be used to set the phase factor \(\delta \) explicitly. The result is
$$\begin{aligned} e^{i\delta }\!=\!\frac{F^{^{_{_{B}}}} \bar{\tilde{q}}_{_{A}}}{\omega \sqrt{2}} \partial _{_{B}} \phi ^{^{_A}}_0 \!=\!\frac{1+q^2+p^2}{4\omega (1\!+\!\tilde{q}^2+\tilde{p}^2)}\sqrt{\frac{2}{K+1}} \left( (K+1)(2+\eth \bar{X}_0+2\bar{X}_0\zeta )-J\bar{\eth }\bar{X}_0\right) .\nonumber \\ \end{aligned}$$
(295)
An alternative approach (Bishop et al. 1997b; Babiuc et al. 2009), to the phase factor \(\delta \) uses the condition that \(m_{_{[G]}}^\alpha \) is parallel propagated along \(\mathcal {J}^+\) in the direction \({n}_{_{[NP]}}^\alpha \), yielding the evolution equation
$$\begin{aligned} 2i(\partial _u +U^{_A}\partial _{_A})\delta = \nabla _{_A} U^{_A} +h_{_{AC}} \bar{F}^{_C} ( (\partial _u +U^{_B} \partial _{_B}) F^{_A} - F^{_B} \partial _{_B} U^{_A}) , \end{aligned}$$
(296)
where \(\nabla _{_A}\) is the covariant derivative with respect to the angular part of the metric \(h_{_{AB}}\).
The gravitational-wave strain
An expression for the contravariant metric \(\tilde{g}^{\alpha \beta }\) in the Bondi gauge is obtained from Eqs. (262) and (269), and each metric variable is expressed as a Taylor series about \(\tilde{\rho }=0\) (e.g., \(\tilde{J}=0+\tilde{\rho } \partial _{\tilde{\rho }}\tilde{J}+{{\mathcal {O}}}(\tilde{\rho }^2)\)). Applying the coordinate transformation (277) we find \(g^{\alpha \beta }\), then use \(\tilde{\rho }=\omega \rho +A^\rho \rho ^2\) to express each component as a series in \(\rho \); note that the coefficients are constructed from terms in the Bondi gauge, e.g., \(\partial _{\tilde{\rho }}\tilde{J}\). Then both sides of
$$\begin{aligned} J=\frac{q_{_{A}} q_{_{B}} g^{^{_{AB}}}}{2\rho ^2}, \end{aligned}$$
(297)
with \(g^{^{_{AB}}}\) given in Eq. (262), are expressed as series in \(\rho \), and the coefficients of \(\rho ^1\) are equated. This leads to an equation in which \(\partial _\rho J\) depends linearly on \(\partial _{\tilde{\rho }} \tilde{J}\) (\(=H=\lim _{\tilde{r}\rightarrow \infty } \tilde{r}\left( h_+ +i h_\times \right) \), the rescaled strain defined in Eq. (273) (Bishop and Reisswig 2014),
$$\begin{aligned} C_1 \partial _\rho J= C_2 \partial _{\tilde{\rho }} \tilde{J} +C_3 \partial _{\tilde{\rho }}\tilde{\bar{J}}+C_4, \end{aligned}$$
(298)
which may be inverted to give
$$\begin{aligned} H=\partial _{\tilde{\rho }} \tilde{J}=\frac{C_1\bar{C}_2\partial _\rho J -C_3 \bar{C}_1 \partial _{\rho }\bar{J} +C_3 \bar{C}_4 -\bar{C}_2 C_4}{\bar{C}_2 C_2-\bar{C}_3 C_3}, \end{aligned}$$
(299)
where the coefficients are
$$\begin{aligned} C_1&=\frac{4\omega ^2 (1+\tilde{q}^2+\tilde{p}^2)^2}{(1+q^2+p^2)^2},\qquad C_2=\omega (2+\eth \bar{X}_0+2\bar{X}_0\zeta )^2, \end{aligned}$$
(300)
$$\begin{aligned} C_3&=\omega (\eth X_0)^2,\qquad C_4=\eth A (4+2\eth \bar{X}_0+4 \bar{X}_0\zeta )+\eth X_0 (2\eth \bar{A}+4\bar{A}\zeta ) +4\eth \omega \eth u_0. \end{aligned}$$
(301)
These results are obtained using computer algebra as discussed in section “Computer algebra” in “Appendix 3”. The above formula for the wave strain involves intermediate variables, and the procedure for calculating them is to solve equations for the variable indicated in the following order: Eq. (281) for \(x_0^{^{_A}}\) and thus \(X_0\), Eq. (287) for \(\omega \), Eq. (280) for \(u_0\), Eq. (283) for \(A^u\), and Eq. (285) for A.
Conformal compactification
Here we give only a brief introduction to this topic, as these matters are discussed more fully in many standard texts and reviews, e.g., Wald (1984) and Frauendiener (2004). We have made a coordinate compactification, resulting in the metric and null tetrad being singular at \(\rho =0\), which is therefore not included in the manifold. Thus, quantities are not evaluated at \(\rho =0\), but in the limit as \(\rho \rightarrow 0\). Introducing a conformal transformation has the advantage that this technical issue is avoided and \(\mathcal {J}^+\) at \(\rho =0\) is included in the manifold; but also that the resulting formulas for \({{\mathcal {N}}}\) and \(\psi ^0_4\) are simpler. (Of course, it should be possible to use the asymptotic Einstein equations to simplify expressions derived in physical space, but due to the complexity of the formulas this approach has not been adopted).
We use the notation \(\hat{ }\) for quantities in conformal space. In the general gauge, the conformal metric \(\hat{g}_{\alpha \beta }\) is related to the metric Eq. (261) by \(g_{\alpha \beta }=\rho ^{-2}\hat{g}_{\alpha \beta }\) so that
$$\begin{aligned} d\hat{s}^2= & {} -\left( e^{2\beta }(\rho ^2+\rho W_c) -h_{_{AB}}U^{^{_A}}U^{^{_{_{B}}}}\right) \, du^2 +2e^{2\beta } \, du \, d\rho -2 h_{_{AB}}U^{^{_{_{B}}}} \, du \, d\phi ^{^{_A}} \nonumber \\&+ h_{_{AB}} \, d\phi ^{^{_A}} \, d\phi ^{^{_{_{B}}}}. \end{aligned}$$
(302)
In a similar way, the Bondi gauge conformal metric \(\hat{\tilde{g}}_{\alpha \beta }\) is related to the Bondi gauge metric \(\tilde{g}_{\alpha \beta }\) by \(\tilde{g}_{\alpha \beta }=\tilde{\rho }^{-2}\hat{\tilde{g}}_{\alpha \beta }\). Thus \(\hat{g}_{\alpha \beta }\) and \(\hat{\tilde{g}}_{\alpha \beta }\) are regular at \(\rho =0\) (or equivalently at \(\tilde{\rho }=0\)) and so in the conformal picture \(\rho =0\) is included in the manifold. The conformal metrics \(\hat{g}_{\alpha \beta }\) and \(\hat{\tilde{g}}_{\alpha \beta }\) are related by
$$\begin{aligned} \hat{g}_{\alpha \beta }=\rho ^2 g_{\alpha \beta }=\rho ^2 \frac{\partial {\tilde{x}}^\gamma }{\partial x^\alpha } \frac{\partial {\tilde{x}}^\delta }{\partial x^\beta }{{\tilde{g}}}_{\gamma \delta } =\frac{\rho ^2}{\tilde{\rho }^2}\frac{\partial {\tilde{x}}^\gamma }{\partial x^\alpha } \frac{\partial {\tilde{x}}^\delta }{\partial x^\beta }{\hat{{\tilde{g}}}}_{\gamma \delta } =\frac{1}{\omega ^2}\frac{\partial {\tilde{x}}^\gamma }{\partial x^\alpha } \frac{\partial {\tilde{x}}^\delta }{\partial x^\beta }{\hat{{\tilde{g}}}}_{\gamma \delta } +{{\mathcal {O}}}(\rho ), \end{aligned}$$
(303)
which at \(\rho =0\) is the usual tensor transformation law with an additional factor \(\omega ^{-2}\). A quantity that obeys this property is said to be conformally invariant with weight n where n is the power of \(\omega \) in the additional factor; thus the metric tensor is conformally invariant with weight \(-2\). In practice, it is not necessary to establish a relation of the form Eq. (303) to prove conformal invariance. The key step is to be able to show that a tensor quantity \(T^{a\cdots }_{b\cdots }\) satisfies \(\hat{T}^{a\cdots }_{b\cdots } =\rho ^{-n}T^{a\cdots }_{b\cdots }\), then conformal invariance with weight n easily follows. In Eq. (303) the error term \({{\mathcal {O}}}(\rho )\) is shown explicitly, although it turns out to be irrelevant since the relation is evaluated at \(\rho =0\). This is generally the case, so from now on the error terms will not be taken into account; the one exception will be in the News calculation Sect. 6.5.1 which in places involves off-\(\mathcal {J}^+\) derivatives (since \(\partial _\rho {{\mathcal {O}}}(\rho ) ={{\mathcal {O}}}(1)\)).
It is important to note that not all tensor quantities are conformally invariant, and in particular this applies to the metric connection and thus covariant derivatives
$$\begin{aligned} \hat{\varGamma }^\gamma _{\alpha \beta }=\varGamma ^\gamma _{\alpha \beta } +\frac{\delta ^\gamma _\alpha \partial _\beta \rho +\delta ^\gamma _\beta \partial _\alpha \rho -\hat{g}_{\alpha \beta }\hat{g}^{\gamma \delta }\partial _\delta \rho }{\rho }, \end{aligned}$$
(304)
and to the Ricci scalar in n-dimensions
$$\begin{aligned} \hat{R}=\rho ^{-2}\left[ R-2(n-1)g^{ab}\nabla _a\nabla _b\ln \rho -(n-1)(n-2)g^{ab}(\nabla _a\ln \rho )(\nabla _b\ln \rho )\right] . \end{aligned}$$
(305)
The Weyl tensor, however, is conformally invariant,
$$\begin{aligned} \hat{C}^\alpha _{\beta \gamma \delta }=C^\alpha _{\beta \gamma \delta }, \qquad \hbox {and}\qquad \hat{C}_{\alpha \beta \gamma \delta }=\rho ^2 C_{\alpha \beta \gamma \delta }, \end{aligned}$$
(306)
so that the forms \(C^\alpha _{\beta \gamma \delta }\) and \(C_{\alpha \beta \gamma \delta }\) are conformally invariant with weights 0 and \(-2\) respectively. The construction of the conformal null tetrad vectors is not unique. It is necessary that orthonormality conditions analogous to Eq. (97) be satisfied, and it is desirable that the component of leading order in \(\rho \) should be finite but nonzero at \(\rho =0\). These conditions are achieved by defining
$$\begin{aligned} \hat{n}_{_{[NP]}}^\alpha =n_{_{[NP]}}^\alpha , \quad \hat{m}_{_{[G]}}^\alpha =\frac{m_{_{[G]}}^\alpha }{\rho }. \end{aligned}$$
(307)
Thus \(\hat{n}_{_{[NP]}}^\alpha \) and \(\hat{\tilde{n}}_{_{[NP]}}^a\) are related by the usual tensor transformation law, and
$$\begin{aligned} \hat{\tilde{m}}_{_{[G]}}^\alpha =\frac{\tilde{m}_{_{[G]}}^\alpha }{\tilde{\rho }}= \frac{1}{\omega \rho }\frac{\partial \tilde{x}^\alpha }{\partial x^\beta } m_{_{[G]}}^\beta =\frac{1}{\omega }\frac{\partial \tilde{x}^\alpha }{\partial x^\beta } \hat{m}_{_{[G]}}^\beta . \end{aligned}$$
(308)
With these definitions, \(n_{_{[NP]}}^\alpha ,m_{_{[G]}}^\alpha \) are conformally invariant with weights 0 and 1 respectively.
Considering the conformally compactified metric of the spherical 2-surface described by the angular coordinates (\(\tilde{\phi }^{^{_A}}\) or \(\phi ^{^{_A}}\)) at \(\mathcal {J}^+\), we have
$$\begin{aligned} ds^2=\tilde{\rho }^{-2} d\hat{\tilde{s}}^2= \tilde{\rho }^{-2}q_{_{AB}}d\tilde{\phi }^{^{_A}}d\tilde{\phi }^{^{_{_{B}}}} =\rho ^{-2}d\hat{s}^2=\rho ^{-2}h_{_{AB}}d\phi ^{^{_A}} d\phi ^{^{_{_{B}}}}, \end{aligned}$$
(309)
so that
$$\begin{aligned} q_{_{AB}}d\tilde{\phi }^{^{_A}}d\tilde{\phi }^{^{_{_{B}}}} =\omega ^2h_{_{AB}}d\phi ^{^{_A}}d\phi ^{^{_{_{B}}}}, \end{aligned}$$
(310)
since \(\omega =\tilde{\rho }/\rho \). The curvature of \(\mathcal {J}^+\) is evaluated in two different ways, and the results are equated. The metric on the LHS is that of a unit sphere, and therefore has Ricci scalar \({\mathcal {R}}(\tilde{x}^{^{_A}})=2\); and the metric on the RHS is evaluated using Eq. (305)with \(n=2\). Thus,
$$\begin{aligned} 2=\frac{1}{\omega ^2}\left( {\mathcal {R}}(\phi ^{^{_A}})-2h^{^{_{AB}}}\nabla _{_{A}}\nabla _{_{B}} \log (\omega )\right) , \end{aligned}$$
(311)
leading to
$$\begin{aligned} 2\omega ^2+2h^{^{_{AB}}}\nabla _{_{A}}\nabla _{_{B}}\log (\omega )= & {} 2K-\bar{\eth }\eth K+\frac{1}{2} \left( \eth ^2\bar{J}+\bar{\eth }^2 J\right) \nonumber \\&+\frac{1}{4K}\left( (\bar{\eth }\bar{J})(\eth J)-(\bar{\eth }{J})(\eth \bar{J})\right) , \end{aligned}$$
(312)
where the relationship between \({\mathcal {R}}(\phi ^{^{_A}})\) and J, K is derived in Gómez et al. (1997), and where \(h^{^{_{AB}}}\nabla _{_{A}}\nabla _{_{B}}\log (\omega )\) is given in terms of the \(\eth \) operator in Eq. (B1) of Bishop et al. (1997b). Eq. (312) is a nonlinear elliptic equation, and in practice is not actually solved. However, it will be used later, when considering the linearized approximation, since in that case it has a simple analytic solution.
The news \({{\mathcal {N}}}\)
A difficulty with evaluating the gravitational radiation by means of Eq. (271) is that it is valid only in a specific coordinate system, so a more useful approach is to use the definition (Penrose 1963; Bishop et al. 1997b; Babiuc et al. 2009)
$$\begin{aligned} {{\mathcal {N}}}= \lim _{\tilde{\rho }\rightarrow 0}\frac{\hat{\tilde{m}}^\alpha \hat{\tilde{m}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho }}{\tilde{\rho }}. \end{aligned}$$
(313)
At first sight Eqs. (271) and (313) do not appear to be equivalent, but the relationship follows by expanding out the covariant derivatives in Eq. (313)
$$\begin{aligned} {{\mathcal {N}}}= \lim _{\tilde{\rho }\rightarrow 0}\frac{\hat{\tilde{m}}^\alpha \hat{\tilde{m}}^\beta (\partial _\alpha \partial _\beta \tilde{\rho }-\hat{\tilde{\varGamma }}^\gamma _{\alpha \beta } \partial _\gamma \tilde{\rho })}{\tilde{\rho }} =-\lim _{\tilde{\rho }\rightarrow 0}\frac{q^{^{_A}} q^{^{_{_{B}}}} \hat{\tilde{\varGamma }}^1_{_{AB}} }{2\tilde{\rho }}, \end{aligned}$$
(314)
then expressing the metric coefficients \(\tilde{J}\) etc. as power series in \(\tilde{\rho }\) as introduced just before Eq. (263). Using the Bondi gauge conditions Eq. (269), it quickly follows that \(-q^{^{_A}} q^{^{_{_{B}}}} \hat{\tilde{\varGamma }}^1_{(0)AB}/2 = \partial _{\tilde{u}} \tilde{J}/2\), which is zero, and the result follows since \(-q^{^{_A}} q^{^{_{_{B}}}} \hat{\tilde{\varGamma }}^1_{(1)AB}/2 = \partial _{\tilde{u}}\partial _{\tilde{\rho }}\tilde{J}/2\). Computer algebra has been used to check that replacing \(\hat{\tilde{m}}^\alpha \) in Eq. (313) by \(\hat{\tilde{m}}_{_{[G]}}^\alpha \) (with \(\tilde{m}^\alpha _{_{[G]}}\) defined in Eq. (293)) has no effect.Footnote 13
Because covariant derivatives are not conformally invariant, transforming Eq. (313) into the general gauge is a little tricky. We need to transform to physical space, where tensor quantities with no free indices are invariant across coordinate systems, and then to conformal space in the general gauge. From Eq. (304), and using \(\tilde{\rho }= \rho \omega \) and \(\tilde{\nabla }_\gamma \tilde{\rho }=\delta ^1_\gamma \),
$$\begin{aligned} \hat{\nabla }_\alpha \hat{\nabla }_\beta \tilde{\rho }&= \nabla _\alpha \nabla _\beta \tilde{\rho } +\frac{\hat{g}_{\alpha \beta }\hat{g}^{\gamma 1}-\delta ^\gamma _\alpha \delta ^1_\beta -\delta ^\gamma _\beta \delta ^1_\alpha }{\rho } \nabla _\gamma (\rho \omega ), \end{aligned}$$
(315)
$$\begin{aligned} \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho }&= \tilde{\nabla }_\alpha \tilde{\nabla }_\beta \tilde{\rho } +\frac{\hat{\tilde{g}}_{\alpha b}\hat{\tilde{g}}^{11}-2\delta ^1_\alpha \delta ^1_\beta }{\tilde{\rho }}. \end{aligned}$$
(316)
Now consider \(\tilde{m}_{_{[G]}}^\alpha \tilde{m}_{_{[G]}}^\beta \times \) Eq. (316) \(-\;m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \times \) Eq. (315). Using the conditions that (a) scalar quantities are invariant in physical space so that \(\tilde{m}_{_{[G]}}^\alpha \tilde{m}_{_{[G]}}^\beta \tilde{\nabla }_\alpha \tilde{\nabla }_\beta \tilde{\rho } - m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \nabla _\alpha \nabla _\beta \tilde{\rho }=0\), (b) \(\hat{\tilde{g}}^{11}\) is zero to \({{\mathcal {O}}} (\tilde{\rho }^2)\), (c) \( m_{_{[G]}}^\alpha \delta ^1_\alpha =0\), and (d) from Eq. (293)
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha \delta ^1_\alpha =e^{-i\delta }\rho ^2 F^A\partial _A\omega = \rho m_{_{[G]}}^\alpha \partial _\alpha \omega , \end{aligned}$$
(317)
it follows that
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha \tilde{m}_{_{[G]}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho } \!=\!m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \left( \hat{\nabla }_\alpha \hat{\nabla }_\beta (\rho \omega ) -\hat{g}_{\alpha \beta }\left( \frac{\hat{g}^{11}\omega }{\rho } +\hat{g}^{1\gamma }\partial _\gamma \omega \right) \!-\!\frac{2\rho \partial _\alpha \omega \partial _\beta \omega }{\omega }\right) . \end{aligned}$$
(318)
Since \(m_{_{[G]}}^\alpha \) and \(\hat{\nabla }_\alpha \rho \) are orthogonal, we may write \(\hat{\nabla }_\alpha \hat{\nabla }_\beta (\rho \omega ) = \omega \hat{\nabla }_\alpha \hat{\nabla }_\beta \rho +\rho \hat{\nabla }_\alpha \hat{\nabla }_\beta \omega \), so that
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha \tilde{m}_{_{[G]}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho } =m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \left( \omega \hat{\nabla }_\alpha \hat{\nabla }_\beta \rho +\rho \hat{\nabla }_\alpha \hat{\nabla }_\beta \omega -\hat{g}_{\alpha \beta }\left( \frac{\hat{g}^{11}\omega }{\rho } +\hat{g}^{1\gamma }\partial _\gamma \omega \right) -\frac{2\rho \partial _\alpha \omega \partial _\beta \omega }{\omega }\right) . \end{aligned}$$
(319)
This expression is simplified by (a) expanding out the covariant derivatives, (b) expressing the metric as a power series in \(\rho \) and using \(m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \hat{g}_{(0)\alpha \beta }=0\), and (c) using Eq. (282),
$$\begin{aligned} \tilde{m}_{_{[G]}}^\alpha \tilde{m}_{_{[G]}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho }\!=\! m_{_{[G]}}^\alpha m_{_{[G]}}^\beta \left( - \omega \hat{\varGamma }^1_{a\beta } +\rho \partial _\alpha \partial _\beta \omega -\rho \partial _\gamma \omega \hat{\varGamma }^\gamma _{\alpha \beta } -\frac{\rho \omega e^{-2\beta }W_c \partial _\rho \hat{g}_{\alpha \beta }}{2} -\frac{2\rho \partial _\alpha \omega \partial _\beta \omega }{\omega }\right) . \end{aligned}$$
(320)
Finally, Eq. (292) is used to replace \(m_{_{[G]}}^\alpha \) in terms of \(F^{^{_A}}\), and the whole expression is divided by \(\tilde{\rho }^3\), yielding (Bishop et al. 1997b; Babiuc et al. 2009)
$$\begin{aligned} {{\mathcal {N}}}&= \lim _{\tilde{\rho }\rightarrow 0} \frac{\hat{\tilde{m}}^\alpha \hat{\tilde{m}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho }}{\tilde{\rho }} = \lim _{\tilde{\rho }\rightarrow 0} \frac{\hat{\tilde{m}}_{_{[G]}}^\alpha \hat{\tilde{m}}_{_{[G]}}^\beta \hat{\tilde{\nabla }}_\alpha \hat{\tilde{\nabla }}_\beta \tilde{\rho }}{\tilde{\rho }}\nonumber \\&= \frac{e^{-2i\delta }}{\omega ^2}\bigg [ -\lim _{\rho \rightarrow 0}\frac{F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}}}{\rho }\nonumber \\&\quad +F^{^{_A}}F^{^{_{_{B}}}}\left( \frac{\partial _{_A}\partial _{_B}\omega }{\omega } -\frac{\hat{\varGamma }^{\gamma }_{_{AB}}\partial _\gamma \omega }{\omega } -\frac{\partial _\rho \hat{g}_{_{AB}}e^{-2\beta }W_c}{2} -\frac{2\partial _{_A}\omega \partial _{_B}\omega }{\omega ^2} \right) \bigg ]. \end{aligned}$$
(321)
The limit is evaluated by expressing each metric coefficient as a power series in \(\rho \), e.g., \(J=J_{(0)}+\rho J_{(1)}\), and then writing \(F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}}= F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}(0)}+ \rho F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}(1)}\). Direct evaluation combined with use of the asymptotic Einstein Eq. (266) shows that \(F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}(0)}=0\) (see section “Computer algebra” in “Appendix 3”), so that the limit evaluates to \(F^{^{_A}}F^{^{_{_{B}}}} \hat{\varGamma }^1_{_{AB}(1)}\). Further evaluation of Eq. (321) into computational \(\eth \) form is handled by computer algebra, as discussed in section “Computer algebra” in “Appendix 3”.
The attentive reader may have noticed that the derivation above used \(\tilde{\rho }=\rho \omega \) rather than \(\tilde{\rho }=\rho (\omega +\rho A^\rho )\), so that \(\partial _\rho \omega \) should not be taken as 0 but as \(A^\rho \). However, the corrections that would be introduced remain \({{\mathcal {O}}}(\rho )\) since (a) \(\hat{m}_{_{[G]}}^1=0\), (b) in Eq. (319) the term \(\hat{g}^{11}A^\rho \) contained in \(\hat{g}^{1\gamma }\partial _\gamma \omega \) is \({{\mathcal {O}}}(\rho ) A^\rho \), and (c) in Eq. (321) the term \(F^A F^B \hat{\varGamma }^1_{_{AB}}A^\rho \) contained in \(F^A F^B \hat{\varGamma }^\gamma _{_{AB}}\partial _\gamma \omega \) is also \({{\mathcal {O}}}(\rho ) A^\rho \).
The Newman–Penrose quantity \(\psi ^0_4\)
The Newman–Penrose quantity \(\psi _4\), and its re-scaled version \(\psi _4^0\), were introduced in Sect. 3.3, and defined there for the case of physical space. Because the Weyl tensor is conformally invariant, it is straightforward to transform the earlier definition into one in the conformal gauge. Thus, in the conformal Bondi gauge,
$$\begin{aligned} \psi ^0_4=\lim _{\tilde{\rho }=0} \frac{\hat{\tilde{C}}_{\alpha \beta \mu \nu } \hat{\tilde{n}}_{_{[NP]}}^\alpha \bar{\hat{\tilde{m}}}^\beta \hat{\tilde{n}}_{_{[NP]}}^\mu \bar{\hat{\tilde{m}}}^\nu }{\tilde{\rho }}, \end{aligned}$$
(322)
and again, as in the case of the news \({{\mathcal {N}}}\), the limiting process means that the metric variables need to be expressed as power series in \(\tilde{\rho }\). Calculating the Weyl tensor is discussed in section “Computer algebra” in “Appendix 3”, and the result is \(\psi ^0_4=\partial ^2_{\tilde{u}}\partial _{\tilde{\rho }}\bar{\tilde{J}}\) as given in Eq. (275). The “Appendix” also checks that replacing \(\hat{\tilde{m}}^\alpha \) in Eq. (322) by \(\hat{\tilde{m}}^\alpha _{_{[G]}}\) (with \(\tilde{m}^\alpha _{_{[G]}}\) defined in Eq. (293)) does not affect the result for \(\psi ^0_4\).
In this case, transforming Eq. (322) to the conformal general gauge is straightforward, since the tensor quantities are conformally invariant and the net weight is 0. The result is
$$\begin{aligned} \psi ^0_4=\frac{1}{\omega }\lim _{\rho =0} \frac{\hat{C}_{\alpha \beta \mu \nu }\hat{n}_{_{[NP]}}^\alpha \bar{\hat{m}}_{_{[G]}}^\beta \hat{n}_{_{[NP]}}^\mu \bar{\hat{m}}_{_{[G]}}^\nu }{\rho }, \end{aligned}$$
(323)
where \(\hat{m}_{_{[G]}}^\alpha =e^{-i\delta }(0,0,F^A)\), and is further evaluated, by directly calculating the Weyl tensor, in section “Computer algebra” in “Appendix 3” (Babiuc et al. 2009) (but note that this reference uses a different approach to the evaluation of \(\psi ^0_4\)).
Linearized case
In the linearized case the Bondi–Sachs metric variables \(\beta ,J,U,W_c\) and the coordinate transformation variables \(u_0,A^u,(\omega -1),A^\rho ,x^{^{_A}}_0,A^{^{_A}}\) are regarded as small. Algebraically, the approximation is implemented by introducing a parameter \(\epsilon =\) max\((|\beta |,|J|,|U|,|W_c|)\) in a neighbourhood of \(\mathcal {J}^+\). Then, the metric variables are re-written as \(\beta \rightarrow \epsilon \beta \) etc., and quantities such as \({{\mathcal {N}}},\psi ^0_4\) are expressed as Taylor series in \(\epsilon \) with terms \({{\mathcal {O}}}(\epsilon ^2)\) ignored, leading to considerable simplifications. It is common practice to assume that the error in the approximation is about \(\epsilon ^2\). While computational results do not contradict this assumption, a word of caution is needed: no work on establishing a formal error bound for this problem has been reported.
Equations (280)–(282) and Eq. (285), simplify to
$$\begin{aligned} \partial _u u_0=(\omega -1)+2\beta ,\qquad \partial _u \phi ^{^{_A}}_0=-U^{^{_A}},\qquad \partial _u \omega =-W_c/2, \;\; A=-\eth u_0. \end{aligned}$$
(324)
It will also be useful to note the linearized form of Eq. (289),
$$\begin{aligned} J=\eth X_0. \end{aligned}$$
(325)
In the linearized case, Eq. (312) takes the form (Bishop 2005)
$$\begin{aligned} 2+4 (\omega -1) +2\bar{\eth }\eth \omega =2+\frac{1}{2} \left( \eth ^2\bar{J}+\bar{\eth }^2 J\right) , \end{aligned}$$
(326)
which is solved by decomposing \(\omega \) and J into spherical harmonic components
$$\begin{aligned} \omega =1+\sum _{\ell \ge 2,|m|\le \ell }Y^{\ell \,m}\omega _{\ell \,m}, \;\;J=\sum _{\ell \ge 2,|m|\le \ell }\,{}_2Y^{\ell \,m}J_{\ell \,m}, \end{aligned}$$
(327)
leading to
$$\begin{aligned} \omega _{\ell \,m}(4-2\varLambda )=-\mathfrak {R}(J_{\ell \,m})\varLambda (2-\varLambda ) \sqrt{\frac{1}{(\ell +2)\varLambda (\ell -1)}}, \end{aligned}$$
(328)
[recall that \(\varLambda =\ell (\ell +1)\)] so that
$$\begin{aligned} \omega _{\ell \,m}=-\frac{\varLambda }{2}\sqrt{\frac{1}{(\ell +2)\varLambda (\ell -1)}}\mathfrak {R}(J_{\ell \,m}). \end{aligned}$$
(329)
Evaluating Eq. (321) for the news, and using Eq. (325), is discussed in section “Computer algebra” in “Appendix 3”. The result is Bishop (2005)
$$\begin{aligned} {{\mathcal {N}}}=\frac{1}{2\rho }\left( \partial _u J+\eth U\right) +\frac{1}{2} \left( \eth ^2 \omega +\partial _u\partial _\rho J+\partial _\rho \eth U\right) . \end{aligned}$$
(330)
Now from the linearized asymptotic Einstein equations, \(\partial _u J+\eth U=0\) and \(\partial _\rho U-2\eth \beta =0\), so we get
$$\begin{aligned} {{\mathcal {N}}}=\frac{1}{2} \left( \eth ^2 \omega +\partial _u\partial _\rho J+2\eth ^2\beta \right) . \end{aligned}$$
(331)
The result is more convenient on decomposition into spherical harmonics, \({{\mathcal {N}}}=\sum \,{}_2Y^{\ell \,m}{{\mathcal {N}}}_{\ell \,m}\),
$$\begin{aligned} {{\mathcal {N}}}_{\ell \,m}=-\frac{\varLambda \mathfrak {R}(J_{\ell \,m})}{4} +\frac{\partial _u\partial _\rho J_{\ell \,m}}{2} +\sqrt{(\ell +2)\varLambda (\ell -1)}\beta _{\ell \,m}. \end{aligned}$$
(332)
In the linearized case, the evaluation of \(\psi ^0_4\) is straightforward, because the Weyl tensor is a first-order term so the null tetrad vectors need be correct only to zeroth order. As discussed in section “Computer algebra” in “Appendix 3”, we find (Babiuc et al. 2009)
$$\begin{aligned} \psi ^0_4=\lim _{\tilde{\rho }\rightarrow 0}\left[ \frac{\partial _u\bar{\eth }\bar{U}+\partial ^2_u\bar{J}}{\tilde{\rho }} +\frac{\rho }{\tilde{\rho }} \left( -\bar{\eth }\bar{U} +\partial _u\partial _\rho \bar{\eth }\bar{U} - \partial _u\bar{J} +\partial ^2_u\partial _\rho \bar{J} -\frac{1}{2} \bar{\eth }^2W_c \right) \right] . \end{aligned}$$
(333)
It would appear that \(\psi ^0_4\) is singular, but applying the asymptotic Einstein equation Eq. (266) we see that these terms cancel; further, to linear order the deviation of \(\omega \) from unity is ignorable, so that
$$\begin{aligned} \psi ^0_4=\left( - \bar{\eth }\bar{U} +\partial _u\partial _\rho \bar{\eth }\bar{U} -\partial _u\bar{J} +\partial ^2_u\partial _\rho \bar{J} -\frac{1}{2} \bar{\eth }^2W_c \right) . \end{aligned}$$
(334)
Eq. (276) stated a relationship between \(\psi ^0_4\) and the news \({{\mathcal {N}}}\) which should remain true in this general linearized gauge. In order to see this, we modify Eq. (334) by applying Eq. (324), Eq. (264) and Eq. (266) to the terms \(W_c\), \(\partial _\rho \bar{U}\) and \(\partial _u\bar{J}\), respectively, yielding
$$\begin{aligned} \psi ^0_4=2\partial _u\bar{\eth }^2\beta + \bar{\eth }^2 \partial _u\omega +\partial ^2_u\partial _\rho \bar{J}, \end{aligned}$$
(335)
from which it is clear that \(\psi ^0_4=2\partial _u\bar{{\mathcal {N}}}\).
In the linearized approximation, the wave strain Eq. (301) simplifies to Bishop and Reisswig (2014)
$$\begin{aligned} H=\partial _\rho J-\eth A, \end{aligned}$$
(336)
and using Eq. (324) to replace A,
$$\begin{aligned} H=\partial _\rho J+\eth ^2 u_0. \end{aligned}$$
(337)
An expression for \(u_0\) is obtained using the first relationship in Eq. (324), \(\partial _u u_0=(\omega -1)+2\beta \), which is integrated to give \(u_0\). It is clear that \(u_0\) is subject to the gauge freedom \(u_0\rightarrow u_0^\prime = u_0 + u_{_G}\), provided that \(\partial _u u_{_G}=0\) so that \(u_{_G}=u_{_G}(x^{_A})\). Thus the wave strain H is subject to the gauge freedom \(H\rightarrow H^\prime = H + H_{_G}\), where \(H_{_G}=\eth ^2 u_{_G} (x^{_A})\). Decomposing H into spherical harmonics, \(H=\sum \,{}_2Y^{\ell \,m}H_{\ell \,m}\), it follows that
$$\begin{aligned} H_{\ell \,m}(u)=\partial _\rho J_{\ell \,m}(u)+\sqrt{(\ell +2)\varLambda (\ell -1)} \int ^u \omega _{\ell \,m}(u^\prime )+2\beta _{\ell \,m}(u^\prime ) du^\prime , \end{aligned}$$
(338)
with \(\omega _{\ell \,m}\) given by Eq. (329). The gauge freedom now appears as a constant of integration for each spherical harmonic mode in Eq. (338). This freedom needs to be fixed by a gauge condition. Normally the spacetime is initially dynamic but tends to a final state that is static, for example the Kerr geometry. In such a case, we impose the condition \(H_{\ell \,m}(u)\rightarrow 0\) as \(u\rightarrow \infty \). The same gauge freedom would occur if the wave strain H were obtained by time integration of the news \({{\mathcal {N}}}\), in this case appearing as an arbitrary “constant” of integration \(f(\tilde{x}^{_A})\).