The structure and properties of asymptotically shear-free NGCs (our main topic) are best understood by first looking at the special case of congruences that are shear-free everywhere (except at their caustics). Though shear-free congruences are also found in algebraically-special spacetimes, in this section only the shear-free NGCs in Minkowski spacetime, \(\mathbb{M}\), are discussed [7]
The flat-space good-cut equation and good-cut functions
In Section 2, we saw that in the NP formalism, two of the complex spin coefficients, the optical parameters ρ and σ of Eqs. (2.23) and (2.24), play a particularly important role in their description of an NGC; namely, they carry the information of the divergence, twist and shear of the congruence.
From Eqs. (2.23) and (2.24), the radial behavior of the optical parameters for general shear-free NGCs, in Minkowski space, is given by
$$\rho = {{i\Sigma - r} \over {{r^2} + {\Sigma ^2}}},\qquad \sigma = 0,$$
(3.1)
where Σ is the twist of the congruence. A more detailed and much deeper understanding of the shear-free congruences can be obtained by first looking at the explicit coordinate expression, Eq. (2.19), for all flat-space NGCs:
$${x^a} = {u_{\rm{B}}}({\hat l^a} + {\hat n^a}) - L{\bar \hat m^a} - \bar L{\hat m^a} + ({r^{\ast}} - {r_0}){\hat l^a},$$
(3.2)
where \(L(u_{\rm{B}},\zeta, \bar{\zeta})\) is an arbitrary complex function of the parameters \({y^w} = ({u_{\rm{B}}},\zeta, \bar \zeta)\); r0, also an arbitrary function of \({u_{\rm{B}}},\zeta, \bar \zeta\), determines the origin of the affine parameter; and r* can be chosen freely. Most frequently, to simplify the form of ρ, r0 is chosen as
$${r_0} \equiv - {1 \over 2}\left(\eth {\bar L + \bar{\eth} L + L\dot \bar L + \bar L\dot L} \right).$$
(3.3)
At this point, Eq. (3.2) describes an arbitrary NGC with \(u_{\rm{B}},\zeta, \bar{\zeta}\) labeling the geodesics and r* the affine distance along the individual geodesics; later \(Lu_{\rm{B}},\zeta, \bar{\zeta}\) will be chosen so that the congruence is shear-free. The tetrad \({\hat l^a},{\hat n^a},{\hat m^a},{\bar \hat m^a}\) is given by Eqs. (1.1–1.2), see [40].
There are several important comments to be made about Eq. (3.2). The first is that there is a simple geometric meaning to the parameters (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)): they are the values of the Bondi coordinates of ℑ+, where each geodesic of the congruence intersects ℑ+. The second concerns the geometric meaning of L. At each point of ℑ+, consider the past light cone and its sphere of null directions. Coordinatize that sphere (of null directions) with stereographic coordinates. The function \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) is the stereographic angle field on ℑ+ that describes the null direction of each geodesic intersecting ℑ+ at the point (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)). The values L = 0 and L = ∞ represent, respectively, the direction along the Bondi la and na vectors. This stereographic angle field completely determines the NGC.
The twist, Σ, of the congruence can be calculated in terms of \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) directly from Eq. (3.2) and the definition of the complex divergence, Eq. (2.20), leading to
$$i\Sigma = {1 \over 2}\left\{{\eth\bar L + L\dot \bar L - \bar \eth L - \bar L\dot L} \right\}.$$
(3.4)
We now demand that L be a regular function of its arguments (i.e., have no infinities), or, equivalently, that all members of the NGC come from the interior of the spacetime and not lie on ℑ+ itself.
It has been shown [12] that the condition on the stereographic angle field L for the NGC to be shear-free is that
$$\eth L + L\dot L = 0.$$
(3.5)
Our task is now to find the regular solutions of Eq. (3.5). The key to doing this is via the introduction of a new complex variable τ and complex function [39, 40],
$$\tau = T({u_{\rm{B}}},\zeta ,\bar \zeta){.}$$
(3.6)
T is related to L by the CR equation (related to the existence of a CR structure on ℑ+; see Appendix B):
$${\eth_{({u_{\rm{B}}})}}T + L\dot T = 0.$$
(3.7)
Remark 5. The following ‘gauge’ freedom becomes useful later. τ → τ* = F(τ), with F analytic, leaving Eq.
(3.7)
unchanged. In other words,
$${\tau ^ {\ast}} = {T^ {\ast}}\left({{u_{\rm{B}}},\zeta ,\bar \zeta} \right) \equiv F\left({T({u_{\rm{B}}},\zeta ,\bar \zeta)} \right),$$
(3.8)
leads to
$$\begin{array}{*{20}c} {{\eth_{({u_{\rm{B}}})}}{T^ {\ast}} = F\prime{\eth_{({u_{\rm{B}}})}}T,}\\ {{{\dot T}^ {\ast}} = F\prime\dot T,}\\ {{\eth_{({u_{\rm{B}}})}}{T^ {\ast}} + L{{\dot T}^ {\ast}} = 0.\quad \quad \quad \quad \quad \quad}\\ \end{array}$$
We assume, in the neighborhood of real ℑ+, i.e., near the real uB and \(\tilde \zeta = \bar \zeta\), that \(T({u_{\rm{B}}},\zeta, \bar {\zeta})\) is analytic in the three arguments (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)). The inversion of Eq. (3.6) yields the complex analytic cut function
$${u_{\rm{B}}} = G(\tau ,\zeta ,\tilde \zeta){.}$$
(3.9)
Though we are interested in real values for uB, from Eq. (3.9) we see that for arbitrary τ it may take complex values. Shortly, we will also address the important issue of what values of are needed for real uB.
Returning to the issue of integrating the shear-free condition, Eq. (3.5), using Eq. (3.6), we note that the derivatives of T, ð(uB)T and Ṫ can be expressed in terms of the derivatives of \(G(\tau, \zeta, \bar {\zeta})\) by implicit differentiation. The uB derivative of T is obtained by taking the uB derivative of Eq. (3.9):
$$1 = G\prime (\tau ,\zeta ,\bar \zeta)\dot T \Rightarrow \dot T = {1 \over {(G\prime)}},$$
(3.10)
while the ð(uB)T derivative is found by applying ð(uB) to Eq. (3.9),
$$\begin{array}{*{20}c} {0 = G\prime (\tau ,\zeta ,\bar \zeta){\eth_{({u_{\rm{B}}})}}T + {\eth_{(\tau)}}G,\;\;}\\ {{\eth_{({u_{\rm{B}}})}}T = - {{{\eth_{(\tau)}}G} \over {G\prime (\tau ,\zeta ,\bar \zeta)}}.\quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$
(3.11)
When Eqs. (3.10) and (3.11) are substituted into Eq. (3.7), one finds that L is given implicitly in terms of the cut function by
$$L({u_{\rm{B}}},\zeta ,\bar \zeta) = {\eth_{(\tau)}}G(\tau ,\zeta ,\bar \zeta),$$
(3.12)
$${u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) \Leftrightarrow \tau = T({u_{\rm{B}}},\zeta ,\bar \zeta){.}$$
(3.13)
Thus, we see that all information about the NGC can be obtained from the cut function \(G(\tau, \zeta, \bar {\zeta})\).
By further implicit differentiation of Eq. (3.12), i.e.,
$$\begin{array}{*{20}c} {{\eth_{({u_{\rm{B}}})}}L({u_{\rm{B}}},\zeta ,\bar \zeta) = \eth_{(\tau)}^2G(\tau ,\zeta ,\bar \zeta) + {\eth_{(\tau)}}G\prime (\tau ,\zeta ,\bar \zeta)\cdot{\eth_{({u_{\rm{B}}})}}T,\;\;}\\ {\dot L({u_{\rm{B}}},\zeta ,\bar \zeta) = {\eth_{(\tau)}}G\prime (\tau ,\zeta ,\bar \zeta)\cdot\dot T,\quad \quad \quad \quad \quad \quad \quad}\\ \end{array}$$
using Eq. (3.7), the shear-free condition (3.5) becomes
$$\eth_{(\tau)}^2G(\tau ,\zeta ,\bar \zeta) = 0{.}$$
(3.14)
This equation will be referred to as the homogeneous Good-Cut Equation and its solutions as flat-space Good-Cut Functions (GCFs). In the next Section 4, an inhomogeneous version, the Good-Cut Equation, will be found for asymptotically shear-free NGCs. Its solutions will also be referred to as GCFs.
From the properties of the ð2 operator, the general regular solution to Eq. (3.14) is easily found: G must contain only l = 0 and l = 1 spherical harmonic contributions; thus, any regular solution will be dependent on four arbitrary complex parameters, za. If these parameters are functions of, τ, i.e., za = ξa (τ), then we can express any regular solution G in terms of the complex world line ξa(τ) [39, 40]:
$${u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) = {\xi ^a}(\tau){\hat l_a}(\zeta ,\bar \zeta) \equiv {{\sqrt 2 {\xi ^0}(\tau)} \over 2} - {1 \over 2}{\xi ^i}(\tau)Y_{1i}^0{.}$$
(3.15)
The angle field \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) then has the form
$$L({u_{\rm{B}}},\zeta ,\bar \zeta) = {\eth_{(\tau)}}G(\tau ,\zeta ,\bar \zeta) = {\xi ^a}(\tau){\hat m_a}(\zeta ,\bar \zeta),$$
(3.16)
$${u_{\rm{B}}} = {\xi ^a}(\tau){\hat l_a}(\zeta ,\bar \zeta){.}$$
(3.17)
Thus, we have our first major result: every regular shear-free NGC in Minkowski space is generated by the arbitrary choice of a complex world line in what turns out to be complex Minkowski space. See Eq. (2.66) for the connection between the l = (0, 1) harmonics in Eq. (3.15) and the Poincare translations. We see in the next Section 4 how this result generalizes to regular asymptotically shear-free NGCs.
Remark 6. We point out that this construction of regular shear-free NGCs in Minkowski space is a special example of the Kerr theorem (cf. [67]). Writing Eqs.
(3.16)
and
(3.17)
as
$$\begin{array}{*{20}c} {{u_{\rm{B}}} = {{a + b\bar \zeta + \bar b\zeta + c\zeta \bar \zeta} \over {1 + \zeta \bar \zeta}},\quad \;}\\ {L = {{(\bar b + c\bar \zeta) - \bar \zeta (a + b\bar \zeta)} \over {1 + \zeta \bar \zeta}},}\\ \end{array}$$
where the (a(τ), b(τ), c(τ), d(τ)) are simple combinations of the ξa(τ), we then find that
$$\begin{array}{*{20}c} {L + {u_B}\bar \zeta = \bar b + c\bar \zeta ,} \\ {{u_B} - L\zeta = a + b\bar \zeta .} \\ \end{array}$$
Noting that the right-hand side of both equations are functions only of τ and \(\bar \zeta\), we can eliminate the τ from the two equations, thereby constructing a function of three variables of the form
$$F(L + {u_B}\bar \zeta ,{u_B} - L\zeta ,\bar \zeta) = 0{.}$$
This is a special case of the general solution to Eq.
(3.5)
, which is the Kerr theorem.
In addition to the construction of the angle field, \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) from the GCF, another quantity of great value in applications, obtained from the GCF, is the local change in uB as τ changes, i.e.,
$$V(\tau ,\zeta ,\tilde \zeta) \equiv {\partial _\tau}G = G\prime .$$
(3.18)
Real cuts from the complex good cuts, I
Though our discussion of shear-free NGCs has relied, in an essential manner, on the use of the complexification of ℑ+ and the complex world lines in complex Minkowski space, it is the real structures that are of main interest to us. We want to find the intersection of the complex GCF with real ℑ+, i.e., what are the real points and real cuts of \({u_{\rm{B}}} = G(\tau, \zeta, \bar {\zeta}),\,\,(\tilde \zeta = \bar \zeta)\), and what are the values of τ that yield real uB. These reality structures were first observed in [7] and recently there have been attempts to study them in the framework of holographic dualities (cf. [8] and Section 8).
To construct an associated family of real cuts from a GCF, we begin with
$${u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) = {{\sqrt 2} \over 2}{\xi ^0}(\tau) - {1 \over 2}{\xi ^i}(\tau)Y_{1i}^0(\zeta ,\bar \zeta)$$
(3.19)
and write
$$\tau = s + i\lambda$$
(3.20)
with s and λ real. The cut function can then be rewritten
$$\begin{array}{*{20}c} {{u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) = G(s + i\lambda ,\zeta ,\bar \zeta)\quad \quad \;}\\ {= {G_R}(s,\lambda ,\zeta ,\bar \zeta) + i{G_I}(s,\lambda ,\zeta ,\bar \zeta),}\\ \end{array}$$
(3.21)
with real \({G_R}(s,\lambda, \zeta, \bar {\zeta})\) and \({G_I}(s,\lambda, \zeta, \bar {\zeta})\). The \({G_R}(s,\lambda, \zeta, \bar {\zeta})\) and \({G_I}(s,\lambda, \zeta, \bar {\zeta})\) are easily calculated from \(G(\tau, \zeta, \bar {\zeta})\) by
$${G_R}(s,\lambda ,\zeta ,\bar \zeta) = {1 \over 2}\left\{{G(s + i\lambda ,\zeta ,\bar \zeta) + \overline {G(s + i\lambda ,\zeta ,\bar \zeta)}} \right\},$$
(3.22)
$${G_I}(s,\lambda ,\zeta ,\bar \zeta) = - {i \over 2}\left\{{G(s + i\lambda ,\zeta ,\bar \zeta) - \overline {G(s + i\lambda ,\zeta ,\bar \zeta)}} \right\}.$$
(3.23)
By setting
$${G_I}(s,\lambda ,\zeta ,\bar \zeta) = 0$$
(3.24)
and solving for
$$\lambda = \Lambda (s,\zeta ,\bar \zeta)$$
(3.25)
we obtain the associated one-parameter, s, family of real slicings,
$$u_{\rm{B}}^{(R)} = {G_R}(s,\Lambda (s,\zeta ,\bar \zeta),\zeta ,\bar \zeta) = {\xi ^a}\left({s + i\Lambda (s,\zeta ,\bar \zeta)} \right){l_a}(\zeta ,\bar \zeta){.}$$
(3.26)
Thus, the values of τ that yield real values of uB are given by
$$\tau = s + i\Lambda (s,\zeta ,\bar \zeta){.}$$
(3.27)
Perturbatively, using Eq. (3.19) and writing \({\xi ^a}(s) = \xi _R^a(s) + i\xi _I^a(s)\), we find λ to first order:
$$\begin{array}{*{20}c} {{u_{\rm{B}}} = {{\sqrt 2} \over 2}\xi _R^0(s) - {{\sqrt 2} \over 2}\xi _I^0(s)\prime \lambda - {1 \over 2}[\xi _R^i(s) - \xi _I^i(s)\prime \lambda ]Y_{1i}^0(\zeta ,\bar \zeta)\quad \quad \quad \quad}\\ {+ i\left[ {{{\sqrt 2} \over 2}\xi _I^0(s) + {{\sqrt 2} \over 2}\xi _R^0(s)\prime \lambda} \right] - i{1 \over 2}\left[ {\xi _I^i(s) + \xi _R^i(s)\prime \lambda} \right]Y_{1i}^0(\zeta ,\bar \zeta),}\\ \end{array}$$
(3.28)
$$\begin{array}{*{20}c} {u_{\rm{B}}^{(R)} = {G_R}(s,\Lambda ,\zeta ,\bar \zeta)\;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {= {{\sqrt 2} \over 2}\xi _R^0(s) - {{\sqrt 2} \over 2}\xi _I^0(s)\prime \lambda - {1 \over 2}\left[ {\xi _R^i(s) - \xi _I^i(s)\prime \lambda} \right]Y_{1i}^0(\zeta ,\bar \zeta),}\\ \end{array}$$
(3.29)
$$\lambda = \Lambda (s,\zeta ,\bar \zeta) = - {{\sqrt 2 \xi _I^0(s) + \xi _I^i(s)Y_{1i}^0(\zeta ,\bar \zeta)} \over {[\sqrt 2 \xi _R^0(s)\prime - \xi _R^i(s)\prime Y_{1i}^0(\zeta ,\bar \zeta)]}}.$$
(3.30)
Continuing, with small values for the imaginary part of \(\xi^{a}(\tau)=\xi_{R}^{a}(\tau)+i\xi_{I}^{a}(\tau)\), (\(\xi_{R}^{a}(\tau),\ \xi_{I}^{a}(\tau)\) both real analytic functions) and hence small \(\Lambda (s,\zeta, \bar {\zeta})\), it is easy to see that \(\Lambda (s,\zeta, \bar {\zeta})\) (for fixed value of s) is a bounded smooth function on the (\(\zeta, \bar {\zeta}\)) sphere, with maximum and minimum values, \({\lambda _{\max}} = \Lambda (s,{\zeta _{\max}},{\bar \zeta _{\max}})\) and \({\lambda _{\min}} = \Lambda (s,{\zeta _{\min}},{\bar \zeta _{\min}})\). Furthermore on the (\(\zeta, \bar {\zeta}\)) sphere, there are a finite line-segments worth of curves (circles) that lie between (\(\zeta_{\min},\overline{\zeta}_{\min}\)) and (\({\zeta _{\max}},{\bar \zeta _{\max}}\)) such that \(\Lambda (s,\zeta, \bar {\zeta})\) is a monotonically increasing function on the family of curves. Hence there will be a family of circles on the (\(\zeta, \bar {\zeta}\))-sphere where the value of λ is a constant, ranging between λmax and λmin.
Summarizing, we have the result that in the complex τ-plane there is a ribbon or strip given by all values of s and line segments parametrized by λ between λmin and λmax such that the complex light-cones from each of the associated points, ξa(s + iλ), all have some null geodesics that intersect real ℑ+. More specifically, for each of the allowed values of τ = s + iλ there will be a circle’s worth of complex null geodesics leaving the point ξa(s + iλ), reaching real ℑ+. It is the union of these null geodesics, corresponding to the circles on the (\(\zeta, \bar {\zeta}\))-sphere from the line segment, that produces the real family of cuts, Eq. (3.26).
The real structure associated with a complex world line is then this one-parameter family of slices (cuts) Eq. (3.26).
Remark 7. We saw earlier that the shear-free angle field was given by
$$L({u_{\rm{B}}},\zeta ,\bar \zeta) = {\eth_{(\tau)}}G(\tau ,\zeta ,\bar \zeta),$$
(3.31)
$${u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) \Leftrightarrow \tau = T({u_{\rm{B}}},\zeta ,\bar \zeta),$$
(3.32)
where real values of ub should be used. If the real cuts, \({u_{\rm{B}}} = {G_R}\left({s,\Lambda (s,\zeta, \bar {\zeta}),\zeta, \bar {\zeta}} \right)\), were used instead to calculate \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\), the results would be wrong. The restriction of τ to yield real uB, does not commute with the application of the ð operator, i.e.,
$$L({u_{\rm{B}}},\zeta ,\bar \zeta) \neq \eth{G_R}.$$
The ð differentiation must be done first, holding τ constant, before the reality of uB is used. In other words, though we are interested in real ℑ+, it is essential that we consider its (local) complexification.
There are a pair of important (dual) results that arise from the considerations of the good cuts [7, 8]. From the stereographic angle field, i.e., L from Eqs. (3.31) and (3.32), one can form two different conjugate fields, (1) the complex conjugate of L:
$$\bar L = {\overline \eth _{\bar \tau}}\bar G = {\bar \xi ^a}(\bar \tau){\bar \hat m_a}$$
(3.33)
and (2) the holomorphic conjugate, \(\tilde L\), given by
$$\tilde L = {\overline \eth _\tau}G = {\xi ^a}(\tau){\bar \hat m_a}{.}$$
(3.34)
The two different pairs, the complex conjugate pair (\(L,\bar L\)) and the holomorphic pair (\(L,\tilde L\)) determine two different null vector direction fields at ℑ+, the real vector field, l*a, and the complex field, \(l_{C}^{\ast a}\), via the relations
$${l^a} \rightarrow {l^{{\ast} a}} = {l^a} - {{\bar L} \over r}{m^a} - {L \over r}{\bar m^a} + O({r^{- 2}}),$$
(3.35)
and
$${l^a} \rightarrow l_C^{{\ast} a} = {l^a} - {{\tilde L} \over r}{m^a} - {L \over r}{\bar m^a} + O({r^{- 2}}).$$
(3.36)
Both generate, in the spacetime interior, shear-free null geodesic congruences: the first is a real twisting shear-free congruence while the latter is a complex twist-free congruence that consists of the light-cones from the world line, za = ξa(τ), i.e., they focus on ξa(τ). It is this fact that they focus on the world line, za = ξa(τ), that is of most relevance to us.
The twist of the real congruence, \(\Sigma ({u_B},\zeta, \bar {\zeta})\), which comes from the complex divergence,
$$\rho = - {1 \over {r + i\Sigma}}$$
(3.37)
$$\begin{array}{*{20}c} {2i\Sigma = \eth\bar L + L{{(\bar L)}^\cdot} - \overline \eth L - \bar L\dot L.\quad \;\;\;} \\ {= ({\xi ^a}(\tau) - {{\bar \xi}^a}(\bar \tau))\;({n_a} - {l_a}).} \\ \end{array}$$
(3.38)
is proportional to the imaginary part of the complex world line and consequently we have the real structure associated with the complex world line coming from two (dual) places, the real cuts, Eq. (3.26) and the twist.
It is the complex point of view of the complex light-cones coming from the complex world line that dominates our discussion.
Approximations
Due to the difficulties involved in the intrinsic nonlinearities and the virtual impossibility of exactly inverting arbitrary analytic functions, it often becomes necessary to resort to approximations. The basic approximation will be to consider the complex world line ξa(τ) as being close to the straight line, \(\xi _0^a(\tau) = \tau \delta _0^a\); deviations from this will be considered as first order. We retain terms up to second order, i.e., quadratic terms. Another frequently used approximation is to terminate spherical harmonic expansions after the l = 2 terms.
It is worthwhile to discuss some of the issues related to these approximations. One important issue is how to use the gauge freedom, Eq. (3.8), τ → τ* = F(τ), to simplify ξa(τ) and the ‘velocity vector’,
$${v^a}(\tau) = {\xi ^{a}\prime}(\tau) \equiv {{d{\xi ^a}} \over {d\tau}}.$$
(3.39)
A Notational issue: Given a complex analytic function (or vector) of the complex variable τ, say G(τ), then G(τ) can be decomposed uniquely into two parts,
$$G(\tau) = {\mathfrak{G}_R}(\tau) + i{\mathfrak{G}_I}(\tau),$$
where all the coefficients in the Taylor series for \(\mathfrak{G}_{R}(\tau)\) and \(\mathfrak{G}_{I}(\tau)\) are real. With but a slight extension of conventional notation we refer to them as real analytic functions.
With this notation, we also write
$$\begin{array}{*{20}c} {{\xi ^a}(\tau) = \xi _R^a(\tau) + i\xi _I^a(\tau)} \\ {{v^a}(\tau) = v_R^a(\tau) + iv_I^a(\tau)}{.} \\ \end{array}$$
By using the reparametrization of the world line, via τ* = F(τ), we choose F(τ) = ξ0(τ), so that (dropping the *) we have
$$\begin{array}{*{20}c} {{\xi ^0}(\tau) = \xi _R^0(\tau) = \tau ,\;\;\;\xi _I^0(\tau) = 0} \\ {{v^0}(\tau) = v_R^0(\tau) = 1,\;\;\;\;v_I^0(\tau) = 0} \\ \end{array}$$
Finally, from the reality condition on the uB, Eqs. (3.23), (3.26) and (3.25) yield, with τ = s+iλ and λ treated as small,
$$u_{\rm{B}}^{(R)} = \xi _R^a(s){\hat l_a} + v_I^a(s){\hat l_a}{{\xi _I^b(s){{\hat l}_b}} \over {\xi _R^{c}\prime (s){{\hat l}_c}}},$$
(3.40)
$$\begin{array}{*{20}c} {\lambda = \Lambda (s,\zeta ,\bar \zeta) = - {{\xi _I^b(s){{\hat l}_b}} \over {\xi _R^{c}\prime(s){{\hat l}_c}}},}\\ {= {{{{\sqrt 2} \over 2}\xi _I^i(s)Y_{1i}^0} \over {1 - {{\sqrt 2} \over 2}\xi _R^i\prime (s)Y_{1i}^0}}.\quad \;\;}\\ \end{array}$$
(3.41)
Within this slow motion approximation scheme, we have from Eqs. (3.40) and (3.41),
$$u_{{\rm{ret}}}^{(R)} = \sqrt 2 u_{\rm{B}}^{(R)} = s - {1 \over {\sqrt 2}}\xi _R^i(s)Y_{1i}^0 + 2v_I^a(s){{\hat l}_a}\xi _I^b(s){{\hat l}_b},$$
(3.42)
$$\lambda \approx {{\sqrt 2} \over 2}\xi _I^i(s)Y_{1i}^0\left({1 - {{\sqrt 2} \over 2}v_R^j(s)Y_{1j}^0} \right),$$
(3.43)
or, to first order, which is all that is needed,
$$\lambda = {{\sqrt 2} \over 2}\xi _I^i(s)Y_{1i}^0.$$
.
We then have, to linear order,
$$\begin{array}{*{20}c} {\tau = s + i{{\sqrt 2} \over 2}\xi _I^i(s)Y_{1i}^0,} \\ {u_{{\rm{ret}}}^{(R)} = s - {1 \over {\sqrt 2}}\xi _R^i(s)Y_{1i}^0.\quad \;\;} \\ \end{array}$$
(3.44)
Asymptotically-vanishing Maxwell fields
A prelude
The basic starting idea in this work is simple. It is in the generalizations and implementations where difficulties arise.
Starting in Minkowski space in a fixed given Lorentzian frame with spatial origin, the electric dipole moment \({\overrightarrow D_E}\) is calculated from an integral over the (localized) charge distribution. If there is a shift, \(\overrightarrow R\), in the origin, the dipole transforms as
$$\overrightarrow {D_E^ {\ast}} = {\vec D_E} - q\vec R.$$
(3.45)
If \({\overrightarrow D_E}\) is time dependent, we obtain the center-of-charge world line by taking \(\overrightarrow {D_E^ \ast} = 0\), i.e., from \(\overrightarrow R = {\overrightarrow D_{\,E}}{q^{- 1}}\). It is this idea that we want to generalize and extend to gravitational fields.
The first generalization is formal and somewhat artificial: shortly it will become quite natural. We introduce, in addition to the electric dipole moment, the magnetic dipole moment \({\overrightarrow D_M}\) (also obtained by an integral over the current distribution) and write
$${\overrightarrow D_{\mathbb{C}}} = {\overrightarrow D_E} + i{\overrightarrow D_M}.$$
By allowing the displacement \(\overrightarrow R\) to take complex values, \(\overrightarrow{R}_{\mathbb{C}}\), Eq. (6.37), can be generalized to
$$\overrightarrow {D_{\mathbb{C}}^ {\ast}} = {\overrightarrow D_{\mathbb{C}}} - q{\overrightarrow R_{\mathbb{C}}},$$
(3.46)
so that the complex center-of-charge is given by \(\overrightarrow{D}_{\mathbb{C}}^{\ast}=0\) or
$${\overrightarrow R_{\mathbb{C}}} = {\overrightarrow D_{\mathbb{C}}}{q^{- 1}}.$$
(3.47)
We emphasize that this is done in a fixed Lorentz frame and only the origin is moved. In different Lorentz frames there will be different complex centers of charge.
Later, directly from the general asymptotic Maxwell field itself (satisfying the Maxwell equations), we define the asymptotic complex dipole moment and give its transformation law, including transformations between Lorentz frames. This yields a unique complex center of charge independent of the Lorentz frame.
Asymptotically-vanishing Maxwell fields: General properties
In this section, we describe how a complex center of charge for asymptotically vanishing Maxwell fields in flat spacetime can be found by using the shear-free NGCs, constructed from solutions of the homogeneous good-cut equation, to transform certain Maxwell field components to zero. Although this serves as a good example for our later methods in asymptotically flat spacetimes, the reader may wish to skip ahead to Section 4, where we go directly to gravitational fields in a setting of greater generality.
Our first set of applications of shear-free NGCs comes from Maxwell theory in Minkowski space. We review the general theory of the behavior of asymptotically-flat or vanishing Maxwell fields assuming throughout that there is a nonvanishing total charge, q. As stated in Section 2, the Maxwell field is described in terms of its complex tetrad components, (ϕ0,ϕ1,ϕ2). In a Bondi coordinate/tetrad system the asymptotic integration is relatively simple [50, 38] resulting in the radial behavior (the peeling theorem):
$$\begin{array}{*{20}c} {{\phi _0} = {{\phi _0^0} \over {{r^3}}} + O({r^{- 4}}),} \\ {{\phi _1} = {{\phi _1^0} \over {{r^2}}} + O({r^{- 3}}),} \\ {{\phi _2} = {{\phi _2^0} \over r} + O({r^{- 2}}),} \\ \end{array}$$
(3.48)
where the leading coefficients of r, \(r,(\phi _0^0,\, \phi _1^0,\, \phi _2^0)\) satisfy the evolution equations:
$$\dot \phi _0^0 + \eth\phi _1^0 = 0,$$
(3.49)
$$\dot \phi _1^0 + \eth\phi _2^0 = 0.$$
(3.50)
The formal integration procedure is to take \(\phi _2^0\) as an arbitrary function of (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)) (the free broadcasting data), then integrate the second, for \(\phi_{1}^{0}\), with a time-independent spin-weight s = 0 function of integration and finally integrate the first, for \(\phi_{0}^{0}\). Using a slight modification of this, namely from the spherical harmonic expansion, we obtain,
$$\phi _0^0 = \phi _{0i}^0Y_{1i}^1 + \phi _{0ij}^0Y_{2ij}^1 + \ldots ,$$
(3.51)
$$\phi _1^0 = q + \phi _{1i}^0Y_{1i}^0 + \phi _{1ij}^0Y_{2ij}^0 + \ldots ,$$
(3.52)
$$\phi _2^0 = \phi _{2i}^0Y_{1i}^{- 1} + \phi _{2ij}^0Y_{2ij}^{- 1} + \ldots ,$$
(3.53)
with the harmonic coefficients related to each other by the evolution equations:
$$\begin{array}{*{20}c} {\phi _0^0 = 2q{\eta ^i}({u_{{\rm{ret}}}})Y_{1i}^1 + Q_{\mathbb{C}}^{ij\,}\prime Y_{2ij}^1 + \ldots ,\quad \quad \quad \quad} \\ {\phi _1^0 = q + \sqrt 2 q{\eta ^i}\prime ({u_{{\rm{ret}}}})Y_{1i}^0 + {{\sqrt 2} \over 6}Q_{\mathbb{C}}^{ij}\prime \prime Y_{2ij}^0 + \ldots ,} \\ {\phi _2^0 = - 2q{\eta ^i}\prime \prime ({u_{{\rm{ret}}}})Y_{1i}^{- 1} - {1 \over 3}Q_{\mathbb{C}}^{ij\,}\prime \prime \prime Y_{2ij}^{- 1} + \ldots \quad \quad} \\ \end{array}$$
(3.54)
The physical meaning of the coefficients are
$$\begin{array}{*{20}c} {q = {\rm{total}}\;{\rm{electric}}\;{\rm{charge}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;}\\ {q{\eta ^i} = D_{\mathbb C}^i = {\rm{complex}}\;({\rm{electric}}\;\& \;{\rm{magnetic}})\;{\rm{dipole}}\;{\rm{moment}}\; = D_E^i + iD_M^i,}\\ {Q_{\mathbb C}^{ij} = {\rm{complex}}\;({\rm{electric}}\;\& \;{\rm{magnetic}})\;{\rm{quadrupole}}\;{\rm{moment}},\quad \quad \quad \quad \quad}\\ \end{array}$$
(3.55)
etc. Recall from Section 1.1 that this electromagnetic quadrupole needs to be rescaled (\(Q_{\mathbb{C}}^{ij} \rightarrow 2\sqrt 2 Q_{\mathbb{C}}^{ij}\)) to obtain the physical quadrupole which appears in the usual expressions for Maxwell theory [43]. For later use, the complex dipole is written as \(D_{\mathbb{C}}^i({u_{{\rm{ret}}}}) = q{\eta ^i}({u_{{\rm{ret}}}})\). Note that the \(D_{\mathbb{C}}^i\) is defined relative to a given Bondi system. This is the analogue of a given origin for the calculations of the dipole moments of Eq. (6.37).
Later in this section it will be shown that we can find a unique complex world line, ξa(τ) = (ξ0, ξi), (the world line associated with a shear-free NGC), that is closely related to the ηi(uret) From this complex world line we can define the intrinsic complex dipole moment, \(D_{\mathcal{I}\mathbb{C}}^i = q{\xi ^i}(s)\).
However, we first discuss a particular Maxwell field, Fab, where one of its eigenvectors is a tangent field to a shear-free NGC. This solution, referred to as the complex Liénard-Wiechert field is the direct generalization of the ordinary Liénard-Wiechert field. Though it is a real solution in Minkowski space, it can be thought of as arising from a complex world line in complex Minkowski space.
A coordinate and tetrad system attached to a shear-free NGC
The parametric form of the general NGC was given earlier by Eq. (3.2),
$${x^a} = {u_B}({\hat l^a} + {\hat n^a}) - L{\overline {\hat m} ^a} - \bar L{\hat m^a} + ({r^*} - {r_0}){\hat l^a}.$$
(3.56)
The parameters (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)) labeled the individual members of the congruence while r* was the affine parameter along the geodesics. An alternative interpretation of the same equation is to consider it as the coordinate transformation between the coordinates, xa (or the Bondi coordinates) and the geodesic coordinates (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)). Note that while these coordinates are not Bondi coordinates, though, in the limit, at ℑ+, they are. The associated (geodesic) tetrad is given as a function of these geodesic coordinates, but with Minkowskian components by Eqs. (3.56). We restrict ourselves to the special case of the coordinates and tetrad associated with the from a shear-free NGC. Though we are dealing with a real shear-free twisting congruence, the congruence, as we saw, is generated by a complex analytic world line in the complexified Minkowski space, za = ξa(τ). The complex parameter, τ, must in the end be chosen so that the ‘uB’ of Eq. (3.19) is real. The Minkowski metric and the spin coefficients associated with this geodesic system can be calculated [40] in the (\({u_{\rm{B}}},{r^ {\ast}},\zeta, \bar {\zeta}\)) frame. Unfortunately, it must be stated parametrically, since the τ explicitly appears via the ξa(τ) and can not be directly eliminated. (An alternate choice of these geodesic coordinates is to use the τ instead of the uB. Unfortunately, this leads to an analytic flat metric on the complexified Minkowski space, where the real spacetime is hard to find.)
The use and insight given by this coordinate/tetrad system is illustrated by its application to a special class of Maxwell fields. We consider, as mentioned earlier, the Maxwell field where one of its principle null vectors, l*a, (an eigenvector of the Maxwell tensor, \({F_{ab}}{l^{{\ast}a}} = \lambda l_b^{\ast}\)), is a tangent vector of a shear-free NGC. Thus, it depends on the choice of the complex world line and is therefore referred to as the complex Liénard-Wiechert field. (If the world line was real it would lead to the ordinary Liénard-Wiechert field.) We emphasize that though the source can formally be thought of as a charge moving on the complex world line, the Maxwell field is a real field on real Minkowski space. It will have a real (distributional) source at the caustics of the congruence. Physically, its behavior is very similar to real Liénard-Wiechert fields, the essential difference is that the electric dipole is now replaced by the combined electric and magnetic dipoles. The imaginary part of the world line determines the magnetic dipole moment.
In the spin-coefficient version of the Maxwell equations, using the geodesic tetrad, the choice of l*a as the principle null vector ‘congruence’ is just the statement that
$$\phi _0^ {\ast} = {F_{ab}}{l^{{\ast} a}}{m^{{\ast} b}} = 0.$$
This allows a very simple exact integration of the remaining Maxwell components [50].
Complex Liénard-Wiechert Maxwell field
The present section, included as an illustration of the general ideas and constructions in this work, is rather technical and complicated and can be omitted without loss of continuity.
The complex Liénard-Wiechert fields (which we again emphasize are real Maxwell fields) are formally given by the (geodesic) tetrad components of the Maxwell tensor in the null geodesic coordinate system (\({u_{\rm{B}}},{r^{\ast}},\zeta, \bar {\zeta}\)), Eq. (3.56). As the detailed calculations are long [50] and take us too far afield, we only give an outline here. The integration of the radial Maxwell equations leads to the asymptotic behavior,
$$\phi _0^ {\ast} = 0,$$
(3.57)
$$\phi _1^ {\ast} = {\rho ^2}\phi _1^{{\ast} 0},$$
(3.58)
$$\phi _2^ {\ast} = \rho \phi _2^{{\ast} 0} + O({\rho ^2}),$$
(3.59)
with
$$\begin{array}{*{20}c} {\rho = - {{({r^ {\ast}} + i\Sigma)}^{- 1}},\quad \quad} \\ {2i\Sigma = \eth\bar L + L\dot \bar L - \overline \eth L - \bar L\dot L.} \\ \end{array}$$
(3.60)
The O(ρ2) expression is known in terms of (\(\phi _1^{{\ast} 0},\phi _2^{{\ast} 0}\)). The function \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) is given by
$$\begin{array}{*{20}c} {L({u_{\rm{B}}},\zeta ,\bar \zeta) = {\eth_{(\tau)}}G(\tau ,\zeta ,\bar \zeta),\quad\quad\quad\quad\quad\quad\quad\quad\quad} \\ {{u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta) = {\xi ^a}(\tau){{\hat l}_a}(\zeta ,\bar \zeta),\quad} \\ \end{array}$$
with ξa(τ) an arbitrary complex world line that determines the shear-free congruence whose tangent vectors are the Maxwell field eigenvectors.
Remark 8. In this case of the complex Liénard-Wiechert Maxwell field, the ξa determines the intrinsic center-of-charge world line, rather than the relative center-of-charge line.
The remaining unknowns, \(\phi _1^{{\ast} 0},\phi _2^{{\ast} 0}\), are determined by the last of the Maxwell equations,
$$\begin{array}{*{20}c} {\eth\phi _1^{{\ast} 0} + L\dot \phi _1^{{\ast} 0} + 2\dot L\phi _1^{{\ast} 0} = 0,\quad \;} \\ {\eth\phi _2^{{\ast} 0} + L\dot \phi _2^{{\ast} 0} + \dot L\phi _2^{{\ast} 0} = \,\dot \phi _1^{{\ast} 0},} \\ \end{array}$$
(3.61)
which have been obtained from Eqs. (2.55) and (2.56) via the null rotation between the Bondi and geodesic tetrads and the associated Maxwell field transformation, namely,
$${l^a} \rightarrow {l^{{\ast} a}} = {l^a} - {{\bar L} \over r}{m^a} - {L \over r}{\bar m^a} + O({r^{{\ast} - 2}}),$$
(3.62)
$${m^a} \rightarrow {m^{{\ast} a}} = {m^a} - {L \over r}{n^a},$$
(3.63)
$${n^a} \rightarrow {n^{{\ast} a}} = {n^a},$$
(3.64)
with
$$\phi _0^{{\ast} 0} = 0 = \phi _0^0 - 2L\phi _1^0 + {L^2}\phi _2^0,$$
(3.65)
$$\phi _1^{{\ast} 0} = \phi _1^0 - L\phi _2^0,$$
(3.66)
$$\phi _2^{{\ast} 0} = \phi _2^0.$$
(3.67)
These remaining equations depend only on \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\), which, in turn, is determined by ξa(τ). In other words, the solution is driven by the complex line, ξa(τ). As they now stand, Eqs. (3.61) appear to be difficult to solve, partially due to the implicit description of the \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\).
Actually they are easily solved when the independent variables are changed, via Eq. (3.15), from \(L({u_{\rm{B}}},\zeta, \bar {\zeta})\) to the complex (\(\tau, \zeta, \bar {\zeta}\)). They become, after a bit of work,
$${\eth_{(\tau)}}({V^2}\phi _1^0) = 0,$$
(3.68)
$${\eth_{(\tau)}}(V\phi _2^0) = \phi _1^{0}\prime,$$
(3.69)
$$V = {\xi ^{a}}\prime(\tau){\hat l_a}(\zeta ,\bar \zeta),$$
(3.70)
with the solution
$$\begin{array}{*{20}c} {\phi _1^{{\ast} 0} = {q \over 2}{V^{- 2}},\quad \quad \quad \quad \quad \;} \\ {\phi _2^{{\ast} 0} = {q \over 2}{V^{- 1}}{{\overline \eth}_{(\tau)}}({V^{- 1}}{\partial _\tau}V){.}} \\ \end{array}$$
(3.71)
q being the Coulomb charge.
Though we now have the exact solution, unfortunately it is in complex coordinates where virtually every term depends on the complex variable τ, via ξa(τ). This is a severe impediment to a full description and understanding of the solution in the real Minkowski space.
In order to understand its asymptotic behavior and physical content, one must transform it, via Eqs. (3.62)–(3.67), back to a Bondi coordinate/tetrad system. This can only be done by approximations. After a lengthy calculation [50], we find the Bondi peeling behavior
$$\begin{array}{*{20}c} {{\phi _0} = {r^{- 3}}\phi _0^0 + O({r^{- 4}}),} \\ {{\phi _1} = {r^{- 2}}\phi _1^0 + O({r^{- 3}}),} \\ {{\phi _2} = {r^{- 1}}\phi _2^0 + O({r^{- 2}}),} \\ \end{array}$$
(3.72)
with
$$\phi _0^0 = q\left({L{V^{- 2}} + {1 \over 2}{L^2}{V^{- 1}}{{\overline \eth}_{(\tau)}}[{V^{- 1}}V\prime ]} \right),$$
(3.73)
$$\phi _1^0 = {q \over {2{V^2}}}(1 + LV{\overline \eth _{(\tau)}}[{V^{- 1}}V\prime ]),$$
(3.74)
$$\phi _2^0 = - {q \over 2}{V^{- 1}}{\overline \eth _{(\tau)}}({V^{- 1}}V\prime),$$
(3.75)
$$V = {\xi ^a}\prime\,{\hat l_a}(\zeta ,\bar \zeta){.}$$
(3.76)
Next, treating the world line, as discussed earlier, as a small deviation from the straight line, \({\xi ^a}(\tau) = \tau \delta _0^a\), i.e., by
$$\begin{array}{*{20}c} {{\xi ^a}(\tau) = (\tau ,{\xi ^i}(\tau)),} \\ {{\xi ^i}(\tau) \ll 1.\quad \quad \quad} \\ \end{array}$$
The GCF and its inverse (see Section 6) are given, to first order, by
$${u_{{\rm{ret}}}} = \sqrt 2 {u_{\rm{B}}} = \sqrt 2 G = \tau - {{\sqrt 2} \over 2}{\xi ^i}(\tau)Y_{1i}^0(\zeta ,\bar \zeta),$$
(3.77)
$$\tau = {u_{{\rm{ret}}}} + {{\sqrt 2} \over 2}{\xi ^i}({u_{{\rm{ret}}}})Y_{1i}^0(\zeta ,\bar \zeta){.}$$
(3.78)
Again to first order, Eqs. (3.73), (3.74) and (3.75) yield
$$\begin{array}{*{20}c} {\phi _0^0 = 2q{\xi ^i}({u_{{\rm{ret}}}})Y_{1i}^1,\quad \quad} \\ {\phi _1^0 = q + \sqrt 2 q{\xi ^i}\prime ({u_{{\rm{ret}}}})Y_{1i}^0,} \\ {\phi _2^0 = - 2q{\xi ^i}\prime \prime ({u_{{\rm{ret}}}})Y_{1i}^{- 1},\quad} \\ \end{array}$$
(3.79)
the known electromagnetic dipole field, with a Coulomb charge, q. One then has the physical interpretation of qξi(uret) as the complex dipole moment; (the electric plus ‘i’ times magnetic dipole) and ξi(uret) is the complex center of charge, the real part being the ordinary center of charge, while the imaginary part is the ‘imaginary’ magnetic center of charge. This simple relationship between the Bondi form of the complex dipole moment, qξi(uret), and the intrinsic complex center of charge, ξi(τ), is true only at linear order. The second-order relationship is given later.
Reversing the issue, if we had instead started with an exact complex Liénard-Wiechert field but now given in a Bondi coordinate/tetrad system and performed on it the transformations, Eqs. (2.10) and (3.65) to the geodesic system, it would have resulted in
This example was intended to show how physical meaning could be attached to the complex world line associated with a shear-free NGC. In this case and later in the case of asymptotically-flat spacetimes, when the GCF is singled out by either the Maxwell field or the gravitational field, it will be referred to it as a UCF. For either of the two cases, a flat-space asymptotically-vanishing Maxwell field (with nonvanishing total charge) and for a vacuum asymptotically-flat spacetime, there will be a unique UCF. In the case of the Einstein-Maxwell fields there will, in general, be two UCFs: one for each field.
Asymptotically vanishing Maxwell fields & shear-free NGCs
We return now to the general asymptotically-vanishing Maxwell field, Eqs. (3.48) and (3.51), and its transformation behavior under the null rotation around na,
$$\begin{array}{*{20}c} {{l^a} \rightarrow {l^{{\ast} a}} = {l^a} - {{\bar L} \over r}{m^a} - {L \over r}{{\bar m}^a} + 0({r^{- 2}}),} \\ {{m^a} \rightarrow {m^{{\ast} a}} = {m^a} - {L \over r}{n^a} + 0({r^{- 2}}),\quad \quad} \\ {{n^a} \rightarrow {n^{{\ast} a}} = {n^a},\quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(3.80)
with \(L({u_{\rm{B}}},\zeta, \bar {\zeta}) = {\xi ^a}(\tau){\hat m_a}\), being one of our shear-free angle fields defined by a world line, za = ξa(τ) and cut function \({u_{\rm{B}}} = {\xi ^a}(\tau){\hat l_a}(\zeta, \bar {\zeta})\). The leading components of the Maxwell fields transform as
$$\phi _0^{{\ast} 0} = \phi _0^0 - 2L\phi _1^0 + {L^2}\;\phi _2^0,$$
(3.81)
$$\phi _1^{{\ast} 0} = \phi _1^0 - L\phi _2^0,$$
(3.82)
$$\phi _2^{{\ast} 0} = \phi _2^0.$$
(3.83)
The ‘picture’ to adopt is that the new ϕ*s are now given in a tetrad defined by the complex light cone (or generalized light cone) with origin on the complex world line. (This is obviously formal and perhaps physically nonsensical, but mathematically quite sound, as the shear-free congruence can be thought of as having its origin on the complex line, ξa(τ).) From the physical identifications of charge, dipole moments, etc., of Eq. (3.54), we can obtain the transformation law of these physical quantities. In particular, the l = 1 harmonic of \(\phi_{0}^{0}\), or, equivalently, the complex dipole, transforms as
$$\phi _{0i}^{0 {\ast}} = \phi _{0i}^0 - 2(L\phi _1^0)\vert _{i} + ({L^2}\phi _2^0)\vert _{i},$$
(3.84)
where the notation W∣
i
means extract only the l = 1 harmonic from a Clebsch-Gordon expansion of W. A subtlety and difficulty of this extraction process is here clarified.
The (non-)uniqueness of spherical harmonic expansions
An important observation, obvious but easily overlooked, concerning the spherical harmonic expansions is that, in a certain sense, they lack uniqueness. As this issue is significant, its clarification is important.
Assume that we have a particular spin-s function on ℑ+, say, \(\eta_{(s)}(u_{\rm{B}},\zeta, \overline{\zeta})\), given in a specific Bondi coordinate system, (\({u_{\rm{B}}},\zeta, \bar {\zeta}\)), that has a harmonic expansion given, for constant uB, by
$${\eta _{(s)}}({u_{\rm{B}}},\zeta ,\bar \zeta) =\underset{l,(ijk \ldots)}{\Sigma} \eta _{(s)}^{l,(ijk \ldots)}({u_{\rm{B}}})Y_{l,(ijk \ldots)}^{(s)}$$
If exactly the same function was given on different cuts or slices, say,
$${u_{\rm{B}}} = G(\tau ,\zeta ,\bar \zeta),$$
(3.85)
with
$$\eta _{(s)}^ {\ast} (\tau ,\zeta ,\bar \zeta) = {\eta _{(s)}}\left({G(\tau ,\zeta ,\bar \zeta),\zeta ,\bar \zeta} \right),$$
the harmonic expansion at constant τ would be different. The new coefficients are extracted by the two-sphere integral taken at constant τ:
$$\eta _{(s)}^{{\ast} l,(ijk \ldots)}(\tau) = \int\nolimits_{{S^2}} {\eta _{(s)}^ {\ast}} (\tau ,\zeta ,\bar \zeta)\bar Y_{l,(ijk \ldots)}^{(s)}dS.$$
(3.86)
It is in this rather obvious sense that the expansions are not unique.
The transformation, Eq. (3.84), and harmonic extraction implemented by first replacing the uB in all the terms of all \(\phi _0^0,\phi _1^0,\phi _2^0\), by \({u_{\rm{B}}} = {\xi ^a}(\tau){\hat l_a}(\zeta, \bar {\zeta}) \equiv {{\sqrt 2 \tau} \over 2} - {1 \over 2}{\xi ^i}Y_{1i}^0\), yields \(\phi _{0i}^{0 {\ast}}(\tau)\) with a functional form [8],
$$\phi _{0i}^{0 {\ast}}(\tau) = {\Gamma _i}(\phi _0^0,\phi _1^0,\phi _2^0,{\xi ^a}(\tau)) = \oint\nolimits_{{S^2}} {[\phi _0^0 - (2cL\phi _1^0 - {c^2}{L^2}\phi _2^0)]} Y_{1i}^{- 1}dS$$
(3.87)
is decidedly nontensorial: in fact it is very nonlocal and nonlinear.
Though it is clear that extracting \(\phi _{0i}^{0 {\ast}}(\tau)\) with this relationship is available in principle, in practice it is impossible to do it exactly and all examples are done with approximations: essentially using slow motion for the complex world line.
Remark 9. If by some accident the Maxwell field was a complex Liénard-Wiechert field, a world line ξa(τ) could be chosen so that from the associated complex null cones we would have \(\phi_{0}^{\ast 0}=0\). However, though this cannot be done in general, the l = 1 harmonics of \(\phi _0^{\ast 0}\) can be made to vanish by the appropriate choice of the ξa(τ). This is the means by which a unique world line is chosen.
The complex center of charge
The complex center of charge is defined by the vanishing of the complex dipole moment \(\phi _{0i}^{0 {\ast}}(\tau)\); in other words,
$${\Gamma _i}(\phi _0^0,\phi _1^0,\phi _2^0,{\xi ^a}) = 0$$
(3.88)
determines three components of the (up to now) arbitrary complex world line, ξa(τ); the fourth component can be taken as τ. In practice we do this only up to second order with the use of only the (l = 0, 1, 2) harmonics. The approximation we are using is to consider the charge as zeroth order and the dipole moments and the spatial part of the complex world line as first order.
From Eq. (3.84),
$$\phi _{0i}^{0 {\ast}} = {\Gamma _i} \approx \phi _{0i}^0 - 2L\phi _1^0\vert _{i} = 0$$
(3.89)
with the identifications, Eq. (3.54), for q and Dℂ, we have to first order (with \(\sqrt 2 {u_B} = {u_{ret}} \approx \tau\)),
$$D_{\mathbb{C}}^i({u_{{\rm{ret}}}}) = q{\eta ^i}({u_{{\rm{ret}}}}) = q{\xi ^i}({u_{{\rm{ret}}}}).$$
(3.90)
This is exactly the same result as we obtained earlier in Eq. (3.47), via the charge and current distributions in a fixed Lorentz frame.
Carrying this calculation [50] to second order, we find the second-order complex center of charge and the relationship between the intrinsic complex dipole, \(D_{{\mathcal I}:\mathbb{C}}^i\), and the complex dipole, \(D_\mathbb{C}^i\),
$$D_{{\mathcal I}:\mathbb{C}}^i = q{\xi ^i}(s),\qquad D_\mathbb{C}^i = q{\eta ^i}(s),$$
(3.91)
$${\xi ^k} = {\eta ^k} - {i \over 2}{\eta ^i}{\eta ^j}\prime {\epsilon _{ijk}} - {{\sqrt 2} \over {10}}{q^{- 1}}Q_{\mathbb{C}}^{ik}\prime \prime {\xi ^i},$$
(3.92)
$${\eta ^k} = {\xi ^k} + i{1 \over 2}{\xi ^i}{\xi ^j}\prime {\epsilon _{ijk}} + {{\sqrt 2} \over {10}}{q^{- 1}}Q_{\mathbb{C}}^{ik}\prime \prime {\xi ^i}$$
(3.93)
The GCF,
$${u_{\rm{B}}} = X(\tau ,\zeta ,\tilde \zeta) = {\xi ^a}(\tau){\hat l_a}(\zeta ,\tilde \zeta),$$
with this uniquely determined world line is referred to as the Maxwell UCF.
In Section 5, these ideas are applied to GR, with the complex electric and magnetic dipoles being replaced by the complex combination of the mass dipole and the angular momentum.