3.4 Asymptotically-vanishing Maxwell fields

3.4.1 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 −→ D E is calculated from an integral over the (localized) charge distribution. If there is a shift, −→ R, in the origin, the dipole transforms as

−→ −→ −→ D ∗E = D E − q R . (3.45 )
If −→ D E is time dependent, we obtain the center-of-charge world line by taking −→ D ∗E = 0, i.e., from −→ −→ −1 R = D Eq. 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 −→ D M (also obtained by an integral over the current distribution) and write

−→ −→ −→ D ℂ = D E + iD M .

By allowing the displacement −→R to take complex values, −→R ℂ, Eq. (6.37View Equation), can be generalized to

−D→∗ = −→D − q−→R , (3.46 ) ℂ ℂ ℂ
so that the complex center-of-charge is given by −→ D ∗ℂ = 0 or
−→ −→ R ℂ = D ℂq−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.

3.4.2 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 [50Jump To The Next Citation Point, 38] resulting in the radial behavior (the peeling theorem):

ϕ00 −4 ϕ0 = -3-+ O (r ), (3.48 ) r0 ϕ1 = ϕ1-+ O (r−3), r2 ϕ02 −2 ϕ2 = ---+ O (r ), r
where the leading coefficients of r, (0 0 0 ϕ0,ϕ1,ϕ2) satisfy the evolution equations:
˙0 0 ϕ0 + ∂ϕ1 = 0, (3.49 ) ˙ϕ01 + ∂ϕ02 = 0. (3.50 )

The formal integration procedure is to take ϕ02 as an arbitrary function of (uB,ζ, ¯ζ) (the free broadcasting data), then integrate the second, for ϕ01, with a time-independent spin-weight s = 0 function of integration and finally integrate the first, for ϕ0 0. Using a slight modification of this, namely from the spherical harmonic expansion, we obtain,

ϕ00 = ϕ00iY11i + ϕ00ijY21ij + ..., (3.51 ) 0 0 0 0 0 ϕ1 = q + ϕ1iY1i + ϕ 1ijY2ij + ..., (3.52 ) ϕ02 = ϕ02iY1−i1 + ϕ02ijY −2i1j + ..., (3.53 )
with the harmonic coefficients related to each other by the evolution equations:
ϕ00 = 2qηi(uret)Y11i + Qiℂj′Y21ij + ..., (3.54 ) √ -- ϕ0 = q + √2q-ηi′(u )Y 0 + --2Qij′′Y 0 + ..., 1 ret 1i 6 ℂ 2ij 0 i′′ −1 1 ij′′′ − 1 ϕ2 = − 2qη (uret)Y1i − 3Q ℂ Y2ij + ...
The physical meaning of the coefficients are
q = total electric charge, (3.55 ) qηi = Diℂ = complex (electric & magnetic ) dipole moment = DiE + iDiM, ij Q ℂ = complex (electric & magnetic ) quadrupole moment,
etc. Recall from Section 1.1 that this electromagnetic quadrupole needs to be rescaled (√ -- Qij → 2 2Qij ℂ ℂ) to obtain the physical quadrupole which appears in the usual expressions for Maxwell theory [43Jump To The Next Citation Point]. For later use, the complex dipole is written as i i D ℂ(uret) = qη (uret). Note that the i Dℂ 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. (6.37View Equation).

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 i η(uret). From this complex world line we can define the intrinsic complex dipole moment, i i D ℐℂ = qξ (s).

However, we first discuss a particular Maxwell field, F ab, 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.

3.4.3 A coordinate and tetrad system attached to a shear-free NGC

The parametric form of the general NGC was given earlier by Eq. (3.2View Equation),

xa = u (ˆla + ˆna) − Lmˆa − ¯L ˆma + (r∗ − r )ˆla. (3.56 ) B 0
The parameters (uB, ζ, ¯ζ) 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 ∗ ¯ (uB,r ,ζ, ζ). 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. (3.56View Equation). We restrict ourselves to the special case of the coordinates and tetrad associated with the L 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, a a z = ξ (τ ). The complex parameter, τ, must in the end be chosen so that the ‘uB’ of Eq. (3.19View Equation) is real. The Minkowski metric and the spin coefficients associated with this geodesic system can be calculated [40Jump To The Next Citation Point] in the (uB,r∗,ζ,ζ¯) frame. Unfortunately, it must be stated parametrically, since the τ explicitly appears via the a ξ (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, Fabl∗a = λl∗b), 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

∗ ∗a ∗b ϕ0 = Fabl m = 0.

This allows a very simple exact integration of the remaining Maxwell components [50Jump To The Next Citation Point].

3.4.4 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 (∗ uB, r ,ζ, ¯ζ), Eq. (3.56View Equation). As the detailed calculations are long [50Jump To The Next Citation Point] and take us too far afield, we only give an outline here. The integration of the radial Maxwell equations leads to the asymptotic behavior,

∗ ϕ 0 = 0, (3.57 ) ϕ ∗= ρ2 ϕ∗0, (3.58 ) 1∗ ∗10 2 ϕ 2 = ρ ϕ2 + O (ρ ), (3.59 )
with
ρ = − (r∗ + iΣ)−1, (3.60 ) -- 2iΣ = ∂¯L + L ˙¯L − ∂L − ¯LL˙.
The O (ρ2) expression is known in terms of (ϕ∗0,ϕ∗0 1 2). The function L (u ,ζ, ¯ζ) B is given by
L(uB, ζ, ¯ζ) = ∂(τ)G (τ,ζ,ζ¯), uB = G (τ,ζ, ¯ζ) = ξa(τ)ˆla(ζ,ζ¯),
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, ϕ∗10,ϕ∗20, are determined by the last of the Maxwell equations,

∂ϕ ∗0+ L ˙ϕ∗0+ 2 ˙Lϕ∗0 = 0, (3.61 ) 1 1 1 ∂ϕ ∗20+ L ˙ϕ∗20+ ˙Lϕ∗20 = ϕ˙∗10,
which have been obtained from Eqs. (2.55View Equation) and (2.56View Equation) via the null rotation between the Bondi and geodesic tetrads and the associated Maxwell field transformation, namely,
a ∗a a ¯L a L a ∗−2 l → l = l − --m − --¯m + O (r ), (3.62 ) r r ma → m ∗a = ma − Lna, (3.63 ) r na → n∗a = na, (3.64 )
with
ϕ ∗0= 0 = ϕ0− 2Lϕ0 + L2 ϕ0, (3.65 ) 0∗0 0 0 0 1 2 ϕ 1 = ϕ 1 − L ϕ2, (3.66 ) ϕ ∗0= ϕ0 . (3.67 ) 2 2

These remaining equations depend only on L (uB,ζ, ¯ζ), 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. (3.61View Equation) appear to be difficult to solve, partially due to the implicit description of the ¯ L (uB,ζ,ζ).

Actually they are easily solved when the independent variables are changed, via Eq. (3.15View Equation), from (u ,ζ,ζ) B to the complex (τ,ζ,ζ). They become, after a bit of work,

2 0 ∂ (τ)(V ϕ1) = 0, (3.68 ) ∂ (τ)(V ϕ02) = ϕ01′ , (3.69 ) a′ V = ξ (τ)ˆla(ζ, ¯ζ), (3.70 )
with the solution
q ϕ ∗10= -V −2, (3.71 ) 2q -- ( ) ϕ ∗20= -V −1∂(τ) V −1∂τV . 2
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 a ξ (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. (3.62View Equation) – (3.67View Equation), back to a Bondi coordinate/tetrad system. This can only be done by approximations. After a lengthy calculation [50Jump To The Next Citation Point], we find the Bondi peeling behavior

ϕ = r− 3ϕ0+ O (r−4), (3.72 ) 0 0 ϕ1 = r− 2ϕ01 + O (r−3), ϕ = r− 1ϕ0+ O (r−2), 2 2
with
( 1 -- ) ϕ00 = q LV −2 + -L2V −1∂(τ)[V −1V ′] , (3.73 ) 2 ϕ0 = -q--(1 + LV ∂- [V− 1V′]) , (3.74 ) 1 2V 2 (τ) 0 q- −1-- −1 ′ ϕ 2 = − 2V ∂(τ)(V V ), (3.75 ) V = ξa′ˆl (ζ, ¯ζ). (3.76 ) a
Next, treating the world line, as discussed earlier, as a small deviation from the straight line, ξa(τ) = τδa0, i.e., by
ξa(τ) = (τ,ξi(τ)), ξi(τ) ≪ 1.
The GCF and its inverse (see Section 6) are given, to first order, by
√ -- √ -- √2-- -- uret = 2uB = 2G = τ − ---ξi(τ)Y01i(ζ,ζ ), (3.77 ) √ -- 2 --2- i 0 -- τ = uret + 2 ξ(uret)Y1i(ζ,ζ). (3.78 )

Again to first order, Eqs. (3.73View Equation), (3.74View Equation) and (3.75View Equation) yield

ϕ00 = 2qξi(uret)Y11i, (3.79 ) ϕ0= q + √2q-ξi′(u )Y 0, 1 ret 1i ϕ02 = − 2qξi′′(uret)Y −1i1 ,
the known electromagnetic dipole field, with a Coulomb charge, q. One then has the physical interpretation of i qξ (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, i ξ (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. (2.10View Equation) and (3.65View Equation) to the geodesic system, it would have resulted in

∗ ϕ0 = 0.

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.

3.4.5 Asymptotically vanishing Maxwell fields & shear-free NGCs

We return now to the general asymptotically-vanishing Maxwell field, Eqs. (3.48View Equation) and (3.51View Equation), and its transformation behavior under the null rotation around a n,

a ∗a a ¯L a L a −2 l → l = l − --m − -m¯ + 0(r ), (3.80 ) r r ma → m ∗a = ma − L-na + 0(r−2), r na → n∗a = na,
with L (u ,ζ,ζ-) = ξa(τ )mˆ B a, being one of our shear-free angle fields defined by a world line, za = ξa(τ) and cut function a ˆ -- uB = ξ (τ)la(ζ,ζ ). The leading components of the Maxwell fields transform as
ϕ ∗0= ϕ0− 2Lϕ0 + L2 ϕ0, (3.81 ) 0∗0 00 01 2 ϕ 1 = ϕ1 − Lϕ2, (3.82 ) ϕ ∗0= ϕ0. (3.83 ) 2 2

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, a ξ (τ).) From the physical identifications of charge, dipole moments, etc., of Eq. (3.54View Equation), we can obtain the transformation law of these physical quantities. In particular, the l = 1 harmonic of ϕ00, or, equivalently, the complex dipole, transforms as

ϕ0∗= ϕ0 − 2(L ϕ0)| + (L2ϕ0)|, (3.84 ) 0i 0i 1 i 2 i
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, -- η(s)(uB, ζ,ζ), given in a specific Bondi coordinate system, -- (uB,ζ,ζ ), that has a harmonic expansion given, for constant uB, by

η (u ,ζ,ζ) = Σ ηl,(ijk...)(u )Y (s) (s) B l,(ijk...) (s) B l,(ijk...)

If exactly the same function was given on different cuts or slices, say,

u = G (τ, ζ,ζ), (3.85 ) B
with
∗ -- ( -- -) η(s)(τ,ζ,ζ) = η(s) G (τ, ζ,ζ),ζ,ζ ,

the harmonic expansion at constant τ would be different. The new coefficients are extracted by the two-sphere integral taken at constant τ:

∫ ∗l,(ijk...) ∗ -- -(s) η(s) (τ ) = η(s)(τ,ζ,ζ )Yl,(ijk...)dS. (3.86 ) S2
It is in this rather obvious sense that the expansions are not unique.

The transformation, Eq. (3.84View Equation), and harmonic extraction implemented by first replacing the uB in all the terms of all 0 0 0 ϕ0,ϕ 1,ϕ 2, by a ˆ -- √2τ- 1 i 0 uB = ξ (τ)la(ζ, ζ) ≡ 2 − 2ξ Y1i, yields 0∗ ϕ0i(τ ) with a functional form [8Jump To The Next Citation Point],

∮ ϕ0∗0i(τ) = Γ i(ϕ00,ϕ01,ϕ02,ξa(τ)) = [ϕ00 − (2cLϕ01 − c2L2ϕ02)]Y1−i1dS (3.87 ) S2
is decidedly nontensorial: in fact it is very nonlocal and nonlinear.

Though it is clear that extracting ϕ0∗(τ) 0i 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 a ξ (τ ) could be chosen so that from the associated complex null cones we would have ∗0 ϕ0 = 0. However, though this cannot be done in general, the l = 1 harmonics of ∗0 ϕ0 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.

3.4.6 The complex center of charge

The complex center of charge is defined by the vanishing of the complex dipole moment ϕ00∗i(τ); in other words,

Γ i(ϕ00,ϕ01,ϕ02,ξa) = 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 q as zeroth order and the dipole moments and the spatial part of the complex world line as first order.

From Eq. (3.84View Equation),

0∗ 0 0 ϕ0i = Γ i ≈ ϕ0i − 2L ϕ1|i = 0 (3.89 )
with the identifications, Eq. (3.54View Equation), for q and D ℂ, we have to first order (with √ -- 2uB = uret ≈ τ),
i i i D ℂ(uret) = qη (uret) = qξ (uret). (3.90 )
This is exactly the same result as we obtained earlier in Eq. (3.47View Equation), 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, i D ℐ:ℂ, and the complex dipole, i D ℂ,

i i i i D ℐ:ℂ = qξ (s), D ℂ = q√η-(s), (3.91 ) k k i i j′ 2 −1 ik′′ i ξ = η − --η η 𝜖ijk − ---q Qℂ ξ , (3.92 ) 2 1√0-- ηk = ξk + i1ξiξj′𝜖 + --2q−1Qik′′ξi (3.93 ) 2 ijk 10 ℂ

The GCF,

uB = X (τ,ζ, &tidle;ζ) = ξa(τ)ˆla(ζ, &tidle;ζ),

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.


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