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Asymptotic directional structure of radiation for fields of algebraic type D

The directional behavior of dominant components of algebraically special spin-s fields near a spacelike, timelike or null conformal infinity is studied. By extending our previous general investigations we concentrate on fields which admit a pair of equivalent algebraically special null directions, such as the Petrov type D gravitational fields or algebraically general electromagnetic fields. We introduce and discuss a canonical choice of the reference tetrad near infinity in all possible situations, and we present the corresponding asymptotic directional structures using the most natural parametrizations.


Introduction
In the series of papers [1,2,3,4] we analyzed the asymptotic directional properties of electromagnetic and gravitational fields in spacetimes with a nonvanishing cosmological constant Λ. It had been known for a long time [5,6,7] that -contrary to the asymptotically flat spacetimes -the dominant (radiative) component of the fields is not unique since it depends substantially on the direction along which a null geodesic approaches a given point at conformal infinity I. We demonstrated that, somewhat surprisingly, such directional structure of radiation can be described in closed explicit form. It has a universal character that is essentially determined by the algebraic type of the field, i.e., by the specific local degeneracy and orientation of the principal null directions.
Our results were summarized and thoroughly discussed in the recent topical review [8]. They apply not only to electromagnetic or gravitational fields but to any field of spin s. In addition, the expression representing the directional behavior of radiation can be written in a unified form which covers all three possibilities Λ > 0, Λ < 0 or Λ = 0, corresponding to a spacelike, timelike or null character of I, respectively. This paper further elaborates and supplements some aspects of our work reviewed in [8]. It extends possible definitions of the canonical reference tetrad for the algebraically simple fields -those which admit an equivalent pair of distinct (degenerate) principal null directions. We systematically describe the most natural choices of the reference tetrad for all possible algebraic structures of type D fields, and for any value of Λ.
The notation used in the present paper is the same as in [8]; in fact, we will frequently employ the material already described and derived there. For brevity, we will refer directly to the equations and sections of the review [8] by prefixing the letter 'R' in front of the reference: equation (R.1.1), section R.2.1, etc.

Summary of general results
We wish to study the behavior of radiative component of fields near conformal infinity I. An overview of the concept of conformal infinity can be found in textbooks (e.g., [9]; our notation is described in section R.2 of the work [8]).
Let us only recall that it is possible to define a normalized vector n normal to the conformal infinity. The causal character of the infinity -spacelike, null, or timelike -is given by the sign of the square of this vector, σ = n · n = −1, 0, +1 (see also figure 1). Here and in the following, the dot '·' denotes the scalar product defined using the spacetime metric g. Typically, the causal character of I is correlated with the sign of the cosmological constant, σ = − sign Λ (see section R.2.2 for details).

Null tetrads
To study various components of the fields we introduce suitable orthonormal, and associated with them null tetrads. We denote the vectors of an orthonormal tetrad as t, q, r, s, where t is a future oriented unit timelike vector. With this tetrad we associate a null tetrad of null vectors k, l, m,m by The normalization conditions are respectively, with all other scalar products being zero. The crucial tetrad in our study is the interpretation tetrad k i , l i , m i ,m i . It is a tetrad which is parallelly transported along a null geodesic z(η), the vector k i being tangent to the geodesic. With respect to this tetrad we define the radiative component of the field. The precise definition and description of asymptotic behavior of the interpretation tetrad was thoroughly presented in sections R.3.3 and R.3.4 where more details can be found.
Here, we are going to concentrate on the reference tetrad k o , l o , m o ,m o . It is a tetrad conveniently defined near the conformal infinity. It serves as the reference frame for parametrization of directions near I. It can be defined using special features of the spacetime geometry (e.g., the Killing vectors, direction toward sources, etc.). Alternatively, it can be adapted to the studied fields -namely, it can be 'aligned' with algebraically special directions of the fields under consideration. A general situation was discussed in sections R.5.4 and R.5.5. In the present work we will offer other possible privileged definitions of the reference tetrad for algebraically simple fields of type D. Fig. 1. The reference tetrad adjusted to conformal infinity I of various character, determined by σ which is a norm of the vector n normal to I. If the vector ko is oriented along an outgoing direction (outward from the physical spacetime M) we have ǫo = +1, if it is ingoing (oriented inward to M) then ǫo = −1.
Following [8] we require that the reference tetrad is adjusted to conformal infinity, i.e., that the vectors k o and l o satisfy the relation where the sign ǫ o = ±1 indicates the outgoing/ingoing orientation of the vector k o with respect to I (see also below). This adjustment condition guarantees that the vectors k o and l o are colinear with the normal n to I, and normalized such that Czech. J. Phys. 55 (2005) All possible orientations of the reference tetrad with respect to I are shown in figure 1. The normalization and adjustment conditions do not fix the reference tetrad uniquely. Additional necessary conditions -the alignment with the algebraically special directions -will be specified in section 3.

Parametrization of null directions
In the following it will be necessary to parametrize a general null direction k near I. This can be done with respect to the reference tetrad by a complex directional parameter R: The value R = ∞ is also permitted -it corresponds to k oriented along l o . In addition, we introduce the orientation parameter ǫ which indicates whether the null direction k is an outgoing direction (pointing outside the spacetime), ǫ = +1, or if it is an ingoing direction (pointing inside the spacetime), ǫ = −1.
When the infinity I has a spacelike character it is also possible to parametrize the null direction k using spherical angles which specify its normalized spatial projection into I. We define the angles θ, φ by where q is the unit vector pointing into the spatial (q · n = 0) direction given by k, see (R.5.5). The complex parameter R of (5) is actually a stereographic representation of the spatial direction q: Near a timelike infinity I we can analogously describe null direction k by pseudospherical parameters ψ, φ of its projection into I. If we label the normalized projection of k by t, cf. (R.5.8), ψ and φ are given by These parameters have to be supplemented by the orientation ǫ of k with respect to I. The parameter R is the pseudostereographic representation of t: In fact, we can introduce both these parametrizations simultaneously, independently of the causal character of the infinity, just with respect to the reference tetrad -the angles θ, φ using a projection onto the 3-space orthogonal to the time vector t o of the reference tetrad, and the parameters ψ, φ using a projection to the 2+1-space orthogonal to q o . In such a case they are related by expressions tanh ψ = sin θ , sinh ψ = tan θ , cosh ψ = cos −1 θ , tanh ψ 2 = tan θ 2 .
If the infinity has a timelike character, all future oriented null directions at one point at I naturally split into two families of outgoing and ingoing directions. The directions of each of these families form a hemisphere which can be projected onto a unit circle parametrized by ρ and φ, such that see also figure R.3.

Asymptotic directional structure of radiation
The main result of the paper [8] is derivation of the explicit dependence of the radiative component of the field on a direction along which the infinity is approached -we call this dependence the asymptotic directional structure of radiation.
To study the asymptotic behavior we evaluated the field with respect to the interpretational tetrad -the tetrad which is parallelly transported along a null geodesic z(η). It turned out that these field components satisfy the standard peelingoff property, namely that they exhibit a different fall-off in η when approaching I, η being the affine parameter of the geodesic. The leading component of the field is the component Υ i 2s with the fall-off of order η −1 , and we call it the radiative component. In sections R.4.3 and R.4.4 we found that the radiative field component depends on the direction R of the null geodesic along which a fixed point at the infinity is approached. This directional structure is determined mainly by the algebraic structure of the field, and it reads The complex constants R 1 , . . . , R 2s represent the principal null directions k 1 , . . . , k 2s of the spin-s field, the sign σ = ±1, 0 specifies the causal character of the conformal infinity, ǫ o denotes orientation of the reference tetrad, and Υ o 2s * is a constant normalization factor of the field evaluated with respect to the reference tetrad, . The principal null directions (PNDs) are special directions along which some of the field components vanish -see [9,10] or section R.4.2 for a precise definition. The field of spin s has 2s PNDs. However, these can be degenerate, and this degeneracy (or, more generally, mutual relations of all the PNDs) is called the algebraic structure of the field. Distinct PNDs are also called algebraically special directions of the field. The classification according to the degeneracy of PNDs for a gravitational field is the well-known Petrov classification.

Fields of type D
In this paper we wish to discuss the situation when the field has two distinct and equivalent algebraically special directions. This may occur only for fields of an integer spin, s ∈ N, with PNDs having the degeneracy The directional structure (12) of such a field takes the form with the constants R 1 and R 2s parametrizing the two distinct PNDs. The directional dependence of the magnitude of a gravitational type D field and of an algebraically general electromagnetic field are thus quite similar. This similarity is even closer if we recall that the square of the electromagnetic component Φ i 2 is proportional to the magnitude of the Poynting vector with respect to the interpretation tetrad, As discussed in section R.4.5, the form of the directional structure (12) depends on the choice of the normalization factor Υ o 2s * . Other choices can sometimes be more convenient, in particular if the factor Υ o 2s * vanishes which happens when one of the PNDs points along the direction l o of the reference tetrad. For the type D fields there exists a more natural 'symmetric' choice of normalization of the directional structure of radiation which is guaranteed to be non-degenerate. For these fields we can define a canonical field component, namely, the only nonvanishing component with respect of the null tetrad associated with the PNDs (13).
Having two distinct algebraic directions k 1 and k 2s we can define algebraically special null tetrad k s , l s , m s ,m s (and associated orthonormal tetrad t s , q s , r s , s s ) by requiring that k s , l s are proportional to the PNDs and future oriented, and that the spatial vector s s is tangent to I, For PNDs which are not tangent to the conformal infinity the normalization of null vectors k s , l s can be fixed by condition where ǫ 1 , ǫ 2s = ±1 parametrize orientations of the PNDs with respect to I. The special case of PNDs tangent to I will be discussed below.
Using the definition of PNDs (see section R.4.2) we find that the field components with respect to the algebraically special tetrad have very special form -only the component Υ s s is nonvanishing (Ψ s 2 for gravitational and Φ s 1 for electromagnetic fields). This component is, in fact, independent of the choice of the spatial vectors r s , s s , and thus it does not depend on the normal vector n which we used in the definition (17). It also does not depend on the normalization (18), provided that the normalization (2) is satisfied. We will use the privileged component Υ s s for the normalization of the directional structure of radiation. However, the algebraically special tetrad is not adjusted to the infinity (cf. condition (3)) since k s and l s are not in general colinear with n and thus it cannot be used as a reference tetrad.
Nevertheless, we can define privileged reference tetrad which is 'somehow aligned' with the algebraically special tetrad, and which shares some of the symmetries of the geometric situation. We will always assume that the reference tetrad satisfies the normalization and adjustment conditions (2), (3), and we set s o = s s . This is, however, still not sufficient to fix the reference tetrad completely. The remaining necessary condition cannot be prescribed in general -we have to discuss separately several possible cases, depending on the character of the infinity I, and on the orientation of the PNDs with respect to I.
Below we will define the reference tetrad for all possible cases. We will present the relation between the component Υ o 2s and the canonical component Υ s s , which can be substituted into the directional structure (14). Finally, we will rewrite the results in terms of the angular variables introduced with respect to the reference tetrad.

Spacelike I
We begin with a spacelike conformal infinity, σ = −1. In this case the two distinct future oriented algebraically special directions are either both ingoing or both outgoing, and we set the orientation ǫ o of the reference tetrad accordingly. We define the reference tetrad by conditions and by adjustment condition (4). It follows that the algebraically special directions k s and l s are parametrized with respect to the reference tetrad by a single parameter θ s as The algebraically special and reference tetrads are thus related by which is actually a boost in t o -r o plane with rapidity parameter ψ s related to θ s by (10), see figure 2.
Inspecting the spatial projections of k s and l s onto conformal infinity I, we find that their angular coordinates with respect to the reference tetrad are θ 1 = θ s , φ 1 = 0, and θ 2s = π − θ s , φ 2s = 0, respectively. It means that the complex parameters R 1 and R 2s of both these algebraic special directions are Straightforward calculation shows that the transformation (21) from the algebraically special to the reference tetrad can be decomposed into boost (R.
For gravitational and electromagnetic fields we can write explicit expressions for all field components with respect to the reference tetrad as Following the discussion in section R.4.4 we assume asymptotic behavior (R.4.18), i.e., Υ o j ≈ Υ o j * η −s−1 with constant coefficients Υ o j * . Similarly, we introduce the coefficient Υ s s * by Υ s s ≈ Υ s s * η −s−1 . Clearly, the relations (23)-(25) hold also in their 'stared' forms.
Substituting (22), the 'stared' version of (23), (7), and σ = +1 into (14) we finally obtain the asymptotic directional structure of radiation for type D fields The null direction along which the field is measured is parametrized by angles θ, φ, the field itself is characterized by the normalization component Υ s s * and by the parameter θ s which encodes the directions of the algebraically special directions with respect to a spacelike infinity I. As discussed in section R.4.5, only the magnitude of this radiative component has a physical meaning. For the magnitude of the Ψ i 4 component of the gravitational field and for the Poynting vector of the electromagnetic field we thus obtain These are exactly the expressions for the asymptotic directional structure of radiation as derived in [11] for test electromagnetic field of accelerated charges in de Sitter spacetime, and for gravitational filed and electromagnetic fields of the C-metric spacetime with Λ > 0, as presented in [1]. This directional structure is illustrated in figure 3.

Timelike I with non-tangent PNDs, ǫ1 = ǫ2s
Now we will study the situation near a timelike conformal infinity (σ = −1) when both distinct algebraic directions are not tangent to I, such that one of them is outgoing and the other ingoing, ǫ 1 = ǫ 2s . In this case we require that the orientation ǫ o of the reference tetrad is adjusted to ǫ 1 , and that t s is aligned along t o , together with the adjustment condition (4). Again, the algebraically special directions k s and l s are parametrized with respect to the reference tetrad by a single parameter θ s : The algebraically special and reference tetrads are thus related by To read out the normalized projection into I of the null vectors k s , l s it is useful to rewrite (30) in a different way where we have used the parameter ψ s instead of θ s related by (10). Comparing the normalized projections of k 1 = k s and k 2s = l s with (8) we find that the pseudospherical parameters ψ, φ, ǫ of the algebraically special directions are ψ 1 = ψ s , φ 1 = 0, ǫ 1 = ǫ o , and ψ 2s = ψ s , φ 2s = π, ǫ 2s = −ǫ o respectively. The corresponding complex parameters are Again, the transformation (31) can be decomposed into boost, null rotation with k fixed, and null rotation with l fixed, given by B = 2 cosh ψ s (1 + cosh ψ s ) −1 , L = 1 2 tanh ψ s , and K = − tanh(ψ s /2). Applying these transformations to the field components we obtain and, in more detail, for gravitational and electromagnetic fields Substituting (33), 'stared' version of (34), (9), and σ = +1 into expression (14) we obtain the asymptotic directional structure of radiation in the form (37) The direction is given by pseudospherical parameters ψ, φ, ǫ, the field is characterized by the component Υ s s * and by the parameter ψ s which fixes the orientation of the algebraically special directions with respect to infinity I. Again, only the magnitude of component Υ i 2s has a physical meaning, so that This directional structure is illustrated in figure 5(a).  (69)). The arrows indicate mirror reflection with respect to I of the algebraically special directions (PNDs). The radiative component evaluated along the geodesics in these directions is (for non-tangent PNDs) asymptotically vanishing. The radiative component diverges for unphysical geodesics tangent to I (the border of the circles) due to fixed normalization of the null directions -see section R.5.5.

Timelike I with non-tangent PNDs, ǫ1 = ǫ2s
In the previous case we have studied the directional structure of radiation near a timelike I with two PNDs oriented in opposite directions with respect to the infinity. Now, we will discuss the situation when both PNDs are outgoing, or both ingoing, ǫ 1 = ǫ 2s . The derivation of the directional structure is similar to the previous case and we will thus sketch it only briefly. The reference tetrad is fixed by the conditions together with the adjustment condition (4). Therefore where parameters θ s and ψ s are again related by equations (10). The algebraically special and reference tetrads are thus see figure 4(b). Pseudospherical parameters ψ, φ, ǫ of projections of the PNDs into I are ψ 1 = ψ s , φ 1 = 0, ǫ 1 = ǫ o and ψ 2s = ψ s , φ 2s = π, ǫ 2s = ǫ o , respectively, i.e., The transformation from the algebraically special to the reference tetrad can be decomposed into boost (R.3.5), null rotation with k fixed (R.3.4), and null rotation with l fixed (R.3.3) with parameters B = 2 tanh(ψ s /2), L = 1 2 coth(ψ s /2), and K = − tanh(ψ s /2). For the field components we obtain and, in more detail, for gravitational and electromagnetic fields Substituting into the expression (14) we finally obtain A phase of this component is unphysical, its magnitude can be put into the form For gravitational and electromagnetic fields it gives which is illustrated in figure 5(b).
3.4 Timelike I, one PND tangent to I Until now we have concentrated on a generic orientation of algebraically special directions with respect to the conformal infinity. In this and the next sections we are going to study the special cases when the PNDs are tangent to I. This can occur only for timelike or null conformal infinity, the later case will be discussed in section 3.6. First, we assume that only one of two distinct PNDs, say k 2s , is tangent to I. Let us note that in such a case we require normalization (18) only for the vector k s ∝ k 1 , the normalization of the other PND l s is fixed by the condition k s · l s = −1. We use the PND k s as the vector k o , i.e., we define the reference tetrad by conditions together with the condition (4). The algebraically special directions k s and l s are then given in terms of the reference tetrad as see figure 4(c). The tetrads are related by (53) Complex parametrizations of k 1 and k 2s are then The transformation from the algebraically special to the reference tetrad is just the null rotation with k fixed with L = −1. Applying this transformation we obtain specifically for s = 2, 1, Substituting into (14) we get the asymptotic directional structure of radiation i.e., for gravitational and electromagnetic field, see figure 5(c) and (R.5.37).
3.5 Timelike I, two PNDs tangent to I Next, let both the PNDs are tangent to a timelike conformal infinity. In such a situation there exists no natural normalization of both PNDs analogous to the condition (18) used above. This is related to an ambiguity in a choice of the timelike unit vector t s -we can choose any of the (future oriented) unit vectors in the plane k 1 -k 2s . However, despite the fact that we cannot fix the algebraically special tetrad uniquely, the nonvanishing component Υ s s is independent of this ambiguity: different choices of the special tetrad differ only by a boost in k 1 -k 2s plane, and Υ s s does not change under such a boost. In the following, we arbitrarily choose one particular algebraically special tetrad with respect to which we define the reference tetrad. The reference tetrad thus shares the same ambiguity as the algebraically special tetrad.
The reference tetrad is fixed simply by conditions and (4). PNDs k s and l s are given by so that the algebraically special and reference tetrads are related by It is just a simple spatial rotation by π/2 in q o -r o plane, as illustrated in figure  4(d). The complex directional parameters of k 1 and k 2s are The transformation can be decomposed into boost B = 2, null rotation with k fixed L = −1/2, and null rotation with l fixed K = 1. Applying these we obtain and Substituting into (14) we get which for gravitational and electromagnetic fields gives see figure 5(d) and (R.5.36).

Null I
Finally, we investigate the case of conformal infinity I of a null character, σ = 0. It can be easily observed from (12) (cf. section R.5.1 for more detail) that the directional structure of radiation near the null infinity is independent of the direction along which the infinity is approached. The only interesting question is whether the dominant field component is vanishing or not -in other words: whether the field is radiative or nonradiative. It follows from the definition of PNDs that the normalization factor Υ o 2s * in (12) is vanishing if and only if one of the PNDs is tangent to I. Thus, the tangency of PNDs to I serves as the geometrical characterization of radiative/nonradiative fields.
Let us first assume that one of the PNDs, say l s , is tangent to I. As in the previous section we cannot fix the normalization of algebraically special tetrad using the condition (18); the algebraically special tetrad cannot be selected uniquely. Nevertheless, the field component Υ s s is still unique. If we choose one algebraically special tetrad we can use it also as the reference tetrad -it satisfies the adjustment condition (4), cf. figure 6. As mentioned above, the component Υ o 2s is then vanishing: If both distinct PNDs are not tangent to I we may normalize them by (18) and fix the reference tetrad by the condition t o = t s , namely, together with the adjustment condition (4), see figure 6. In terms of the reference tetrad the PNDs k s and l s are given by and their complex parameters are The relation between the algebraically special and reference tetrads is thus the same as in section 3.5. Using (12), σ = 0, and relation (65) we find that the radiative component has no directional structure: (75) Fig. 6. Algebraically special and reference tetrads at a null infinity. Vectors ko, lo (ks, ls) of the null tetrad, and to, q o , ro (ts, q s , rs) of the orthonormal reference (algebraically special, respectively) tetrad are shown; the direction so = ss tangent to I and orthogonal to PNDs is not plotted. The vectors ks, ls are aligned with algebraically special directions. The diagram (a) depicts the situation with one PND tangent to I. In this case the algebraically special tetrad can be used as the reference tetrad. In the diagram (b) neither of both distinct PNDs is tangent to I. The algebraically special tetrad can be then obtained from the reference tetrad by a spatial rotation in q o -ro plane by π/2.
In particular, for gravitational and electromagnetic field we obtain

Conclusions
We have analyzed the asymptotic directional structure of fields that are characterized by the existence of two distinct but equivalent algebraicaly special null directions. This involves a generic electromagnetic field (s = 1), the Petrov type D gravitational fields (s = 2) having double degenerate principal null directions, and other possible fields of an integer spin s which admit a pair of s-degenerate PNDs.
The structure of such fields near the conformal infinity depends on the specific orientation of these algebraically special directions with respect to I, and on the causal character of I. In case of a spacelike conformal infinity (Λ > 0) there is essentially only one possibility which is described in section 3.1, whereas for a timelike conformal infinity (Λ < 0) four different situations may occur that have to be discussed separately, see sections 3.2-3.5. For the conformal infinity having a null character (Λ = 0) the asymptotic directional structure disappears: when one of the (degenerate) PNDs is tangent to I the radiative component vanishes, otherwise the radiation is present and it is independent of the direction along which the infinity is approached, cf. section 3.6.
In all such cases we have introduced the privileged 'symmetric' reference tetrad which is naturally adapted to the algebraically special directions and to I. These are illustrated in figures 2, 4, and 6. With respect to these reference tetrads it is possible to characterize any null direction by standard (pseudo)spherical parameters. The corresponding explicit directional structure of radiation for a spacelike I is presented in expression (26), and the four possibilities for a timelike I are given by (37), (47), (58), (68).
These results generalize our previous study of the asymptotic directional structure of gravitational and electromagnetic radiation in the C-metric spacetimes [1,2] to other fields which are of type D. On the other hand, the expressions presented here are more detailed and more explicit than those given in the review article [8]. It would now be an interesting task to apply them on particular exact model spacetimes of type D. This may provide a deeper insight into the geometric relation between the structure of the sources and the properties of radiation generated by them, as observed at spacelike or timelike conformal infinities.