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Kerr-Schild Approach to the Boosted Kerr Solution

Using a complex representation of the Debney-Kerr-Schild (DKS) solutions and the Kerr theorem we analyze the boosted Kerr geometries and give the exact and explicit expressions for the metrics, the principal null congruences, the coordinate systems and the location of the singularities for arbitrary value and orientation of the boost with respect to the angular momentum. In the limiting, ultrarelativistic case we obtain light-like solutions possessing diverging and twisting principal null congruences and having, contrary to the known pp-wave limiting solutions, a non-zero value of the total angular momentum. The implications of the above results in various related fields are discussed.


Introduction
Recently, boosted black hole solutions attracted a renewed interest in connection with numerical simulations of black hole interactions [1]. On the other hand, the problem of finding the ultrarelativistic limit of exact, particlelike solutions of the Einstein field equations received considerable attention in connection with non-trivial gravitational effects which are expected to occur in the interparticle interactions at extreme energies due to the presence of gravitational shock waves [2,3].
First results in this field were obtained by Aichelburg and Sexl [4], who considered the behaviour of the Schwarzschild metric under ultrarelativistic boost. Because of spherical symmetry the results are in this case independent on the direction of the boost.
A similar treatment in the case of the Kerr geometry, which can be considered as a model of a spinning particle in General Relativity, has to take into account the orientation of the angular momentum with respect to the boost. Unlike the simplest twist free case the problem exhibits extra difficulties which enforces quite complicated and refined methods of analysis and usually leads to very complicated or approximate expressions before taking the ultrarelativistic limit [5,6,7]. In particular, difficulties appear when ultrarelativistic limits are involved, due to the singular character of Lorentz transformations at v = c. We should note that this singularity a priori can lead to different limiting results depending on the performed limiting procedure.
The approach which we are going to formulate here deals with a class of sourceless gravitational solutions of Kerr-Schild class and it is based on the Debney, Kerr and Schild formalism [8] (DKS) and on the Kerr theorem [9,10,11]. It gives the possibility of obtaining exact and explicit expressions for the boosted Kerr geometry by arbitrary values and orientations of the boost with respect to the angular momentum. As a result, in the general cases of the boost, including the ultrarelativistic cases, we determine the exact expressions for the metric, coordinate system, principal null congruence, and location of singularity. In the ultrarelativistic cases this method leads to a DKS-class of solutions possessing diverging and twisting principal null congruences contrary to the pp-wave limiting solutions which posses zero total angular momentum [7].
The paper is organized as follows. First, we briefly recall the DKS-formal-ism and the Kerr theorem in a form, in which the Kerr solution is represented as being generated by a complex source. This approach was initiated by Lind and Newman [12,13] and was considered in the DKS-formalism in [14] (a more complete description of this approach and of its geometrical basis can be found in [15]). Using this representation, the boosted Kerr solutions can be constructed simply by considering straight lines in complexified Minkowski space as complex world lines of the sources. In this way, we obtain explicit expressions for the metric and the singular regions in the most representative cases. We discuss various physical applications and some unusual features in the limiting behaviour of the Kerr singular ring.

The DKS formalism and the Kerr theorem
In the notation we follow the work of Debney, Kerr and Schild [8] (see also [9,16] for a review on DKS solutions). In the four-dimensional space-time with signature (− + ++), let e 1 , e 2 , e 3 , e 4 be a null tetrad satisfying The vectors e 3 and e 4 are real null vectors, while e 1 and e 2 are complex conjugates. The general Kerr-Schild metric can be written as where h is a scalar function and the principal null direction e 3 is null also with respect to the auxiliary Minkowski space with metric In the above formula, the null coordinates ζ,ζ, u, v are related to the Cartesian coordinates by A general field of null directions in Minkowski space can be defined by where Y (x) is a complex function. The Kerr theorem gives a rule to construct the geodesic, shear free congruences: the general geodesic, shear-free null congruence in Minkowski space is defined by a function Y which is a solution of the equation where F is an arbitrary analytic function of the projective twistor coordinates (the above parameters can be written in twistor notation (µ A , ψȦ), µ A = x µ σ AȦ µ ψȦ as (λ 1 , λ 2 , Y, 1) = (µ 0 , µ 1 , ψ˙0, ψ˙1)/ψ˙1.) The Kerr theorem also allows to obtain other important parameters of the solution. In particular, the quantitỹ is a complex radial distance, which is connected with the complex representation of the Kerr solution, as will be explained below.
The singularities of the metric can be defined as the caustics of the congruence given by the system of equations The Kerr solution belongs to the sub-class of solutions having singularities contained in a bounded region [14,17]. In this case the function F must be at most quadratic in Y : where the coefficients c and p are real constants and a 0 , a 1 , a 2 , q,q, are complex constants. The solutions of the equations (8) can be found in this case in explicit form. It can be shown that they correspond to the Kerr solution up to a Lorentz boost and a shift of the origin.

Congruences generated by a complex world line
Our approach is based on the complex world-line representation of the Kerr solution which was initiated by Newman and Lind [12,13]. It allows to represent the Kerr solution as a retarded-time field, generated by a complex source propagating along a complex world-line. The structure of this representation in DKS-formalism gives a very convenient form of the function F , which turns out to be dependent on the coordinates of the world-line [14,15]. For the aim of convenience we give here a short review of this approach. Let x µ 0 (τ ) be a complex world line parameterized by a complex time parameter τ = t + iσ. The coordinates of this world line are complex, 4 , so thatζ 0 and ζ 0 are not necessarily complex conjugates.
The function F can be expressed in the form where the twistor components with zero indices denote the values of λ 1 and λ 2 on the points of the complex world-line x 0 (τ ), while K is a Killing vector of the solution, whose action on a scalar f is defined by (a dot denotes derivative with respect to τ ). It has been shown [15] that the form of F given by (10) is equivalent to (9). In this representation, the Kerr congruence can be described via a retardedtime construction. For example, the Schwarzschild and the Kerr metrics correspond to the world line of a particle at rest at the origin and to the world line of a particle at rest "at a distance ia from the origin", respectively [13]. The general case of a solution with a boost may be obtained considering a straight complex world line with 3-velocity V in CM 4 Writing the function F in the form where from F = 0 we obtain the following explicit solutions for the function Y (x): where ∆ = (B 2 − 4AC) 1/2 . On the other hand from equations (7) and (14) one obtains and This relation reflects the "twofoldedness" of the Kerr geometry: the complex radial coordinater can be expressed as r + ia cos θ and the double sign corresponds to a transition from the "positive" r sheet of the metric to the "negative" one where r ≤ 0.
In the DKS notation the metric can be written as where The field e 3 can be normalized by introducing l µ = e 3µ /P so thatẋ µ 0 l µ = 1, and this yields the following, equivalent form of the metric: The complex radial distance (7) is given by and the twistor parameters λ 1 and λ 2 may be represented in the form (the explicit form of the DKS-tetrad e a is given in the Appendix).

Boosted Kerr solution: examples and behaviour of singularities
As we have seen, there is a one-to-one correspondence between straight lines in complex Minkowski space and the class of the DKS solutions having singularities contained in a bounded region. Since such solutions are equivalent to the Kerr one up to Poincaré' transformations, it is clear that "boosting" the Kerr solution by a velocity V is equivalent to consider a DKS solution generated by a particle moving with a speed V in complex Minkowski space. This motion can be represented in the form x µ 0 (τ ) = {τ, x 0 (0) + V τ }, and the complex parameter τ may be always chosen in such a way that Reτ corresponds to the "real time"t (see [13] for details on complex Minkowski space). The complex initial displacement can be decomposed as In the case of a boost orthogonal to the direction of d, this vector is not altered by Lorentz contraction This shows that the parameter a coincides with its rest value a 0 if d and V are orthogonal, while if V and d are collinear.
In order to calculate the parameters A, B, C it is convenient to express the complex world line in null coordinates The Killing vector of the solution will then be while the function P takes the form The complex radial coordinater ≡ P Z −1 is given by (18). As for the standard Kerr solution, one can representr as a "sum" of the real radial distance r and an angular coordinate. Then equation (18) can be used to fix the relation between the polar coordinates r, θ, φ and the null Cartesian coordinates (26) through the expressions (15) for the coefficients A, B, C. Due to the formula (7), the singular regions are defined by the zeros of the functionr. In what follows, we present some examples of boosted Kerr solutions and then discuss the general features exhibited by them.

Example I
A spinning particle moves with the speed of light in the positive direction of the z-axis, and the 3-vector d = (0, 0, a) is also directed along the z-axis. The complex world line is therefore given by t 0 (τ ) = τ , z 0 (τ ) = ia + τ, x 0 (τ ) = y 0 (τ ) = 0. In null coordinates this yields

Formula (3.3) implies that the coefficients A, B, C are given by
As a result the function F acquires the form and the solution of the equation F = 0 is and therefore the metric has no singularities (there is no real solution to the system of equations (8)). On the other hand, setting a = 0 we obtain the case of spinless particle, and a moving singular plane is placed at z = t [4]. Therefore there is no smooth limit as a → 0.

Example II
A spinning particle moves with the speed of light in the positive direction of the x-axis, orthogonal to the 3-vector d which defines the direction and the value of the angular momentum J = m(0, 0, a), a = | d|. We have the complex world line t 0 (τ ) = x 0 (τ ) = τ, y 0 (τ ) = 0, z 0 (τ ) = ia . Correspondingly, the world line in null coordinates is and the velocities are The functionr ≡ P Z −1 takes the form This solution is, therefore, singular: there is a moving singular plane placed at x = t.

Example III
The absence of a smooth limit in the first example may be better understood considering the general case in which the value of the velocity is arbitrary as well as its direction with respect to the angular momentum. Without loss of generality, we can consider the boost performed with a parameter α in the zdirection (α = v z /c), and a parameter β in the x-direction (β = v x /c), while the angular momentum is defined by d = (0, 0, a). Denoting w 2 = α 2 + β 2 the following general formula for the coordinate relations can be obtained: The singular region r = 0, cos θ = 0 is placed on the plane z = αt and is described by Let us first consider the cases corresponding to the two examples above. If β = 0 (boost in the direction of the angular momentum) the singularity is a ring of radius a 0 = a/ √ 1 − α 2 located on the moving plane z = −αt. In this case, the coordinate relations are the following: The radius of the ring grows as α increases, and for α → 1 the singularity goes at infinity. At first sight, this result looks strange. However, it is easy to check that being to reexpressed via a 0 , the rest value of a, the position of singular region becomes a ring of constant radius a 0 . However, keeping a = const. = 0 and taking the limit α = 1 we obtain that singular region is going to infinity, so there will not be singularity in finite region, in agreement with results of Example I. We will discuss this situation later in connection with the problem of renormalization of parameters.
In the case α = 0, the boost is performed with a speed orthogonal to the direction of angular momentum. The singular region is a moving ring oblate in the x direction with a Lorentz factor √ 1 − β 2 and located on the plane z = 0. In this case the coordinate relations are: As in the previous example, the limit β → 1 is not smooth since the singular region is a line parallel to the y axis placed at x = t, z = 0.
In the general case described by equation (37) the singularity is a moving ring on the z = αt plane, deformed in the x direction by a factor (1 − β 2 )/(1 − α 2 ) and in the y direction by a factor (1 − β 2 )/(1 − w 2 ). The ultrarelativistic limit corresponds to w = 1 and the singular region is a couple of straight lines parallel to the y axis. Therefore, we can conclude that the non-smoothness and the non-commutativeness of the limiting procedure is a general feature of the boosted Kerr solutions. Another peculiarity, which can be seen by the analysis of the above examples is a non-trivial coordinate dependence of the function Y which forms the principal null congruence. As a consequence the congruence itself acquires a non-trivial coordinate dependence and a non-zero expansion θ and twist ω. This property is conserved even in the ultrarelativistic limit. For instance in the case of Example I the expansion and twist of the congruence are defined by Z = θ + iω [8] and are given by . One sees that there is no singularity in this case, and expansion tends to zero only at the z = t plane where the twist takes the constant value 1/a.

Concluding remarks
There are three different physical situations which should be described by the boosted Kerr solution.
The first is connected with the original Aichelburg-Sexl problem, namely the description of the gravitational field of light-like particles with or without spin. For this case the problem of "renormalization" of the parameters of the solution has been discussed by many authors [5,7]. Indeed the light-like particle must have a infinitely small rest mass in such a way that the boosted momentum will be finite. Similarly, "renormalization" of other parameters, such as charge and angular momentum, has been discussed [6,7], and there is no yet an unique agreement concerning this renormalization procedure. The above considerations on the behaviour of the singular ring under the boost in the orthogonal direction suggest, however, that the physical most satisfactory way to perform the "renormalization" in this case should be to keep J = ma =const. In fact in this way the projection of the angular momentum on the direction of the boost is invariant with respect to the value of the boost. One can come to this conclusion also by considering spinning particles in a quantum context, since the projection of the spin on the boost direction is the helicity which indeed must be considered as a constant. In terms of the rest values a 0 and m 0 we have to put J = ma = m 0 a 0 = const. As far as m = m 0 / √ 1 − v 2 by the boost, this yields a 0 = a/ √ 1 − v 2 . Therefore, for finite values of a the rest value a 0 and the location of the singularity tend to infinity in the ultrarelativistic limit, explaining the results of Example III, and clarifying the absence of the singularity in the ultrarelativistic limit of Example I.
One should note that, keeping the value a 0 = const. during the limit, one enforces a fixed size of the singular ring, so that putting in the limit m 0 = 0 one obtains in fact a limiting twist-free solution with J = m 0 a 0 → 0. This corresponds to the known results on pp-wave limiting metrics [5,6,7,18,19] with a finite size of the singular ring and, correspondingly, vanishing total angular momentum [7].
These arguments are not valid for Example II since the corresponding projection of angular momentum is initially zero, and we have here a twistfree solution with a finite location of singularity.
The second application, which has lead to the recently renewed interest in this problem, consists in modelling the gravitational field of elementary particles with finite rest mass under the boost. A specific feature of this case is that the rest mass m 0 as well as the projection of the angular momentum J = m 0 a 0 = ma have to be kept constant. This leads to m 0 = const., and consequently we obtain a finite position of singularity which is determined by the value of a 0 . 3 However, the parameter of the solution a has to be scaled by the boost as a = a 0 √ 1 − v 2 . In both problems described above one deals with naked singularities rather than black holes, since the values of mass, spin and charge of elementary particles typically correspond to this kind of solutions.
The third physical problem is connected with astrophysical applications [1] of the boosted black hole solutions. In this case also the behavior of the horizon and of the ergosphere under the boost are of interest. A simple analysis using the above suggested coordinates shows that the horizon as well as the ergosphere are simply given by the known formulae for the Kerr case where m must be the relativistic mass parameter.
The method proposed here allows to describe in explicit form the metric and the behaviour of the singular region of the Kerr solution under arbitrary boost and with arbitrary orientations of the angular momentum. In particular, we have shown that the Kerr theorem automatically allows to obtain the exact form of the boosted solution in an asymptotically flat coordinate system and the equations describing the singularities in these coordinates. The ultrarelativistic limit is a singular point of the Lorentz transformations, and we have obtained a quite general picture of the non-smoothness and non-commutativeness of the limits a → 0, v → 1 and r → 0. The method shows that light-like limits of the Kerr geometry exist which belong to DKSclass and have twisting principal null congruences and non-zero total angular momentum J. The results can be easily extended also to the boost of the Kerr-Newman solution and the Kerr-Sen [20] solution generalizing the Kerr solution to low energy string theory. It was shown in [21], the one of the principal null congruences retains its properties to be geodesic and shear free , and that the Kerr theorem remains valid for the Kerr-Sen solution too [22].
The inverse tetrad has the form It was shown in [8] that