The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Comparison of the Bondi{Sachs and Penrose Approaches to Asymptotic Flatness J. Tafel S. Pukas Vienna, Preprint ESI 734 (1999) August 5, 1999 Supported by Federal Ministry of Science and Transport, Austria Available via

2 COMPARISON OF THE BONDI-SACHS AND PENROSE APPROACHES TO ASYMPTOTIC FLATNESS J. Tafel and S. Pukas Insitute of Theoretical Physics, University of Warsaw, Ho_za 69, Warsaw, Poland, Abstract. Relations between the Bondi-Sachs approach and the Penrose conformal technique for asymptotically at metrics are reviewed. Conditions on the conformal factor and the Ricci tensor are examined in order to compare the two approaches. The Bondi- Sachs coordinates are constructed (up to 0( 3 )) for a class of Robinson-Trautman metrics. Some solutions within this class (with pure radiation elds) are given. 1.Introduction At present there are two main approaches to the problem of gravitational radiation from bounded sources in general relativity; that of Bondi [1,2] and Sachs [3] and that of Penrose [4] in the framework of his conformal technique. The rst approach was preceded by pioneering work of Trautman [5] on boundary conditions appropriate for gravitational radiation. Bondi, van der Burg, Metzner [2] and Sachs [3] analyzed the Einstein equations under the assumption that the spacetime metric admits a power series expansion in a distance from sources (measured along a system of null geodesics) with the leading term being the Minkowski metric. The main result was a simple denition of the total gravitational energy (the Bondi mass), which, due to the Einstein equations, is a nonincreasing function of a retarded time (a similar result was obtained earlier in [5], see [6]). Asymptotic properties of the Riemann tensor were obtained in the form of the 'peeling o' theorem [3], which generalizes a description found by Trautman [5]. An important observation was that the asymptotic symmetry group (the Bondi-Metzner-Sachs group) is much larger than the Poincare group. A rst explicit model of gravitational radiation from bounded sources was proposed by Robinson and Trautman [7]. In this model the vacuum Einstein equations reduce to a single equation. The Robinson-Trautman metrics admit a shear free congruence of null geodesics (with vanishing twist). The assumption that such a congruence (with or without twist) exists proved to be very eective in solving the Einstein equations (especially when combined with the Newman-Penrose calculus [9], see [8] and references therein). Unfortunately, none of known vacuum solutions represents (to the knowledge of the authors) a realistic model of gravitational radiation from bounded sources. Penrose [4] encoded the Bondi-Sachs conditions into conformal properties of spacetime. This approach turned out to be very useful in studying asymptotic properties of gravitational elds (see e.g. [10]). An important link between the two approaches was found by Tamburino and Winicour [11] who gave a construction of the Bondi-Sachs coordinates for 1

3 metrics satisfying Penrose's assumptions and the vacuum Einstein equations. The latter assumption was weakened in the work of Persides [12], who investigated conditions on the conformal factor (without referring to the Einstein equations) which are responsible for particular properties of the Bondi-Sachs metrics. This paper was a motivation for the present work since, in our feeling, some statements in [12] need a clarication. In Sec 2 we shortly review the Bondi-Sachs considerations in slightly more geometrical terms. In Sec 3 we present a variant of the Penrose conformal approach which is very close to that of Bondi and Sachs. In Sec 4 we perform the Tamburino-Winicour construction in a compact form and we nd conditions on the conformal factor equivalent to the main assumptions about the Bondi-Sachs metrics. In Sec 5 we relate these conditions with properties of the Ricci tensor. In Sec 6 we consider the Robinson-Trautman metrics, their conformal properties and the existence of the corresponding Bondi-Sachs coordinates. We assume the metric signature +???. The physical spacetime and its metric are denoted by ~ M; ~g as the unphysical (partially compactied) spacetime and its metric are M; g. Coordinates on M and the scri J + are given, respectively, by x and x i, = 0; 1; 2; 3 and i = 0; 2; 3. Equations to be satised only on J + are denoted with 'hat'. Partial derivatives of a eld are denoted by ; and covariant derivatives with respect to g are given by j. 2. Bondi-Sachs metrics In the Bondi-Sachs description of gravitational radiation from bounded sources [2,3] one assumes that spacetime contains a domain ~ M 0, which is dieomorphic to the exterior of the timelike cylinder r = r 0 in the Minkowski space (here r is the radial distance measured by some inertial observer and r 0 is a constant). The only boundary of ~ M 0 is the cylinder. It is assumed that ~ M 0 admits a foliation by null surfaces u = const, which is a smooth deformation of the foliation of Minkowski space by future null cones and it tends to the latter far away from the cylinder. The manifold ~ M 0 is the product RRS 2, S 2 is the 2-dimensional sphere. Its metric takes the form ~g = du(~g 00 du + 2~g 01 dr + 2~!) + r 2 g 2 ; (1) u; r are coordinates of R R and ~g 00 ; ~g 01 ; ~!; ~g 2 are, respectively, two functions, a 1-form and a metric on S 2. All these objects may depend on u and r. They should satisfy the following asymptotic conditions at future null innity (i.e. when r! 1 and u stays bounded): ~g 00! 1; (2) ~g 01! 1; (3) r?1 ~!! 0; (4) g 2!?g S ; (5) g S is the standard metric of S 2. These conditions guarantee the existence of coordinates in which ~g tends to the Minkowski metric (in at coordinates). 2

4 In order to study the Einstein equations in the asymptotic region one assumes that all components of the metric (1) allow expansions into (negative or positive) powers of r (see [13] for a more general approach). By virtue of (2)-(5) this means that ~g 00 ; ~g 01 ; ~!; g 2 depend smoothly on = r?1 up to = 0: (6) In addition to (1)-(6) one usually assumes that the coordinate r is so called luminosity distance [3]. In our setting this requirement reads det g 2 = det g S ; (7) determinants are taken with respect to any system of coordinates on S 2. Condition (7) assures that the area of any 2-surface given by u = const; r = const is equal to 4r 2. For metrics satisfying (1)-(5) equality (7) can be achieved by a transformation of r provided r@ u (det g 2 )! 0: The latter condition is important for a proof of nitness of the total gravitational energymomentum of a surface tending asymptotically to u = const. It is satised if ~g is a solution of the Einstein equations in vaccum (or the Ricci tensor of ~g falls down suciently fast at null innity, see Sec 5). If (1)-(6) are satised then one can relate with metric (1) a conformally equivalent metric g g = 2 ~g = du( 2 ~g 00 du? 2~g 01 d ~!) + g 2 ; (8) which is smooth and nondegenerate up to = 0. Thus, one can attach to ~ M 0 a boundary J +, given by = 0 and equipped with the structure of RS 2, such that metric (8) extends to J +. Substituting = 0 in (8) yields the standard metric of the sphere (with the minus sign because of our choice of the metric signature). It follows that surface J + is null with respect to the metric (8). The existence of a conformal compactication of spacetime with the above properties is a basic assumption in the Penrose description of asymptotically at elds [4]. 3. Conformal properties of spacetime Guided by the conformal approach of Penrose and taking into account the properties of the Bondi-Sachs metrics we make the following assumptions about spacetime. (a) The physical spacetime ~ M is a submanifold of an unphysical spacetime M. Metric g of M is conformally equivalent (on ~ M) to the metric ~g of ~ M g = 2 ~g: (9) (b) A boundary of ~ M in M contains a 3-dimensional surface J + such that, on J +, ^=0, d 6= 0 and g ; ; ^=0: (10) 3

5 The function is smooth up to J +. (c) The boundary J + is dieomorphic to the product J + = R S 2 ; the lines R fpointg are integral lines of the vector eld v = : (11) (d) The pullback of g under the natural imbedding : J +! M is (minus) the standard metric g S of the sphere g =?g S : (12) Above assumptions are closely related to the notion of weak asymptotic simplicity [4,10] restricted to J +. We discuss them shortly in this section and in the next section we nd supplementary conditions on, which guarantee that ~g has properties (1)-(5) and, possibly, (7). Under a conformal transformation of the sphere its standard metric g S = 2(1 + =2)?2 dd (we follow the convention in [8]) is multiplied by a 2 0, a 0 = 2 + ; c = const 2 SL(2; C): (13) 2jc 11 + c 12 j 2 + jc 21 + c 22 j2 It follows from (9)-(13) that the conformal factor and the unphysical metric g are dened up to the transformation 0 = a; g 0 = a 2 g; (14) a is a positive function such that (on J + ) a ^=a 0 ; a 0 being given by (13). On J + one can introduce coordinates x i = u; x A, A = 2; 3, such that v = u (15) and x A are (local) coordinates on S 2. Given equation (15) denes u up to the transformation u 0 = u + f(x A ); f is a function on S 2. Due to (14) this freedom extends to the Bondi-Metzner-Sachs transformation [2,3,14] u 0 = a?1 u + 0 f(xa ); a 0 is given by (13). 4

6 Solutions of the Einstein equations are usually obtained in local coordinates. In order to nd J + and the conformal factor one has to investigate global properties of the solution. It seems that there is no simple method of localizing J + however an analysis of invariants of the Riemann tensor (they should vanish on J + if spacetime is to be asymptotically at) might be helpful. Assume that conditions (a) and (b) are satised. Then is given up to the transformation 0 = a; a > 0 on J + : (16) It follows from (10) that the vector eld v given by (11) is tangent to J +. Assumption (c) is always satised locally. A neccessary condition to obtain (d) (even locally) is that for an in the class dened by (16) the metric g is u-independent L v ( g) = 0 (17) (L v denotes the Lie derivative along v). When (17) is satised and condition (c) is true globally then g must be proportional to the standard metric g S and an appropriate rescaling of yields (12). In order to nd the proportionality factor one can proceed as follows (at least in principle). First one should solve a system of linear equations for a complex function (coordinate) such that g = 2f(1 + =2)?2 dd ; (18) f = f(; ). This function is dened up to the transformation 0 = h(); (19)) h is a holomorphic function of. Among these locally equivalent coordinates one should nd one for which f(; ) is smooth and positive for all values of including 1. Then can be interpreted as the complex stereographic coordinate of S 2. (Note that if such coordinate does not exist then assumption (c) cannot be satised.) Transformation (16) with a satisfying a ^=f?1 on J + allows to obtain (12). Let x = ; x i be coordinates in a neighbourhood of J +. Metric g can be written in the form g = d + g 3 ; (20) g 3 = g ij dx i dx j and is a 1-form. The Lie derivative of (20) yields L v g = (L v )d + d[v()] + L v g 3 : It follows from (10) and (17) that L v g = 0 (21) 5

7 (note that L v g 3 = L v ( g 3 ) = L v ( g) since v is tangent to J + ). Thus, equation (21) is equivalent to L v g ^=2d; (22) = dx is a 1-form. In terms of covariant (with respect to the unphysical metric g) derivatives of equation (22) reads j ^= ( j) : (23) Condition (23) (or (22)) is invariant under transformation (14) as it converts into j ^= ( j) + g (24) (with some function ) under (16). The last condition can be easily veried when a conformal factor, satisfying assumptions (a) and (b), is proposed. One does not have to do this if the Ricci tensor falls down suciently fast at null innity (see Sec 5). 4. Existence of Bondi-Sachs coordinates Assume that conditions (a)-(d) are satised and the coordinates u, x A on J + (see (15)) are chosen. In order to obtain the Bondi-Sachs form of ~g one denes coordinates x in the following way [11]. From each point of J + we send into M ~ a null geodesic in the direction orthogonal to the section u = const of J + passing through this point. These geodesics form a null congruence K emanating from J +. A point p in a neighbourhood M 0 of J + can be described by a value of the conformal factor at the point and by position of the corresponding point in J + which is linked to p by one of the geodesics. In this way one obtains, as domain in the physical spacetime, M ~ 0 = M 0 \ M ~ = R R S 2 with coordinates x = u; ; x A extendable to J +. In order to nd these coordinates explicitely one has to nd solutions u, x A of equations u ; u ; = 0; u ; x A ; = 0 (25) such that condition (15) is satised on J +. By construction the vector eld k tangent to the null congruence K. It satises equations is g(k; k) = 0; (26) ^ L k = 0; (27) = g k dx. It follows from (26) and (27) that (in coordinates u; ; x A ) g 11 = 0; g 1i = pn i n i = 0; (28) p is a function. The orthogonality of k to the spheres u = const on J + implies pn A ^=0: 6

8 The function p must not vanish on J + (otherwise the metric tensor g would be degenerated on J + due to (28) and the orthogonality u to all vectors tangent to J + ). Hence n A = 0 and g = du(g 00 du + 2g 01 d + 2!) + g 2 ; (29)! =! A dx A ; g 2 = g AB dx A dx B are, respectively, a 1-form and a metric on the sphere S 2 (in general, they depend on and u). Substituting = r?1 into the physical metric ~g =?2 g (on M ~ 0 ) yields the Bondi-Sachs form (1) however not all of the properties (2)-(5) are satised. It follows from (29) and (12) that! ^=0; g 2 ^=? g S : (30) In order to investigate other asymptotic properties of (29) it is useful to express components of g in the following form It follows from (10) and (15) that Hence, due to (30) and (31), g 0i = 0; g 01 = n; g 11 = n 2 (! A! A? g 00 ); g 1A =?n! A ; (31) n = (g 01 )?1 ; g AC g CB = A B;! A = g AB! B : g 11 ^=0; g 01 ^=? 1; g 1A ^=0: g 00 ^=0; g 01 ^=? 1: (32) Properties (30), (32) represent only a part of conditions (2)-(5). In order to express lacking conditions in terms of let us calculate j and j j in the coordinates u; ; x A. Using the standard formula for the Christoel coecients? 1 (note that j = ) 1 and taking their limits on J + shows that equation (24) is satised with ^=@ g 0. In order to have g 00 = 0( 2 );! = 0( 2 ) (as in (8)) one should require i = 0 what leads to the condition q is a function on J + related to g 01 by j ^=q j j ; g 01 ^=q: 7

9 If (33 ) is satised then using (31) yields j j = g 11 =?g ( 3 ): Hence, to obtain?2 g 00 ^=1 (as in (2)) one should require?2 j j ^=? 1: (34) Equation (33) is invariant under the transformation (14) as the l.h.s. of (34) transforms according to 0?2 0j 0 j ^=a?2 0?2 j j? 2a?3 0 a ;u + a?4 0 aja 0 a 0jA (this formula is true for any boundary value a 0 of a such u a 0 = 0). It follows that in order to satisfy (34) one can replace by 0 = (1 + a 1 ); (35) 2@ u a 1 ^=1 +?2 j j : Under transformation (16) equation (33) changes into j ^=q j j + g : (36) Thus, if satises assumptions (a), (b) of Sec 3 then equation (36) is a neccessary condition to obtain properties (c), (d) and the Bondi-Sachs conditions (2)-(5). On the other hand, if (36) is satised then expanding j into powers of and taking the covariant derivative of this expansion shows that?1 j ^=2: (37) Transformation (16) with a given by (ln a) ^=? leads to new which already satises (33). Then?2 j is nite on J + (due to (37)) and one can obtain (34) by means of transformation (35). Finally, let us consider condition (7) xing r =?1 as the luminosity distance. If conditions (a)-(d), (33), (34) are satised then from (31) and the standard formula = (det g) 1 2, one obtains j =?1 (g ; ) ; ; j =?2? 1 u ln(det g 2 ) + 0( 2 ): 8

10 Condition (7) implies?1 j ^=? 2: (38) On the other hand, if equality (38) is true then one can use transformation (14) with a = ( det g S det g 2 ) 1=4 (39) to dene a new coordinate r for which condition (7) is already satised (note that a ^=1 and from (38) it follows that a ;u ^=0, hence (34) is preserved). In this sense equations (7) and (38) are equivalent. We summarize these results in the following proposition. Proposition 1. Assumptions (1)-(6) (and (7)) within the Bondi-Sachs approach are equivalent to the assumptions (a)-(d), (33), (34) (and (38)) within the Penrose approach. Equations (33), (34) can be obtained by means of transformation (16) provided (a), (b), (36) are satised. 5. Properties of the Ricci tensor The conformal transformation (9) induces the following relation between the Ricci tensor R of g and the Ricci tensor ~ R of the physical metric ~g ~ R = R + 2 j + ( j? 3?1 j j )g : (40) Components of R (in contrary to ~ R ) has to be nite on J +, hence ~ R ^=2 j + ( j? 3?1 j j )g : (41) Due to (41) any condition on j on J + impose restrictions on boundary values of ~ R. For instance, if assumptions (a)-(d) of Sec. 2 are satised then in virtue of (23) one obtains ~ R ^=2 ( j) + g ; (42) is a function on J +. In terms of coordinate independent objects equation (42) reads ~ Ricci ^=2d + g; (43) ~Ricci = ~ R dx dx : Condition (43) must be satised if the property (12) is required. Obviously, it is satised if the physical metric ~g is a solution of the vacuum Einstein equations. If (33) is satised then, due to (10) and (37), j ^=0 (44) 9

11 and?1 j j ^=0: (45) In this case equation (41) yields ~ R ^=2q j j (46) or, equivalently, ~ Ricci ^=2qd 2 : (47) The same equation follows from (36) (due to (37)). Since the inverse implication is also true we obtain the following proposition which completes Proposition 1 of Sec 4. Proposition 2. Under assumptions (a) and (b) condition (36) is equivalent to (46). In order to introduce the luminosity distance as the coordinate r =?1 one needs property (38) which does not follow from (46). If assumptions (a)-(d) and equations (33), (34) are satised then an explicit calculation of R AB on J + and using (40) shows that (38) is equivalent to ~R AB ^=0: (48) Since equations ~R 00 ^=0; ~ R0A ^=0 (49) are trivially satised equation (48) can be conveniently written as ( ~ Ricci) = 0; (50) is the imbedding of J + into M, ~ Ricci satises (47) and we set (?1 d 2 ) = 0. Hence we obtain another completion of Proposition 1. Proposition 3. One can replace condition (38) in Proposition 1 by condition (50). Equations (47), (50) are neccessary conditions for transforming a metric with properties (a)-(d) into the Bondi-Sachs form satisfying conditions (1)-(7). They are automatically satised if ~g is a solution of the vacuum Einstein equations (then also q = 0). In a presence of matter, in order to satisfy (47) and (50) one has to impose appropriate conditions on the energy momentum tensor on J Robinson-Trautman metrics In this section we investigate asymptotic properties and the Bondi-Sachs coordinates for the Robinson-Trautman solutions [7] and give examples of the solutions (with pure radiation elds) which admit a regular scri J + (see [15] and [16] for recent studies of this subject). In the case of vacuum or pure radiation the Robinson-Trautman metrics read [8] ~g = 2Hdu 2 + 2dudr? 2r 2 P?2 dd ; (51) 10

12 u; r; (complex), are coordinates, 2H = 2P 2 (ln P ) ;? 2r(ln P ) ;u? 2mr?1 ; m = m(u) and P is an r-independent function subject to further condition following from the Einstein equations. In general, these coordinates are not the Bondi-Sachs coordinates. In this section the latter coordinates will be denoted with prime. The structure of ~g and that of invariants of the Weyl tensor suggests the following form of the conformal factor = f r ; (52) f = f(u; ; ). On J + one has = 0 as the coordinates u; are arbitrary (real and complex, respectively). The unphysical metric g and its pullback to J + read g = 2du(H 2 du + df? fd)? 2f 2 P?2 dd ; (53) g =?2f 2 P?2 dd : (54) If the assumption (c) of Sec 3 is to be satised should be a holomorphic function of the stereographic coordinate on S 2. Since holomorphic transformations of preserve (51) we can assume that on J + is already the stereographic coordinate such that g S = 2(1 + =2)?2 dd : Then condition (12) yields On J + one obtains hence one can take f = (1 + =2)?1 P: (55) ^=? u ; (56) u 0 = Z fdu (57) and x A = ; as coordinates on J + compatible with (15). The Robinson-Trautman solutions satisfy the condition?1 ~ Ricci ^=0 (58) which implies (47) and (50). Thus, these metrics can be put into the Bondi-Sachs form (1) satisfying conditions (2)-(7) provided the functions f and m are smooth on R S 2. According to (35) condition (34) can be achieved by means of transformation of. Up to 0( 3 ) this transformation yields a new conformal factor given by = ( r f + u 0 2 )?1 : (59) 11

13 In order to extend the coordinates u 0 ; ; on J + to the Bondi-Sachs coordinates u 0 ; 0 ; 0 one has rst to solve the nonlinear equation (see (25)) with the boundary condition 2H 2 0 (u0 ; 0 ) 2 + 2u 0 ;u u0 ; 0 + f 2 (ru 0 ) 2 = 0 (60) u 0 ^= Z fdu: Here u 0 is considered as a function of u, 0, and, 0 = r?1, and (ru 0 ) 2 is the square of the gradient of u 0 with respect to g S (ru 0 ) 2 = 2(1 + =2) 2 u 0 ;u 0 ; : Given a solution u 0 of (60) the Bondi-Sachs coordinate 0 is dened by the linear equation 2H 2 0 u0 ; 0 0 ; 0 + u 0 ;u 0 ; 0 + u 0 ; 0 0 ;u + f 2 ru 0 r 0 = 0; (61) ru 0 r 0 = (1 + =2) 2 (u 0 ; 0 ; + u0 ; 0 ;) and 0 is assumed to satisfy the boundary condition 0 ^=: In most cases one cannot nd solutions of equations (60), (61) in a closed form. Approximate solutions of these equations (for small values of ) read u 0 = u 0 + u (u 1u 0? ru 0 ru 1 ) 2 + 0( 3 ); (62) 0 = ( 1u 0? ru 0 r 1? rru 1 ) 2 + 0( 3 ); (63) u 1 =? 1 2 (ru 0) 2 ; 1 =?rru 0 and is given by (59). They can be used to obtain an approximate expression for the Robinson-Trautman metrics in terms of the Bondi-Sachs coordinates. Some characteristics of this expression (especially the Bondi mass) will be published else. In the case of vacuum the requirement of regularity of f on RS 2 is a strong condition on solutions of the reduced Einstein equations. Then f has to be an every positive solution of the equation f 2 (f 2 ln f + f 2 ) + 12m(ln f) ;u? 4m ;u = 0; (64) is the Laplace operator on S 2 = 2(

14 (see [17] for an advanced analysis of equation (64)). In the case of pure radiation the only requirement is For instance, this condition is satised if and a and b are functions of u given by f 2 (f 2 ln f + f 2 ) + 12m(ln f) ;u? 4m ;u 0: (65) m 0 (66) m ;u a + bm; (66) a = min[f (f 2 ln f + f 2 )] ; b = 3 min(ln f) ;u : (67) S2 S2 Due to this observation solutions to (65) can be obtained as follows. Let f be a positive function on R S 2 such that for u! 1 lim a 0; lim b 0 (68) (e.g. lim f = 1 ). Choose functions ~a(u), ~ b(u) satisfying conditions and Let m 0 be a constant such that ~a(u) a(u); ~ b(u) b(u) (69) Z ~a < 1; Z R R ~ b < 1: (70) Then Z 1 m 0 max ( ~ac); (71) u u c = exp( Z 1 u m = c?1 (m 0? ~ b): (72) Z 1 u ~ac) (73) satises conditions (65) and (66). Hence, any m given by (73) and f satisfying (68) dene a Robinson-Trautman solution (with pure radiation eld) with a regular scri J +. This solution can be transformed into the Bondi-Sachs form (1) satisfying conditions (2)-(7). If f! 1 when u! 1 then it tends to to the Schwarzschild metric with the mass m 0 when u! 1 and to another Schwarzschild metrico when u!?1. 13

15 Acknowledgements. Main part of this paper was completed during the workshop "Spaces of geodesics and complex structures in general relativity and dierential geometry" held in the Erwin Schrodinger Institute in Vienna. We have beneted from discussions with P. Chrusciel, L. Mason, R. Penrose, D. Robinson, I. Robinson and A. Trautman. This work was partially supported by the Polish Committee for Scientic Research (KBN). References. [1] H. Bondi, Nature (London) 186 (1960), 535. [2] H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Proc. R. Soc. Lond. A269 (1962), 21. [3] R.K. Sachs, Proc. R. Soc. Lond. A270 (1962), 103. [4] R. Penrose, Phys. Rev. Lett. 10 (1963), 66; Proc. R. Soc. Lond. A284 (1965), 159. [5] A. Trautman, Bull. Acad. Pol. Sci., Serie sci. math. astr. et phys.vi (1958), 407; King's College lecture notes on general relativity, mimeographed notes (unpublished), [6] P.T.Chrusciel, J. Jezierski and M.A.H. MacCallum, Phys. Rev. D58 (1998), [7] I. Robinson and A. Trautman, Phys. Rev. Lett. 4 (1960), 461; Proc. R. Soc. Lond. A265 (1962), 463. [8] D. Kramer, H. Stephani, E. Herlt and M.A.H. MacCallum, Exact solutions of Einstein's eld equations, Cambridge University Press, [9] E.T. Newman and R. Penrose, J. Math. Phys. 3 (1962), 566. [10] R. Penrose and W. Rindler, Spinors and spacetime, vol. II, Cambridge Univerity Press, [11] L.A. Tamburino and J.H. Winicour, Phys. Rev. 150 (1966), [12] S. Persides, J. Math. Phys. 20 (1979), [13] P.T.Chrusciel, M.A.H. MacCallum, D.B. Singleton, Phil. Trans. R. Soc. A350 (1995), 113. [14] R.K. Sachs, Phys. Rev. 128 (1962), [15] U. von der Gonna and D. Kramer, Class. Quantum Grav. 15 (1998), 215. [16] F.H.J. Cornish and B. Micklewright, Class. Quantum Grav. 16 (1999), 611. [17] P.T.Chrusciel, Proc. R. Soc. Lond. A436 (1992),