Invariant differential operators and the Karlhede classification of type N vacuum solutions

Transcription

1 Class. Quantum Grav. 13 (1996) Printed in the UK Invariant differential operators and the Karlhede classification of type N vacuum solutions M P Machado Ramos and J A G Vickers Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK Received 27 November 1995 Abstract. A spacetime calculus based on a single null direction, and which is therefore invariant under null rotations, is employed to show that a type N vacuum solution of Einstein s equations requires the calculation of at most five covariant derivatives of the curvature for its complete Karlhede classification. PACS numbers: 0420, 0420C 1. Introduction The equivalence problem, that is, the problem of deciding whether or not two given spacetime metrics expressed in different coordinates give rise to locally equivalent geometries, is an important problem in general relativity. The solution, which goes back to the work of Cartan, involves the calculation of successive covariant derivatives of the Riemann tensor up to a certain order called the bound. Considerable progress in reducing the required order of differentiation was made by Karlhede [1], who devised an algorithm for classifying geometries. Karlhede reduced the theoretical bound on the highest order of covariant derivative required from ten to seven in the worst three cases (non-vacuum types N and D and vacuum type N) and five in all other cases (vacuum and non-vacuum types I, II, III and vacuum type D). More recently these bounds have been reduced by Collins et al to three for vacuum type D [2] and six for non-vacuum type D [3] and vacuum type N [4]. An important aspect of the approach used by Collins et al in the case of type D (vacuum and non-vacuum) is the use of the GHP formalism [5]. Vacuum type D spacetimes have a Weyl spinor which admits spin and boost transformations as its invariance group and the GHP spin coefficients and operators are covariant under this same group. It then turns out that at all orders of covariant differentiation the dyad components of the Weyl spinor and its derivatives can be expressed completely in terms of GHP notation which makes the classification process easier. In the case of vacuum type N, where the invariance group is that of null rotations, Collins makes use of the NP formalism [4] to express the dyad components of the Weyl spinor and its derivatives. However the use of this notation is not as productive as one would like since terms which are not invariant under null rotations appear in the Karlhede algorithm. It then seems natural, as in the type D case, to use a notation covariant under such transformations in order to simplify the classification process and therefore to be able to reduce the bound /96/ $19.50 c 1996 IOP Publishing Ltd 1589

2 1590 M P Machado Ramos and J A G Vickers In what follows we show that, by applying a new generalized GHP type formalism invariant under null rotations [6] to the equivalence problem of vacuum type N spacetimes, one can reduce the bound by one. We should, however, advise the reader that the calculations involved, although straightforward, are quite extensive. In section 2, we start off by giving the algorithm developed by Karlhede for the complete classification of the Riemann tensor and its successive covariant derivatives. This procedure uses the tetrad components of the Weyl spinor and its derivatives (in vacuum) and determines the corresponding invariance group H n at each stage of the classification. The idea of the present paper is to use the generalized GHP formalism to obtain spinors which are invariant under null rotations. It turns out that these spinors encode all the coordinate information contained in the derivatives of the curvature in a way which does not depend upon the choice of iota. We can therefore count the coordinate functional information in these spinors without fixing iota. In sections 3 and 4 we proceed to calculate all terms relating to the first and second covariant derivative of the Weyl spinor in generalized GHP language. We then determine the corresponding invariance group H n and the number of functionally independent terms in the spinors. At both first and second order we are forced to allow the possibility of continuing the procedure. Section 5 gives a general expression for the nth covariant derivative of Ψ in the generalized GHP formalism; in particular we explicitly write out the expressions for the third, fourth and fifth derivative. Finally, in section 6 we analyse the upper bound on the order of covariant differentiation of the Weyl spinor required in the classification and show how one can then lower the bound from 6 to 5. We comment on why the Bianchi identities in the case of vacuum type N are not of much use in restricting the number of functionally independent terms one obtains at each stage since there is no constraint on Ψ 4 as this term does not feature in the Bianchi identities. Hence, unlike in the vacuum type D case, one can not use these methods to lower the bound any further. 2. The Karlhede classification A practical procedure to investigate the equivalence of metrics was developed by Karlhede [1] in the form of an algorithm which starts by calculating the Riemann tensor in a particular frame (tetrad) and then calculates successively higher covariant derivatives together with possible changes of frame until a complete classification is obtained. If we denote frame components of the Riemann tensor and its covariant derivatives up to qth order by R q, then the procedure is as follows: (1) Let q = 0. (2) Compute R q. (3) Fix up the frame as much as possible by choosing a canonical form for R q. (4) Find the invariance group H q of the frame which leaves R q invariant. (5) Find the number of functionally independent components t q amongst the set R q. (6) If t q t q 1 or dim(h q ) dim(h q 1 ) then set q = q + 1 and go to (2). (7) Otherwise the set H p,t p,r p, p =1,...,q classifies the solution. Then if we wish to compare for equivalence two given metrics g and g, we start by completing the above classification for each metric. The rest of the procedure is contained in the following steps: (8) If the two sequences H 0,t 0 ;H 1,t 1 ;...;H q,t q for g and g differ, then so do the metrics.

3 Karlhede classification of type N solutions 1591 (9) If the set of simultaneous algebraic (but possibly transcendental) equations R 0 = R 0,R 1 =R 1,...,R q =R q admit a coordinate transformation x i = x i (x i ), i = 1,...,n as a solution then the metrics are equivalent, otherwise they are inequivalent. Unfortunately step (9) is not algorithmic, since there is no constructive procedure for solving simultaneous algebraic equations. Notice also that in the vacuum case R q can be substituted by C q where C q denotes the frame components of the Weyl tensor and its covariant derivatives up to qth order. Here we choose to work with spinor notation. So instead of working with tetrad components of the Weyl tensor one uses the dyad components of the equivalent Weyl spinor and instead of considering Lorentz transformations of the frame one considers SL(2,C) transformations of the dyad. We are able to do this since the tetrad components of a tensor in a Newman Penrose null tetrad are the same as the dyad components of the equivalent spinor [1]. We introduce a spin frame {o A,ι A } which satisfies: o A ι A = 1. (1) Because the Weyl spinor is totally symmetric it has only five independent dyad components which, in standard notation, are labelled: 0 = ABCD o A o B o C o D 1 = ABCD o A o B o C ι D 2 = ABCD o A o B ι C ι D (2) 3 = ABCD o A ι B ι C ι D 4 = ABCD ι A ι B ι C ι D. The Weyl spinor of a Petrov type N spacetime has the form: 0 = 1 = 2 = 3 = 0; 4 = 0 (3) which is preserved under the invariance group H 0 of null rotations defined by: o A o A ι A ι + ao A. (4) The generalized GHP formalism [6] involves invariants which are symmetric spinors rather than scalars, so that if one is to apply this formalism to the classification procedure, instead of considering the terms 0, 1, 2, 3, 4 and the respective invariance group of null rotations we consider the spinors Ψ 0, Ψ 1, Ψ 2, Ψ 3, Ψ 4 defined by: Ψ 0 = ABCD o A o B o C o D (Ψ 1 ) A = ABCD o B o C o D (Ψ 2 ) AB = ABCD o C o D (5) (Ψ 3 ) ABC = ABCD o D (Ψ 4 ) ABCD = ABCD and the group that leaves these terms invariant. The Weyl spinor for vacuum type N in this notation has the general form given by: Ψ 0 = (Ψ 1 ) A = (Ψ 2 ) AB = (Ψ 3 ) ABC = 0 (Ψ 4 ) ABCD = o A o B o C o D. (6) Notice that (6) is invariant under the four- (real) parameter group of transformations: o A λo A ι A λ 1 ι + ao A (7) where λ is a nowhere vanishing complex scalar field.

4 1592 M P Machado Ramos and J A G Vickers It is however convenient to use the simplest form (canonical form) possible in order to simplify the calculations and hence the classification procedure. By taking a suitable dyad as basis we may scale to one and obtain the following canonical form for the Petrov type N Weyl spinor: Ψ 0 = (Ψ 1 ) A = (Ψ 2 ) AB = (Ψ 3 ) ABC = 0 (Ψ 4 ) ABCD = o A o B o C o D. (8) Notice that now the invariance group H 0 is the two- (real) parameter group of null rotations: o A o A ι A ι + ao A. (9) In the generalized GHP formalism, the Bianchi identities in vacuum under condition (6) become þ A (A BCDE) = R (A BCDE) ō A (10) ð (A B )(A BCDE) = (BCDE T A)(A ō B ) (11) S A = 0 (12) K = 0 (13) which in compact notation is given by þψ 4 = RΨ 4 (14) ðψ 4 = T Ψ 4 (15) S = 0 (16) K = 0. (17) Using the fact that Ψ 4 is a spin and boost weighted object of weight {0, 0}, and that (E F )(EF (o A o B o C o D) ) = δ E (EF (o A o B o C o D) ) = δ (E F )(E(o A o B o C o D) ) = D E (E(o A o B o C o D) ) = 0 we have by the definition of the generalized GHP operators þ, ð, ð, þ: þ A B (AB CDEF) = 4o (C o D o E o F G AB)A (18) ð A (AB CDEF) = 4A (AB o C o D o E o F) ō A (19) ð (A B )(A BCDE) = 4ō (B B A )(Ao B o C o D o E) (20) þ A (A BCDE) = 4E (A o B o C o D o E) ō A. (21) Or in compact form þ Ψ 4 = 4GΨ 4 (22) ðψ 4 = 4BΨ 4 (23) ð Ψ 4 = 4AΨ 4 (24) þψ 4 = 4EΨ 4. (25) Comparing with the Bianchi identities gives: R A = 4E A (26) T AA = 4B AA. (27) The Ricci equations become: þ A (A R B) = R (A R B) ō A (28) þ AA T BB = R (A T B)(B ō A ) (29) ð (A B )(AR B) =ō (B T A )(AR B) T AA R B ō A (30) ð AA B T BC = T AA T BB ō C (31) þ (A B )(AB R C) ð ABA T CB = T AA T BB o C (32)

5 Karlhede classification of type N solutions 1593 with their compact form being given by: þr = R 2 (33) þt = RT (34) ðr = TR TR (35) ðt = T 2 (36) þ R ð T = TT. (37) Finally we write the commutators which for simplicity we give only in the compact form: (þþ þ þ)φ = (T ð + T ð )φ (38) (þð ðþ)φ = Rðφ (39) (ðð ð ð)φ = (R R)þ φ (40) (þ ð ðþ )φ = Tþ φ. (41) Note that the GHP vacuum field equations contain the same information as Einstein s vacuum field equations [7] and therefore the same is also true for the equations given above since they are complete in the sense that all such GHP identities can be obtained from them. 3. First covariant derivative We now proceed to calculate the first covariant derivative of ABCD which we will denote by ( Ψ) ABCDF F. It follows from the Bianchi identities in spinor form ɛ AF ABCD;FF = 0 that the first covariant derivative of the Weyl spinor is symmetric on all primed and unprimed indices so that the application of the generalized GHP notation is viable. One can easily deduce the general expression giving the dyad components of the first covariant derivative [2] which is given by: ( ) µ;ff = ( µ ) ;ff µɣ 11ff µ 1 + (2µ 4)Ɣ 01ff µ + (4 µ)ɣ 00ff µ+1 (42) with µ {0,1,2,3,4}. Let ζa A = {oa,ι A } be a normalized spinor dyad with dual ζa A so that bold indices represent dyad terms, for example λ A = λ A ζa A is a scalar and not a spinor. In order to write (42) in terms of the invariant formalism we first introduce some notation. Let Ψ, with from zero to five unprimed indices, be defined by: ( Ψ) Ap...A 5 A = A 1...A 4 ;A 5 A oa 1...o A (p 1). (43) Then in the invariant formalism of [8], equation (42) becomes: o AN+1 ( Ψ) A1...A N FF { (ΨA1...A N ) ;FF o NƔ AN+1 A 1 A 2 FF Ψ A 3...A N+1 = sym + (2N 4)Ɣ 0A1 FF Ψ A 2...A N+1 + (4 N)Ɣ 00FF } ΨA1...A N A N+1 (44) with N {0,1,2,3,4}and where sym indicates symmetrization on all free primed and unprimed indices.

6 1594 M P Machado Ramos and J A G Vickers One then finds that all the non-vanishing terms from equation (44) are given by: o (A5 ( Ψ) A1 A 2 A 3 A 4 F )(F ō G ) = 4ō (G G F )(FA 1 o A2 o A3 o A4 o A5 ) = þ (F G )(A 1 A 2 Ψ A3 A 4 A 5 F) (45) o (A4 ( Ψ) A1 A 2 A 3 F )(F ō G ) =ō (G T F )(F o A1 o A2 o A3 o A4 ) = ð (G F )(A 1 Ψ A2 A 3 A 4 F) (46) o F o (A5 Ψ) A1 A 2 A 3 A 4 F) = A (FA1 o A2 o A3 o A4 o A5 )o F = ð F (FA 1 Ψ A2 A 3 A 4 A 5 ) (47) ō F o (F ( Ψ) A1 A 2 A 3 A 4 ) = R (F o A1 o A2 o A3 o A4 )o F = þ F (F Ψ A 1 A 2 A 3 A 4 ) (48) and in compact notation we have ( Ψ) = 4G = þ Ψ 4 (49) ( Ψ) o = 4A = ð Ψ 4 (50) ( Ψ) ō=4b=ðψ 4 =T (51) ( Ψ) o ō=4e=þψ 4 =R (52) where we recall that in the compact notation a dot denotes a contraction and that one may have to multiply terms by suitable factors of o A and ō A and then symmetrize to obtain expressions such that the indices balance. It is important to note that equations (49) (52) form an inverted hierarchical system. Since κ = 0 the only functional information in E is given by ɛ. Again since σ vanishes and ɛ is known from (52), the only new functional information in B is given by β. Since by (26) ρ = 4ɛ and ɛ is known from (52) the only new functional information in A is given by α. Finally since by (27) τ = 4β and all the other terms are known the new functional information in G is given by γ. Thus (49) (52) encode all the coordinate functional information at first order, which we can extract without making a particular choice of iota. It is clear that the spinors (49) (52) obtained at first order are invariant under null rotations so that the dimension of the invariance group H 1 remains 2. Note that as elsewhere in this paper we have used a bold font to indicate that we are dealing with a spinor quantity rather than a scalar quantity. However it is important to realise that the components of these spinors may not be invariant under null rotations so that H 1 may well have a smaller dimension. Indeed the components of the spinors will depend upon the choice of iota unless the spinors can be written entirely in terms of omicrons. We must consider the possibility of there being at least one new functionally independent piece of information at this stage and proceed to the next derivative. 4. Second covariant derivative The calculation leading to the second covariant derivative, which we will denote by ( 2 Ψ) ABCDF GF G, is similar to the one performed to obtain the first covariant derivative. The general expression giving the dyad components of the second covariant derivative is calculated in [2] and is as follows: ( 2 ) µf ;gg = [( ) µf ] ;gg µɣ 11gg ( ) (µ 1)f + (2µ 5)Ɣ 10gg ( ) µf + (5 µ)ɣ 00gg ( ) (µ+1)f Ɣ f 1 g g( ) µ0 + Ɣ f 0 g g( ) µ1 (53) with µ {0,1,2,3,4,5}.

7 Karlhede classification of type N solutions 1595 In terms of the invariant formalism one obtains: o A o AN+1 ( 2 Ψ) A1...A N F GG { [( Ψ)A1...A N F ] ;GG o A o A NƔ N+1 A 1 A 2 GG ( Ψ) A 3...A o N+1F A = sym + (2N 5)Ɣ A1 0GG ( Ψ) A 2...A N+1 F o A + (5 N)Ɣ 00GG ( Ψ) A1...A N+1 F o A Ɣ F A G G( Ψ) A1...A N o AN+1 + Ɣ F 0 G G( Ψ) A1...A N A o A N+1 }. (54) At all orders of covariant differentiation of the Weyl spinor, we only need to consider the symmetric parts since only these terms are algebraically independent [9]. One then obtains all symmetric non-zero terms corresponding to the second covariant derivative of Ψ 4 from expression (54), which in compact notation are given by: ( 2 Ψ) = þ þ Ψ 4 (55) ( 2 Ψ) ō=ð þ Ψ 4 +(þ Ψ 4 )T (56) ( 2 Ψ) ō ō=ð ð Ψ 4 (57) ( 2 Ψ) o = ðþ Ψ 4 + (þ T)Ψ 4 +2(þ Ψ 4 )T (58) ( 2 Ψ) o ō =þþ Ψ 4 + ðð Ψ 4 + ð T + þ R + 2(ð Ψ 4 )T + (þ Ψ 4 )R + 2(þ Ψ 4 )R + (þ Ψ 4 )R + 5/4T T (59) ( 2 Ψ) o ō ō=þð Ψ 4 +ð R+2(ð Ψ 4 )R (60) ( 2 Ψ) o o = ðt + (ðψ 4 )T (61) ( 2 Ψ) o o ō=þ T+ðR+9TR+TR (62) ( 2 Ψ) o o ō ō=þr+3r 2. (63) Notice that all the above equations can be obtained from equation (55) by contraction with omicrons, and as at first order form an inverted hierarchical system. These secondcovariant-derivative terms encode the same coordinate functional information as the secondcovariant-derivative terms obtained by Collins [4] but one can extract this information without making a choice of iota. Note that one can obtain the expressions above by translating Collins terms into generalized NP language (for example: ρ becomes R) and leaving out all terms that transform badly under null rotations. Because they have been obtained using invariant operators we again obtain spinors that are invariant under null rotations so that the dimension of H 2 is 2. By considering equations (22) (25), we see that equations (55) (63) tell us that the potentially new functionally independent information can only come from the following 16 terms: þþψ 4, þðψ 4, þð Ψ 4, þþ Ψ 4, þ þψ 4, þ ðψ 4, þ ð Ψ 4, þ þ Ψ 4, ðþψ 4, ððψ 4, ðð Ψ 4, ðþ Ψ 4, ð þψ 4, ð ðψ 4, ð ð Ψ 4, ð þ Ψ 4. However, the commutator limits the number of functionally independent terms to 10, which are obviously given by: þþψ 4, þðψ 4, þð Ψ 4, þþ Ψ 4, þ ðψ 4, þ ð Ψ 4, þ þ Ψ 4,

8 1596 M P Machado Ramos and J A G Vickers ððψ 4, ðð Ψ 4, ð ð Ψ 4. And by means of (22) (25) and the Ricci equations, we are left with: þð Ψ 4, þþ Ψ 4, þ ðψ 4, þ ð Ψ 4, þ þ Ψ 4, ð ð Ψ 4, (64) as our possibly functionally independent terms. Unfortunately we are unable to relate these invariants to the invariants obtained at first order of covariant differentiation, because the Bianchi identities do not relate þ Ψ 4 and ð Ψ 4 to Ψ 4, R and T. In fact the term Ψ 4 does not even feature in the Bianchi identities. As a result, and unlike the vacuum type D case [2] where these identities are used to relate higher-order derivatives of 2 to lower-order derivatives of 2 and hence limit the number of functional information obtained at each step, here we must consider the possibility of there existing at least one new functionally independent term among the invariants given by (64). We must therefore continue the Karlhede algorithm. 5. Higher-order derivatives The calculation of third, fourth,..., etc covariant derivatives of the Weyl spinor is lengthy but straightforward and can be seen as an extension of the calculation performed for lowerorder derivatives. The general expressions giving the symmetric non-vanishing terms relating to the third, fourth and fifth covariant derivatives are given by the following: Third covariant derivative o A o AN+1 ( 3 Ψ) A1...A N F GG HH = { [( 2 Ψ) A1...A N F GG ] ;HH o A o A N+1 sym NƔ A1 A 2 HH ( 2 Ψ) A3...A N+1 F GG o A + (2N 5)Ɣ A1 0HH ( 2 Ψ) A2...A N+1 F GG o A + (5 N)Ɣ 00HH ( 2 Ψ) A1...A N+1 F GG o A Ɣ F A H H( 2 Ψ) A1...A N GG o A N+1 + Ɣ F 0 H H( 2 Ψ) A1...A N A GG o A N+1 Ɣ GAN+1 HH ( 2 Ψ) A1...A N F G o A + Ɣ G0HH ( 2 Ψ) A1...A N F A N+1 G o A Ɣ G A H H( 2 Ψ) A1...A N F Go AN+1 + Ɣ G 0 H H( 2 Ψ) A1...A N F GA o A N+1 }. (65) Fourth covariant derivative o A o AN+1 ( 4 Ψ) A1...A N F GG HH MM = { [( 3 Ψ) A1...A N F GG HH ] ;MM o A o A N+1 sym NƔ A1 A 2 MM ( 3 Ψ) A3...A N+1 F GG HH o A +(2N 5)Ɣ A1 0MM ( 3 Ψ) A2...A N+1 F GG HH o A

9 Karlhede classification of type N solutions (5 N)Ɣ 00MM ( 3 Ψ) A1...A N+1 F GG HH o A Ɣ F A M M( 3 Ψ) A1...A N GG HH o A N+1 +Ɣ F 0 M M( 3 Ψ) A1...A N A GG HH o A N+1 Ɣ GAN+1 MM ( 3 Ψ) A1...A N F G HH o A +Ɣ G0MM ( 3 Ψ) A1...A N F A N+1 G HH o A Ɣ G A M M( 3 Ψ) A1...A N F GHH o A N+1 + Ɣ G 0 M M ( 3 Ψ) A1...A N F GA HH o A N+1 Ɣ HA1 MM ( 3 Ψ) A2...A N+1 F GG H o A + Ɣ H0MM ( 3 Ψ) A1...A N+1 F GG H o A Ɣ H A M M ( 3 Ψ) A1...A N F GG Ho AN+1 + Ɣ H 0 M M ( 3 Ψ) A1...A N F GG HA o A N+1 }. (66) Fifth covariant derivative o A o AN+1 ( 5 Ψ) A1...A N F GG HH MM NN = { [( 4 Ψ) A1...A N F GG HH MM ] ;NN o A o A N+1 sym NƔ A1 A 2 NN ( 4 Ψ) A3...A N+1 F GG HH MM o A +(2N 5)Ɣ A1 0NN ( 4 Ψ) A2...A N+1 F GG HH MM o A +(5 N)Ɣ 00NN ( 4 Ψ) A1...A N+1 F GG HH MM o A Ɣ F A N N( 4 Ψ) A1...A N GG HH MM o A N+1 +Ɣ F 0 N N( 4 Ψ) A1...A N A GG HH MM o A N+1 Ɣ GAN+1 NN ( 4 Ψ) A1...A N F G HH MM o A +Ɣ G0NN ( 4 Ψ) A1...A N F A N+1 G HH MM o A Ɣ G A N N( 4 Ψ) A1...A N F GHH MM o A N+1 +Ɣ G 0 N N( 4 Ψ) A1...A N F GA HH MM o A N+1 Ɣ HA1 NN ( 4 Ψ) A2...A N+1 F GG H MM o A +Ɣ H0NN ( 4 Ψ) A1...A N+1 F GG H MM o A Ɣ H A N N( 4 Ψ) A1...A N F GG HMM o A N+1 +Ɣ H 0 N N( 4 Ψ) A1...A N F GG H A MM o A N+1 Ɣ MA1 NN ( 4 Ψ) A2...A N+1 F GG HH Mo A +Ɣ M0NN ( 4 Ψ) A1...A N+1 F GG HH M o A Ɣ M A N N( 4 Ψ) A1...A N F GG HH Mo AN+1 +Ɣ M 0 N N( 4 Ψ) A1...A N F GG HH MA o A N+1 }. (67) One can deduce from these expressions and the general expressions for the first and second derivative that the highest-order spinor term corresponding to the nth covariant derivative (in compact notation) is given by: ( n Ψ 4 ) = þ n Ψ 4 (68) with all other terms corresponding to the nth derivative obtained by contracting (68) with omicrons, as we have seen to be the case at first and second order.

10 1598 M P Machado Ramos and J A G Vickers 6. Upper bounds In this final section we analyse the upper bound on the order of covariant differentiation of the Weyl spinor required in the Karlhede classification. We have seen from equation (68) above that all the required functional information about the coordinate system may be obtained by successively applying the differential operator þ to Ψ 4 and taking all the possible contractions with omicrons. The resulting spinors are therefore all invariant under null rotations and the invariance group H n remains two dimensional. Furthermore at each stage the spinor equations form a hierarchical system in which one can extract coordinate functional information without making a choice of iota. At most the Karlhede classification will produce four functionally independent pieces of information from the set {( n Ψ 4 ),...,( n Ψ 4 ) o... o o... o}. So that in the worst case when one obtains only one new piece of functional information at each derivative one would obtain no new coordinate functional information from the fifth derivative onwards. Thus there would be some functional relationship between the components of the spinors obtained at fifth order and those already obtained whatever the choice of iota. Although this shows that at fifth order one has obtained all the possible coordinate information, in principle, one might have to go to higher order to fix the frame and complete the Karlhede classification. However, one can show that any gauge freedom remaining by the fourth derivative will always remain (see, for example, Collins [4] and below). Thus no new coordinate or frame information can be obtained at the fifth derivative and the Karlhede bound is therefore five. The approach adopted in this paper has been to use a new connection based on the invariant differential operators which enables one to extract the coordinate functional information without making a choice of gauge. One then uses a different argument to see how the frame is fixed. Since the Karlhede bound is five, in those cases where the frame is fixed one must obtain both coordinate and frame information at some stage of the process. In fact one can show that this must occur at the first derivative. At first order the terms obtained are given by þ, ð, ð, þ acting on Ψ 4 which give G, A, B and E. Since κ vanishes E A = ɛo A has components invariant under null rotations. However even though σ vanishes B AA = βo A ō A ɛo A ῑ A is not invariant unless ɛ = 0. Similarly A AB is not invariant unless ρ + ɛ vanishes, but since ρ = 4ɛ by (26), this implies that ρ = 0. Finally even if κ, ρ, ɛ and σ all vanish we have G ABA = γo A o B ō A (β + τ)o (A ι B) ō A αo A o B ῑ A and this will only be invariant if α = 0 and β + τ = 0. But by (27) τ = 4β, so we must have τ = 0. Hence the dimension of the invariance group of the curvature components at first order H 1 will not remain two unless ρ = τ = α = 0 (69) and Collins [4] has proved that when (69) is satisfied the upper bound is two. Furthermore this is precisely the condition that must hold if there is to be no coordinate information at this stage since this would require G, T = 4B, A, and R = 4E to have constant components, i.e. γ, τ, α, and ρ would have to be constants. However if we apply the NP Ricci equations [10] to these constants, we get (69). Thus if a consideration of the components of the spinors obtained at first order fixes the frame at all, then one must also obtain some coordinate information. But if the frame is not fixed at all at this stage no further information concerning the frame or coordinates will be obtained at the next order and the Karlhede bound in this case will be two in agreement with Collins result.

11 Karlhede classification of type N solutions 1599 References [1] Karlhede A 1980 A review of the geometrical equivalence of metrics in general relativity Gen. Rel. Grav [2] Collins J M, d Inverno R A and Vickers J A 1990 The Karlhede classification of type D vacuum spacetimes Class. Quantum Grav [3] Collins J M and d Inverno R A 1993 The Karlhede classification of type D non-vacuum spacetimes Class. Quantum Grav [4] Collins J M 1991 The Karlhede classification of type N vacuum spacetimes Class. Quantum Grav [5] Geroch R, Held A and Penrose R 1973 A space-time calculus based on pairs of null directions J. Math. Phys [6] Machado Ramos M P and Vickers J A 1996 A space-time calculus based on one null direction Class. Quantum Grav [7] Penrose R and Rindler W 1984 Spinors and Space-time vol 1 (Cambridge: Cambridge University Press) [8] Machado Ramos M P and Vickers J A 1995 A space-time calculus invariant under null rotations Proc. R. Soc. A [9] MacCallum M A H and Åman J E 1986 Algebraically independent nth derivatives of the Riemannian curvature spinor in general spacetime Class. Quantum Grav [10] Newman E and Penrose R 1962 An approach to gravitational radiation by a method of spin coefficients J. Math. Phys