Riemannian Curvature and the Petrov Classification G. S. Hall Department of Mathematics, The Edward Wright Building, Dunbar Street, University of Aber- deen, Aberdeen AB9 2TY, Scotland

Z. Naturforsch. 33a, 559-562 (1978); received November 11, 1977

A connection is shown between the Riemannian Curvatures of the wave surfaces of a null vector I at a point p in space-time and the Petrov type of the space-time at p. Some other results on Riemannian Curvature are discussed.

1. Introduction at the point p under consideration. (A full discussion of the Petrov classification and the geometry of Let M be a Lorentzian space-time manifold. If null congruences and wave surfaces can be found p G 31, let TV(M) denote the tangent space to M in the works of Ehlers and Kundt [5], Sachs [6, 7], at p and let I e Tp (M) be a null vector. Because of Pirani [8, 9] and Petrov [10]. The importance of the the identification of radiation in General Relativity Petrov classification in General Relativity in the with null geodesic congruences of curves in M, study of exact solutions of Einstein's equations is one is led to study the geometrical properties of the dealt with thoroughly in Ref. [5] and in a more wave surfaces of I. These are the two dimensional recent paper by Kinnersley [11].) subspaces of TP(M), each member of which is spacelike and orthogonal to I. There is in fact a two parameter family of such wave surfaces for a 2. Auxiliary Results given null vector leTv(M) which, from the physical viewpoint, might be thought of as the The notation will be the usual one. The Weyl totality of instantaneous wave surfaces of all tensor, Riemann tensor, Ricci tensor and Ricci observers with all possible velocities at p.* This scalar are, in local coordinates, connected by the two parameter family of wave surfaces can also be component relation described as the orbit of one particular such surface Cabcd = Rabcd + Rc[a9b]d + Rd[b9a]c under the action of a two parameter null rotation + h K 9c[bga]d • (l) subgroup of the proper Lorentz group about the null vector I. These transformations are those null Denoting the duality operator by an asterisk, one rotations about I for which I is the only fixed null also has direction and they constitute an abelian subgroup *R*bcd + Rabcd = %9a[dSc]b + 2 9b[c^d]a of the proper Lorentz group, being in fact iso- (Sab — Rab — j R gab) • (2) morphic to the (translation) subgroup of the

Möbius group which have only one fixed point Equation (1) shows that Cabcd= Rabcd if and only

(the point at infinity) on the extended complex if Rab = 0, whilst Eq. (2) yields the equivalences plane. (See for example [2].) *R*bcd = — Rabcd ^ * Rabcd This note will investigate the Riemannian = Rabcd o Sab — 0 o Rab (3) (sectional) curvature of the wave surfaces of a given null vector I and under what conditions the Rie- = i Rgab • mannian curvature is the same for all wave surfaces The last condition in Eq. (3) is equivalent to M of I [c.f. 3, 4]. These conditions turn out to be being an Einstein space. associated with the Petrov type of the Weyl tensor If V is a non-degenerate (non-null) two dimensional

subspace of TV(M), its Riemannian curvature K(V) Reprint requests to G. S. Hall, Department of Mathematics, The Edward Wright Building, University of Aberdeen, is given by Aberdeen AB9 2TY, Scotland. * It should be pointed out that if the null vector I is K(V)= Rabcd^arjbJcyl_ part of a geodesic congruence of curves in M, then this a b c d congruence need not possess finite wave surfaces. This will 2ga[cgd]b£ rj £ r] occur when and only when the congruence is hypersurface Rabcd l[v] i[cVd] orthogonal (twistfree) [1]. ~ (^U){r]b rib)-(^ria)2 ( ] 560 G. S. Hall • Riemannian Curvature and the Petrov Classification where £a and t]a are the components of any basis where for V, K(V) being independent of the basis chosen. Fiab = 2xlaybl, F2ab = 2l\-axbl, Finally it is convenient to introduce at p a * complex null tetrad of vectors (I, m, t, t) where I F3ab = 2UayW = — F2ab . (9) and m are real null vectors, t and t are complex The final equality in (9) gives the orientation of x null vectors and the only non-zero inner products and y. If one then defines the tensor between the tetrad members are la ma = tata= 1. The complex vector t is assumed oriented so that Tab — Rcadb 1° lA the complex bivectors at p, there results Vab — 21 [atb], Tab x*xb = Tabyayb = Tab x"yb = 0, Mab = 21 [am6] + 21 [atb] (5) Tablb = 0. (10) Uab = 2 m [atf,] There then follows the existence of real numbers are self dual; * * * fx, v, A such that Kb = i Vab Mab = iMab Vab = i Uab . T ab = /ulah + vl (axb) + A I (t&b) • (11) The complex self dual Weyl tensor + ^ * From the definition of Tab this implies

Cabed — ^abed + i Qabcd lblcl[eRa)be[dlf] = 0, Rabl"lb = 0. (12) can now be expanded in terms of the bivectors (5) The Eqs. (12), (1) and (6) and the fact that I is null l and five complex scalars C (1 [6] + then imply that l[e^a]bc[dh]= which means that cabed = avab Vcd + C*(Vab Mcd + Mab Vcd) either the Weyl tensor is zero or I is a Debever- + C3 (Uab Vcd + Vab Ucd + Mab Mcd) Penrose vector at p [6].

+ CHUabMcd+ Mab Ucd) Now suppose that I is an eigenvector of the

+ C* Uab uca . (6) Ricci tensor at p. From the symmetries of the Riemann tensor, one has 3. The Petrov Types 12 3 la Rabcd — h Hcd + Hcd + yb Hcd . (13) LeUe TP(M) be a null vector and let x,y e TP(M) be spacelike orthogonal unit vectors spanning a Then the Eqs. (8) imply various algebraic relations k wave surface of I. Then any other wave surface of I on the Hab They imply in particular can be obtained by performing one of the null 2 3 that the bivectors H and H are linear combinations rotations mentioned earlier on this wave surface. * * The transforms x and y' of x and y respectively of the bivectors Fi, F2 and F2. The statement will span the new wave surface, where in com- that I is a Ricci eigenvector at p then shows that ponents [2] i * * H is a linear combination of F\, Fi, F2 and F2. x a = xa + y la , y'a = ya -f d la . (7) Finally, by contracting (13) with lc and noting The two parameter family of wave surfaces of I is * that T[ab] = 0, one sees that the coefficient of Fi in thus reflected in the two real parameters y and d. 2 3 Suppose now that the Riemannian curvatures of the expressions for H and H is zero. This gives all wave surfaces of I at p are equal. By using (4) lalcRabc[dh] = 0 which, because of (1) and (6) and and (7) and the arbitrariness of y and <5, one finds since I is a null Ricci eigenvector, then yields in terms of the vectors x and y that this is equiv- + alent to lalcVabc[dh] = 0. So the Weyl tensor is either zero or else algebraically special at p with I as a repeated Rabcd Fxab F2cd = Eabcd F^b F2cd principal null direction. ab = RabedFi F3 ^ Conversely suppose that the Weyl tensor is

= RabedF3^F3cd (8) either zero or else algebraically special at p with I

= RabedF2^F3cd = 0 as a repeated principal null direction and that I is 561 G. S. Hall • Riemannian Curvature and the Petrov Classification a Ricci eigenvector at p. Then in (6), using a null has all its wave surfaces of the same Riemannian tetrad based on I, one finds that C4 = C5 = 0 and so curvature since, in this case, Rab a la h and the Weyl tensor is either zero (whence one readily Cabed Vab Vcd = iabcd Vcd = 0 . (14) finds la Rabcd — 0 and so (8) holds) or else it is algebraically special with I as a repeated principal If one converts from the null tetrad used in (14) to null direction [12]. a real null tetrad under the identification

j/2 ta = xa + iya , 4. The Riemannian Curvature of Orthonormal then because *Cabed = Q*bcd and because I is a Tetrad Planes Ricci eigenvector, the Eqs. (8) for the Riemann tensor follow from (1). Hence the Riemann cur- On many occasions in the literature, restrictions vatures of all the wave surfaces of I are equal. on the Riemannian curvature of certain 2-spaces It is thus shown that for non conformally flat have been used in order to impose "symmetry space-times: conditions" on a manifold. As an example, it is (i) If the Riemannian curvatures of the wave recalled that if, say, M is a space-time manifold surfaces of I are all equal, then I is a Debever- and if one asks that all 2-spaces at any given point Penrose vector and (12) holds. p e M have the same Riemannian curvature

Let M be a four dimensional Lorentzian manifold so that in the new tetrad (x', y', z, t), Kx z = Ky'Z = which is an Einstein space. If p e M, let (x, y, z, t) «eR. Then Kx-y- = — (A + 2a). Now perform a be an orthonormal tetrad at p with similar rotation, keeping t and y' fixed, to a tetrad

(x", y', z", t) where Ky'Z" = Ky'X". It then follows xa xa = ya ya = 2a Za = — ta ta = 1 , that KX"Z" = cn and Ky'Z>> = Ky'X" = — |(A + a). let PXy denote the two dimensional subspace of Now consider the function which associates with TV(M) spanned by x and y and so on for the other each tetrad (x, y, z, t) a real number according to pairs of tetrad members and let Kxy denote the the scheme Riemannian curvature of Pxy. The following (*, y, z, t) -> (Kxy + a X) (Kxz + 1 A) (Kyz + * X). theorem can now be proved. (15) At any point p in a general 4 dimensional Einstein space which has either positive definite or Lorentzian Here, only the restriction of this (continuous) signature one can always choose an orthonormal tetrad function to the tetrads obtainable from the original at p (in fact infinitely many) such that all the one by spatial orientation preserving rotations will

Riemannian curvatures Kxy, Kyz, Kxz, Kxt, Kyt, be considered. Since the associated subgroup of the

Kzt are equal. Lorentz group is connected, the intermediate value To prove this, note that for coordinates about p theorem may be applied in the following way. If in such that xa = dia, ya = b2a, za = d3a and ta = the tetrad (x', y', z,t) a= — then clearly ab one has gra& = diag(1, 1, 1, — \) = g and so the KX'y' --- KX'Z — Ky'Z — ^ X Einstein space condition at p, Rab = Xgab (A £ 1R), gives for a = b = 4. Kn X24 + -^34 = — Next, and the conclusion of the theorem holds. If howrever Eq. (3) shows that the Riemannian curvature of arp+0, then the function (15) maps the tetrads any 2-space is equal to that of its complement*. (x', y', z, t) and (x", y', z", t) onto real numbers of

Hence K\2-\-K23-\-K%i= — X. To establish the opposite sign. One concludes the existence of a above mentioned result, it is then sufficient to tetrad (x, y, z, t) in which one of the canonical prove that a tetrad may be chosen at p for which 2-spaces, say Pxy, has Kxy— — Another the Riemannian curvatures of P12, P23 and P31 rotation keeping z and t fixed yields a tetrad with are all equal. To do this one chooses an arbitrary the desired properties and completes the proof. tetrad at p, say (x, y, z, t) and performs a spatial The above argument also shows that infinitely orientation preserving rotation keeping t and 2 fixed many such tetrads at p exist because the first tetrad was arbitrary and only spatial transforma- * For four dimensional positive definite Einstein spaces the Riemann tensor satisfies *i?abcd = -^abcd and so this tions were used. The proof for the positive definite result still holds. case is similar.

[1] W. Kundt, Z. Phys. 163, 77 (1961). [9] F. A. E. Pirani, Lectures on General Relativity, [2] A. Trautman, Lectures on General Relativity, Chap- Brandeis Summer Institute in Theoretical Physics, ter 3. Brandeis Summer Institute in Theoretical Prentice Hall, London 1964. Physics, Prentice Hall. London 1964. [10] A. Z. Petrov, Einstein Spaces, Pergamon Press, New [3] G. S. Hall, J. Phys. A6, 619 (1973). York 1969. [4] J. A. Thorpe, J. Math. Phys. 10, 1 (1969). [11] W. Kinnersley, General Relativity and Gravitation, [5] J. Ehlers and W. Kundt, Gravitation, ed. L. Witten, ed. G. Shaviv and J. Rosen, John Wiley, New York John Wiley, New York 1962. 1975. [6] R, K. Sachs, Proc. Roy. Soc. A264, 309 (1961). [12] J. N. Goldberg and R, K. Sachs, Acta Phys. Polon. [7] R. K. Sachs, Relativity Groups and Topology, ed. Suppl. 22, 13 (1962). De Witt and De Witt, Gordon and Breach, London, [13] L. P. Eisenhart, Riemannian Geometry, Princeton Glasgow 1964. University Press 1966. [8] F. A. E. Pirani, Phys. Rev. 105, 1089 (1957). [14] Yi-Chiang Yen, Proc. Amer. Math. Soc. 40, 554 (1973).