Volume 56A, number 1 PHYSICS LETTERS 23 trebruary 1976

R I C C I R E C U R R E N T S P A C E T I M E S

G.S. HALL Department of Mathematies, University of Aberdeen,

The Edward Wright Building, Aberdeen AB9 2TY, Scotland

Received 15 December 1975

The algebraic restrictions on the Ricci tensor in a Ricci-recurrent spaoe-time are determined. The restrictions im- posed on the Petrov type of the Weyl tensor are also given.

Let M be a Lorentzian space-time manifold and let p E M. The Ricci tensor at p considered as a linear transformation: Tp(M) -+ Tp(M), where Tp(M) is the tangent space to M and p, may take one of four alge- braic types namely the Segr6 types{ l , 1,1,1},{2, 1,1}, {3, 1}, {z ,2 , 1,1), (or their degeneracies) [ 1 - 3 ] . In the first three types, all the eigenvalues are real whilst a pair of complex conjugate eigenvalues appears in the fourth. Attent ion will be focussed on an open subset U of M in which it is supposed the eigenvectors and eigenva- lues of the Ricci tensor yield differentiable vector fields and functions respectively. The (covariant) Ricci ten- sorR is called recurrent on U i fR is non zero on U and VR = R ® 0 on U where 0 is a differentiable one

form on U called the recurrence one f o rm and where V and ® denote the covariant derivative operator and tensor product symbol respectively. I f U is connec ted and R recurrent on U the Segr~ type o f R is the same at a l lpoin ts o f U. To see this, note that since all mani- folds are locally path connected, U is path connected since it is connected. If a, b, EM, choose a piecewise smooth part c: a ~ b. By considering parallel propaga- tion of the Ricci tensor along c, the recurrence condi- tion can'be used to show that to each spacelike (resp. timelike, null) eigendirection of R at a there corresponds a spacelike (resp timelike, null) eigendirection of R at b and conversely *. Suppose now that R is recurrent on U and let C be a chart domain of M. One can choose a pseudo-orthonormal tetrad of vector fields with com- ponents x i , y i, z i, t i in C satisfying x i x i = y i y i = ziz i =

- t i t i = 1 (all other inner products zero) and a null te- trad of vector fields l i, m i, a i, b i satisfying limi = aiai=

* Similarly one can show the invariance of the algebraic (Petrov) types for complex recurrent Weyl tensors.

bib i = 1 (all other inner products zero) such that the components Rij of the Ricci tensor for the above men- tioned Segrd types take the respective canonical forms

[1-31:

Rii = P 1xixf +P2YiY/+ P3ZiZ/ - P4 ti tj , (1)

Ri j= 2Ol l ( imj ) + Xlilj+ oza ia j+o3b ib j , (X:/=0), (2)

Ri /= 2 a l l ( i m j ) + 21Jl(ia/) * a l a i a j + a 2 b i b j , (t~=/:O), (3)

Rij = 2fill(ira/) + (32(1il j - m i m j ) + {J3aiaj + ~4bibj , (4)

~2 :¢: 0 ) ,

where the o's, a 's , lYs, X and/ l are real valued differen- tiable functions on C with X, ~ and/3 2 never zero on C. One may always choose/1 = 1 in (3). In (2), if X > 0 ( X < 0 ) one may set X = 1, (X = - 1 ) . The orthogonali ty relations on the tetrad fields give

li; / = liP / + aiq / + bir ] ,

mi;j = - m i P j +ais j + biu i ,

ai;j= - l i s j - miq] + bio/,

bi;j= - l i u j - m i r j - air / ,

xi;F-yi /zi4+ti i, Yi; j = -Xil}] + z i s f + t iu ] ,

(5) zi; j= - -Xigl / - y i ~ / + tiU ] ,

ti;j= x i r ] + Y i~tj + zio j ,

for vector fields with components pl~Zq~tl e.t.c, on C. In components, the recurrence condition is Ri/;k = Ri]O k, where O k are components of the one form 0, and this equation, together with (5) and the Ricci identity can be used to show that no Ricci tensor of the form (3) or (4) can be recurrent, tha t ' i f (2) holds then o 1 = o2= o 3 = 0 and that i f ( l ) h o l d s then every- where in C the O's satisfy one of the following condi- tions,

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Volume 56A, number 1 PHYSICS LETTERS 23 February 1976

0 ~ Pa = Pb = Pc = Pd " (6)

O=/=pa=Pb=Pc P d = O , (7)

0 4- Pa = Pb 4:Pc = Pd 4 : 0 , (8)

O ~ P a =pb Pc =pd = 0 ' (9)

where (a, b, c, d) is any permutat ion of (1 ,2 , 3,4) . So under the above conditions a recurrent Ricci tensor is (some degeneracy) o f type (1, 1, 1, 1 } or {2, 1, 1 }. In the (2, 1,1) case, the Ricci tensor has zero eigen- values and takes the form R 6 = Xlil/. The null vector 1 i is recurrent (li; j = liP~). It is easily shown that the Weyl tensor is of Petrov type III, N of O with I i as repeated principal null direction if the Weyl tensor is non zero. In the type N case, the Weyl tensor is either complex recurrent or constant (for definitions see [4]). The Riemann tensor is not necessarily recur- rent or constant, but if it is either, then the Weyt tensor is necessarily Petrov type N or O. The general form of the Riemann tensor is readily evaluated, hr the {1, 1, I, 1) case, equation (6) represents an Einstein space. The p 's and the Ricci tensor are constant on C. If eq. (7) or (8) holds, the Bianchi identities can be used to show that the p 's and the Ricci, Riemann and Weyl tensors are all constant. In all cases the Riemann tensor is readily constructed. If (9) holds, the Riemann tensor is again easily constructed and the Ricci, Riemann and Weyl tensors are all recurrent and are constant if and only if the non-vanishing p's are con- stant. The one form 0 is a gradient, proport ional to

the gradient of the non-vanishing/9 and the associated vector field with components O i in C is a Ricci eigen- vector [c.f. 5 ]. The non-vanishing of the Ricci tensor means that when (9) holds, the Weyl tensor is Petrov type D. If (8) holds, the Weyl tensor is type D (type O)

Pa + Pc 4= 0 (Pa + Pc = 0) and retains the same Petrov type throughout C. If eq. (7) holds, the Weyl tensor is zero.

From these results one further concludes that if the Ricci scalar is nowhere zero, then; (i) a recurrent Ricci tensor with nowhere zero recurrence one-form im- plies a recurrent Riemann tensor with nowhere zero recurrence one form (ii) if the space-time is not an Einstein space, a constant Ricci tensor implies a con- stant Riemann tensor [c.f. 6]. Finally, it is remarked that the two algebraic types of Ricci tensor allowed by the recurrence condition are the only two types of Ricci tensor which contain members satisfying the "energy condi t ion" in general relativity theory [2, 3 ].

References

[1 ] J. Plebanski, Acta Phys. Polon. 26 (1964) 963. [2] C.D. Collinson and R. Shaw, Int. J. Theor. Phys. 6

(1972) 347. [3] G.S. Hall, Phys. A (1976). To be published. [4] R.G. McLenaghan and J. Leroy, Proc. Roy. Soc. A327

(1972) 229. [5] W. Roter, Bull. Acad. Polon. Sci. 10 (1962) 533. [6] A.H. Thompson, Bull. Acad. Polon. Sci. 17 (1969) 661.

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