General Relativity and Gravitation, Vol. 23, No. 7, 1991

Simply-Transitive Homothety Groups J o h n D. Steele 1 Received October 8, 1990

In this paper the spin-coefficient formalism of Penrose and Rindler is applied to the problem of finding space-times that admit a simply-transitive group of homotheties. The method is illustrated by an application to algebraically special vacuum space-times. Vacuum metrics with a fivedimensional homothety group are also considered.

1. I N T R O D U C T I O N As is well-known, classifying those space-times in General Relativity that admit a simply-transitive group G4 of motions is an algebraic problem; this follows from the result that if a second countable Lie group G of transformations of a Hausdorff manifold M acts transitively and freely, then M and G are diffeomorphic. In practice, one then does one of two (essentially equivalent) things to find all such solutions. Following Hiromoto and Ozsvs [5] one can put a suitable metric on the Lie group G4 with respect to which a choice of the generators of the Lie algebra ~4 of ieft-invariant vector fields on G4 are a pseudo-orthonormal (or real null or Newman-Penrose) tetrad, and find the connection coefficients and Ricci tensor in terms of the structure constants of the Lie algebra, which is classifted with respect to Lorentz transformations of the tetrad. Alternatively, we can use the Newman-Penrose formalism applied in a tetrad formed from generators of the reciprocal group of G4, which is generated by the right invariant vector fields on G4, [9]. In this latter method, which is best suited to algebraically special space-times, all the spin coefficients and the Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia 811 000L7701/91/0700-0811506.50/0 ~ 1991 P]e~um PubllshJ~ Co~poratlon

812

Steele

tetrad components of the Weyl and Ricci tensors are constant. The two methods give the same results; the only difference between them arises in the actual group operation on the Lie group, as the left-invariant vector fields on a Lie group are the right-invariant vector fields of the opposite Lie group (i.e. the Lie group with the same underlying set but reversed group operation, see e.g. Ref. 13), and the actual form of the group operation does not concern us. We note that a Lie group and its reciprocal are isomorphic, as opposite Lie groups are. There is no reason in particular why the transitive group has to consist of motions: the manifolds are diffeomorphic whatever type of transformation one uses. The spin coefficients and tetrad components of the Weyl and Ricci tensors are constant in the case of motions because motions are symmetries of the connection and the respective tensors. But so are homotheties - - in fact homotheties are the most general type of transformation that are both conformal and affine, and therefore symmetries of the Weyl and Ricci tensors and the connection. Thus it is to be expected that finding simply-transitive homothety groups is susceptible to the same, or similar, techniques to those used for motions. This is indeed the case, as this paper intends to show. The one major difference we encounter is that as homotheties are not symmetries of the metric we cannot impose on the Lie group a metric whose tetrad components in the tetrad of generators are constant, as was done in [5], or equivalently the reciprocal generators cannot be taken t o form a normalised tetrad, where by normalised we mean that the nonzero inner products are q-l, i.e. constants. We will deal with this problem in Section 2, where it will become apparent that the generalised spin-coefficient formalism given by Penrose and Rindler [11] is the tool we require. Section 3 deals with an application of the technique to algebraically special vacuum space-times. In Section 4 we use the results of Section 3 to find all vacuum space-times with an 7/5. Throughout this paper (M,g) will denote a space-time with metric g of signature - 2 , and conventions will follow Penrose and Rindler [11]. For basic results on homotheties, see for example [4] and the references therein. 2. B A S I C F O R M A L I S M

We denote the simply-transitive group of homotheties that the spacetime ( M , g ) is assumed to admit by H4, and its associated Lie algebra of homothetic vectors on M by 7t4 (a vector field X on M is homo~hetic if Lxgab = 2r for /~ denoting Lie derivative and r a constant called

Simply-Transitive H o m o t h e t y G r o u p s

813

the divergence or homothetic constant). In the sequel, sans-serif indices a,b .... will denote generators of (four-dimensional) Lie algebras and will range and sum from 1 to 4. Suppose we have generators {Xa} of this Lie algebra, and assume, as we may without loss of generality (see e.g. Ref. 4), that X4 is a proper homothety with non-zero divergence r and the other three generators are Killing vectors, so that

~x4g = 2r

l~xlg = s

= f~x3g = O.

We will denote the derivative s by /:a in the sequel. Note that the generators X1, X2, X3 generate an ideal of the algebra 7"/4. The reciprocal algebra 7i4 of 7/4 is the Lie algebra consisting of vector fields Y such that [Y, Xa] = 0 for all a, and is, as suggested by the notation, also four dimensional. Let Ya be generators of 7-/4, and define gab = j so that gab are the tetrad components of the metric in the tetrad gijY;i Yg, {Ya}- Then by the choice of generators/~cgab = gab,j X j is identically zero unless c=4, when it equals 2r It follows that the gradient of log Igabl is the same for each pair ab, so if f = 89 [ga0bo[ for some a0b0 we can write gab -----kab e2l for a matrix of constants kab and s = 0 unless c=4, when it is the constant r Thus the tetrad components of the metric are not constant. We could follow the methods of [5] and introduce a torsion-free connection o n / / 4 (which is diffeomorphic to M) by ~xaXb

= FabCXc

where 2F[ab] c = CCb and the CCb are the structure constants of the algebra 7-/4, and then introduce a metric f on H4 with which this connection is compatible. If we have g'(Xa,Xb) = gab then the compatability requirement is [5] Fabc = 1 (X gbc t + Xbg , c -- XCg b)

+

1

(Cbc + Cc b - C br

where Fab c = Fabcfg~c and Cabc = C~bgfc. f ' In this situation the reciprocal algebra 7-14 consists of homothetic vectors of g' and comparing with the previous paragraph we see that Xcfa b -- /_tCfab for constants /tc - - which will all be zero only in the motion case. Thus we cannot simplify the expressions for the connection coefficients as in the motion case, which makes the formulae for the Riemann and Ricci tensors much more complicated.

814

Steele

Rather than use this idea, we will apply the formalism for generalised spin-coefficients given by Penrose and Rindler (__Ref. 11, pp.226ff). Firstly, as the generators of both 7-14 and 7-/4 span the tangent spaces at each point of M, there are functions Fab such that Yb = F a b X a 9 Thus

~:cFab = - F d bC~da But the subalgebra of Killing vectors form an ideal of 7/4, so that for all cd, Cc4d = 0, and thus F4b is a constant, ~b say. From the above results we see that Y ~ f , i = F a b X ai f , , . = Ab, and hence the directional derivatives of f along the tetrad members Ya are constant. We still have the choice of the matrix kab, which corresponds to the choice of generators. In order that we may apply the spin coefficient formalism, we take the tetrad so that it forms a (non-normalised) complex null tetrad, ]I1 ~ ~, Y2 --+ n, Y3 + iY4 --* m , where ~a na = - m a ma = e 2J . If we form a dyad of spinors oA and t A in the~usual way, so that g a ~ o A o A etc. (see Ref. 11, p.l19), then we will assume that the "normalisation" is such that o A L A = e ] = X" We can then define the spin coefficients for the tetrad (or dyad) as in [11] as follows. Let D, 6, 8', D t denote directional derivatives along s m, rh and n respectively. Then the spin coefficients are defined as the coefficients in the following equations [Ref. 11, equation (4.5.26)] D o A = eo A _ ~tA

D t A = ,~t t A __ TI oA

~t o A = 0r A - - p t A

6t t A = 13it A - - 0.'0 A

50 A = ~0 a -- r A

D ' o A = 7o A - .i-ta

6t A = ~' t A _ pl oA

Dt tA = CI ~A __ Iq.lo A.

Let gab = e - 2 f g a b . Then the homothetic vectors of g are Killing vectors of ~', for f.agab = 2(r -- f~af)gab = 0, where Ca is the divergence of Xa with respect to g. We use the spinors ~A = o A, ~dA = t A as a dyad for (M,~), since these spinors are essentially determined by the algebra 7/4, but with OA = DOA, TA = ChA for ~ = e - f = X-1. T h e s e s p i n o r s are normalised with respect to ~', i.e. form a s p i n - f r a m e for ~. Clearly admits a simply-transitive group of motions and therefore, as is wellknown, the spin-coefficients and tetrad components of the Weyl and Ricci tensors of ~ in the tetrad above are constant. Furthermore, the formulae given in Ref. 11, p.357-9 for the behaviour of the spin-coefficients and tetrad components of the Weyl and Ricci tensors under a conformM change show that these scalars are constant for g (as one would expect). Also, since we have aspin-frame for ~, the spin-coefficients for ~ are such that ~"= - { ' , ~ = - ~ ' , g~ = - ~ , H' = - ~ . Thus from [11] the spin-coefficients for g satisfy e+7'

=of

c~+fl'=6'f

ce'+~=6f

7+e'=O'f.

(1)

Simply-Transitive I-][omothety Groups

815

These equations prove useful. The unknown constants D f etc. correspond to the constants Pc in the "Hiromoto and Osvs style" method above. We can also give formulae that relate the tetrad components of the Weyl and Ricci tensors for ~ and g by speciMising from the formulae given in [11]. The choice of a real X shows that ~j = ~j for each of the Weyl scalars. For the Ricci tensor the components are related as follows (see Ref. 11 for the definition of these scalars):

Df.Df D f .6' f ~'f.6'f D f .6f ~m = ~tl + 89( D f . D ' f + 6f.6'f). ~oo = ~1o = ~2o = ~o1=

~oo + ~1o + ~2o + ~ol +

+ D~f.D'f ~12 = ~12 + D' f.6 f ~o2 = ~o2 + ~f.~f '~21= ~ 1 + D' f.c~'f ~22 = r

Finally, from the expression for the Ricci scalar R we find that the scalar II = x~ R/24 transforms as 1] = II + ~1 ( D f . D ' f - 5f.6'f) Our basic method in applying this formalism is to use the Bianchi identities (Ref. 11, p.259), the curvature or "Newman-Penrose" equations (Ref. 11, equations (4.11.12) p.248-9) and the above equations (1) to find the spin-coefficients in terms of the directional derivatives D f, D~f etc. and as few other constants as possible. This will then lead to the possible structures of the Lie algebra 7-/4, and thus of 714, via the commutators of the tetrad vectors, which may be checked against the Jacobi identities to ensure we have a Lie algebra. We can then find expressions for the tetrad vectors in suitable coordinates and give the function f in these coordinates. The metric will follow from the completeness relation

gob =

(2e(onb) _ 2m( mb)).

The homothetic algebra will then be the reciprocal algebra of the algebra formed from the tetrad vectors. 3. A L G E B R A I C A L L Y SPECIAL V A C U U M SOLUTIONS

We illustrate the use of the technique with algebraically special vacuum solutions. The generators of the algebra 7/4 will be assumed to form a non-normalised Weyl canonical tetrad, with g a repeated Debever-Penrose

816

Steele

vector - - the constancy of the tetrad components ensures this can be done. This implies that 90 = 91 = 0, and from the Goldberg-Sachs theorem ~ = ~r = 0. The latter can be proved directly in this formalism just as easily as in the original Newman-Penrose formalism. Note that we cannot choose the parameterisation of the geodesic congruence ~ to be affine (which would give e + g = x - 1 D x = D f), as its parameterisation is fixed by the algebra. If we use the above equations (1) the Bianchi identities (Ref. 11 p.259) are

(3p + 2Df)gg2

(2a)

D ~ 3 - 6'~2 =

(2p + 27' + Df)9a - (3r' + 26'f)~112

(2c)

D94

3cr'92 -- (4r' + 2/3' + 6 ' f ) ~ 3

D92 =

- 6tga :

+ (p - e + 37' + Df)~4

(2d)

D~ga - 6~4 = - 3 ~ ' 9 2 + (4p' + 2e' + D'f)~3 D ' ~ 2 - 6~3 =

- (v - / 3 + 3c~' + 6 f ) ~ 4

(2e)

(Up' + 2D'f)92 - (2r + 2 d + 6f)@3

(2f)

The constancy of the scalars ~i implies that the left hand sides of these equations are zero. We will consider the Petrov types in turn, and simply denote the curvature equations (4.11.12a)-(4.11.12/') of Ref. 11 by ([11]a)([ll]l'). The formalism applies to finding space-times admitting a simplytransitive group of motions, of course: we would just assume f to be constant. The results of [5] show that the only possible algebraically special vacuum metrics with such a group of motions are plane waves, when the Petrov type is N, and thus these results will serve as a partial check on the calculations that follow. T h a t no such type II or D metrics exist is obvious, as then we have ~ # 0 and equations (2a) and (2b) would imply that g is recurrent, which is not possible [3]. Petrov Type N In a canonical tetrad we have ~2 = 93 = 0 and ql4 a non-zero constant, and we also have the freedom to perform constant null rotations about without disturbing these relations. The Bianchi identities reduce to

(a) p - e + 3 7 ' + D f = O

(b) r - ] 3 + 3 o / + 6 f = O .

(3)

From ([ll]a) we have p(p + e + g) = O, and we consider two cases: p # 0 and p = O.

Simply-Transitive Homothety Groups

817

If p # 0 then p = - r - g is real. E q u a t i o n (3a) then implies t h a t 2 "7' = 2 ( e - 7".1, f r o m which, using the first of eqs. (1), r = -gDf, 7' = 8 9 and thus p = - 4 O f . Now as p # 0 we can p e r f o r m a n u l l r o t a t i o n a b o u t / ? to set r = 0; this r o t a t i o n will be constant as p and 7" are, it just corresponds to a change of generators in 7-/4. But now ([11]e) implies t h a t r ' = 0 and ( [ l l ] c ' ) implies ~' = 0. However f r o m ( [ l l ] f ' ) pl = 0, and then ( [ l l ] a ' ) shows t h a t ~' = 0. These results contradict ([ll]b') (and the Goldberg-Sachs theorem, as n cannot be a repeated Debever-Penrose vector). Hence p = 0. W i t h p = 0 eq. (3a) gives r - 33" = D f which equals e + 7 I, so 71 = 0 and e = D r . Now we have ([ll]c) reducing to r e = 0. I f r # 0, then e = 0 and ([ll]e) and ( [ l l ] f ) give r = - ( 6 + c~') = - ( f l + fl') and ( [ l l ] e ' ) and ( [ l l ] f f ) give

7'+

+

+

= 7-1(r +

+

= o,

so t h a t either 7-1 = 0 or 7-I = ~. T h e latter possibility contradicts ( [ l l ] k ' ) (which would show t h a t 7-% = Irl S = 0) and hence 7-' = 0. B u t now ([11]k)-([11]/1) b e c o m e

fl~ + aT- = 0 + o/) =

fll7- + ~1~ = 0 +

+

=

l(s +

But these equations and the above expression for 7- imply t h a t c~ -- ~ / = fl = fll = 0, and hence 7- = 0, a contradiction. Hence 7- = 0 and e is recurrent. T h u s the space-time is a p.p. wave: is parallel to a covariantly constant vector. T h e only possible metrics are then h o m o g e n e o u s plane waves, some of which do a d m i t such a group, and the p.p. wave

ds 2 = dx 2 + dy 2 + 2dudv + 2 log(x 2 + y2)du2,

(4)

see [12]. T h e three Killing vectors t h a t this metric admits can be taken as 0,, a~ and ya~: - xO u and the proper h o m o t h e t y as x0~ + y0~ + ua~ + (v 2u)Oo. This metric a d m i t s an Abelian G3 with time-like orbits, but does not seem to be covered by Table 11.3 of [9]. Petrov Type III In a Weyl canonical t e t r a d we have ~2 = gl4 = 0 and ~3 a non-zero real constant. However, it will be more convenient not to assume t h a t ~4 = 0, in which case we will have the freedom to perform null rotations a b o u t g w i t h o u t disturbing the assumed conditions on the ~i. T h e Bianchi identities (2) for type I I I are then 0 = 2(p + 7') + O f

(ha)

818

Steele 0 -- (4r' + 2/3' % 6'f)ffla - (p - e + 37' + Df)kll4

(5b)

0 = (4p ~ + 2e' + D'f)oJ3 - (v - / 3 + 3c~' + 6 f ) ~ 4

(5c)

0 = 2(v + ~') + 6 f

(hd)

Again from ([ll]a) either p = 0 or p = - e - g # 0. If p = 0 then ( [ l l ] f ) shows that either r = 0 or r = - a ' - ~. In the former case, t is recurrent, and from [7] such space-times admit at most two Killing vectors, contradicting the existence of an 7-/4. Hence r # 0 and a similar argument to the type N case above leads to a contradiction of ( [ l l ] k ) - ( [ l l ] / ' ) . Hence p # 0. As p = - ( e + g) # 0 we can perform a (constant) null rotation to ensure that v = 0, and then (Ill]c) implies that r ' = 0 and ([ll]e) that a' = 0. Reality of p and ([ll]d') shows that p' # O, whence from ( [ l l ] f ' ) P = - ( 7 ' + 7') and so from (5a) 7' = e -- 8 9 so that p = - O f . (Hence there can be no type III vacuum metric with a transitive G4.) Also, from (hd), ~' = - 8 9 so/3 = { 6 f . But ([ll]d) implies that ()+/3 = 0, thus a = -a--6'f2 and/3' = 5-6'f2. It now follows from ([ll]c') and ([ll]d') that ~3 = ~ ' D f = 2p'6'f and from ([ll]b') 0/4 = 6~'6'f, and eq. (5b) is identically satisfied. T h e n ([ll]a') and ( [ l l ] f ) imply that - p ' = e' + g = 7 + 7 which must equal D ' f . If we now use ([ll]h'), the two expressions for 0/a above and the fact that 6 f # 0 we find that e' = 7 = 1D' f and pl = - D ' f . Thus 6 f is real and non-zero. We have now expressed every spin-coefficient in terms of the directional derivatives of f . The only remaining curvature equations or Bianchi identities that give any further information are ([11]/) and ([11]/'), which both reduce to ( D f ) ( D ' f ) = 6 ( 6 f ) ( 6 ' f ) = 6(6f) 2. We now know enough to determine the structure of the Lie algebra 7-/4. From the formulae on p.228 of Ref. 11, the commutators of the tetrad vectors are [g,n] = D f n D'ft [e,m]

= 0

[u, m] =

- 26/n

[m, rh] ----2(6'f m -- 6f rh)

and the appropriate complex conjugates. But using the two expressions for O/a and transforming the generators we can simplify the structure so that the only non-zero commutators are

where ):I = :.~Dr = -i(m

]:2 = n - (D' f)Y1 -

= (m +

-

Simply-Transitive Homothety

Groups

819

and a = 2 5 f 5s O. Now from Lie's third theorem, see e.g. [9], any Lie algebra of vector fields is the Lie algebra of some Lie group, at least locally. So, putting coordinates {u, v, x, y} on the Lie group we choose as generators o f ~ 4 the vector fields

Y1 = ucg~

]I2 = UOv

]I3 = yOz

Y4 ": - y (Oy

from which, if k = D f and k ~ = D~ f we have ~a = ( k u , O,O,O)

n a ---- (ktzt, u,O,O)

m a : (au, O, i y , - a y )

and f = log(uyl/2). Hence from the completeness relation

ds 2 = -kl ( 2 y d u d v _ 2 k , y d v 2 + 2 u d v d y ) - ~

~t2

( a 2 d 2 + d y 2) ,

where 2kk ~ = 3a 2 and the homothetic algebra is generated by 0~, 0~, uO~, + vOo and xOz + yOu. A change in coordinates such that v ~ = uy, u ~ = v / k , y~ = 2a3x and z ~ = - 2 a 2 y puts this metric, after dropping the primes, into the form 3

2

V2

ds ~ = 2dudv + -~ x du + ~-ff (dz ~ + dye),

(6)

which we reeognise as the singular type III Robinson-Trautman metric discussed by Kerr and Debney and others [6]. Petrov Type II For type II we have ~2 and ~4 non zero. The tetrad could be arranged so that k~3 5s 0, but as in the type III case it will prove useful not to assume this, which will give us the freedom to perform null rotations about g. Again we consider essentially two cases: p 5s 0 and p = 0. In the former case, after a null rotation to set r = 0, we find, via the Bianchi identities and eqs. ([ll]e) and (Ill]e) a contradiction of ([ll]a). Thus p = 0, and clearly r 5s 0, else g would be recurrent. We now perform the null rotation to set ~3 to zero. Chasing through the Bianchi identities and the curvature equations we find that D f = D I f = 0 and the only non-zero spin coefficients are v = ~ = - ~2S f , a I, fl, fl', c~ and ~'. Now eqs. ([11]/) and ([ll]l') reduce to, respectively,

3flS' f -- 3o~5f + 6 fS' f = 0 3fl'S f - 3o/6' f -t- 6 f 6 ' f = O.

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Steele

It follows from these equations t h a t a and fl' are real multiples of a ' f and a ' and fl are real multiples of 6f. For if a ' = A6f + r for A E I~ and r E C then fl = ( 1 - A ) 6 f - r , /3' = ( A - ~)6 1 , f + ~ and a = (4 - A ) 6 ' f - ~, the equation derived above from ([11]/) then reduces to r 6 ' f = ~6f. So r 6 ' f is real, so r is a real multiple of 6 f which can be taken as zero. Equation ( [ l l ] a ' ) now shows t h a t either A = 2/3 and ~' = # 6 ' f or A = 1/3 and ~' = p i 6 ' f where in each case p E ~ is non-zero. No other information is obtained from the rest of the curvature equations. We consider the first case in detail, the second is similar. We calculate the brackets of the tetrad vectors in terms of/~ and the real and imaginary parts of 6f. Let 89 = a + ib for a, b ~ 1~. T h e n we

h ave

It, r d = + b)t It, n] = o [n, rn] = - 3 ~ ( a + ib)g - (a + ib)n [m, rh] = 4(a + ib)fn - 4(a - ib)m, and appropriate complex conjugates. At least one of a and b is non-zero; we assume, using a rotation in the plane of m and rh if necessary, t h a t b 5s 0, put m = x + iy and arrange that generators so t h a t Yx = g, Y~ = n, Y3 = bx - ay and Y4 = by and we find t h a t [111, Y4] = -Y1

[Y2, Y4] = -Y2 - 3#Y1

[Ya, II4] = - 4 Y a

and all other c o m m u t a t o r s are zero. We see that there is an three dimensional Abelian ideal which equals the c o m m u t a t o r subalgebra. Hence there is an Abelian G3 of motions, which must equal the complete group of motions because of the Petrov type. Given that metric (4) is omitted from Table 11.3 of Ref. 9, we will find the metric directly, rather than use this table. If we use coordinates on the Lie group {x, y, z , w } then the vector fields Y1 = e-~~

Y2 -" - 3 # w e - ' ~

+ e-~~

Ya = e - 4 ~ &

Y4 = -0~,

satisfy the commutation relations above. If we let u = c -4w SO f = 3 log U and using the completeness relation we find, after suitable trivial coordinate changes, that the metric is ds 2

u - 1 / 2 ( d u 2 + dz 2) - 2 u d x d y 4 - u l o g u d y 2

(7)

(the 4- is to account for the sign of/*), with Killing vectors 0~, 0y and 0z and h o m o t h e t y (x + 2y)O~ + yOy q- 4UO~ + 4zOz. This is a special case of van Stockum's stationary axisymmetric metric, (see Ref. 9, p.205).

Simply-Transitive Homothety Groups

821

For the second case, where A = 1/3, a similar a r g u m e n t leads to the metric ds 2 = x - 1 / 2 ( d x 2 + dy 2) - 2 x d u ( d v + # y du) (8) for a non-zero real c o n s t a n t / t . This metric, which is another special case of van S t o c k u m ' s metric, a d m i t s the Killing vectors Or, c~u and (gy - pU(gv and the h o m o t h e t y 4yOy + 4xc9~ + 3rOy - uO~,. Note t h a t in this case the G3 is not Abelian, but of Bianchi type II acting on time-like orbits Petrov Type D As all v a c u u m type D metrics were given by Kinnersley [8], a s y s t e m a t i c search t h r o u g h his metrics would solve the problem. However, before using Kinnersley's results we will a p p l y our m e t h o d to narrow down the field. We note t h a t as all v a c u u m type D metrics a d m i t an even n u m b e r of Killing vectors, there must be an e x t r a Killing vector in all cases, i.e. there is a 7i5 if there is an 7-/4. We assume t h a t b o t h s and n are r e p e a t e d Debever-Penrose vectors, so t h a t only k~2 r 0. T h e Bianchi identities give 3p + 2 D f = 3p' + 2 D ' f = 3 7 + 2 5 f = 3r ~ + 251f = O.

Hence p and p~ are real and r = ~ once more. T h e r e are two cases to consider: p -- 0 and p r 0. However, ([11]c) implies that 7"('~'+e) = 0, so if 7- :~ 0 then "~ = - e , and hence f r o m the first of eqs. (1) D f = 0. Similarly, from ([11]c') either 7"' --- 0 or D ' f -- 0. T h u s p # 0 if and only if 7" = 0 (and similarly with primes). However the Bianchi identities above show t h a t v -- ~', and thus f r o m ( [ l l ] f ' ) and ([11]f), p # 0 if and only i f p ' ~ 0, as e x p e c t e d from [8]. In order to c o m p a r e with Kinnersley's results, which are in the s t a n d a r d N e w m a n - P e n r o s e formalism, we calculate the N e w m a n Penrose coefficients with respect to the same metric for the normalised t e t r a d { e - Y g a , e - / n a, e - Y m a , e - / ~ a } . T h e effect of this change is t h a t to, a, p, 7" and td, cr~, p~, 7-~ are all scaled by e - / . Hence if the PenroseRindler p is zero then in t e r m s of N e w m a n - P e n r o s e quantities we have = a = p = v = A = p = 0 and r + ~" = 0. T h u s the only possibility is Kinnersley's metric I V B [2,8] which, as can be checked directly, m a y a d m i t a h o m o t h e t y . We find t h a t the Kinnersley constant C must be zero, so the metric is ds 2 = 2 d u d r - 4r v d u d v - ~ vm dr2 - -2-m dy2 v

for m a non-zero constant. C h a n g i n g coordinates so t h a t 2t = u + v - 2 r , 2z = v - 2 r - u, x = v 3/2 gives the metric as ds 2 =- - c d x 2 - E - l x - 2 / 3 d y 2 + x4/3(dt 2 - d 2 ) ,

(9)

822

Steele

where e = 1 / 2 m can be normalised to +1 and thus accounts for the signature. Hence this metric is a type D Kasner metric, e.g. [9], with time-like 2-surfaces of constant curvature. The Killing vectors are Or, Oy, Oz and zOt +tOz and the homothetic vector can be taken as 3z0~ +4yOy + zOz +tO~. Now consider the p # 0 case, so that pl # 0 and r = r ' = ~ f = 0. We find t h a t c~ = c~' = / 3 = / 3 ' = 0 and

= p(v' + 7 + : --pp'

= v'(p + 7' +

= C(7 + ~) - 7(7' + 7') = ~'(7' + 7') - 7'(7 + 7)" and 7 ' + 7' = 4 D f . The only spin coefficients It follows t h a t 7 + 7 = 4_D'f 3 yet to be expressed in terms of the directional derivatives of f are the imaginary p a r t s of c, ~', 7 and 7'. Equations (1) show t h a t 9 c = - 9 7 ' and 9~' = - 9 7 , where 9~ is the imaginary part of ~ etc. and the above equations for qz2 show t h a t ( g r = ( 9 7 ) D f . Thus in addition to the directional derivatives there is only one unknown, corresponding to the imaginary p a r t of ~ (or 7 etc.). There is no more information gained from the rest of the curvature equations or from the commutators. We now calculate the brackets of the tetrad vectors. If we let A = D f~3 and # = D ' f / 3 , z and y be real space-like vectors such that m = z + i y and set t~ = 9.%7 and ~r = 9~ we have lr# = 1/A. Adjusting the generators of the algebra so that ]I1 = An - #s Y2 = z, ]I3 = Y and ]I4 = - s the only non-zero commutators are []I1, Y4] = 4111

[Y2, ]I4] : ]/2 - kY3

[Y3, ]I4] : Yz + kY2

where k = 2 r / A . Hence the algebra 7/4 has a three-dimensional Abelian ideal, and as in the type II case above, there must be an Abelian G3 of motions, this time not maximal of course. Using coordinates {w, v,y, z} on the Lie group the vector fields Y1 = e4~0v

Y2 = - 0 ~

Y3 = e'~(sinkwOu + coskwO~) Y4 = e ~ (cos kw by - sin kw O~) satisfy the commutators. We find t h a t f = 3w and the metric is

ds 2 = 2e2Wdwdv + e-le-2Wdv 2 _ e4W(dy2 + dz2), where e-1 = - A # . Changing coordinates such t h a t z = v + ee4'~/4 and = e 3 w gives the metric as

ds 2 = -ed~ ~ + et-2/3d~: 2 - t4/3(dy 2 + d 2 ) ,

(10)

Simply-Transitive I-Iomothety Groups

823

after suitably normalising c to -t-1, which is a type D Kasner metric with space-like 2-surfaces of constant curvature. The Killing and homothetic vectors are similar to those in the p = 0 case, m u t a t i s mutandis.

4. METRICS ADMITTING AN 7/5 In Hall and Steele [4] a study was made of space-time metrics with a homothetic isotropy, i.e. metrics admitting an r-dimensional Lie algebra 7/r of homothetic vectors with at least one (and of course essentially only one) proper homothetic vector whose orbit was no greater than (r - 1)dimensional. In t h a t paper all such metrics for r > 6 were found, and for r = 5 it was shown that the 7/5 is transitive and the ideal G4 of Killing vectors is necessarily intransitive on orbits that were, if everywhere of the same type and dimension, non-null; the Petrov type is therefore D, N or O. We use the results of section 3 to study this r = 5 case in vacuum. Firstly, in the type N case the ~3 orbit must be timelike as the isotropy is a null rotation (about the quadruple Debever-Penrose direction). From Kramer et al. [9] the two possibilities are the Barnes metric, with a ~3 on null 2-space orbits, or the metric ds2=dw2+A2[dy2-2eYdv(du+BeYdv)]

(11)

with A, B functions of w only. However if a Barnes metric is vacuum it is a plane wave [1], which admits at least an 7/6, and metric (11) can only be Ricci flat if it has signature (+, + , - , - ) . To prove this we define a tetrad for this metric by 0~ 1 = A ( d u + BeY dv), ~2 = AeY dr, w 3 = A d y and w4 = dw, so t h a t ds ~ = - 2 w l w 2 + (w3) 2 + (w4) 2. Then calculating the Ricci tensor we find t h a t A = ztziw/2, B = c l w 2 + c2w -4 for constants Cl and c2 and that in this case the metric has signature (+, + , - , - ) . Hence there are no type N vacuum metrics with an 7/5. In the type D case there are two possibilities for the isotropy. Either the isotropy is a spatial rotation (so the metric is an L.R.S. metric in the terminology of Ref. 9) or it is a boost; we consider these two cases separately. Our intention is to use the Schmidt method on the full algebra to show t h a t in either case there is in fact a transitive 7/4. For b o t h cases let X1, . . . , X4 be generators of the Killing algebra and X5 a proper homothety. We choose any event p and suppose that X4(p) = 0 and the other generators are linearly independent at p L.R.S. metrics Let { x a, ya, z a , t a} be an othonormal tetrad at p such that x ~, y~ and z ~ are tangent to the orbit of the ~4 at p, x a and ya are space-like and zaza =

824

Steele

- t a t ~ = e where r = 1 if the orbit is space-like and e = - 1 if the orbit is time-like. Now VX4(p) is a space-like bivector, so for a suitable such tetrad X4~;b(p) = 2z[ayb], and we can arrange that Z ~ ( p ) = z", X~(p) = ya, X ~ ( p ) = z ~ and X g ( p ) = t ~ T h e n calculating the commutators of each of the linearly independent generators with X4 we have

[X1, X4] = - X 2 + aX4 [x~, x4] = -rx4

[X2, X4] = X1 + fiX4 [xs, x4] = ~x4

for constants a , . . . , 6 where we can re-arrange the generators so that a -fl = 0. T h e other commutators will be arbitrary Killing vectors, i.e. the generator )(5 will not occur in any of them. Applying the Jacobi identities to the algebra we find t h a t the only non-zero commutators are then, for constants A , . . . , E, A, [X1, X2] = A X 3 A- AX4

[X2, X3] = - B X I [X2, Xh] -- - D X 1 + C X 2

IX1, X4] = - X 2

[X1, X3] = B X 2 IX1, Xh] -- CX1 + D X 2 [Xa, Xh] = E X 3 + E B X 4 [X2, X4] = X1

and the remaining Jacobi identities are A ( 2 C - E) = 2CA - A E B = O. A change of basis such t h a t Xa --+ X3 + B X 4 effectively sets B = 0, and thus CA = 0. I f A = 0 then we are through, for there is no X4 in the c o m m u t a t o r subalgebra and so {X1, X2, Xa, X5 } generate a transitive 7-/4. Otherwise C = 0 and the remaining Jacobi identity A E = 0 implies t h a t after changing the basis again by X5 ---* X5 + DX4, effectively setting D = O, { X 1 , X 2 , AX4 + A X 3 , X h } generates a transitive 7-/4. Non-L.R.S. metrics Here VX4(p) is a time-like bivector, so we let { e a , n a , z a , y a} be a halfnull tetrad at p such that X4a;b(p ) = 2~[anbl and arrange that X I ( p ) = l, X~(p) = n, X3(p) = x, Zh(p) = y so that ~, n and z are tangent to the orbit. We can then calculate the commutators as in the L.R..S. case above and nearly identical calculations ensue and show that there is a transitive 7t4 in this case also. It now follows from the results of Section 3 that the only vacuum metrics with a proper maximal 7/5 are type D Kasner solutions. 5. C O N C L U S I O N In conclusion, we have shown t h a t algebraically special vacuum spacetimes admitting a simply-transitive homothety g r o u p s / / 4 are more numerous than those with simply-transitive motion groups. For Petrov type N

Simply-Transitive I-Iomothety Groups

825

the possibilities are plane waves and the p.p.wave (4), for type III the only possibility is the singular Robinson-Trautman metric (6), for type II there are the van-Stockum metrics (7) and (8) and for type D we have the two Kasner solutions (9) and (10). A perusal of these metrics shows that apart from the type III solution and the second of the type II solutions, the metrics admit an Abelian G3 which are, except possibly for the plane waves, a subgroup of the H4, and have hypersurfaee orbits that are, again except for the plane waves, non-null. Exactly why this is so is not clear a priori, at least not to the author. All vacuum metrics with an Abelian G3 on non-null orbits and a n / / 4 are given in [10], where they are derived by a technique better suited to that particular form of the algebra. The algebraically special cases in that paper tally with the results here. ACKNOWLEDGEMENTS

Coordinate and tetrad calculations in this paper were checked using packages written by J. R. Pulham of the University of Aberdeen. I wish to express my thanks to Dr Pulham, and also to Drs. C. B. G. McIntosh and A. W.-C. Lun for useful discussions. Finally, I acknowledge the financial support of the Australian Research Council. REFERENCES 1. Barnes, A. (1979). J. Phys. A 12 1149. 2. Flaherty, E. J. (1976). Hermitian and K~hlerian Geometry in Relativity (Lecture Notes in Physics, Springer-Verlag, Berlin). 3. Goldberg, J. N., and Kerr, R. P. (1961). J. Math. Phys., 2, 327. 4. Hall, G. S., and Steele, J. D. (1990).Gen. Rel. Gray., 22, 457. 5. Hiromoto, R. E., and Oszvs I. (1978).Gen. Rel. Gray., 9,229. 6. Kerr, R. P., and Debney, G. C. (1970). J. Math. Phys., 11, 2807; Halford, W. D., and Kerr, R. P. (1980) J. Math. Phys., 21, 120; McIntosh, C. B. G. (1976). Gen. Rel. Gray., 7,215. 7. Kerr, R. P., and Goldberg, J. N. (1961). J. Math. Phys., 2,332. 8. Kinnersley, W. (1969). J. Math. Phys., 10, 1195. 9. Kramer, D., Stephanl, H., MacCallum, M. A. H., and Herlt, E. (1980). Exact Solutions of Einstein's .Field Equations (Cambridge University Press, Cambridge) 10. McIntosh, C. B. G., and Steele, J. D. (1990). Class. Quant. Gravity, to appear. 11. Peurose, R., and Rindler, W. (1984). Spinors and Space-time (Cambridge University Press, Cambridge), vol. 1. 12. Salazar, H. I., Garc~a, A. D., and Plebafiski, J. F. (1983). J. Math. Phys., 21, 2191. 13. Spivak, M. (1970). A comprehensive introduction to Differential Geometry (Pubfish or Perish, Berkeley), vol. 1.