Physics Letters B 308 (1993) 240-246 North-Holland
PHYSICS LETTERS B
( 1 + 1 )-dimensional field theories and space-times that contain a twist-free null vector field J.H. Y o o n 1 Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul, 151-742, South Korea Received 30 November 1992; revised manuscript received 12 April 1993 Editor: M. Dine
We present a ( 1+ 1)-dimensional description of 4-dimensional space-times that contain a twist-free null vector field. It is described by the ( 1+ 1)-dimensionalYang-Mills action interacting with matter fields, with spatial diffeomorphisms of 2-surface as the gauge symmetry. The constraint appears polynomial in part, whereas the non-polynomial part is a non-linear sigma model type in ( 1+ 1)-dimensions. The representations of woo-gravity are shown to appear naturally as special cases of this description, and the geometry of wo~-gravity is discussed briefly in term of the fibre bundle.
Lower dimensional field theories have received considerable attention in connection with self-dual spaces of 4-dimensions [1] and the physics of black holes [2,3]. Recently it was realized that general relativity itself can be also viewed as a ( 1 + 1 )-dimensional field theory, where the other two space-like dimensions are regarded as the fibre ¢1, and the action principle from the ( 1 + 1 )-dimensional perspective was obtained in [5]. In contrast to the cases of the selfdual spaces and black-hole space-times, however, the (1 + 1 )-dimensional action principle in general appears to be rather formal and consequently, of little practical use. In this letter we therefore draw attention [4] to a specific class of space-times for which a twist-free null vector field exists, and interpret the entire class from the ( 1 + 1 )-dimensional point of view. Namely, we shall show that space-times of this class can be regarded as (1 + 1 )-dimensional local gauge theories, where the gauge group is the diffeomorphism
l E-mail address:
[email protected]. #1 Here we are regarding space-times as a fibre bundle, treating the ( 1+ 1)-dimensionalsection as the base manifold and the remaining two space-like dimensions as the fibre. For the class of space-times we consider here, the 2-dimensional fibre may be interpreted as the transverse wave-surface [4].
240
group of the 2-dimensional fibre *2 . We emphasize that the only restriction on space-times is that a twistfree null vector field exist, and therefore our result is completely general for this class of space-times. As a bonus of this ( 1 + 1 )-dimensional analysis of space-times, we find the fibre bundle as the natural framework for the geometric description of the so-called woo-gravity, whose geometric understanding was lacking so far [7,8]. In this picture the local gauge fields for woe-gravity are the connections valued in the infinite dimensional Lie algebra associated with the area-preserving diffeomorphisms of the 2-dimensional fibre. Due to this picture of woogeometry, we are able to construct field theoretic realizations of woo-gravity in a straightforward way, as we shall show later. Let us consider those space-times that contain a twist-free null vector field k A (A, B, . . . = O, l, 2, 3). Being twist-free, the null vector field may be chosen to be a gradient field, so that/cA = OAu for some function u. The null hypersurface N2 defined by u = constant spans the 2-dimensional subspace for which we introduce two space-like coordinates ya (a, b , . . . = 2, 3).
#2 It is worth mentioning that Einstein's field equations can be "derived" from the source-free Yang-Mills equations [6]. Here, however, we are essentially making the (2 + 2)-decomposition of vacuum GR.
0370-2693/93/$ 06.00 © 1993 Elsevier SciencePublishers B.V. All fights reserved.
Volume 308, number 3,4
PHYSICS LETTERS B
1 July 1993
The general line element for this class has the form [4,9]
D l t 4 a b = Ol~4ab -
d s 2 -- 4ab d Y a d Y b
where the bracket means the Lie derivative here (and afterwards), an infinite dimensional generalization of the finite dimensional matrix commutators, which acts only on the "internal" indices a, b, etc. 2. The field strength Fu, a is defined as usual,
-2du(dv
+ m a d y a -I- H d u ) ,
(1)
where v is the affine parameter for the null vector field kA, and 4ab, ma and H are functions of all of the four coordinates (u, V,y a), as we assume no Killing vector fields here (and afterwards). The line element (1) is completely general, apart from the assumption that a twist-free null vector field exists. For the class of space-times ( 1 ), we shall find the ( 1 + 1 )-dimensional action principle defined on the (u, v)-surface. Namely, we wish to show that space-times of the above type can be viewed as (1 + 1 )-dimensional gauge theories, where the gauge fields are valued in the infinite dimensional Lie algebra associated with diffNz. To show how this works, let us first recall that the general line element of space-time, as viewed as a local product of two 2dimensional submanifolds M~ +1 × N2, can be written as, at least locally [5],
(2)
where 7u~ resp. 4ab is the metric on the ( l + l ) dimensional surface MI+I resp. the 2-dimensional surface Nz spanned by O / O x u resp. O / O y ~. In this (2 + 2)-decomposition, the Einstein-Hilbert action for the line element (2) can be written as [5] lt~l ¢ £ = - X / -p--ST_, ~Vcpt~UR~u
+ ~qgabRab q- g1~.1.a b r12 ~ u a rr p v b
1 #u ab cd
×{(DuCat) ( D v 4 b d ) I
-- ( D I L 4 a b ) (
- {A~Oc4ab + (OaACu)4cO + (ObA;)4ac},
Flzu a = O u A f - O ~ A u -
(4)
(5)
[Au, A , ] a .
3. The Levi-Civita connections F, and Ru~ and Rac are defined as 1 aft Fu~a = ~Y (DuY~# + D~Yu# - D f T u ~ ) ,
l~ab c = ~14 cd (Oa4bd "[- Ob4ad -- O d 4 a b ) ,
g u , = D u F ~ ~ - D~Fu~ ~ + Fup~F~, p - Fp BFu, ~ , R a b = Oal-cb c _ OCi.ab c ..[_ FadCFcb d _ Fdcdi.ab c '
(6)
where DuT-/~ = 0u7~13 - [Au, Y~B ]
d s 2 = 4 a b d y a d Y b + (Tuu + 4 a b A ~ A u b) d x U d x ~
+ 24~bA f d x u dy a ,
[All, 4 ]ab = O#Oab
D~4cd)}
(7)
= OuT,p - AuOaT, p ,
since y~p contains no "internal" indices. Notice that here Ru~ is covariantized, which we might call the "gauged" Ricci tensor [ 5, l 0]. (The review ends here.) Although the action (3) brings general relativity (up to a surface term) into a form of ( l + l ) dimensional field theory for the line element (2), it appears rather formal. For those space-times that we wish to consider in this letter, however, the action reduces to a remarkably simple form. In order to show this, let us first introduce the "light-cone" coordinates (u, v) such that
ab lau a #
+z4r
r
x { ( O a Y u ~ ) ( O b T ~ # ) - (OaTu~)(ObT~p)}],
u=
(x°+xt), v=-~(x°-x~),
(8)
(3)
up to the surface terms. Here is the summary of our notations (for details, see [ 5 ] ): 1. The covariant derivative Du4ab is defined by
and define Aua and Ava Aua = - ~ ( A o a + A l a ) ,
Av a =
(Aoa-Ala).
(9)
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Volume 308, number 3,4
F o r 7u,, we choose the Polyakov gauge [ 11 ]
Yu~ =
(-2h -1
;1)
(? '
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PHYSICS LETTERS B
7u" =
1
-1) 2h
(det ~'u~ = - 1 ) ,
,
Au'a(y
,u,v)
Oy 'a
= --~
AuC(y,u,v)-Ouy
which become, under the infinitesimal variations, 6 y a = Ca ( y , u , v ) and ~u = S v = 0 ,
in the (u, v )-coordinates. Then the line element (2) becomes
5h = - [~, h ] = -~aOah,
ds2 = Oab dY a dY b - 2 du dv - 2h (du) 2
6(~ab "~ --
+ ~)ab ( A a du + A v a dv ) (Au b du + Av b dv )
If we choose, at least locally, the "light-cone" gauge #3 Av a = 0, then this becomes ds2 ~-" ~ab dY a dY b - 2 du [ dv - ~babAub dy a
(12)
A quick glance at (1) and (12) tells us that if the following identifications, H = h -- i,eab~u 1 #. A a .au A b
ma = -(babau b ,
(13)
are made, then the two line elements are the same. This shows that the Polyakov gauge (10) amounts to the restriction (modulo the gauge choice Av a = 0) to those s p a c e - t i m e s for which a twist-free null vector field exists. Let us now examine the transformation properties o f h, qSab, Aua, and Ava under the diffeomorphism o f
N2, U I = U,
ZP = W.
(14)
U n d e r these transformations, we find that h'(y',u,v)
Oy'a Oy'b
(Oa~C)~cb-
(Ob~C)f~ac,
a = - O u ~ a + [ h u , ~ ] a,
(16b) (16c) (16d)
¢~A v a = - O v ~ a .
This shows that h and ~ab a r e a scalar and tensor field, respectively, and A u a and h v a are the gauge fields valued in the infinite dimensional Lie algebra associated with the group o f diffeomorphisms o f N2. That A a is a pure gauge is clear, as it depends on the gauge function Ca only. Therefore it can always be set to zero, at least locally, by a suitable coordinate transformation (14). To maintain the explicit gauge invariance, however, we shall work with the line element ( 11 ) in the following, with the understanding that A v a is a pure gauge. Let us now proceed to writing down the action principle for ( 11 ) in terms o f the fields h, Cbab,Au a, and A v a . F o r this purpose, it is convenient to decompose the 2-dimensional metric ~ab into the conformal classes
(I2 > 0 and det Pab
= 1 ).
(17)
The kinetic term K o f q)ab in (3) then becomes K = l~-'~W~lln/oab(ficd × { (DuqSac) ( Du(gbd ) -- ( D # ~ a b ) ( DuqScd ) }
= h(y,u,v),
dP'ab(y', U, V ) -- Oyc Oyd d~ca(y, u, v )
(15)
#3 Here we are referring to the disposable gauge degrees of freedom in the action. The justification of this gauge choice comes from the transformation properties of (A a , Av a ) under diffN2, which allow us to identify them as gauge fields valued in the Lie algebra ofdiffN2. There could be topological obstruction against globalizing this choice, as the general coordinate transformation of 5/2 corresponds to the gauge transformation. 242
OA a = -Du~
(Oab = g-2Pab
yta = yla(yb, u,v),
(16a)
[~, O ]ab
= --~COc~ab-
( 11 )
+ ( h -- gq~abAu l aA u b ) d u ] .
(15 cont'd)
A v 'a (y', u, v ) = - O v y 'a ,
(10)
+ 2C~ab ( A a du + Av a dv ) dy b .
'a,
-
( Du o )2 2------~--- + ¼E2yU~'pabpcd(Dupac)(D~'Pbd)
--
½eO(Daa) 2 1
ct
Itv ab cd
+ -~e 7' P
P
(DuPac)(D,,pbd),
(18)
where we defined e by tr = In 12, and the covariant derivatives DuO, DaPab, and D u a are Dul2 = Ou~ - A ~ O a f 2 - ( O ~ A ~ ) I 2 ,
(19a)
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PHYSICS LETTERS B
DuPab = OuPab- [Au,P]ab + (OcA~)Pab,
(19b)
Dua = Oua - AuOaa
(19c)
- OaAfl,
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F r o m ( 10 ) and (24), the "gauged" Ricci scalar yU~Ru ~ is given by
respectively, where [Au, P ]~b is given by
yU~Ruv = 2 y + - R + _ = 2D2_h,
[Au, Plab = AuOcPab + (OaACu)Pcb
since ~++ = R _ _ = 0. Putting together (21), (22), and (25) into (3), the action becomes
+ (ObAu)pa c .
(25)
(20)
The inclusion o f the divergence term OaA~ in (19) is necessary to ensure (19) transform covariantly (as the tensor fields) under diffN2, t2 and Pao being densities o f weight - 1 and + 1, respectively. Using the gauge (10), the kinetic term (18) becomes
•2
= - - ~l ~ 2 a ~P a b ~r + - ra+ E, -b -I-
ea ( D + a ) ( D _ a )
- ½e~pabp cd ( D+ Pac ) ( D - Pbd ) +heO{ ½pab pcd ( D - p a c ) ( D-Pbd ) -- ( D - t r ) 2 } + 2e a D2_h.
K = e~(D+a)(D-a) 1 a ab cd
- ~e p
p
The last term in (26) can be expressed as
(D+pac)(D-Pbd)
e ~ D Z h = e~(0_ - A_bOb)(O_h -- A_aOah)
- h e a { ( D _ a ) 2 _ ½p ab pcd( D - pac ) ( D - Poa ) } , (21) where + ( - ) stands for u ( v ) . The Polyakov gauge (10) simplifies enormously the remaining terms in the action (3), as we now show. Let us first notice that detYu~ = - 1 . Therefore the term X / - ~ V / ~ a b R a b can be removed from the action being a surface term. Moreover, we have that yu~Oayu~ = 2(-~)-1/20~(-7)1/2 = 0. Furthermore, one can easily verify t h a t ~ ) a b ~ u ~ a f l ( O a ~ # a ) (Ob~ufl) vanishes identically. The only remaining terms that contribute to the action (3) are thus the (1 + 1 )-dimensional Y a n g - M i l l s action and the "gauged" gravity action. The Y a n g - M i l l s action becomes 1.~
~
a~lwb
l_a^
Zq~abru~ r
F
ar
b
= --~c IJab + _ r + _ .
(22)
To express the "gauged" Ricci scalar 7U~Ru~ in terms o f h and A~ a, etc., we have to compute the Levi-Civita connections first. They are given by F+~ = -D_h, F+-- = F+-
(26)
= e°{O2h - O_ (A_aOah) = -e~(D_a)(D_h)
--
AaOa ( D - h ) }
+ 0- (e~D_h)
- Oa(e~A_a D _ h ) ,
(27)
using the Stokes theorem, where the last two terms in (27) are the surface terms which we may drop. The first term in (27) can be written as e° ( D _ a ) ( D _ h ) = - h e a (
D_a
)2 _
he a D2_a
+ O_{eah D _ a } - Oa{e~hA_a D _ a } ,
(28)
where the last two terms in (28) are also the surface terms. F r o m (27) and (28), we therefore have e ~ D 2 h _~ h e ~ { ( D _ a ) 2 + D2_a},
(29)
neglecting the surface terms. The resulting (1 + 1 )dimensional action principle therefore becomes
F++ = D + h + 2 h D _ h , = D_h,
(23)
f~2 = - ~ l ~ 2 a . P a bgr, + a- rL'+ - b + eo ( D + t r ) ( D _ t r )
and vanishing otherwise. Thus the "gauged" Ricci tensor becomes
- leapabpcd ( D+ pac ) ( D - Pbd )
R+_ =R-+
+ ½pabpcd(D_pac)(D-pbd)},
=-D2h,
R _ _ = 0.
(24)
+ he~{2DZ_a + ( D - t r ) 2 (30) 243
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PHYSICS LETTERS B
up to the surface terms. Notice that h is a Lagrange multiplier, whose variation yields the constraint Ho = D 2 - a + 1 ^ab ^cd
+ Zp p
½(D-a) 2 (D-Pac)(D-pbd)
.~0.
(31)
From this ( 1 + 1 )-dimensional point of view, h is the lapse function (or a pure gauge) that prescribes how to "move forward in u-time", carrying the surface N2 at each point of the section u = constant. The constraint, Ho ~ 0, is polynomial in a and A a, and contains a non-polynomial term of the non-linear sigma model type but in (1 + 1 )-dimensions, where it often admits exact solutions. This allows us to view the problem o f the constraints of general relativity [ 12 ] from a new perspective. We now have the ( 1 + 1 )-dimensional action principle for space-times for which a twist-free null vector field exists. It is described by the Yang-Mills action, interacting with the fields a and Pab on the "fiat" ( 1 + 1 )-dimensional surface, which however must satisfy the constraint H0 ~ 0. (The flatness of the ( 1 + 1 )dimensional surface can be seen from the fact that the lapse function, h, can be chosen as zero, provided that H0 ~ 0 holds.) The infinite dimensional group of diffeomorphisms of N2 is built-in as the local gauge symmetry, via the minimal couplings to the gauge fields. Having formulated this class of space-times as a ( 1 + 1 )-dimensional field theory, we may wish to apply varieties of field theoretic methods developed in ( 1 + 1 )-dimensions. For instance, the action (30) can be viewed as the bosonized form [ 13 ] of some version of the ( 1 + 1 )-dimensional Q C D in the infinite dimensional limit of the gauge group [ 14 ]. For small fluctuations of a, the action (30) becomes £2 = - - ~ t1~ ' a b l~+ _aE" 1 + _b + ( D + o ) ( D _ a ) I ^ab ^cd, D+ Pac ) ( D - Pbd )
(32)
modulo the constraint H0 ~ 0. It is beyond the scope of this letter to investigate these theories as (1 + 1 )dimensional quantum field theories. However, this formulation raises many intriguing questions such as: would there be any phase transition in quantum gravity as viewed as the (1 + 1 )-dimensional quantum 244
1 July 1993
field theories? If it does, then what does that mean in quantum geometrical terms? Thus, general relativity, as viewed from the ( 1 + 1 )-dimensional perspective, renders itself to be studied as a gauge theory in full sense [15], at least for those space-times discussed in this letter. So far we derived the action principle on the flat ( 1 + 1 )-dimensions as the vantage point of studying general relativity for this class o f space-times. We now ask different but related questions: what kinds of other ( 1 + 1 )-dimensional field theories related to this problem can we study? For these, let us consider the case where the local gauge symmetry is replaced by the area-preserving diffeomorphisms of N2. (For these varieties of field theories, we shall drop the constraint (31 ) for the moment. It is at this point that we are departing from general relativity.) This class of field theories naturally realizes the so-called w ~ gravity [7,8 ] in a linear and geometric way, as we now describe. The area-preserving diffeomorphisms are generated by the vector fields ~a, tangent to the surface N2 and divergence-free, Oa~ a = O.
(33)
Let us find the gauge fields A:~ compatible with the divergence-free condition (33). Taking the divergence of both sides of (16c) and (16d), we have Oa(~A:~ ~- -04- (Oa~ a) ÷ Oa [A4-, ~]a.
(34)
This shows that the condition OaA~ = 0 is invariant under the area-preserving diffeomorphisms, and characterizes a special subclass of the gauge fields, compatible with the condition (33). Moreover, when OaA:~ = 0, the fields Pab and a behave under the areapreserving diffeomorphisms as a tensor and a scalar field, respectively, as (19b) and (19c) suggest. Indeed, the Jacobian for the area-preserving diffeomorphisms is just 1, disregarding the distinction between the tensor fields and the tensor densities. The ( 1 + 1 )dimensional action principle now becomes £2t = - ~ e1
20
~, a ~ - b pabr+-~++e~(D+a)(D-o)
- i1eopabpcd (D+Pac) ( D - P b d ) , where Dug, Dupab, and F+ a are
(35)
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D + a = Oa:a - Ae~Oaa,
(36a)
D+Pab = O+Pab -- [ A + , P ] a b ,
(36b)
F+ ~_ = O+A_a - O - A + -
(36e)
[ A + , A - ] a.
U n d e r the infinitesimal variations 6 y a = Ca ( y , u , v ) ,
6u = dv = 0
( Oa~a = 0 ) ,
(37)
the fields transform as 617 = -- [ ~, 17] = --~aOat7 ,
(38a)
6p~a = --[~,P]ab = --{CO¢Pab- (Oa{C)Pca- (Ob~C)Pac,
(38b)
a A + = - D + ~ ~ = - 0 + ~ ~ + [A+,~] a,
(38e)
~A a = - 0 - ~ a,
(38d)
which shows that it is a linear realization of the areapreserving diffeomorphisms. The geometric picture o f the action principle (35) is now clear: it is equipped with the natural bundle structure, where the gauge fields are the connections valued in the Lie algebra associated with the area-preserving diffeomorphisms of the fibre N2. Thus the action principle (35) provides a field theoretical realization of w ~ - g r a v i t y [7,8] in a linear and geometric way, with the built-in areapreserving diffeomorphisms as the local gauge symmetry. With this picture o f w ~ - g e o m e t r y at hands, we may construct as m a n y different realizations o f w ~ - g r a v i t y as one wishes. The simplest example would be a real scalar field representation, which we may write ~t
alt7 a = - ~1, ~, +_--+_ + (D+tr)(D_a),
(39)
where we used 5ab in the summation, and D ± a and F+ a are as given in (36a) and (36c). By choosing the gauge A a = 0 and eliminating the auxiliary field A.~ in terms o f tr using the equations o f m o t i o n of A.~, we recognize (39) a real scalar field realization o f w~-gravity. In the presence o f the auxiliary field A ~ , (39) provides an example of the linearized realization o f w ~ - g r a v i t y for a real scalar field. It would
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be interesting to see if the representation (39) is related to the ones constructed in the literature [7,8]. N o w we summarize our discussions. In this letter, for the class of space-times which contain a twist-free null vector field, the ( 1 + 1 )-dimensional field theory description was presented. The local gauge symmetry o f the ( 1 + 1 )-dimensional theory originates from the infinite dimensional group o f diffeomorphisms of the 2-dimensional transverse surface. Parts o f the constraints are identified as the gauge fixing condition, and the constraint appears in a manageable form (for the class o f space-times investigated here). As a related problem, a special subclass o f (1 + 1 )dimensional field theories associated with the areapreserving diffeomorphisms was also discussed in connection with the geometrical formulation of w ~ gravity, There seem to be many interesting questions to be asked about general relativity from this lower dimensional perspective, which we might be able to address somewhere else. The author thanks Q.H. Park and S.K. N a m for many enlightening discussions. This work is supported in part by the Ministry o f Education and by the Korea Science and Engineering F o u n d a t i o n through the SRC program.
References [1] Q.H. Park, Phys. Lett. B 238 (1990) 287. [2] C.W. Misner, Phys. Rev. D 18 (1978) 4510; V.E. Belinskii and V.E. Zakharov, Sov. Phys. JETP 48 (1978) 985; D. Maison, Phys. Rev. Lett. 41 (1978) 521. [3] E. Witten, Phys. Rev. D 44 (1991) 314; S.B. Giddings and A. Strominger, Phys. Rev. Left. 67 (1991) 2930; J.A. Harvey and A. Strominger, Quantum aspects of black holes, lectures presented at the 1992 Trieste Spring School on String theory and quantum gravity (September 1992), and references therein. [4] D. Kramer, H. Stephani, E. Herr and M. MacCallum, Exact solutions of Einstein's field equations (Cambridge University Press, 1979). [5] Y.M. Cho, Q.H. Park, K.S. Soh and J.H. Yoon, Phys. Lett. B 286 (1992) 251; J.H. Yoon, ( 1+ I )-dimensional methods for general relativity, to appear in The Brill Festschrift, ed. B.L. Hu and T. Jacobson (Cambridge University Press, May 1993). 245
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[6] L.J. Mason and E.T. Newman, Comm. Math. Phys. 121 (1989) 659; S. Chakravarky, S. Kent and E.T. Newman, J. Math. Phys. 33 (1992) 382. [7] E. Bergshoeff, C.N. Pope, L.J. Roman, E. Sezgin, X. Shen and K.S. SteUe, Phys. Lett. B 243 (1990) 350; K.S. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Phys. Lett. B 251 (1990) 355; E. Sezgin, Area-preserving diffeomorphisms, woo algebras and woo gravity, lectures presented at the Trieste Summer School in High energy physics (July 1991); J.-L. Gervais and Y. Matsuo, Phys. Lett. B 274 (1992) 309. [8] I. Bakas, Phys. Lett. B 228 (1989) 57; C.M. Hull, Phys. Lett. B 269 (1991) 257. [9]W. Kundt, Z. Phys. 163 (1961) 77. [10] R. Wald, Phys. Rev. D 33 (1986) 3613; S. Deser, J. McCarthy and Z. Yang, Phys. Lett. B 222 (1989) 61;
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C.N. Pope and P.K. Townsend, Phys. Lett. B 225 (1989) 245; M.P. Bleneowe, Class. Quantum Grav. 6 (1989) 443. A.M. Polyakov, Mod. Phys. Lett. A 2 (1987) 893; V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819. K. Kucha~, Canonical quantization of gravity, in Relativity, astrophysics and cosmology, ed. W. Israel (1973). E. Witten, Comm. Math. Phys. 92 (1984) 455; V. Baluni, Phys. Lett. B 90 (1980) 407. D.J. Gross and E. Witten, Phys. Rev. D 21 (1980) 446; I. Bars, Phys. Lett. B 245 (1980) 35; A. Jevicki and B. Sakita, Phys. Rev. D 22 (1980) 467; A.A. Migdal, Phys. Rep. 102 (1983) 199. Y.M. Cho, Phys. Rev. D 14 (1976) 2521; A. Trautman, Rep. Math. Phys. 1 (1970) 29.
