A SPACE-TIME CALCULUS BASED
ON A SINGLE NULL DIRECTION
M P Machado Ramos
Faculty of Mathematical Studies
University of Southampton
Southampton SO17 1BJ
England
J A G Vickers
Faculty of Mathematical Studies
University of Southampton
Southampton SO17 1BJ
England
Short Title: Space-time calculus with a single null direction
Abstract
A space-time calculus based on a single null direction is developed. The
fundamental objects are totally symmetric spinors formed from the Ricci rotation
coe cients together with four new di erential operators which only depend upon
the choice of o up to a complex scaling. The formalism combines features of the
A
GHP formalism with those of one whichis invariant under null rotations. Einstein's
equations and the full Bianchi identities are given in this new formalism.
1. Introduction
Certain physical problems in general relativity are often best described by using
a formalism adapted to the geometry of the particular situation. For example
the Geroch-Held-Penrose (GHP) formalism (Geroch, Held & Penrose 1973) or
compacted spin-coe cient formalism (Penrose and Rindler 1984) best describe the
a a
geometry of a null 2-surface S where one can choose ` and n to point along the
a
null normals of S and the real and imaginary parts of m are tangent to S . The
remaining gauge freedom in the choice of tetrad is the 2-dimensional subgroup of
the Lorentz group representing a boost in the normal directions and a rotation in
the tangential directions. In terms of spinors this is equivalent to the rescaling
A A A ;1 A
o ! o ! (1)
7 7
The GHP formalism works with those Ricci rotation coe cients which simply
rescale under (1) and combines the others with directional derivatives to form new
0 0
o o
operators g , g , I and I which also just rescale under (1).
Recently a new formalism has been introduced in which the generalised spin
coe cients and di erential operators transform in a simple way under a null ro-
tation (Machado Ramos & Vickers 1995). In terms of spinors a null rotation is
given by
A A A A A
o ! o ! + (2)
7 7 ao
where a is an arbitrary complex scalar eld. The resulting formalism uses spinors
formed from the Ricci rotation coe cients whose components transform covari-
antly under null rotations, and four new di erential operators which are formed
from the directional derivatives and the remaining Ricci rotation coe cients which
Section 1: Introduction 2
transform `badly' under null rotations. These operators act on totally symmet-
ric spinors and produce totally symmetric spinors (of higher valence) and when
applied to a spinor whose components transform covariantly under null rotations
produce one whose components also transform covariantly. Because only totally
symmetric spinors are used there is no need to explicitly use indices in any of the
equations.
In this paper we are interested in developing a formalism which only depends
a
upon the choice of a single null direction, ` . If we choose the agpole of the spinor
A A A A
o to point in this null direction then o is determined up to rescalings o ! o
7
where is a complex nowhere vanishing scalar eld (and the magnitude as well
a
as the direction of ` is xed if we require j j =1 ). The other spinor in the spin
A
frame is arbitrary so for convenience we normalise it so that o = 1 , although
A
A
it would not be hard to generalise the formalism to allow o = as one has in
A
the compacted spin coe cient formalism. Thus specifying a single null direction
is equivalent to specifying a spin frame up to the gauge freedom
A A A ;1 A A
o ! o ! + (3)
7 7 ao
which can be thought of as a rescaling of the spinor dyad followed by a null rotation
or vice-versa. It is therefore not surprising that the resulting formalism combines
features of the GHP and null rotation invariant formalisms.
We will use the null rotation invariant formalism as our starting point. In
x2 we nd that the generalised spin coe cients K , R , S and T transform well
under (3) but that the others transform badly. However the ones that transform
badly may be combined with the di erential operators D , , and to form
0 0
o o
o o o o
o o
I I g g0
new operators I , I , g and g in the same way that the GHP operators I , I , g
I I g g
0
and g are constructed from the Newman-Penrose (NP) operators D , , and .
As in the null rotation invariant formalism the new operators are distinguished by
Section 1: Introduction 3
being written in bold type, and act on symmetric spinors producing new symmetric
spinors of higher valence. The Ricci equations and commutator equations for these
new operators are easily obtained and may be written in an index free notation.
These equations are equivalent to the Einstein equations. The full and vacuum
Bianchi identities are also easily obtained in this new formalism.
A formalism in which a single spinor is speci ed up to rescaling has also been
given by Penrose. However although it is good for describing the geometry of null
hypersurfaces, where it is natural to choose oA in such a way that = 0 , it is not
so convenient to give the Einstein equations and Bianchi identities in terms of his
formalism. An important application of our present formalism which needs this is
the Karlhede classi cation of type N spacetimes, the details of which will be given
elsewhere (Machado Ramos & Vickers 1995). In x3 we examine the relationship
between Penrose's formalism and that of the present paper.
In what follows we shall adopt the convention that lower case Latin indices
run from 0 to 4. Upper case Latin indices will be spinor indices which run from 0
to 1. The abstract index convention will be used to enable us to write equations
such as = . Boldface indices as in will be used to denote the scalar
AA0 a A
eld obtained by contracting the spinor eld with an element of the the spin
A
A A A
frame =(oA ) so that = .
A A
A A
A totally symmetric spinor eld A1 : ::AN A0 :: :A0 with N unprimed indices
0
1
N
and N0 primed indices will be said to be a type (N N0) -spinor. Following Geroch
Held and Penrose we say a scalar eld which rescales under the transformation
(1) according to
p
! q (4)
7
has weight fp qg . We will also apply this idea to type (N N0) -spinor elds
which will be said to have weight fp qg if they rescale according to
Section 2: The New Compacted Formalism 4
p
7 q A1 : ::AN A0 :: :A0 (5)
A1 : ::AN A0 : : :A0 !
1 0 1 0
N N
2. The New Compacted Formalism
In our previous paper we showed that
K = (6)
0 0
SA = oA ; A0 (7)
RA = oA ; A (8)
0 0 0
TAA = oAoA ; oA ; oA + (9)
A0 A A A0
0 0 0
BAA = oAoA ; oA ; oA + (10)
A0 A A A0
AAB = oAoB ; ( + )o( A + (11)
B) A B
EA = oA ; A (12)
0 0 0 0
GABA = oAoB oA ; ( + )o( A oA + oA
B) A B
; oAoB +( + )o( A ; (13)
A0 B) A0 A B A0
were invariant under null rotations given by (2). Furthermore under spin and
boost transformations given by (1)
3
! (14)
7
3
! ;1 (15)
7
! (16)
7
! ;1 (17)
7
while the other spin coe cients in (6)-(13) transform in a way that involves deriva-
Section 2: The New Compacted Formalism 5
tives of . Hence under the full gauge freedom (3)
3
K ! K (18)
7
3
0 0
SA ! SA (19)
7
2
RA ! RA (20)
7
2
0 0
TAA ! TAA (21)
7
These quantities therefore have weight fp qg given by
K : f3 1g
S : f3 0g
R : f2 1g
T : f2 0g
As might be expected the null rotation invariant di erential operators D , ,
and do not produce objects with a well de ned spin and boost weight when
acting on a totally symmetric spin and boost weighted spinor eld, but just as in
the GHP case they can be combined with the generalised spin coe cients A , B ,
G and E which transform badly under (3) to produce new derivative operators
0
o o
o o
o o
I g g0 I
I , g , g and I which do have a proper spin and boost weight. (N.B. recall the
I g g I
g
use of boldface to distinguish g and g .
g
o
o
o
I
The new operator I acts on a totally symmetric (N N0) -spinor A1 :: :AN A0 :: :A0
I
0
1
N
of weight fp qg to produce a (N +1 N0 + 1) -spinor of type fp +2 q +2g and
is given by
Section 2: The New Compacted Formalism 6
o
o
o
I
(I )AA : : :AN A0 A0 =(D )AA : ::AN A0 A0 ; (p ; N)E( A A1 : : :AN )( A0 :: :A0 oA0
I
:: :A0 : ::A0 )
1 1
1 0 1 0
N N 1 0
N
; (q ; N0)o( A A1 :: :AN )( A0 : ::A0 EA0 (22)
)
1 0
N
Since every term in the above expression is a totally symmetric spinor the order
of the indices does not matter and we may introduce the compact notation of
(Machado Ramos & Vickers 1995) and write this equation as
o
o
o
I
I = D ; (p ; N)E ; (q ; N0)E (23)
I
Note that to obtain an expression with indices from one in the compact form one
multiplies the terms in the sum by appropriate factors of o 's and o 's to make
every term in the sum a spinor of the same type (so that the indices balance) and
then symmetrise over the primed and unprimed indices.
g
In a similar way we may de ne the operator g which acts on a type (N N0)-
g
spinor of weight fp qg to produce a type (N +1 N0 + 2) -spinor of weight
g
fp +2 q +1g . In compact notation g is given by
g
g
g = ; (p ; N)B ; (q ; N0)A (24)
g
0
g
The operator g acts on a type (N N0) -spinor of weight fp qg to produce a type
g
g0
(N +2 N0 + 1) -spinor of weight fp+1 q +2g . In compact notation g is given by
g
0
g
g = ; (p ; N)A ; (q ; N0)B (25)
g
0
o
o
o
I
Finally we may de ne the operator I operator which acts on a type (N N0) -spinor
I
of weight fp qg to produce a type (N +2 N0 + 2) -spinor of weight fp +1 q +1g .
Section 2: The New Compacted Formalism 7
0
o
o
o
I
In compact notation I is given by
I
0
o
o
o
I
I = ; (p ; N)G ; (q ; N0)G (26)
I
A somewhat simpler de nition of all the new di erential operators will be given
in the next section in terms of a single auxiliary di erential operator D .
ABA0 B0
We next note that the null rotation invariant curvature spinors have proper
weight fp qg given by
0 0
= oAoB oA oB : f2 2g (27)
00 ABA0 B0
0
0
( )B = ABA0 B0 oAoB oA : f2 1g (28)
01
0
( )A B0 = ABA0 B0 oAoB : f2 0g (29)
02
0 0
( )B = oAoA oB : f1 2g (30)
10 ABA0 B0
0
0
( )BB = ABA0 B0 oAoA : f1 1g (31)
11
0
( )BA B0 = ABA0 B0 oA : f1 0g (32)
12
0 0
( )AB = ABA0 B0 oA oB : f0 2g (33)
20
0
0
( )ABB = ABA0 B0 oA : f0 1g (34)
21
0
( )ABA B0 = ABA0 B0 : f0 0g (35)
22
= oAoB oC oD : f4 0g (36)
0 ABC D
( )A = oB oC oD : f3 0g (37)
1 ABC D
( )AB = ABC D oC oD : f2 0g (38)
2
( )ABC = ABC D oD : f1 0g (39)
3
( )ABC D = ABC D : f0 0g (40)
4
= : f0 0g (41)
Section 2: The New Compacted Formalism 8
We are now in a position to translate various equations into this new formalism.
We begin by considering the Ricci equations. Guided by the GHP example we
0
o
o
o
I g
start by considering the expression I R ; g K . Note that both terms are (2 1) -
I g
spinors of weight f4 3g so it makes sense to consider their di erence. An explicit
calculation shows that
0
o
o
o
0
I 0 g 0 0 0
(I R)ABA ; (g K)ABA = R( ARB)oB + SB S( AoB) ; Ko( B TA)A0 + oAoB oB
I g
00
(42)
which may be written in the compact notation as
0
o
o
o
I g
I R ; g K = R2 + SS ; KT + 00 (43)
I g
The other Ricci equations are similarly found to be
o
o
o
I g
I S ; g = RS ; RS ; KT + 0 (44)
I gK
0
o o
o o
o o
I I
I T ; I K = RT + TS + + (45)
I I
1 01
0
o
o
o
g I
g ; I S = T2 + 02 (46)
gT I
0 0
o
o
o
I g
I R ; g T = ;TT ; ; 2 (47)
I g
2
0
g g
gR ; g S =(R ; R)T ; + (48)
g g
1 01
Note that one can take the complex conjugate of these equations but there are no
primed versions since such equations would involve derivatives of spin coe cients
which transform badly under the null rotation part of (3).
We next consider the commutator equations. A rather long but straightfor-
ward calculation shows that these are given by
Section 2: The New Compacted Formalism 9
0 0 0
o o o o
o o o o
o o o o
I I I I g g
(I I ; I I ) = fTg + Tg ; (p ; N)( + ; )
I I I I g g
2 11
; (q ; N0)( 2 + ; )g +( + )( o) + ( + 3)( o)
11 21 3 12
(49)
0 0
o o o
o o o
o o o
I g gI g g I
(I g ; gI ) = fRg + Sg ; KI ; (p ; N)( ) ; (q ; N0)( )g
I g gI g g I
1 01
+(2 + )( o) + ( o) (50)
2 02
0 0 0
o
o
o
gg g g I
(g g ; g g = f(R ; R)I +(p ; N)( ; ; )
gg g g) I
2 11
; (q ; N0)( 2 ; ; )g ; ( ; )( o) ; ( ; 3)( o)
11 3 21 12
(51)
0 0 0
o o o
o o o
o o o
I g gI I
(I g ; gI ) = f;TI ; (p ; N)( 3) ; (q ; N0)( )g
I g gI I
12
; ( o) ; 4( o) (52)
22
0 0 0 0
o o o
o o o
o o o
I g g I g g I
(I g ; g I ) = fRg + Sg ; KI ; (q ; N0)( 1 ) ; (p ; N)( )g
I g g I g g I
10
+(2 + 2)( o) + ( o) (53)
20
0 0 0 0 0
o o o
o o o
o o o
I g g I I
(I g ; g I ) = f;TI +(p ; N)( ) + (q ; N0)( )g
I g g I I
3 21
; ( o) ; ( o) (54)
22 4
N
where ( o) is the (N ; 1 N0) -spinor A1 :: :AN A0 : ::A0 oA and o is the (N N0 ;
0
1
N
0
0
N
1) -spinor A1 :: :AN A0 : ::A0 oA , and if the contraction is not possible then these
0
1
N
terms are set to zero.
Finally we consider the Bianchi identities which in the compact notation take
the form
0
o o
o o
o o
I g I g
I ; g ; I + g =4R ; 3K ; 2R ; 2S +2K + K
I g I g
1 0 01 00 1 2 01 10 11 02
(55)
Section 3: Relationship to the Penrose operators 10
0 0 0
o o o
o o o
o o o
I g g I I
I ; g ; g +I +2I =3R ; 2K ; 2T +2T +2R + S
I g g I I
2 1 01 00 2 3 01 10 11 02
(56)
0 0
o o
o o
o o
I g I g g
I ; g ; I + g ; 2g = 2R ; K ; 2R + K (57)
I g I g g
3 2 21 20 3 4 21 22
0 0 0
o o
o o
o o
I g g I 21 22
I ; g ; g +I = R ; 2T + S (58)
I g g I
4 3 21 20 4
0 0 0
o o
o o
o o
I g I g
I ; g ; I + g = ;T + T (59)
I g I g
3 4 21 22 4 22
0 0
o o o
o o o
o o o
I g g I I
I ; g ; g +I +2I = S ; 2T + R (60)
I g g I I
2 3 21 22 4 3 22
0 0 0
o o
o o
o o
I g I g g
I ; g ; I + g ; 2g = 2S ; 3T ; 2R +2T + T
I g I g g
1 2 01 02 3 2 12 11 02
(61)
0
o o
o o
o o
I g g I
I ; g ; g +I =3S ; 4T ; 2k +2S + R (62)
I g g I
0 1 01 02 2 1 12 11 02
The contracted Bianchi identities take the form
0 0
o o o
o o o
o o o
I I g g I
I +I ; g ; g +3I =2R +2R ; 2T ; 2T 10
I I g g I
11 00 10 01 11 11 01
; K ; K + S + S (63)
12 21 20 02
0 0
o o
o o
o o
I I g g g
I +I ; g ; g +3g =2R + R ; T ; 2T 11
I I g g g
12 01 11 02 12 12 02
; K + S (64)
22 21
0 0 0
o o o
o o o
o o o
I I g g I
I +I ; g ; g +3I = R + R ; T ; T (65)
I I g g I
11 22 12 21 22 22 21 12
3. Relationship to the Penrose operators
In a paper on the geometry of impulsive gravitational waves Penrose (1972)
introduces di erential operators g , g and I which act within a null hyper-
A0 A AA0 o
0 0 be
surface N and act upon weighted scalar and spinor elds. Let C1 :: :CN C1 : ::CN 0
Section 3: Relationship to the Penrose operators 11
a (N N0) -spinor of weight fp qg . Then g 0 0 is a (N +1 N0 +1)-
AA0 C1 : ::CN C1 : ::CN 0
spinor of weight fp +2 qg which is de ned by
o( B0 g C1 : ::CN C1 : : :CN 0 0 0
0 0 = oAoB o( B0 r C1 : : :CN C1 :: :CN 0
A0 )A A0 )B
; (poB o( B0 r oB + qoAoB r oB0 ) C1 :: :CN C1 : ::CN 0
0 0
A0 )A B( A0 )
(66)
0 0
Contracting the above expression with oA oB gives
0 0
0 = q oA C1 : : :CN C1 : ::CN 0 (67)
So the above expression is only well de ned when =0 . This condition is given in
terms of the I-geometry in Penrose's terminology, and is of course very natural in
the context of null hypersurfaces since it expresses the condition that the direction
of the ag pole of oA (and not its extent) is parallelly propagated along the null
geodesic generators of N.
A similar expression for g may be obtained by taking the complex con-
AA0
jugate of the above expression (and swapping p and q ). The operator for di er-
o
entiating along the null geodesic generators of N is given by I .
0
o
0
0 0 = oAoA rAA C1 : ::CN C1 : ::CN 0 ; 0 0 (68)
I C1 : ::CN C1 : : :CN 0 0 0 (p + q ) C1 : : :CN C1 : ::CN 0
In general this expression depends upon the choice of since is not invariant
A
under null rotations, but if oA is again chosen in such a way that = 0 then
does become invariant under null rotations and the above expression de nes a
(N N0) -spinor of weight fp +1 q +1g . Thus with this choice of oA the Penrose
operators are properly weighted operators which are invariant under null rotations.
Section 3: Relationship to the Penrose operators 12
The operators that we have de ned are more general since they make no
assumptions about the choice of oA . Furthermore in order to be able to introduce
a compact index free notation our operators act on totally symmetric spinors and
produce totally symmetric spinors. However in situations where oA is chosen so
that vanishes, our operators are closely related to those of Penrose. Since both
sets of operators obey the Leibnitz property it is enough to give the relationship
between the operators when acting on scalars and spinors with a single index.
We give below the relationship between the operators when oB DoB = =0 .
(i) For a scalar eld
o
o o
o
I 0 0
(I )AA =(I )oAoA (69)
I
g 0
(g )AA B0 = o( B0 g (70)
g
A0 )A
0
g 0
(g )ABA = o( B g (71)
g
A)A0
(ii) For a (1 0) -spinor eld
o
o o
o
0
I 0 ( A
(I )ABA =(I )oB) oA (72)
I
g 0
(g )ABA B0 = o( B0 g B) (73)
g
A0 )( A
0
g
(g )ABC A0 =(g )oC ) (74)
g
A0 ( A
B
(iii) For a (0 1) -spinor eld
o
o o
o
I 0 oB0
(I )AA B0 = oA(I ) (75)
I
)
( A0
g 0 0 oC
(g )AA B0 C =(g ) 0 (76)
g
A( A0 B0 )
0
g 0
(g )ABA B0 = o( B g (77)
g
A)( A0 B0 )
Section 3: Relationship to the Penrose operators 13
The relationship between the various de nitions for edth and thorn can be
0
seen more easily if we introduce the auxiliary di erential operator DABA B0 which
is de ned by
0 0 0 = oAoA rBB C1 :: :CN C1 :: :CN 0
0 0
DABA B0 C1 :: :CN C1 : ::CN 0 0 0
0 0 0 0
; (p rBB oA + qoA rBB oA ) C1 : : :CN C1 : : :CN 0
oA 0 0
(78)
In terms of this operator the standard de nitions of edth and thorn are given by
o
0
0 0 = oB oB DABA B0 C1 : ::CN C1 : ::CN 0
I C1 : ::CN C1 : ::CN 0 A A0 0 0 0 (79)
0 0 = oB DABA B0 C1 : ::CN C1 : ::CN 0
g C1 : ::CN C1 : : :CN 0 A A0 B0 0 0 0 (80)
0
0 0 = oB DABA B0 C1 :: :CN C1 : ::CN 0
g C1 : ::CN C1 : ::CN 0 A B A0 0 0 0 (81)
0
o
0
0 0 = DABA B0 C1 :: :CN C1 :: :CN 0
I C1 : ::CN C1 : ::CN 0 A B A0 B0 0 0 (82)
On the other hand our new derivative operators are given by
Xo oB
0
B
o
o
o
I 0 0 0 0 0
(I )AC : ::CN A0 C1 : ::CN = DABA B0 C1 :: :CN C1 :: :CN 0 (83)
I
1
0
sym
Xo
B
g 0 0 0 0 0
(g )AC : ::CN A0 B0 C1 : ::CN = DABA B0 C1 :: :CN C1 : ::CN 0 (84)
g
1
0
sym
Xo
0 B0
g 0 0 0 0 0
(g )ABC : ::CN A0 C1 : ::CN = DABA B0 C1 :: :CN C1 :: :CN 0 (85)
g
1
0
sym
X
0
o
o
o
I 0 0 0 0 0
(I )ABC : ::CN A0 B0 C1 : ::CN = DABA B0 C1 : ::CN C1 : ::CN 0 (86)
I
1
0
sym
X
Where indicates symmetrisation over all primed and unprimed indices.
sym
Section 3: Relationship to the Penrose operators 14
Finally in the case that =0 the Penrose derivative operators are given by
0
o
0 0
oAoA I C1 :: :CN C1 : ::CN 0 0 0 (87)
0 0 = oB oB DABA B0 C1 :: :CN C1 :: :CN 0
o( B0 g C1 :: :CN C1 : ::CN 0 0 0 (88)
0 0 = oB DAB( A0 B0 ) C1 : ::CN C1 : ::CN 0
A0 )A
0
0 0 = oB D( AB)A0 B0 C1 :: :CN C1 : ::CN 0
o( B g C1 :: :CN C1 :: :CN 0 0 0 (89)
A)A0
0
Now let us suppose that N is a null hypersurface chosen so that oAoA
is tangent to the null generators. Then any tangent vector to N has the form
0
A0
A
va = oA + oA , where is a (1 0) -spinor eld of weight f0 ;1g . Then the
o
o
o
I g
components of Dva , va and va may be obtained from the components of I , g
I g
0 0
A
0
0 0
g
and g . However if one contracts va = vAA with oA one obtains = vAA oA
g
A
which is a (1 0) -spinor eld of weight f2 ;1g . Note that = oA where
= vama is the component of va in the ma direction. If one then applies the
0
gg0 g g A
commutator g g ; g g to using equation (50) and noting that R = R since `a
gg g g
is hypersurface orthogonal, one obtains a totally symmetric (4 3) -spinor
f( ; ; ) + ( 2 ; ; )g (90)
2 11 11
A
If one now makes some choice for then one can calculate the components of
this spinor and one nds for example that
0 0
gg g g 430
(g g ; g g = mambmcmdRabcd (91)
gg g g)
which is just proportional to the sectional curvature in the m ^ m direction.
A
Suppose now that is chosen so that the real and imaginary parts of ma are
tangent to a two surface S and applies the commutator of the GHP operators g
Section 3: Relationship to the Penrose operators 15
A
0
and g to the weight f1 ;1g scalar eld = then one obtains
A
0 0
(gg ; g g) = R (92)
where R is the Gaussian curvature of S . Thus the new commutator involves
the projection of the spacetime curvature into S rather than the curvature of the
projected connection as is the case with the GHP formalism. This is not surprising
A
when one considers that the induced connection depends upon the choice of in
a non-trivial way and hence the curvature of the connection does not transform
at all nicely under null rotations. On the other hand our new formalism has
been designed so that the components transform covariantly under null rotations.
Indeed the di erence between the projection of the spacetime curvature and the
curvature of the projected connection consists of spin coe cients which transform
badly. This explains why our commutators appear somewhat simpler than the
GHP commutators since the terms that transform badly under null rotations have
been incorporated into our di erential operators. Of course a price must be paid
in the correspondingly more complicated de nitions of the new operators.
However as already noted above a general vector in N (rather than in S ) is
determined by the weight f0 ;1g spinor eld rather than the weight f2 ;1g
spinor eld . Applying our commutator to gives
f( ; ; ) ; ( 2 ; ; )g (93)
2 11 11
and involves terms representing the extrinsic curvature of S which arise due to
the `a component in va normal to S . Thus the complex curvature (2 1) -spinor
0 0
; ; plays the same role in this formalism as the term K = ; ;
2 11
+ + does in the GHP formalism.
2 11
References 16
References
Geroch, R. Held, A. & Penrose, R. 1973, A space-time calculus based on pairs of
null directions, J. Math. Phys. 14, 874-81.
Machado Ramos, M. P. & Vickers, J. A. G. 1995, A space-time calculus invariant
under null rotations, Proc. R. Soc. Lond. A 450, 1-17.
Newman, E. T. & Penrose, R. 1962, An approach to gravitational radiation by a
method of spin coe cients, J. Math. Phys. 3, 566-78.
Penrose, R. 1972, The geometry of impulsive gravitational waves, in General Rel-
ativity: papers in honour of J L Synge ed. L O' Raifeartaigh, Oxford University
Press.
Penrose, R. & Rindler, W. 1984, Spinors and space-time, Cambridge University
Press.