CBPF-MO-001/93
Physical and Geometrical Classi cation
Graham Hall
Centro Brasileiro de Pesquisas Fsicas | CBPF/CNPq
Rua Dr. Xavier Sigaud, 150
22290-180 { Rio de Janeiro, RJ { Brasil
1 Intro duction
The ob ject of these notes is to discuss brie y some classi cation pro cedures in general
relativity theory. The geometrical ob jects to b e classi ed here are: (i) bivectors, (ii) the
Weyl tensor, (iii) symmetric second order tensors and (iv) the space-time connection.
Topics (i) and (ii) will b e mentioned only brie y since they are well-known. Topic (iii)
will b e dealt with in more detail since it has a numb er of useful applications. Topic (iv)
which will b e describ ed in terms of the holonomy group of space-time is, mathematically,
little more than classifying the subgroups of the Lorentz group and is p erhaps of less
interest. However in a numb er of problems recently it has proved useful and deserves
some consideration.
Throughout these notes a standard notation is used, (M; g) denotes our space-time
with Lorentz metric g of signature ( + ++), round and square brackets denotes the
usual symmetrisation and skew-symmetrisation of indices, a semicolon denotes a covari-
ant derivative and a comma a partial derivative. The Riemann, Ricci, Weyl and energy-
a c
momentum tensors are denoted by R , R ( R ), C and T and the Ricci scalar
ab abcd ab
bcd acb
by
ab
R = R g .
ab
Permanent Address:
Department of Mathematical Sciences,
UniversityofAberdeen,
Edward Wright Building,
Dunbar Street,
Ab erdeen AB9 2TY, Scotland, U.K.
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2 The Classi cation of General Second Order Ten-
sors
Let p 2 M and let S b e a second order covariant tensor at p. The fact that S is a tensor
ab
means that it makes sense to ask for the eigenvectors and eigenvalues of S , that is, to ask
foravector k at p and a real number such that
b b
S k = k (= g k ) (1)
ab a ab
Note that it makes sense b ecause the answer would b e indep endent of the co ordinate
system used in (1). One should also note that in general it matters over which index of
S the contraction in (1) is p erformed. Here the second index is chosen. Actually since
we will only consider the cases when S is either symmetricorskew-symmetric this will
hardly matter in practice, there just b eing a sign change if S is skew.
The problem in solving (1), that is, nding the eigenvectors and corresp onding eigen-
values, is that since g is of Lorentz signature the problem is not p osed in the usual form
dealt with in the standard algebra texts. One can rectify this by simply rewriting (1) in
the form
b b a a a
k ) (2) k = k (= S
b b
a
The rather than the g on the R.H.S. of (2) casts the problem into the standard
ab
b
a
form. However, wepay a price for this b ecause the tensor S will have no sensible sym-
b
metric or skew-symmetric prop ertyeven if the original S did. Further, the eigenvalues
ab
in (1) or (2) may b e real or complex. Thus the p ossibilities for solving (1) are to either
(i) work with (2) and su er the attendant problems but where at least the problem is in
standard form or (ii) to work with (1) and try to develop a new technique in order to
determine the algebraic structure or (iii) nd an alternative representation of S for which
classi cation techniques exist. We will explore these p ossibilities in these notes.
3 Preliminary Remarks
At p 2 M we can intro duce twotyp es of tetrad bases for the tangent space T M to M at
p
a a a a
t = x x = y y = z z =1 p.A(pseudo-) orthonormal tetrad (t; x; y ; z ) satis es t
a a a a
a a a a a
and a real nul l tetrad (`; n; x; y ) satis es ` ` = n n =0 ` n = x x = y y = 1 with
a a a a a
all other inner pro ducts of tetrad memb ers zero in b oth cases. We can also intro duce a
complex nul l tetrad (`; n; m; m )at p where, ` and n are as in the real null tetrad case
p
and m and (its conjugate)m are de ned from the real null tetrad by 2m = x + iy ,
p
2m = x iy . Thus m andm are complex null vectors and the only non vanishing
a a
inner pro ducts amongst `, n, m,m are ` n =1 m m = 1. The following completeness
a a
relations then hold at p
g =2` n + x x + y y =2` n +2m m = t t + x x + y y + z z (3)
ab a b a b a b a b a b a b
(a b) (a b) (a b)
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It will b e convenient to know the set of changes of one real null tetrad (` ; n; x; y )
0 0 0 0
to another suchtetrad(` ;n;x;y) for which the direction of ` is preserved, that is,
0
` = A` (A 2R;A > 0). This condition on ` together with the orthogonality relations on
0 0 0 0
(` ;n ;x ;y ) for it also to b e a real null tetrad means that this set of tetrad transformations
(which is just a particular subgroup of the Lorentz group and will b e set into prop er
context later) is given by
0
` = A`
p
0
x = x y A 2( + )` (4)
p
0
2( )` y = x + y + A
p
0 1 2 2
n = A n + 2( x y) A( + )`
2 2
where A, , , , 2Rwith A>0 and + =1.Thus there are 4 free parameters in
(4). These transformations can b e more simply expressed in terms of the corresp onding
0 0 0 0
;n;m; m ) again with the direction of ` complex null tetrad change (`; n; m; m ) ! (`
0
preserved, ` = A` (A 2R;A > 0)
0
` = A`
0 i
m = e (m AB`) (5)
0 1
n = A n + Bm + B m AB B`
where A; 2R;A > 0;B 2C { again 4 free parameters. In what follows, these tetrad
changes will b e used to replace a real (or complex) null tetrad with another in order to
simplify the expression for a certain tensor expressed in such a tetrad.
At p the set of second order symmetric tensors is a 10-dimensional vector space which
can b e spanned by the 10 basis symmetric tensors (expressed in terms of a real null tetrad
(`, n; x; y )at p) given by
` ` ; n n ; 2` n ; x x ; y y ; 2x y ;
a b a b (a b) a b a b (a b)
2` x ; 2` y ; 2n x ; 2n y : (6)
(a b) (a b) (a b) (a b)
Similarly,at p,thesetofbivectors at p is a 6-dimensional vector space which can b e
spanned by the 6 basis bivectors (expressed in terms of a real null tetrad at p) given by
2` n ; 2x y ; 2` x ; 2` y ; 2n x ; 2n y : (7)
[a b] [a b] [a b] [a b] [a b] [a b]
Thus any second order symmetric tensor at p can b e written as a linear combination
of the basis memb ers (6) and any bivector at p can b e written as a linear combination of
the memb ers (7).
Now let F b e a (real) bivector at p 2 M and let denote its dual. Then we can
F
ab ab
+
called the complex self dual of F . It has = F + i construct a complex bivector
F F
ab ab ab ab
+ +
the prop erty that ( ) = i . Conversely any complex bivector P at p satisfying
F F
ab ab ab
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the self dual condition = iP can b e written as P = F + i for a real bivector
P F
ab ab ab ab ab
F .Now the set of all complex bivectors at p forms a 6-dimensional complex (or a 12-
ab
dimensional real) vector space. However the set of all complex self dual bivectors at p is a
3-dimensional complex (or a 6-dimensional real) vector space. Noting that the 1st and 2nd
real bivectors in (7) form a dual pair as do the 3rd and 4th (with a minus sign inserted in
front of the 4th) and the 5th and 6th we can de ne a basis for the 3-dimensional complex
vector space of self dual complex bivectors by
V =2` m M =2` n +2m m U =2n m : (8)
ab ab ab
[a b] [a b] [a b] [a b]
p
that is, 2V = ` x + i` x = ` x i` y , etc. In (8) the basis bivectors V , M and
ab
[a b] [a b] [a b] [a b]
U have b een built from a complex null tetrad. When that complex null tetrad is changed
under the scheme (5) the corresp onding complex bivectors V , M and U change according
to
0 i
V = Ae V
ab
ab
0
M = 2ABV + M (9)
ab ab
ab
0 2 i i 1 i
U = AB e V + Be M + A e U
ab ab ab
ab
4 The Classi cation of 2-spaces in T M
p
The set of 2-dimensional subspaces (2-spaces) of T M can b e classi ed according as they
p
contain no null directions (in which case they are called spacelike) exactly one null direction
(in which case they are called nul l)orexactly two null directions (in which case they are
called timelike). It is easy to show that there are no other p ossibilities. For example, let
(`; n; x; y ) b e a real null tetrad at p and if u and v are anytwo indep endentvectors at p
let (u; v ) denote the 2-space that they span (ie determine) at p.Then(x; y ) is spacelike,
(`; n) is timelike and (`; x), (`; y ), (n; x) and (n; y ) are null.
Now given a 2-space at p the set of vectors which are orthogonal to each member of
this 2-space is also a 2-space called its orthogonal complement. Thus (x; y ) and (`; n)
are orthogonal complements as are (`; x) and (`; y ) and also (n; x) and (n; y ). Note
that the orthogonal complement of a spacelike 2-space is timelike and vice-versa and the
orthogonal complementofanull 2-space is null (and they contain the same null direction).
It is then easy to show that every member of a spacelike 2-space is a spacelikevector
and that it can b e spanned by an orthogonal pair of spacelikevectors. Also a nul l 2-space
contains a null vector, say `, and all its multiples and all its other memb ers are spacelike
a a
and orthogonal to `. It can thus b e spanned byvectors ` and x with ` ` = ` x =0,
a a
a
x x =1.A timelike 2-space contains timelike, spacelike and null memb ers and can b e
a
a
spanned by indep endentnull vectors, say ` and n, normalised so that ` n =1.
a
5 The Classi cation of Bivectors
Let p 2 M and F abivector at p. Regarding F as a skew matrix, its rank is therefore
ab ab
an even numb er 0,2 or 4. If it is 0 then F =0. Suppose then that the rank of F is 2.
ab ab
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b
Then there exists a 2-space U T M such that if k 2 U then F k =0.If U is spacelike
p ab
and spanned by orthogonal unit vectors x and y , one completes them to a null tetrad
(`; n; x; y ) and expands F at p in terms of the basis (7)
ab
F =2 ` n +2 x y +2 ` x +2` y +2n x +2n y : (10)
ab
[a b] [a b] [a b] [a b] [a b] [a b]
b b
for ; ; ; ;; 2R. The conditions F x = F y = 0 in (10) then showthat = =
ab ab
= = = 0 and so
!
2R
(11) F =2 ` n
ab
[a b]
6=0
a
Similarly if U is timelike and spanned bynull vectors ` and n (scaled so that ` n =1)
a
a similar argument shows that
!
2R
(12) F =2 x y
ab
[a b]
6=0
and if U is nul l and spanned bya null vector ` and a unit spacelikevector y (so that
a
` y = 0) then
a
!
2R
F =2 ` x (13)
ab
[a b]
6=0
There are no other p ossibilities if rank F = 2 and in each of the ab ove rank 2 cases
ab
F can always b e written in the form F = p q q p and then F is called simple. Note
ab a b a b
that the vectors p and q are not uniquely determined by F .However, the 2-space spanned
by the vectors p and q in any (simple) form for F is well de ned and is called the blade
of F . Simple bivectors are then classi ed as spacelike, timelike or nul l according as their
blade is a spacelike, timelikeornull 2-space. Further, it is easy to show that the dual
F
of a simple bivector F is also simple and its blade is the orthogonal complementofthe
blade of F .Thus the dual of a spacelike (simple) bivector is timelike and vice versa and
the dual of a null bivector is null. As an example note that the bivectors in (7) are all
simple and are, resp ectively, timelike, spacelike, null, null, null and null.
If F is a nul l bivector then it takes a form like (13) and `, which is the only null
direction in its blade, is called the principal nul l direction of F. It follows in this case
that / 2` y and ` is also the principal null direction of . If F is timelike then it
F F
[a b]
takes a form like (11) and then ` and n are called the principal nul l directions of F.If
F is spacelike then although its blade contains no null directions, it uniquely determines
its (timelike) orthogonal complement (ie the blade of its dual). The two principal null
directions of its dual are then called the principal nul l directions of F.Thus if F takes a
form like (12) its principal null directions are ` and n.
Now supp ose the rank of F is 4. First construct the corresp onding self-dual (complex)
ab
bivector F and expand in terms of the basis (8)
ab
+
= V + M + U ; ; 2 C (14)
F
ab ab ab ab
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The idea is to showthatbychanging the basis in (14) (i.e. bychanging the null tetrad
(`; n; m; m ) according to (5) and hence the basis (8) according to (9)) wecanmake and
equal to zero in (14). After making the change (5) and hence (9) weget
+
0 0 0 0
F = V + M + U
ab
ab ab ab
i
= (Ae V )+ (2ABV + M )
ab ab ab
2 i i 1 i
+ (AB e V + Be M + A e U )
ab ab ab
so
+
0 i 2 i
F = ( Ae +2 AB + AB e )V
ab ab
i 1 i
+ ( + Be )M +( A e )U (15)
ab ab
Now , and are given by (14) and wehave the freedom to cho ose A, and B in
(5) and (9) (and clearly one of , and must b e non-zero otherwise wewould have
F = 0). First, x A and and cho ose B so that the co ecientof V in (15) vanishes.
ab ab
+
0
In this new tetrad F has no co ecientof V (and by switching ` and n we can think of
ab ab
+
0
this as F having no co ecientof U ). So wecancho ose the tetrad so that (dropping
ab ab
primes)
+
= V + M (16)
F
ab ab ab
Now in (16) wemust have 6= 0 otherwise, by taking real parts wewould get F
ab
simple and hence of rank 2 { a contradiction. Using (16), we rep eat the tetrad change (5)
and (9) and (15) b ecomes
+
0 i
F =( Ae +2 AB )V + M (17)
ab ab ab
+
i
= M and wecho ose B so that Ae +2 AB = 0. Then (dropping primes again)
F
ab ab
and taking real parts werecover F in the form
ab
F =2a` n +2bx y (18)
ab
[a b] [a b]
where = a ib, a; b 2R. In other words, if F has rank 4 it can b e written (as in
ab
(18)) as the sum of two simple bivector one spacelike and one timelike (in fact they are
duals). Then when F has rank 4 (i.e. F not simple) it is not hard to show that the
ab ab
two 2-spaces (x; y ) and (`, n) represented by (18) are uniquely determined by F .
ab
This completes the classi cation of bivectors. It can b e summarised as follows:
Case I F =0:
ab
Case I I F r ank 2(, F simple) and either F spacelike, timelike,
ab ab ab
null { canonical forms(11); (12); (13)
Case I I I F rank 4(F non-simple) { canonical form (18):
ab ab
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Remarks
a
(i) Since the rank of F is even then the existence of one vector k (6=0)suchthat
ab
b
F k = 0 implies that F is simple (if F 6= 0).
ab ab ab
b b a
(ii) If F satis es F k = k = 0 then F is necessarily null and k is the (unique
F
ab ab ab ab
up to scaling) principle null direction of F (and ).
F
ab ab
(iii) Any bivector (simple or non-simple) which is not null is usually called non-nul l .
The eigenvector structure of these bivector typ es is easily describ ed in terms of a real
null tetrad (`, n; x; y ).
1. F spacelike
ab
Canonical form F = x y y x
ab a b a b
a
The eigenvectors are ` (eigenvalue 0)
a
n (eigenvalue 0)
a a
x + iy (eigenvalue i)
a a
x iy (eigenvalue i)
2. F timelike
ab
Canonical form F = ` n n `
ab a b a b
a
The eigenvectors are ` (eigenvalue 1)
a
n (eigenvalue 1)
a
x (eigenvalue 0)
a
y (eigenvalue 0)
3. F null
ab
Canonical form F = ` x x `
ab a b a b
a
The eigenvectors are ` (eigenvalue 0)
a
y (eigenvalue 0)
4. F non-simple
ab
Canonical form F =2a` n 2bx y
ab [a b] [a b]
a
The eigenvector are ` (eigenvalue a)
a
n (eigenvalue a)
a a
x + iy (eigenvalue bi)
a a
x iy (eigenvalue bi)
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The Segre typ es here are:
F spacelike | f(11)z zg
ab
F timelike | f11(11)g
ab
F null | f(31)g
ab
F non-simple | f11 z zg
ab
6 The Petrov Classi cation of the Weyl Tensor
The Weyl tensor C , b ecause of its symmetries, can b e representedasa6 6 symmetric
abcd
matrix C according to the blo ck index scheme 1 $ 23, 2 $ 31, 3 $ 12, 4 $ 10, 5 $ 20,
AB
6 $ 30. One can then classify the Weyl tensor by lo oking for eigenbivectors of C ,that
abcd
is, solutions for F and of the equation
ab
cd
C F = F (19)
abcd ab
A
and of the equation which is like nding eigen-6 vectors of C , that is, solutions for F
AB
B
C F = F (where blo ck indices are raised and lowered with the \bivector metric"
AB A
g $ G = g g g g ). The resulting p ossible Segre typ es are just the Petrov
AB abcd ac bd ad bc
canonical typ es. However a simple approach is made p ossible by the Weyl symmetry
c
C = 0 This means that we can write C as
AB
abc
! !
A B A symmetric
C =
AB
T
B A A and B tracefree
and so we can describ e C by the 3 3 real matrices A and B or by the complex 3 3
matrix D = A + iB . There are three p ossible Jordan forms for the matrix D over C ,
namely
1 1 0 1 0 0
0 1 0 1 0 0 0
C C B C B B
0 0 1 0 0 0 0
A A @ A @ @
0 0 0 0 0 2 0 0
Segre f111g Segre f21g Segre f3g
PetrovTyp e I PetrovTyp e I I PetrovType III
where the tracefree condition on A and B (and hence on D ) is used. The Segre and Petrov
typ es are indicated. The Petrovtyp e I has a sub case typ e D (where = ) and the typ e
I I has a sub case typ e N (where = 0). They are
0 1 0 1
0 0 0 1 0
B C B C
0 0 0 0 0
@ A @ A
0 0 2 0 0 0
Segre f11)1g (Segre f21)g
PetrovTyp e D PetrovTyp e N
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A particularly imp ortant role is played bynull eigenbivectors and by their principal
null directions. In fact they lead to an alternative and very useful version of the Petrov
classi cation. This classi cation is essentially the work of L. Bel and is referred to as the
+
Bel criteria. First de ne the complex self dual Weyl tensor by
C
abcd
+
= C + i (20)
C C
abcd abcd abcd
Here is the dual of C (and it do es not matter whether the right or left dual is
C
abcd abcd
taken since, for the Weyl tensor, they are equal). If C 6= 0 then the Bel criteria are as
abcd
follows
+
d
(i) C is (Petrov) typ e N , k = 0 for some vector k .Thevector k is necessarily
C
abcd abcd
null and unique up to a scaling and is called the rep eated principal null direction of
C .
abcd
+
a c
k k =0for (ii) Given that condition (i) do es not hold then C is type III ,
C
abcd abcd
some vector k .Thevector k is null and unique up to a scaling and is called the
rep eated principal null direction of C .
abcd
+
a c
(iii) Given that (i) and (ii) do not hold the C is of typ e I I, k k = k k has a
C
abcd abcd b d
unique (up to a scaling) solution for k with 2C, 6= 0. The vector k is necessarily
null and is called the rep eated principal null direction of C .
abcd
(iv) The typ e D characterisation is as ab ove for typ e I I except that now there are required
two indep endent (necessarily null) vectors k satisfying the condition given there and
are each referred to as rep eated principal null directions.
(v) Otherwise the Petrovtyp e is I and there are exactly 4 distinct null directions k
satisfying
b c
k C k k k =0 (21)
[e a]bc[d f ]
and called principal null directions of the Weyl tensor.
7 The Classi cation of Second Order Symmetric Ten-
sors
Now return to section 2 and assume the tensor S to b e symmetric. So wehave equation
ab
(2) with S = S . This equation is, at p 2 M
ab ba
a b a
S k = k (22)
b
and, as it stands, contains no reference to the signature of the metric. However, wehave
the \consistency condition" that the resulting tensor is symmetric
c c
g S = g S (23)
ac bc
b a
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In other words if no metric with Lorentz signature at p can b e found to satisfy (23)
a
for a particular algebraic typ e for S then that algebraic typ e is imp ossible for S in a
b
space-time. This technique reveals one metho d of nding the p ossible algebraic typ es for
S [1]. First supp ose that all the eigenvalues of S are real (we will deal with the case of
a
complex eigenvalues later). Then the p ossible Jordan forms for the matrix S are
b
1 0 1 0 1 0
1 0 0 1 0 0 0 0 0
1 1 1
C B C B C B
0 1 0 0 0 0 0 0 0
C B C B C B
1 1 2
C B C B C B
A @ A @ A @
0 0 0 0 0 0 0 0 0
1 2 3
0 0 0 0 0 0 0 0 0
2 3 4
Segre f1111g Segre f211g Segre f31g
1 1 0 0
1 0 0 1 0 0
1 1
C C B B
0 1 0 0 0 0
C C B B
1 1
C C B B
A A @ @
0 0 1 0 0 1
1 2
0 0 0 0 0 0
1 2
Segre f22g Segre f4g
In each case one checks the consistency with (23) and it turns out that for Segre typ es
f22g and f4g the only solutions of (23) have det g > 0 and hence they are inconsistent
ab
with Lorentz signature. The other Segre typ es f1111gf211g and f31g turn out to b e
consistent. If the eigenvalues of S are not all real then it turns out [1,2] that S must
havetwo real and two complex (i.e. a conjugate pair of ) eigenvalues and this is indicated
by a \Segre" symbol fz z11g.Thus the only p ossibilities for S are f1111g, f211g, f31g,
fz z11g.
It is straightforward to work out canonical forms for each of these typ es at p 2 M in
terms of a real null tetrad at p.
Segre typ e f1; 111g
Here a real null tetrad (`, n; x; y )at p can b e chosen suchthat
S =2 ` n + (` ` + n n )+ x x + y y (24)
ab 1 2 a b a b 3 a b 4 a b
(a b)
a
Here S is diagonisable over R and 4 indep endent eigenvectors and corresp onding
b
a a
eigenvalues are represented by the timelikevector ` n (eigenvalue ) and the
1 2
a a a a
spacelikevectors ` + n ( + ), x ( ) and y ( ). These eigenvectors are mutually
1 2 3 4
orthogonal and , , , 2R.
1 2 3 4
Segre typ e f211g
Here one can cho ose a real null tetrad (`, n; x; y )atp so that
S =2 ` n ` ` + x x + y y (25)
ab 1 a b 2 a b 3 a b
(a b)
a
Here the eigenvectors (and corresp onding eigenvalues) can b e represented by ` (with
a a
eigenvalue ) x ( ) and y ( ) and are mutually orthogonal and , , 2R.
1 2 3 1 2 3
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Segre typ e f31g
Here one can cho ose a real null tetrad (`, n; x; y )at p so that
S =2 ` n +2` x + x x + y y (26)
ab 1 1 a b 2 a b
(a b) (a b)
a a
Here the eigenvectors can b e represented as ` (with eigenvalue )and y ( ) and
1 2
are orthogonal, and ; 2R.
1 2
Segre typ e fz z11g
Here one can cho ose a real null tetrad (`; n; x; y )atp so that
S =2 ` n + (` ` n n )+ x x + y y (27)
ab 1 2 a b a b 3 a b 4 a b
(a b)
a a
The eigenvectors can b e represented by the complex conjugate pair ` in (with re-
a a
sp ective eigenvalues i ) and the real eigenvectors x ( ) and y ( ), ; ;; 2
1 2 3 4 1 2 3 4
R.
The equations (24), (25), (26), (27) are thus canonical forms for the four p ossible
algebraic forms for S . An alternative form for the diagonisable case (24) is
ab
S =( )t t +( + )z z + x x + y y (28)
ab 2 1 a b 1 2 a b 3 a b 4 a b
where a pseudo-orthornormal tetrad (t; z ; x; y ) is used and which is related to (`; n; x; y )
p p
a a a a a a
by 2t = ` n and 2 z = ` + n . In this form the eigenvectors (and corresp onding
a a a a
eigenvalues) can b e represented by t ( ), z ( + ), x ( )and y ( ).
1 2 1 2 3 4
Notes on the classi cation [1,2,3]
(i) The typ e f1111g is the only type that admits a timelike eigenvector. The corresp ond-
ing eigenvalue is usually separated from the spacelike eigenvalues by a comma i.e.
f1; 111g.
(ii) Eigenvalue degeneracy is usually indicated by enclosing the \equal" eigenvalues inside
round brackets. For example, in (24), if = + (, =0)wewould
1 2 1 2 2
write f(1; 1)11g and if = wewould write f1; 1(11)g.IfS has typ e f(1; 111)g
3 4 ab
then S / g .
ab ab
(iii) The typ es f211g; f31g and their degeneracies have unique eigendirections (spanned
by ` in (25) and (26)). There are no (real) null eigendirections in typ e fz z11g
or fz z(11)g,noneintyp es f1; 111g, f1; 1(11)g and f1; (111)g, exactly 2 in typ es
f(1; 1)11g and f(1; 1)(11)g and in nitely manyintyp es f(1; 11)1g and f(1; 111)g.
(iv) It should b e stressed that for this (and the other classi cations discussed so far) that
the classi cation applies to the tensor S at the p oint p and will, in general, vary as
p changes in M .
There is an alternative approach to nding the canonical forms for S whichdoesnot
ab
involve the theory of Jordan forms (in fact there are several). One writes S as a linear
ab
combination of the basis symmetric tensors (6) and considers whether a nul l eigenvector
is or is not admitted. One then simpli es the tetrad (`, n; x; y ) gradually by means of the
{12{ CBPF-MO-001/93
\null rotations" (4) to achieve the canonical forms (24)-(27). However one really needs
to know that these canonical forms are distinct in the sense that one cannot transform
one into another by tetrad rotations. This is, p erhaps, b est done byshowing that they
corresp ond to di erent Segre typ es. The details are in [1,2].
8 Alternative Approaches
There are several alternative approaches to the problem of classifying second order sym-
metric tensors at a p oint in space-time. Churchill suggested manyyears ago [4] that
a
one might consider the invariant 2-spaces asso ciated with the matrix S (rather than
b
eigenvectors which are the invariant 1-spaces of this matrix). Aconvenientwayto
study this problem is, starting from S , to de ne the corresp onding tracefree tensor
ab
cd
~
S = S 1=4(S g )g and then the tensor S where [5,6]
ab ab cd ab abcd
~ ~
S = S g + S g (29)
abcd
a[c d]b b[d c]a
+ +
and nally the tensor = S + iS (right dual!). The tensor is clearly
S S
abcd abcd abcd
abcd
+ +
c
~ ~ ~
uniquely determined by S and conversely uniquely determines S since S = .
S S
ab abcd ab ab
acb
+
The invariant 2-spaces of S are closely related to the eigenbivectors of and the
S
abcd
+
classi cation of by eigenbivectors then recovers the original classi cation of S
S
abcd ab
~
(more precisely of S ).
ab
Other approaches to the problem involve a spinor approach based on equation (29) [7].
Also one can consider a \pro jective" typ e of classi cation based either on the tensor S
ab
or the tensor S [1,5] or by direct techniques from algebraic geometry and the theory
abcd
of quadric surfaces [6,7].
It is remarked that the metho d mentioned in the rst paragraph of this section and
+
[8]. It is also based on equation (29) leads to the idea of Bel-typ e criteria for
S
abcd
noted that one of the rst complete solutions to this classi cation problem was given by
Plebanski [15].
9 Physical Applications
In this section the symmetric tensor S will b e taken to b e the energy-momentum tensor
ab
T , and it should b e noted that the Ricci tensor and the energy-momentum tensor will
ab
have the same Segre typ e, including degeneracies, b ecause of Einstein's equations R
ab
1=2Rg = kT .
ab ab
(a) Energy conditions
a b a b
If one imp oses the dominant energy conditions T u u 0andT u non-spacelikefor
ab
b
a
each timelikevector u at p 2 M ,thenT is forbidden from having either the Segre
ab
typ e f31g or fz z11g or their degeneracies at p and restricts the other twotyp es bythe
eigenvalue inequalities [1,3]
0; 0; ;
1 2 1 2 3 2 1 1 2 4 2 1
{13{ CBPF-MO-001/93
for typ e f1; 111g or its degeneracies, and
0; ;
1 1 2 1 1 3 1
together with p ositive in (25) for typ e f211g or its degeneracies.
(b) Perfect uids
a a
Here wehave pressure p, density , uid owvector u (u u = 1) and
a
T =(p + )u u + pg (30)
ab a b ab
a a
The eigenvectors and eigenvalues are u ( ) and anyvector orthogonal to u (eigen-
value p). Hence wehave Segre typ e f1; (111)g (one can recover the form (28) from (30)
by using the third completeness relation in (3)). The energy conditions give p
(c) Null Einstein-Maxwell elds
a
is the \radiation" Here the Maxwell tensor F isanull bivector of the form (13) where `
ab
(null) direction. The energy-momentum tensor at p 2 M is of the form
2 1
c dc
T = (F F F F g )=v` ` (31)
ab ac cd ab a b
b
k 4
for v 2R;v > 0. From (25) the Segre typ e is f(211)g with all eigenvalues zero and the
energy conditions are satis ed since v>0.
(d) Non-null Einstein-Maxwell elds
Here the Maxwell tensor is non-null and so takes the form (11), (12), or (18) and so
T = (2` n x x y y ) (32)
ab a b a b
(a b)
for 2R, <0. The Segre typ e is f(1; 1)(11)g with eigenvalues and and the
energy conditions are satis ed since <0.
(e) General uid space-times
Here the energy-momentum tensor is
T =(p + )u u +(p )g 2 +2u q (33)
ab a b ab ab
(a b)
a a a
where u is the uid owvector, u u = 1, energy density with resp ect to u , dynamic
a
viscosity , bulk viscosity , isotropic pressure p, shear tensor (= ), expansion and
ab ba
a a b a
heat owvector q .Alsou q =0, u = 0 and = 0. Here the situation requires a
a ab
a
more detailed analysis which can b e found in [3,9].
(f ) Perfect uid and null Einstein-Maxwell eld combined
Here one has [3,9]
T =(p + )u u + pg + v` ` (34)
ab a b ab a b
and the Segre typ e is f1; 1(11)g
{14{ CBPF-MO-001/93
(g) Perfect uid and non-null Einstein-Maxwell eld combined
Here one has [3,9]
T =(p + )u u + pg + (2` n x x y y ) (35)
ab a b ab a b a b
(a b)
a a a
If ` ;n and u are coplanar the Segre typ e is f1; 1(11)g and otherwise it is f1; 111g.
(h) Static space-times
Here M must admit a timelikehyp ersurface orthogonal Killing vector eld X whichthen
turns out to b e a (timelike)eigenvector of T [10]. It follows that T has Segre typ e
ab ab
f1; 111g or some degeneracy of this typ e [2].
There are many other physical applications of this classi cation including a description
of a complete set of algebraic Rainich conditions, applications to the \inheritance of
symmetries" problem, geo desic deviation and scattering problems, \p eeling" problems
(see app endix) and the algebraic degeneracies pro duced in T at p oints in M where there
ab
is a Killing, homothetic or ane isotropy. More details and references can b e found in
[3]. Again it should b e stressed that the classi cation applies to T at some point p 2 M
ab
and the Segre typ e will, is general, change as p changes.
10 Holonomy and the Classi cation of Connections
in Space-Time
k
Let p 2 M and for a xed k; 1 k 1let C (p) denote the set of all piecewise C
k
closed curves starting and ending at p. Each c 2 C (p)gives rise by means of parallel
k
transp ort along c, using the unique symmetric Levi-Civita connnection asso ciated with
the space-time metric g , to an isomorphism of the tangent space T M to M at p onto
p
itself. If, for c 2 C (p), f (c) denotes the asso ciated isomorphism of T M then, using a
k p
1 1
standard notation, f (c )= f (c) and f (c c )=f (c ) f (c )(c ;c 2 C (p)) and
1 2 1 2 1 2 k
it follows that the set of isomorphisms of T M arising from all memb ers of C (p) is a
p k
subgroup of the Lorentz group L. This subgroup is called the k-holonomy group of M at p.
Since M is connected it is necessarily path connected and it follows that the k -holonomy
groups at distinct p oints of M are isomorphic (in fact, conjugate) to each other. Thus
it makes sense to sp eak of the k -holonomy group of M . Finally it can b e shown that
the k -holonomy group of M is indep endentofk (1 k 1) and so one sp eaks of the
holonomy group of M .
It can b e shown that if M is simply connected then the holonomy group is connected
0 0
and hence a connected Lie subgroup of the prop er Lorentz group L (L is the comp onent
of the identityof L). Since there is a one-to-one corresp ondence b etween connected
subgroups of a Lie group G and the subalgebras of the Lie algebra of G it follows that
if one wishes to classify simply connected space-times by their holonomy group, it is
0
sucient to classify the subalgebras of the Lie algebra A of L and then to see which
of them give (connected) subgroups of L which can actually b e a holonomy group of
some (simply connected) space-time. [Alternatively one could drop the simply connected
restriction and concentrate on the restricted holonomy group (rather than the holonomy
group) where the curves c are homotopic to zero [11]]. If we represent L by
{15{ CBPF-MO-001/93
Table 1:
Typ e Subalgebra Con Rec ConstantTensors
R ` ^ n x; y `; n g , x x , y y , x y
2 ab a b a b
(a b)
R ` ^ x `; y { g , ` ` , y y ` y
3 ab a b a b
(a b)
R x ^ y `; n { g , ` ` , n n , ` n
4 ab a b a b
(a b)
R ` ^ n + (x ^ y )
5
R ` ^ n; ` ^ x y ` g , y y
6 ab a b
R ` ^ n,x ^ y { `; n g , ` n
7 ab
(a b)
R ` ^ x,` ^ y ` { g , ` `
8 ab a b
R ` ^ n,` ^ x,` ^ y { ` g
9 ab
R ` ^ n,` ^ x, n ^ x y { g ;y y
10 ab a b
R ` ^ x,` ^ y , x ^ y ` { g ;` `
11 ab a b
R ` ^ x, ` ^ y , ` ^ n + (x ^ y ) { ` g
12 ab
R x ^ y , x ^ z , y ^ z u { g ;u u
13 ab a b
R ` ^ n, x ^ y , ` ^ x, ` ^ y { ` g
14 ab
R A { { g
15 ab
0
L = fT 2 GL(4; R):T T = g (36)
0
where \stroke" means \transp ose" and = diag ( 1; 1; 1; 1) then the Lie algebra A of L
(which equals the Lie algebra of L) can b e represented by the vector space of matrices
which are skew self adjoint with resp ect to , together with the \multiplication" given by
matrix commutation. Roughly sp eaking we can thus represent A by the 6-dimensional
0 0
vector space of bivectors from which L can b e recovered by exp onentiation. [That L is
an \exp onential" Lie group, i.e. it can b e \obtained" by exp onentiating its Lie algebra is
0
sp ecial for L and not a general result for (connected) Lie groups].
The subalgebras of A are well known and have b een conveniently lab elled R R
1 15
in [12]. In terms of a null tetrad (`; n; x; y ) (or in the case of R an orthonormal tetrad
13
(u; x; y ; z )) they are listed in the table. In this table eachtyp e is given with its subalgebra,
the indep endent global covariantly constantvector elds admitted (under \Con"), the
indep endent (prop erly) recurrent global vector elds admitted (under \Rec") and the
indep endent global covariantly constant second order symmetric tensors admitted (under
\ConstantTensors"). In the R and R cases, 2 R , 6=0. However R cannot be the
5 12 5
holonomy group of any space-time and hence the line! [11] [Incidentally, a global vector
eld k on M is called recurrent if k = k q for some 1-form q . It is called properly
a;b a b a
recurrent if it cannot b e globally scaled so as to b e covariantly constant.] All the typ es in
the table except R can berealised as the holonomy group of some space-time [11]. The
5
trivial ( at) case R is omitted.
1
In the table, the typ es R ;R ;R ;R ;R ;R ;R are local ly decomposable. The typ e
2 3 4 6 7 10 13
R is generic in a top ological sense and the holonomy groups of the other typ es can b e
15
describ ed in terms of standard groups [11]. Only the typ es R ;R and R can apply to
8 14 15
non- at vacuum space-times [11,12]. A particularly useful application of this classi cation
{16{ CBPF-MO-001/93
is to the study of ane collineations on space-times.
App endix
A\Peeling" Theorem for Segre Typ es
A rather nice interpretation of the Petrovtyp es is given bythewell known \p eeling"
theorem of Sachs and Penrose. They showhow in an asymptotic expansion of the (vac-
uum) Riemann tensor for a b ounded source the Petrovtyp es \p eel o " in p owers of 1=r
along some null ray with ane parameter r . A simular calculation can b e done for Segre
typ es at least for the case of an Einstein-Maxwell eld due to a b ounded charge-current
distribution [13]. In this case a similar expansion for the Maxwell electromagnetic tensor
F can b e written down [14]
ab
N L G H
ab ab ab ab
5
F = + + + + O (r ) (37)
ab
2 3 4
r r r r
where the expansion is along some null ray with ane parameter r , where N isanull
b
bivector with principal null direction `, L is a bivector satisfying L ` / ` and G and H
ab a
are bivectors ab out which no further information is required. Now compute the energy-
momentum tensor using the rst (general) equation is (31) to get
2 3 4 5
T T T T
ab ab ab ab
6
T = + + + + O (r ) (38)
ab
2 3 4 5
r r r r
3 2
has Segre typ e has Segre typ e f(2111)g with zero eigenvalue, where one nds that
T T
ab ab
4 4 5 5
b a b
f(31)g with zero eigenvalue, satis es ` / ` and satis es ` ` =0.Thus
T T T T
ab ab a ab ab
one has a similar \p eeling" of Segre typ es. The connection b etween this result and the
ab ove one for the Riemann tensor can b e clari ed byintro ducing the fourth order tensor
c
~
T asso ciated with T through its trace-free part T = T 1=4T g
abcd ab ab ab ab
c
~ ~
T = T g + T g (39)
abcd
a[c d]b b[d c]a
One then nds from (38) and (39)
2 3 4 5
T T T T
abcd abcd abcd abcd
6
T = + + + + O (r ) (40)
abcd
2 3 4 5
r r r r
i
(2 i 5) satisfy where the
T
abcd
3 2
d d
` = P ` ` =0
T T
abcd ab c abcd
(41)
4 5
a c b c
` ` / ` ` ` ` ` ` =0
T T
abcd b d [e a]bc[d f ]
where P isanull bivector with principal null direction `. If one recalls the Sachs-Penrose
ab
p eeling theorem and how the individual \p eeled o " typ es are describ ed using the Bel
criteria, one sees a direct analogy with equations (40) and (41) where the \Bel criteria"
for T are used [1,8]. abcd
{17{ CBPF-MO-001/93
Acknowledgements
The author wishes to p oint out that these lecture notes were informally written and
intended as a summary of the wayhelikes to do things. Hence the lack of references to
other researchers in this area. His indebtedness to others has already b een recorded in
the more extended bibliographies in references [1,3].
He would also liketotake this opp ortunity to thank Prof. Marcelo Reb oucas, Filip e
Paiva and Jo el B. Fonseca-Neto for their kindness and hospitality and also for many
valuable discussions during his most enjoyable stay at C.B.P.F. in Rio de Janeiro.
References
[1] G.S. Hall. Banach Centre lectures (1979) (Published 1984).
[2] G.S. Hall. J. Phys. A 9 (1976) 541.
[3] G.S. Hall. Arab. J. Sci. Eng. 9 (1984) 87.
[4] R.V. Churchill. Trans. Am. Math. So c. 34 (1932) 784.
[5] W.J. Cormack and G.S. Hall. Int. J. Theor. Phys. 18 (1979) 279.
[6] R. Penrose, in Petrov Memorial Volume (1972).
[7] R.F. Crade and G.S. Hall. Acta. Phys. Polon. B13 (1982) 405.
[8] W.J. Cormack and G.S. Hall. J. Phys. A 12 (1979) 55.
[9] G.S. Hall and D.A. Negm. Int. J. Theor. Phys. 25 (1986) 405.
[10] J. Ehlers and W. Kundt in \Gravitation: An Intro duction to Current Research", ed.
L. Witten. Wiley (1962).
[11] G.S. Hall. Lectures at International Di erential Geometry Conference, Opava,
Czechoslovakia (1992).
[12] J.F. Schell. J. Math. Phys. 2 (1961) 202.
[13] G.S. Hall in Pro ceedings of 1st. Hungarian RelativityWorkshop Balatonszeplak
(1985).
[14] J.N. Goldb erg and R.P. Kerr. J. Math. Phys. 5 (1964) 172.
[15] J.F. Plebanski. Acta. Phys. Polon. 26 (1964) 963.