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1 Introduction CBPF-MO-001/93 Physical and Geometrical Classi cation in General Relativity 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. {2{ CBPF-MO-001/93 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) {3{ CBPF-MO-001/93 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 {4{ CBPF-MO-001/93 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).
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