Some developments of Petrov's work on conformal and projective structure Текст научной статьи по специальности «Математика»

_____________УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Том 153, кн. 3 Физико-математические пауки

2011

UDK 514.763

SOME DEVELOPMENTS OF PETROV’S WORK ON CONFORMAL AND PROJECTIVE STRUCTURE

G. Hall

Abstract

This paper discusses the achievements of A.Z. Petrov in the area of conformal and projective structure of space-times. In fact, it will be mostly concerned with the latter topic but points out uses of the former in developing the projective theory of space-times. Some new developments in this area of Petrov’s research will be given.

Key words: projective structure, conformal structure. Petrov classification, curvature

Introduction

Petrov’s work on the algebraic classification of the Weyl tensor of the gravitational field in Einstein’s general theory of relativity was a major breakthrough in achieving a fuller understanding of this theory through its applications to the generation of much-needed exact solutions of Einstein’s field equations. The final form of Petrov’s work first appeared in fl] (and in English translation in [2]) and also in his book [3]. The essence of this classification, for Petrov, was an algebraic classification of the Riemann tensor at a point of a space-time which was itself an Einstein space. However, in the important special case when the space-time is a vacuum space-time, the associated Riemann tensor has identical algebraic properties to the Weyl tensor of any space-time and so Petrov’s algebraic classification is usually taken to apply, quite generally, to the Weyl tensor. Petrov showed that there were essentially three distinct algebraic types for such a Riemann tensor (and hence for the Weyl tensor of any space-time) at a particular space-time point. Within a decade of his original paper, many other workers had realised its usefulness and extended his ideas to a comprehensive theory of what is now referred to as the Petrov Classification. Two of Petrov’s types have been divided by eigenvalue degeneracy, and so one now speaks of the Petrov types I, D, II, N Mid III, with type

O reserved for the trivial case when the Weyl tensor vanishes at the point in question. Further details can be found in the above mentioned works of Petrov and also in many other places, for example. [4 6].

In his book [3]. Petrov also discussed the idea of “geodesic mappings” of gravitational fields, that is. roughly speaking, “when two gravitational fields have the same (unparametrised) geodesics”. This problem has been revived recently [7 13] and has interested both geometers (for obvious reasons) and relativists (because of the application to the Newton-Einstein principle of equivalence in general relativity theory). The main part of this paper will be concerned with this problem. In fact. Petrov’s classification of the Weyl tensor is rather useful in some of the ways of attacking it. In addition. Petrov’s algebraic ideas suggested to the present author the idea of the curvature map [6] and this also turns out to be useful in the study of projective structure.

1. The Petrov classification

Let M be a smooth, connected, Hausdorff 4-dimensional manifold with smooth Lorentz metric g of signature (—, +, +, +) so that (M, g) is a space- time. Let V denote the Levi-Civita connection arising from g and Riem the associated curvature tensor. The components of Riem we Rabcd', the associated Ricci tensor, Ricc, has components Rab = Rcacb and the Ricci scalar is R = Rab gab ■ The Weyl tensor is denoted by C and has components Cabcd. Using the usual algebraic symmetries of the Riemann tensor components Rabcd at some point m G M, Petrov [1-3] introduced the well-known 6 x 6 notation for this tensor to turn it into a 6 x 6 symmetric matrix, Rap m

indices according to Petrov’s scheme 1 ^ (14), 2 ^ (24), 3 ^ (34), 4 ^ (23), 5 ^ (31), and 6 ^ (12). Thus, R1234 ^ R(i2)(34) = R63, etc. Petrov also noticed that the symmetric matrix Rap possessed certain other convenient symmetries because of the

R

fact that (M,g) was assumed to be an Einstein space (Ricc= ~^g)- Nowadays, these

are usually expressed, using the duality operator *, in terms of the left and right “duals” Riem

of the argument will be given in terms of it. The dual relation referred to above is then given by

*Cabcd = C*abcd (1)

One then considers the matrix Cap and studies its possible Jordan canonical forms with respect to the (6 x 6) form of the bivector metric Gabcd = gacgbd — gadgbc (which is permitted by the algebraic symmetries of Gabcd) and is non-degenerate with signature ( —, —, —, + , + , + ). This Petrov did by first transferring attention to what is essentially + + * the complex tensor C derived from C and with components Cabcd = Cabcd + iCabcd *

Cabcd C

3 x 3

derived from and containing all the information in the original one. (It is remarked that the trace-free condition arises from the condition Cabab = 0 and is stronger that the

Riem Rabab = R

Thus the possible Segre types (over C) of the original (6 x 6 Weyl) matrix Cap are those of a complex 3 x 3 matrix and are {111}, {21} Mid {3}. These are Petrov’s three types of gravitational field at m G M. By refining the classification on the basis

{111} I {(11)1}

(type D), {21} (type II) {(21)} (type N) and {3} (type III) together with type O C(m) = 0 D N

“null” (this latter term arising from certain algebraic similarities between this type and the Maxwell-Minkowski tensor in “pure radiation” electromagnetic fields). The trace-

6 x 6

3 x 3 N III O

If the Petrov type at m G M is D, II, N III or O, the Weyl tensor is said to be m I m

It is remarked that Petrov’s classification is pointwise and can vary from point MM disjointly in the forms M = I U D U II U N U III U O = I U int D U int II U int N U int III U int O U F where a Petrov symbol now refers to all those points of M which

int

MF

closed and can be shown to have empty interior [6].

Of the main developments of Petrov’s work since its announcement in Kazan one should mention the algebraic work of Bel [14], Geheniau [15], and Debever [16], who discovered the beautiful reformulation of Petrov’s classification using the idea of principal (or repeated principal) null directions (the Del criteria) and the resultant canonical C

elegant treatment of this problem). A comprehensive spinor treatment of similar matters was also given by Penrose [18] and a discussion of the physical interpretation of the Petrov classification was given by Pirani [19].

2. Projective relatedness

Now let M be a manifold of dimension n > g a smooth metric on M of any

V

denoted as before. Suppose g' is another smooth metric on M of arbitrary signature whose associated structures are denoted by adding a prime to the corresponding ones for g. Cal 1 V and V' (or g and g', or (M, g) and (M, g')) projectively related, if the

V V' M

an exact global 1-form 0 such that, in any coordinate domain of M, the Christoffel

V V'

r'abc — rac = Sab^c + 5ac^b (2)

(M, g) (M, g')

V'g' = 0

equivalent form

g'a b;c = 2g'a b0c + g'a c0b + g'b c0a (3)

V

(1, 3) Riem Riem' V

V'

R'abcd = Rabcd + S^bc — Sac^bd R'ab = Rab — 30ab) (4)

where 0ab = 0a;b — 0a0b. Sinee 0 is exact, 0 = d\ for some smooth function x on M

and then 0ab = 0ba. The problem thus becomes that of solving (3) for g' and 0.

Petrov studied this problem in some detail (see [3]). He approached it as a problem

g g'

g' with respect to g. In Petrov’s work, (M, g) was a space-time and g' also had Lorentz

g' {1, 111} {211} {31}

{zz11} together with their degeneracies. Here, {1,111} means that g' is diagonalisable over R (with a comma separating the “timelike eigenvalue” from the “spacelike” ones), and {zz11} necessarily occurs when g' is diagonalisable over C but not R. Petrov then proceeds to solve (an equivalent form of) (3) for each of these Segre types using a method based on the Ricci rotation coefficients.

An alternative approach was suggested by the Russian mathematician Sinjukov [20] (and for the remainder of this section the manifold M is of any dimension n > 2 and g g'

approach contained in (2) and (3) by drawing attention away from the pair (g', 0) to the pair (a, A) where a is a (necessarily) non-degenerate, smooth, type (0, 2) symmetric tensor and A a smooth (necessarily) exact 1-form, on M given by

aab — e Xg gac gbd, Aa — e ^0bg gac (^ Aa — aab0 )

(5)

(where g'ab are the contravariant components of g' and not g'ab with indices raised g

g'ab = e-2xacd gac gbd, 0a = — e-2xAb gbc g'ac. (6)

The condition (3) for projective relatedness is now. from (5) and (6). equivalent to the

a

aab;c = gac Ab + gbc Aa • (7)

The idea then is to solve (7) for a Mid A and convert back, using (6), to find g' and 0. With a Mid A thus found, one first defines a type (2,0) tensor a-1 on M which is, at each m G M, the inverse matrix of a (aac a-1cb = SOb ). Then one defines a related

type (0, 2) tensor on M by a- = gac gbd a-1cd. Finally, one defines a global function

1 ( det g

X = x l°g "T—

2 \ det a

M0

solution on M (see, e.g., [13, 21]; the expression here for g'ab corrects a typographical error in [13]).

(M, g) (1, 3)

projective tensor W with components given by

W\cd = R\cd - -^—(SacRbd - SadRbc). (8)

n— 1

This tensor was discovered by Weyl [22] and has the property that if (M, g) and (M, g')

are projectively related, the Weyl projective tensors associated with g and g' are equal

M

In the remaining sections, the aim is to survey and extend some of Petrov’s results on projective relatedness using the Sinjukov transformation, Petrov’s classification of the Weyl conformal tensor C, the curvature map [6] and holonomy theory.

and a global exact 1-form 0 = dx on M. Then g'ab = e2x aab ,

3. First order systems, curvature maps and holonomy

For the remainder of this paper, let (M, g) be a space-time and let g' be any other (not necessarily Lorentz) metric on M so that (M, g) and (M, g') are projectively related. Then (2)-(7) hold (and n = 4 in (8)). On applying the Ricci identity to the a

(aab;cd aab:dc )aaeR bcd + abeR acd gac Abd + gbc Aad gad Abc gbd Aac, (9)

where Aab = Aa;b(= Aba). On applying certain standard procedures to this equation and with repeated use of (7) one can show [12, 21] that if $ is any of the components of a, the components of A or the scalar Aaa, then, in any coordinate do main of M, $ satisfies a first-order differential equation of the form $,a = Fa where a comma denotes a partial derivative and the quantities Fa depend only on the various quantities that $ can represent and those describing the geometry of M. Thus, any global solution for a Mid A of (7) is uniquely determined by giving the quantities a, A Mid Aaa at any point m G M. From this it follows [21] that if the pairs (a, A), (b, ^) are global solution pairs of (7) and if there exists a non-empty open subset U C M such that b = a + ag (a G R) on U then b = a + ag and A = on M. In particular, if a = b on U, then a = b and A = ^ on M and so (a, A) = (b, ^). Thus, if M admits a non-empty, open subset U such that the only solution of (7) on U is A = 0 and a = ag (0 = a G R), the only solution of (7) on M is A = 0 and a = ag (and so V = V' on M).

This last result is useful in getting global solutions for (7) from local ones. To attack the local problem, the following construction is helpful. Let Am denote the set (2, 0) m G M

f : Am ^ Am constructed from the curvature tensor Riem of (M, g) where

f : Fab ^ RabcdFcd (Fab = — Fba). (10)

(The similarity with Petrov’s 6 x 6 matrix and its associated linear map is clear.) Let f f f f f m ( M, g) m

together with the nature of rgf contain much useful information. Clearly, if F Gkeif and G Grgf, FabGab = 0 and so F and G are orthogonal with respect to the bivector metric and dim(kerf) + dim(rgf) = 6. However, (kerf)n(rgf) need not consist only of the zero bivector (as it would if the bivector metric were positive definite). In fact, for vacuum metrics of Petrov type I, D, II, N III and O, one has, for the pair (dim(rgf), dim(kerf)), the respective values (6,0) (or (4, 2)), (6, 0), (6, 0), (2, 4), (4, 2) Mid (0, 6) and for the types N Mid III, (rgf )n(kerf) is 2-dimensional (and

f

g Riem = C

of vacuum metrics, the Petrov canonical types trivially give a complete classification f

To achieve this, it is convenient to introduce five curvature classes in the following way. In these definitions if F G Am is of matrix rank 2 it is called simple. In this case it maybe written as Fab = paqb — qapb = p A q for tangent vectors p, q at m and the 2-space

spanned by p and q is called the blade of F. Otherwise F is non-simple and gives rise

m

Class A

This covers all possibilities not covered by classes B, C, D and O below. For this

m

Class B

dim f) = 2 f

pair of simple bivectors with orthogonal blades (chosen so that one is the dual of the other). In this case, one can choose a null tetrad l,n,x,y G TmM such that these

* *

F = l A n F = x A y F F

then (using the algebraic identity Ra[bcd] = 0 to remove cross terms) one has, at m,

* *

Rabcd = aFabFcd + ftF abF cd (H)

for a, ft G R, a = 0 = ft.

Class C

In this case dim(rgf) = 2 or 3 and rgf may be spanned by independent simple bivectors F and G (or F, G and H) with the property that the re exists 0 = r G TmM

such that r lies in the blades of F and G (or F, G and ^^^us, Fabrb = Gabrb(=

Habrb) = 0 Mid r is then unique up to a multiplicative non-zero real number.

Class D

dim( f) = 1 f F m

Rabcd aFabFcd (1^)

for 0 = a G R and Ra[bcd] = 0 implies that Fa[bFcd] = 0 from which it may be checked

F

Class О

In this case Riem vanishes at m.

This curvature classification is. like the Petrov classification, pointwise. One may topologically decompose M into its curvature classes in a similar way to that done for Petrov’s types [6].

There is a particularly useful result regarding the curvature rank of /.If F is a simple bivector in ker / (for (M, g)), then the blade of F is a 2-dimensional eigenspace of the symmetric type (0, 2) tensor VA, whilst if F is a non-simple bivector in ker/ the canonical blade pair of F give two g-orthogonal 2-dimensional eigenspaces of VA. It can then be shown [7, 13] that if ker/ is such that the tangent space TmM to M at m e M is an eigenspace of VA at each point of M then, on M.

(i) Aa;b cgab; (ii) AdR abc 0; (iii) aaeR bcd + abeR acd ° (13)

where c is constant in (i). Then A is a homothetic (со)vector field on M and if A van-

MM

in parts (ii) and (iii) may be solved algebraically for A and a at any m e M if the curvature class is known at m [6]. Part (iii) is also usefully related to the curvature class at the appropriate point.

The final technique required in the study of the projective problem is the holonomy (M, g)

( M, g)

algebra of the Lorentz group and can be classified conveniently into fifteen types [24] which are labeled R1,..., R15. Of course, the holonomy group of (M, g) depends not

M

holonomy algebra is all that will be required here. The type Ri is the trivial fiat case, R5 is impossible for a space-time, R15 is the general case and the holonomy types R2 — R4 and R6 — R14 reflect the number of independent (locally) covariantly

VM

(M, g)

(M, g)

M

in solving the projective problem.

4. Main results

(M, g)

(M, g)

in the notation established above, the following result [7 9, 11 13].

Theorem 1. Let (M, g) be a space-time which is an Einstein space and, let g' be Mg Then either V = V' or (M, g) and, (M, g') each have constant curvature. If (M, g) is vacuum and not flat, then (M, g') is also vacuum (and not flat) and, further, g' = cg for constant с except possibly when (M,g) is a pp-wave space-time (when the simple relation between g and g' can easily be found).

Proof. A very brief sketch of several proofs will be given (Petrov’s approach has already been mentioned). The first approach, given in [7], was actually given only for vacuum space-times but is easily extended to Einstein spaces. This approach relies on (3) and (8) and makes no use of the Sinjukov transformation. First, one disjointly M

/ Riem

established and canonical forms for each Petrov type are written down for each region in the above decomposition with use being made of the equality of the tensors W in (8) for g and g'. In an improved proof given in [8], use is made of the Sinjukov equations (5)-(7). Here, one is able to show that either V' = V or the Weyl tensor and any solution pair (a, A) of (7) satisfy

aaeCebcd + abeCeacd = 0, Cabcd Ad = 0 (14)

on M. One then decomposes M as M = A U B where A = {m G M : C(m) = 0} and B = {m G M : C(m) = 0} so that A is open and B is closed in M. (In [8] this argument

Aa

M

M it vanishes on M (and then from (6) and (2) 0 = 0 and so V = V', on M). (In fact, this result is essentially a consequence of the first order system described in Section 3.) If m G A Mid A(m) = 0, it follows from the second equation in (14) together with the Bel criteria (Section 1) that, at m and in some open neighbourhood W of m, the Petrov-type of (M, g) is N Mid A spans a (repeated) principal null direction at m. Then one derives the contradiction that A vanishes on W and so A vanishes on A. Since A is open it follows that, if A = 0, A again vanishes on M. Finally, if A = 0, M = B and (M, g) (M, g)

if g and g' are not of constant curvature they have the same signatures (something which was assumed by Petrov) but that (M, g') may not be an Einstein space. Other

pp

of the theorem only seems to have been pointed out clearly in [7]. (In theorem 1, if g is of signature (+, + , + , +) or (+, + , —, —) again one gets the result that either each of (M, g) and (M, g') is of constant curvature or V = V', that g' need not represent

g

curvature case) Ricci flatness is preserved [8, 9]). □

For a general space-time, it is convenient to proceed by considering the holonomy ( M, g)

theorem summarizes part of the situation.

Theorem 2. Let (M, g) and, (M, g') be projectively related space-times.

(i) If g and g' are (locally) conformally related on M, then they are globally conformally related on M and further V = V' and g' = cg on M for c constant.

(ii) If (M, g) has holonomy type R2, R3, R4, R§, R7, R8 or R12, the n V = V' and the relation between g and g' can be calculated easily using holonomy theory.

(iii) If (M, g) has holonomy type Rio, R11 or R13 and with curvature rank > 1 at some m G M, the n V = V' and the relation be tween g and g' can be calculated easily using holonomy theory.

(i)

mGM

open neighbourhood U of M on which g' = ^g for ^ : U ^ R and substituting into gab gac

0 = 0 and x = const on U. The result follows from a topological argument using the connectedness of M. For parts (ii) Mid (iii), one first uses a result mentioned just

ff of the holonomy types in parts (ii) Mid (iii) TmM is an eigenspace of VA and hence AM

m G M U m

and to write a canonical form for the tensor a in U, which is determined by part (iii)

of (13). in terms of a null tetrad chosen to “fit” the holonomy invariant distributions and/or vector fields. One then substitutes into (7) and performs certain contractions to see that A vanishes on U and hence, since it is homothetic, on M.

This solves the problem for all holonomy types except types R1o, Rn or R13 (and with curvature rank < 1 at each point of M), and for types Rg, R14 and R15. The solution in these cases is more complicated and can be found in [13, 21] (and further holonomy details are available in [27]). For these cases, one does not necessarily achieve

V = V' but the relationship between g and g' can still be found. The general case R15 is not completely solved (although some progress can be made [21]). In particular, in the

(M, g)

model (necessarily of type R15), it has been completely solved [10] and (M,g') must also be an FRWL metric.

( M, g)

and (M, g') are projectively related, how are their holonomy groups related? Clearly, from the above results, there is a close link between such holonomy groups (often equality), as Theorem 2 shows, but it does not follow that they are the same and examples of non-equality have been given. Further, the Petrov type and the curvature class at m e M and the holonomy type of (M, g) are, as may be expected, closely related. In fact, it can be shown [6] that if the holonomy type of (M, g) is R2 or R4, the Petrov-type at any point is either О or D and similarly for holonomy type R3 it is О or N, for R7 it is О or D, for R13 it is О, I or D anf[ for аЦ other holonomy types except R10 and R15 it is algebraically special. In addition, for the Petrov types R2, R3 and R4 О m Riem(m) = 0

type is R2, R3 or R4, the curvature class at any m e M is O or D, for holonomy

types Re, Rg, R10, R11 and R13 it is O, C or D, for R7 it is O, B or D, for Rg and R12, it is O, C, D or A and for R14 or R15, it could be any curvature class (but, if R14 B

esting for physicists, is the problem when the original (M, g) satisfies dim M = 4 and g

this problem in all but the most general holonomy case [28]. Similarly, the case when

dim M = 4 and g has neutral signature (+, +, —, —) has been considered1. Further

details for space-times can be found in [29].

The author wishes to thank David Lonie for many illuminating discussions and collaborations.

Резюме

Г. Холл. Развитие исследований А.З. Петрова по конформной и проективной структурам.

В статье обсуждаются достижения А.З. Петрова в области исследования конформной и проективной структур прострапства-времепи. Основное внимание уделено последней, одпако показано и значение первой в развитии проективной теории прострапства-времепи. Приведены некоторые новые результаты в данной области исследования А.З. Петрова.

Ключевые слова: проективная структура, конформная структура, классификация А.З. Петрова, карта кривизны.

1 Wang Z., Hall G.S. Projective Structure in 4-Dimensional Manifolds with Metric of Signature (+, +, —, —). - Submitted.

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Поступила в редакцию 16.12.10

Hall, Graham PhD. Professor Emeritus of Mathematics, Institute of Mathematics, University of Aberdeen, Scotland, UK.

Холл, Грэм доктор паук, почетный профессор математики Института математики Абердинского университета, Шотландия, Великобритания.

E-mail: у.halMiabdn.ac.uk