Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the hol...

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Datum:	2009
Hauptverfasser:	Hall, G.S., Lonie, D.P.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2009
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1491372019-02-20T01:27:28Z Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds Hall, G.S. Lonie, D.P. A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided. 2009 Article Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds / G.S. Hall, D.P. Lonie // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 26 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 53C29; 53C22; 53C50 http://dspace.nbuv.gov.ua/handle/123456789/149137 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided.
format	Article
author	Hall, G.S. Lonie, D.P.
spellingShingle	Hall, G.S. Lonie, D.P. Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Hall, G.S. Lonie, D.P.
author_sort	Hall, G.S.
title	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
title_short	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
title_full	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
title_fullStr	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
title_full_unstemmed	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
title_sort	holonomy and projective equivalence in 4-dimensional lorentz manifolds
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149137
citation_txt	Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds / G.S. Hall, D.P. Lonie // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 26 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT hallgs holonomyandprojectiveequivalencein4dimensionallorentzmanifolds AT loniedp holonomyandprojectiveequivalencein4dimensionallorentzmanifolds
first_indexed	2025-07-12T21:28:41Z
last_indexed	2025-07-12T21:28:41Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 066, 23 pages Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds? Graham S. HALL † and David P. LONIE ‡ † Department of Mathematical Sciences, University of Aberdeen, Meston Building, Aberdeen, AB24 3UE, Scotland, UK E-mail: g.hall@abdn.ac.uk ‡ 108e Anderson Drive, Aberdeen, AB15 6BW, Scotland, UK E-mail: DLonie@aol.com Received March 18, 2009, in final form June 11, 2009; Published online June 29, 2009 doi:10.3842/SIGMA.2009.066 Abstract. A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided. Key words: projective structure; holonomy; Lorentz manifolds; geodesic equivalence 2000 Mathematics Subject Classification: 53C29; 53C22; 53C50 This paper is dedicated to the memory of Élie Cartan 1 Introduction Élie Cartan (1869–1951) was one of the world’s leading geometers and it is to his memory that the authors dedicate this paper. One of Cartan’s main interests lay in the crucially important study of connections on manifolds and its applications to theoretical physics. The present paper will proceed in a similar vein by presenting a discussion of holonomy theory on a 4-dimensional manifold which admits a Lorentz metric. This leads naturally to applications in Einstein’s general theory of relativity. Here the application will be to the projective structure of space- times and which itself is closely related to the principle of equivalence in Einstein’s theory. A few weeks before this paper was begun, the authors learned, sadly, of the death of Élie Cartan’s son, Henri, at the age of 104. Henri Cartan was also a world leader in geometry and this father and son combination laid down the foundations for a great deal of the research presently undertaken in differential geometry. This paper will be arranged in the following way. Section 2 will be used to introduce general notation and in Section 3 a classification of the curvature tensor will be introduced and which will prove useful in what is to follow. Also included in Section 3 is a discussion of certain relationships between the metric, connection and curvature structures on a space-time. In Section 4 a review of holonomy theory will be given and, in particular, as it applies to 4-dimensional Lorentz manifolds. In Section 5, a discussion of projective structure will be presented. This will be followed, in Sections 6 and 7, with several theorems which show a tight relationship between projective relatedness and holonomy type. A brief summary of the paper is given in Section 8. ?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection is available at http://www.emis.de/journals/SIGMA/Cartan.html mailto:g.hall@abdn.ac.uk mailto:DLonie@aol.com http://dx.doi.org/10.3842/SIGMA.2009.066 http://www.emis.de/journals/SIGMA/Cartan.html 2 G.S. Hall and D.P. Lonie 2 Notation and preliminary remarks Throughout this paper, M will denote a 4-dimensional, Hausdorff, connected, smooth manifold which admits a smooth metric g of Lorentz signature (−,+,+,+). The pair (M, g) is called a space-time. It follows that the (usual manifold) topology on M is necessarily second coun- table [1] and hence paracompact. All structures on M will be assumed smooth (where this is sensible). The unique symmetric Levi-Civita connection arising on M through g is denoted by ∇ and, in a coordinate domain of M , its Christoffel symbols are written Γa bc. The type (1, 3) curvature tensor associated with ∇ is denoted by Riem and its (coordinate) components are written Ra bcd. The Ricci tensor, Ricc, derived from Riem, has components Rab = Rc acb and R = Rabg ab is the Ricci scalar. The Weyl type (1, 3) conformal tensor C has components Ca bcd given by Ca bcd = Ra bcd − Ea bcd − R 12 ( δa cgbd − δa dgbc ) , (2.1) where E is the tensor with components Ea bcd = 1 12 ( R̃a cgbd − R̃a dgbc + δa cR̃bd − δa dR̃bc ) , (2.2) and where R̃icc is the trace-free Ricci tensor with components R̃ab = Rab− R 4 gab. At any m ∈M , the tensors E and R̃icc uniquely (algebraically) determine each other and E = 0 ⇔ R̃icc = 0 ⇔ the Einstein space condition holding at m. For m ∈ M , TmM denotes the tangent space to M at m and this will, for convenience, be identified with the cotangent space T ∗ mM to M at m, through the metric g(m) by index raising and lowering. Thus the liberty will be taken of using the same symbol for members of TmM and T ∗ mM which are so related and similarly for other tensor spaces. A tetrad (that is, a basis for TmM) u, x, y, z ∈ TmM is called orthonormal if and only if the only non-vanishing inner products between tetrad members are −g(u, u) = g(x, x) = g(y, y) = g(z, z) = 1 and a tetrad l, n, x, y ∈ TmM is called null if and only if the only non-vanishing inner products between tetrad members are g(l, n) = g(x, x) = g(y, y) = 1. In this case, l and n are null vectors. Another condition, the non-f lat condition, will be imposed on (M, g), meaning that Riem does not vanish over any non-empty open subset of M . This is a physical requirement and is there to prevent gravitational shielding in general relativity theory. Let ΛmM denote the 6-dimensional vector space of all tensor type (2, 0) 2-forms at m. This vector space can be associated, using the metric g(m), with the vector spaces of tensor type (0, 2) 2-forms at m and of type (1, 1) tensors at m which are skew-self adjoint with respect to g(m), through the component identifications (F ∈ ΛmM) F ab(= −F ba) → Fab ≡ gacgbdF cd → F a b = gcbF ac. Any member F of any of these vector spaces will be referred to as a bivector (at m) and written symbolically as F ∈ ΛmM . Any F ∈ ΛmM , F 6= 0, has (matrix) rank 2 or 4. If F has rank 2 it is called simple and may be written in components as F ab = paqb − qapb for p, q ∈ TmM . Although F does not determine p and q it does uniquely determine the 2-dimensional subspace (referred to as a 2-space) of TmM spanned by p and q and which is called the blade of F . A simple bivector F at m is then called timelike, spacelike or null according as its blade is, respectively, a timelike, spacelike or null 2-space at m. If F ∈ ΛmM has rank 4 it is called non-simple and may be written as F = G + H where G and H are simple bivectors with G timelike and H spacelike and where the blades of G and H are uniquely determined by F and are orthogonal complements of each other. They will be collectively called the canonical pair of blades of F . The Hodge duality operator on bivectors is denoted by ∗. Then F is simple if and Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 3 only if ∗ F is simple and the blades of F and ∗ F are orthogonal complements of each other. In this case F is spacelike, respectively, timelike or null, if and only if ∗ F is timelike, respectively, spacelike or null. If F is simple and either spacelike or timelike, the blades of F and ∗ F are complementary (that is, their union spans TmM). This is not the case if F is null. In the above expression for a non-simple bivector F , G and H are (multiples of) duals of each other. For any bivector F at m, F and ∗ F are independent bivectors at m and ∗∗ F = −F . If F is a simple bivector at m with F ab = paqb − qapb, p, q ∈ TmM , then F (or its blade) is sometimes written as p ∧ q. It is convenient, on occasions, to use round and square brackets to denote the usual symmetrisation and skew-symmetrisation of indices, respectively. As a general remark on notation, both coordinate and coordinate-free notation will be used, depending on relative convenience. More details on these aspects of Minkowski geometry may be found, for example, in [2]. 3 Curvature structure of space-times Because of the algebraic symmetries of Riem, one may introduce the curvature maps f and f̃ from the vector space of bivectors to itself at m (recalling the liberties taken with this vector space in Section 2) by f : F ab → Ra bcdF cd, f̃ : F ab → Rab cdF cd. (3.1) The maps f and f̃ are linear maps of equal rank, the latter being referred to as the curvature rank at m. Let Bm denote the range space of f or f̃ at m (according to the agreed identification) so that dimBm equals the curvature rank atm and which, in turn, is≤ 6. This leads to a convenient algebraic classification of Riem at m into five mutually exclusive and disjoint curvature classes (for further details, see [2]). Class A This covers all possibilities not covered by classes B, C, D and O below. For this class, the curvature rank at m is 2, 3, 4, 5 or 6. Class B This occurs when dimBm = 2 and when Bm is spanned by a timelike-spacelike 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 ∈ TmM such that these bivectors are F = l ∧ n and ∗ F = x ∧ y so that F is timelike and ∗ F is spacelike and then (using the algebraic identity Ra[bcd] = 0 to remove cross terms) one has, at m, Rabcd = αFabFcd + β ∗ F ab ∗ F cd (3.2) for α, β ∈ R, α 6= 0 6= β. Class C In this case dimBm = 2 or 3 and Bm may be spanned by independent simple bivec- tors F and G (or F , G and H) with the property that there exists 0 6= r ∈ TmM such that r lies in the blades of ∗ F and ∗ G (or ∗ F , ∗ G and ∗ H). Thus Fabr b = Gabr b(= Habr b) = 0 and r is then unique up to a multiplicative non-zero real number. Class D In this case dimBm = 1. If Bm is spanned by the bivector F then, at m, Rabcd = αFabFcd (3.3) for 0 6= α ∈ R and Ra[bcd] = 0 implies that Fa[bFcd] = 0 from which it may be checked that F is necessarily simple. Class O In this case Riem vanishes at m. 4 G.S. Hall and D.P. Lonie It is remarked that this classification is pointwise and may vary over M . The subset of M consisting of points at which the curvature class is A is an open subset of M [2, p. 393] and the analogous subset arising from the class O is closed (and has empty interior in the manifold topology of M if (M, g) is non-flat). It is also useful to note that the equation Rabcdk d = 0 at m has no non-trivial solutions if the curvature class at m is A or B, a unique independent solution (the vector r above) if the curvature class at m is C and two independent solutions if the curvature class at m is D (and which span the blade of ∗ F in (3.3)). If dimBm ≥ 4 the curvature class at m is A. If (M, g) has the same curvature class at each m ∈ M it will be referred to as being of that class. Related to this classification scheme is the following result which will prove useful in what is to follow. The details and proof can be found in [3, 4, 5, 2]. Theorem 3.1. Let (M, g) be a space-time, let m ∈M and let h be a second order, symmetric, type (0, 2) (not necessarily non-degenerate) tensor at m satisfying haeR e bcd+hbeR e acd = 0. Then, with all tensor index movements and all orthogonality statements made using the metric g; (i) if the curvature class of (M, g) at m ∈ M is D and u, v ∈ TmM span the 2-space at m orthogonal to F in (3.3) (that is u ∧ v is the blade of ∗ F ) there exists φ, µ, ν, λ ∈ R such that, at m, hab = φgab + µuaub + νvavb + λ(uavb + vaub); (3.4) (ii) if the curvature class of (M, g) at m ∈M is C there exists r ∈ TmM (the vector appearing in the above definition of class C) and φ, λ ∈ R such that, at m, hab = φgab + λrarb; (3.5) (iii) if the curvature class of (M, g) at m ∈ M is B there exists a null tetrad l, n, x, y (that appearing in the above definition of class B) and φ, λ ∈ R such that, at m, hab = φgab + λ(lanb + nalb) = (φ+ λ)gab − λ(xaxb + yayb); (3.6) (iv) if the curvature class of (M, g) at m ∈M is A there exists φ ∈ R such that, at m, hab = φgab. (3.7) The proof is essentially based on the obvious fact that the range Bm of the map f in (3.1) which, by the algebraic symmetries of the curvature Riem of (M, g), consists entirely of members which are skew-self adjoint with respect to g, must likewise consist entirely of members skew-self adjoint with respect to h. Thus each F ∈ Bm satisfies gacF c b + gbcF c a = 0, hacF c b + hbcF c a = 0. (3.8) It is a consequence of (3.8) that the blade of F (if F is simple) and each of the canonical pair of blades of F (if F is non-simple) are eigenspaces of h with respect to g, that is, for F simple, any k in the blade of F satisfies habk b = ωgabk b where the eigenvalue ω ∈ R is independent of k, and similarly for each of the canonical blades if F is non-simple (but with possibly different eigenvalues for these blades) [2, 3, 4]. It is remarked that if g′ is another metric on M whose curvature tensor Riem′ equals the curvature tensor Riem of g everywhere on M then the conditions of this theorem are satisfied for h = g′(m) at each m ∈ M and so the conclusions also hold except that now one must Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 5 add the restriction φ 6= 0 in each case to preserve the non-degeneracy of g′ at m and maybe some restrictions on φ, µ, ν and λ if the signature of g′ is prescribed. If (M, g) is of class A, (3.7) gives g′ = φg and the Bianchi identity may be used to show that φ is constant on M [4, 2]. When Theorem 3.1 is applied to another metric g′ on M in this way, it consolidates the curvature classification scheme which preceded it. To see this note that g′ need not have Lorentz signature (−,+,+,+) but, if this is insisted upon, the curvature classification scheme, including the nature (timelike, spacelike or null) of F and ∗ F in class B, r in class C and F in class D is the same whether taken for (the common curvature tensor) Riem with g or Riem with g′. In this sense it is a classification of Riem, independent of the metric generating Riem [6]. 4 Holonomy theory Let (M, g) be a space-time with Levi-Civita connection ∇ and let m ∈M . The connection ∇ is a complicated object but one particularly pleasing feature of it lies in the following construction. Let m ∈M and for 1 ≤ k ≤ ∞ let Ck(m) denote the set of all piecewise Ck closed curves starting and ending at m. If c ∈ Ck(m) let τc denote the vector space isomorphism of TmM obtained by parallel transporting, using ∇, each member of TmM along c. Using a standard notation associated with curves one defines, for curves c, c0, c1, c2 ∈ Ck(m), with c0 denoting a constant curve at m, the identity map τc0 on TmM , the inverse τ−1 c ≡ τc−1 and product τc1 · τc2 ≡ τc1·c2 to put a group structure on {τc : c ∈ Ck(m)}, making it a subgroup of G ≡ GL(TmM)(= GL(4,R)), called the k-holonomy group of M at m and denoted by Φk(m). In fact, since M is connected and also a manifold, it is also path connected and, as a consequence, it is easily checked that, up to an isomorphism, Φk(m) is independent of m. Less obvious is the fact that Φk(m) is independent of k (1 ≤ k ≤ ∞) and thus one arrives at the holonomy group Φ (of ∇) on M . Further details may be found in [7] and a summary in [2]. One could repeat the above operations, but now only using curves homotopic to zero. the above independence of m and k still holds and one arrives at the restricted holonomy group Φ0 of M . It can now be proved that Φ and Φ0 are Lie subgroups of G, with Φ0 connected, and that Φ0 is the identity component of Φ [7]. Clearly, if M is simply connected, Φ = Φ0 and then Φ is a connected Lie subgroup of G. The common Lie algebra φ of Φ and Φ0 is called the holonomy algebra. The connection ∇ can then be shown to be flat (that is, Riem vanishes on M) if and only if φ is trivial and this, in turn, is equivalent to Φ0 being trivial. For a space-time (M, g), however, one has the additional information that ∇ is compatible with the metric g, that is, ∇g = 0. Thus each map τc on TmM preserves inner products with respect to g(m). It follows that Φ is (isomorphic to) a subgroup of the Lorentz group L, where L = {A ∈ GL(4,R) : AηAT = η} where AT denotes the transpose of A and η is the Minkowski metric, η = diag(−1, 1, 1, 1). Now Φ is a Lie subgroup of GL(4,R) and L can be shown to be a 6-dimensional Lie subgroup of GL(4,R) which is a regular submanifold of GL(4,R) (that is, the (sub)manifold topology on L equals its induced topology from GL(4,R)). It follows that Φ is a Lie subgroup of L (see e.g. [2]) and hence that Φ0 (or Φ, if M is simply connected) is a connected Lie subgroup of the identity component, L0, of L. Thus the holonomy algebra φ can be identified with a subalgebra of the Lie algebra L of L, the Lorentz algebra. The one- to-one correspondence between the subalgebras of L and the connected Lie subgroups of L0 shows that the Lie group Φ0 (or Φ, if M is simply connected) is determined by the subalgebra of L associated with φ. Fortunately, the subalgebra structure of L is well-known and can be conveniently represented as follows. Let m ∈ M and choose a basis for TmM (together with its dual basis) for which g(m) has components equal to, say, gab. Then L is isomorphic to {A ∈ GL(4,R) : Ag(m)AT = g(m)} and L can then be represented as the subset of M4R each member of which, if regarded as the set of components of a type (1, 1) tensor at m in this 6 G.S. Hall and D.P. Lonie Table 1. Holonomy algebras. For types R5 and R12 0 6= ω ∈ R. Every potential holonomy algebra except R5 (for which the curvature tensor fails to satisfy the algebraic Bianchi identity) can occur as an actual holonomy algebra, see, e.g. [2]). There is also a type R1, when φ is trivial and (M, g) is flat, but this trivial type is omitted. Type R15 is the “general” type, when φ = L. Type Dimension Basis Curvature Recurrent Constant Class vector fields vector fields R2 1 l ∧ n D or O {l}, {n} 〈x, y〉 R3 1 l ∧ x D or O – 〈l, y〉 R4 1 x ∧ y D or O – 〈l, n〉 R5 1 l ∧ n+ ωx ∧ y – – – R6 2 l ∧ n, l ∧ x C, D or O {l} 〈y〉 R7 2 l ∧ n, x ∧ y B, D or O {l}, {n} – R8 2 l ∧ x, l ∧ y C, D or O – 〈l〉 R9 3 l ∧ n, l ∧ x, l ∧ y A, C, D or O {l} – R10 3 l ∧ n, l ∧ x, n ∧ x C, D or O – 〈y〉 R11 3 l ∧ x, l ∧ y, x ∧ y C, D or O – 〈l〉 R12 3 l ∧ x, l ∧ y, l ∧ n+ ω(x ∧ y) A, C, D or O {l} – R13 3 x ∧ y, y ∧ z, x ∧ z C, D or O – 〈u〉 R14 4 l ∧ n, l ∧ x, l ∧ y, x ∧ y any {l} – R15 6 L any – – basis, is skew-self adjoint with respect to the matrix gab(m) (that is, its components F a b satisfy an equation like the first in (3.8) with respect to g(m)). Thus one can informally identify L with this collection of bivectors, usually, written in type (2, 0) form. The binary operation on L is that induced from the Lie algebra M4R of GL(4,R) and is matrix commutation. Such a representation of L is well-known and has been classified into fifteen convenient types [8] (for details of the possible holonomy types most relevant for the physics of general relativity see [9]). It is given in the first three columns of Table 1 using either a null tetrad l, n, x, y or an orthonormal tetrad u, x, y, z to describe a basis for each subalgebra. Now suppose M is simply connected. This condition is not always required but is imposed in this section for convenience. It can always be assumed in local work, for example, in some connected, simply connected coordinate domain. In this case Φ = Φ0 and is connected and Φ will be referred to according to its Lie algebra label as in Table 1. Two important results can now be mentioned in connection with the first column of Table 1. First, it turns out [10, 11, 2] that if m ∈ M there exists 0 6= k ∈ TmM such that F a bk b = 0 for each F ∈ φ if and only if M admits a global, covariantly constant, smooth vector field whose value at m is k. A basis for the vector space of such vector fields on M for each holonomy type is given inside 〈〉 brackets in the final column of Table 1. Second, there exists 0 6= k ∈ TmM such that k is an eigenvector of each F ∈ φ but with at least one associated eigenvalue not zero if and only if M admits a global smooth properly recurrent vector field X whose value at m is k, that is, a global nowhere zero vector fiel X on M satisfying ∇X = X ⊗ w for some global, smooth covector field w on M (the recurrence 1-form) and such that no function α : M → R exists such that α is nowhere zero on M and αX is covariantly constant on M . It follows from the existence of one non- zero eigenvalue at m in the above definition that X is necessarily null (by the skew-self adjoint property of the members of φ). The independent properly recurrent vector fields are listed for each holonomy type in { } brackets in the second from last column of Table 1. (It is remarked here that a nowhere zero vector field Y on M is called recurrent if ∇Y = Y ⊗ r for some global covector field r on M . In this case r could be identically zero and so (non-trivial) covariantly constant vector fields are, in this sense, recurrent. In fact, any non-null recurrent vector field or Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 7 any recurrent vector field on a manifold with positive definite metric can be globally scaled to be nowhere zero and covariantly constant because if Y is any such vector field and ∇Y = Y ⊗ r, αY is covariantly constant, where α = exp(−1 2 log(g(Y, Y ))).) A recurrent vector field is easily seen to define a 1-dimensional distribution on M which is preserved by parallel transport. There is an important generalisation of this concept. Let m ∈ M and V a non-trivial proper subspace of TmM . Suppose τc(V ) = V for each τc arising from c ∈ Ck(m) at m. Then V is holonomy invariant and gives rise in an obvious way to a smooth distribution on M which is, in fact, integrable [7]. Clearly, if V ⊂ TmM is holonomy invariant then so is the orthogonal complement, V ⊥, of V . If such a V exists the holonomy group Φ of M is called reducible (otherwise, irreducible). (This concept of holonomy reducibility is a little more complicated in the case of a Lorentz metric than in the positive definite case due to the possibility of null holonomy invariant subspaces giving a weaker form of reducibility. This will not be pursued any further here, more details being available in [12] and summaries in [2, 9, 11].) Thus, for example, in the notation of Table 1 the holonomy type R2 admits two 1-dimensional null holonomy invariant subspaces spanned by l and n and which give rise to two null properly recurrent vector fields and infinitely many 1-dimensional spacelike holonomy invariant subspaces spanned by the infinitely many covariantly constant vector fields in 〈x, y〉. For the holonomy type R7, two 1-dimensional null holonomy invariant subspaces exist and which give rise to two independent properly recurrent null vector fields as in the previous case, together with a 2-dimensional spacelike one orthogonal to each of the null ones. For holonomy types R10, R11 and R13 one has a 1-dimensional holonomy invariant subspace, spanned by a covariantly constant vector field in each case, together with its orthogonal complement. These holonomy decompositions will be useful in Sections 6 and 7. In the event that M is not simply connected, the vector fields determined by the holonomy and described above may not exist globally but do exist locally over some open, connected and simply connected neighbourhood of any point. It is remarked here that if (M, g) is of curvature class B it can be shown that it must in fact be of holonomy type R7 [13, 2]. It is useful, at this point, to introduce the infinitesimal holonomy group Φ′ m, of (M, g) at each m ∈ M . Using a semi-colon to denote a ∇-covariant derivative, consider, in some coordinate neighbourhood of m, the following matrices for (M, g) at m Ra bcdX cY d, Ra bcd;eX cY dZe, . . . (4.1) for X,Y, Z, . . . ∈ TmM . It turns out that the collection (4.1) spans a subalgebra of the holonomy algebra φ (and hence only a finite number of terms arise in (4.1) [7]). This algebra is called the infinitesimal holonomy algebra at m and is denoted by φ′m. The unique connected Lie subgroup of Φ that it gives rise to is the infinitesimal holonomy group Φ′ m at m. This is useful in that it says that the range space of the map f in (3.1) is, at each m ∈ M , (isomorphic as a vector space to) a subspace of φ. This gives a restriction, when φ is known, on the expression for Riem at each m and hence on its curvature class at m. This restriction is listed in the fourth column of Table 1. Thus if the holonomy type of (M, g) is R2, R3 or R4 its curvature class is O or D at each m ∈ M whilst if it is R6, R8, R10, R11 or R13 it is O, D or C, if R7 it is O, D or B, if R9 or R12 it is O, D, C or A and if R14 or R15 it could be any curvature class. A useful relationship between the various algebras φ′m, the algebra φ and the curvature class (through the range space Bm) at each m ∈M is provided by the Ambrose–Singer theorem [14] (see also [7]). 5 Projective structure One aspect of differential geometry that has been found interesting both for pure geometers and physicists working in general relativity theory is that of projective structure. For general 8 G.S. Hall and D.P. Lonie relativity it is clearly motivated by the Newton–Einstein principle of equivalence. In this section it will take the form described in the following question; for a space-time (M, g) with Levi- Civita connection ∇, if one knows the paths of all the unparameterised geodesics (that is, only the geodesic paths in M) how tightly is ∇ determined? Put another way, let (M, g) and (M, g′) be space-times with respective Levi-Civita connections ∇ and ∇′, such that the sets of geodesic paths of ∇ and ∇′ coincide (and let it be agreed that ∇ and ∇′ (or g and g′) are then said to be projectively related (on M)). What can be deduced about the relationship between ∇ and ∇′ (and between g and g′)? (And it is, perhaps, not surprising that some reasonable link should exist between projective relatedness and holonomy theory.) In general, ∇ and ∇′ may be expected to differ but it turns out that in many interesting situations they are necessarily equal. If ∇ = ∇′ is the result, holonomy theory can also describe precisely, the (simple) relationship between g and g′ [15, 2]. If ∇ and ∇′ are projectively related then there exists a uniquely defined global smooth 1-form field ψ on M such that, in any coordinate domain of M , the respective Christoffel symbols of ∇ and ∇′ satisfy [16, 17] Γ′a bc − Γa bc = δa bψc + δa cψb. (5.1) It is a consequence of the fact that ∇ and ∇′ are metric connections that ψ is a global gradient on M [16]. Equation (5.1) can, by using the identity ∇′g′ = 0, be written in the equivalent form g′ab;c = 2g′abψc + g′acψb + g′bcψa (5.2) (recalling that a semi-colon denotes covariant differentiation with respect to ∇). Equation (5.1) reveals a simple relation between the type (1, 3) curvature tensors Riem and Riem′ of ∇ and ∇′, respectively, given by R′a bcd = Ra bcd + δa dψbc − δa cψbd (⇒ R′ ab = Rab − 3ψab), (5.3) where ψab ≡ ψa;b − ψaψb = ψba and where R′ ab ≡ R′c acb are the Ricci tensor components of ∇′. It can now be shown that if ∇ and ∇′ are projectively related, the following type (1, 3) Weyl projective tensor, W , is the same for each of them [18] W a bcd = Ra bcd + 1 3(δa dRbc − δa cRbd). (5.4) A particularly important case of such a study arises where the original pair (M, g) is a space- time which is also an Einstein space so that the tensor E in (2.2) is identically zero on M . Such a situation has been discussed in several places [19, 20, 21, 22, 23, 24] in connection with the principle of equivalence. The particular case which is, perhaps, of most importance in general relativity arises when the Ricci scalar vanishes and then (M, g) is a vacuum (Ricci flat) space-time and this is discussed in [22, 24]. It turns out that if (M, g) is a space-time which is a (general) Einstein space (and with the non-flat assumption temporarily dropped) and if g′ is another metric on M projectively related to g then either (M, g) and (M, g′) are each of constant curvature, or the Levi-Civita connections ∇ and ∇′ of g and g′, respectively, are equal. In the event that neither space-time is of constant curvature, (and so ∇′ = ∇) an argument from holonomy theory can be used to show that, generically, (M, g′) is also an Einstein space and that g′ = cg (0 6= c ∈ R). Although examples exist where each of these conclusions fail, g′ always has Lorentz signature (up to an overall minus sign). If, in addition, (M, g) is assumed vacuum and the non-flat condition is imposed, then, necessarily, ∇ = ∇′ and, with one very special case excluded, g′ = cg on M (0 6= c ∈ R) (and so (M, g′) is also vacuum). For this case g′ has the same signature as g (up to an overall minus sign [22, 24]). This result is relevant for general relativity theory. A similar restrictive result for space-times of certain holonomy types will be established in the next two sections. Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 9 The formalism described above characterises the relationship between the connections and metrics of two projectively related space-times (M, g) and (M, g′). However, it is convenient to rewrite them in a different way using the Sinyukov transformation [20]. Thus, with this projective relatedness assumed, one takes advantage of the fact that the 1-form ψ in (5.1) is necessarily a global gradient by writing ψ = dχ for some smooth function χ : M → R. Then the pair g′ and ψ above are replaced by a type (0, 2) symmetric tensor field a and a 1-form field λ on M which are given in terms of g′ and ψ by aab = e2χg′cdgacgbd, λa = −e2χψbg ′bcgac (⇒ λa = −aabψ b), (5.5) where an abuse of notation has been used in that g′ab denotes the contravariant components of g′ (and not the tensor g′ab with indices raised using g) so that g′acg ′cb = δ b a . Then (5.5) may be inverted to give g′ab = e−2χacdg acgbd, ψa = −e−2χλbg bcg′ac. (5.6) The idea is that if g and g′ are projectively related metrics on M , so that (5.2) holds for some 1-form ψ(= dχ), then a and λ as defined in (5.5) can be shown, after a short calculation, to satisfy Sinjukov’s equation aab;c = gacλb + gbcλa. (5.7) From (5.5) it follows that a is non-degenerate and from (5.7), after a contraction with gab, that λ is a global gradient on M (in fact, of 1 2aabg ab). In practice, when asking which pairs (g′,∇′) are projectively related to some original pair (g,∇) on M , it is often easier to use (5.7) to attempt to find a and λ rather than (5.2) to find g′ and ψ. But then one must be able to convert back from a and λ to g′ and ψ. To do this one first assumes that such a non-degenerate tensor a and a 1-form λ are given on M and which together satisfy (5.7) and so one may define a symmetric non-degenerate type (2, 0) tensor a−1 on M which, at each m ∈M , is the inverse of a (aac(a−1)cb = δ b a ). Then raising and lowering indices on a and a−1 with g in the usual way, so that a−1 ac a cb = δ b a , one defines a global 1-form ψ on M by ψa = −a−1 ab λ b (and so λa = −aabψ b). It follows that ψ is a global gradient on M . To see this first differentiate the condition a−1 ac a cb = δ b a and use (5.7) to find, after a short calculation, a−1 ab;c = a−1 ac ψb + a−1 bc ψa. (5.8) Now define a smooth, symmetric connection ∇′′ on M by decreeing that, in any coordinate domain, its Christoffel symbols are given by Γ′′ where Γ′′a bc = Γa bc − ψagbc. (5.9) Then it is easily checked from (5.8), (5.9) and the equation λa = −aabψ b that ∇′′a = 0 and so if the everywhere non-degenerate tensor a is regarded as a metric on M , ∇′′ is its Levi-Civita connection. But a contraction of (5.9) over the indices a and c gives, from a standard formula for Christoffel symbols Γ′′a ba − Γa ba = ∂ ∂xb ( 1 2 ln ( \|det a\| \|det g\| )) = −ψb (5.10) and so ψ is seen to be a global gradient on M . (The authors have recently discovered that Sinyukov [20] had established the same result by a method involving the direct construction of the connection of the tensor a−1.) Writing χ for this potential function, so that ψ = dχ, one defines a metric g′ on M by g′ = e2χa. Then (5.8) can be used to show that g′ and ψ 10 G.S. Hall and D.P. Lonie satisfy (5.2) and hence that g′ is projectively related to g. It is easily checked that the tensors ψ and g′ thus found satisfy (5.5) and (5.6). [It is remarked that if ψ is replaced by λ and the metric g by the tensor a−1 in (5.9) the connection ∇′′′ thus defined satisfies ∇′′′a−1 = 0. In fact, ∇ = ∇′′ ⇔ ∇ = ∇′′′ ⇔ ∇ = ∇′.] Since any solution pair (g′, ψ) of (5.2) leads to a pair (a, λ) satisfying (5.7) it follows that all projectively related metrics g′ together with their attendant 1-forms ψ will be found if all pairs (a, λ) can be found and which, together, satisfy (5.7). For the finding of the general solution of (5.7) a useful result arises by applying the Ricci identity to a and using (5.7) to get (aab;cd − aab:dc =)aaeR e bcd + abeR e acd = gacλbd + gbcλad − gadλbc − gbdλac, (5.11) where λab = λa;b = λba. This leads to the following lemma (which is a special case of a more detailed result in [22]) and for which a definition is required. Suppose m ∈M , that F ∈ ΛmM and that the curvature tensor Riem of (M, g) satisfies Rab cdF cd = αF ab (α ∈ R) at m so that F is a (real) eigenvector of the map f̃ in (3.1). Then F is called a (real) eigenbivector of Riem at m with eigenvalue α. Lemma 5.1. Let (M, g) and (M, g′) be space-times with g and g′ projectively related. Suppose at m ∈ M that F ∈ ΛmM is a (real) eigenbivector of Riem of (M, g) with zero eigenvalue (so that F is in the kernel, ker f̃ , of f̃ in (3.1)). Then the blade of F (if F is simple) or each of the canonical pair of blades of F (if F is non-simple) is an eigenspace of the symmetric tensor λab with respect to g at m. (That is, if p ∧ q is in any of these blades (p, q ∈ TmM) there exists µ ∈ R such that for any k ∈ p ∧ q, λabk b = µgabk b). In particular, (i) suppose that, at m, the collection of all the blades of all simple members of ker f̃ and all the canonical blade pairs of all the non-simple members of ker f̃ (and which are each eigenspaces of λa;b) are such that they force TmM to be an eigenspace of λa;b. Then λa;b is proportional to gab at m. (ii) If condition (i) is satisfied at each m′ in some connected open neighbourhood U of m, then, on U and for some c ∈ R, (a) λab = cgab, (b) λdR d abc = 0, (c) aaeR e bcd + abeR e acd = 0. Proof. First contract (5.11) with F cd and use Ra bcdF cd = 0 to get gaeF ceλbc + gbeF ceλac − gaeF edλbd − gbeF edλad = 0, (5.12) which rearranges, after cancellation of a factor 2, as λacF c b + λbcF c a = 0, (5.13) where F a b = F acgcb is skew-self adjoint with respect to g. The argument following (3.8) applied to hab = λab then completes the proof of the first part of the lemma. Part (i) of the lemma then also follows if TmM is an eigenspace of λab with respect to g. [It is remarked here that, for such a symmetric tensor, the eigenspaces corresponding to distinct eigenvalues are orthogonal.] For part (ii) one notes that, from part (i), λab = σgab on U for some smooth function σ : U → R. The Ricci identity on λ then shows that λdR d abc = λa;bc − λa;cb = gabσ,c − gacσ,b, (5.14) where a comma denotes a partial derivative. Now for m′ ∈ U there exists 0 6= F ∈ ker f̃ at m′ and a contraction of (5.14) with F bc gives gab(F bcσ,c)− gac(F bcσ,b) = 0 (5.15) Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 11 from which it follows that gabF bcσ,c = 0 and so F abσ,b = 0. So either F is non-simple (and hence dσ = 0) at m′ or each member of ker f̃ is simple and the 1-form dσ is g-orthogonal to its blade. In the latter case, if dim(ker f̃) = 1, and ker f̃ is spanned by a single simple bivector the conditions of (i) are not satisfied, whereas, if dim(ker f̃) = 2, ker f̃ must be spanned by two simple bivectors whose blades intersect in a 1-dimensional subspace of Tm′M and which again results in condition (i) failing. If dim(ker f̃) = 3 it can be checked that either ker f̃ is spanned by three simple bivectors whose blades intersect in a 1-dimensional subspace of Tm′M (in which case dσ is forced to be zero) or ker f̃ is spanned by three simple bivectors the blades of whose duals intersect in a 1-dimensional subspace of Tm′M (and (i) fails). If dim ker f̃ ≥ 4, ker f̃ must contain a non-simple member and dσ = 0 must hold (see [2, p. 392]). It follows that dσ ≡ 0 on U and, since U is connected, the result (ii)(a) follows. The result (ii)(b) is then immediate from (5.14) and the result (ii)(c) follows from (5.11). � It is remarked that, in the construction of examples, the concept of local projective relatedness will be required. For a space-time (M, g) let U be a non-empty connected open subset of M and let g′ be some metric defined on U . Then g and g′ (or their respective Levi-Civita connections) will be said to be (locally) projectively related (on U) if the restriction of g to U is projectively related to g′ on U . 6 Projective structure and holonomy I The relationship between holonomy type and projective relatedness can now be given. Amongst the holonomy types studied are several which include significant solutions to Einstein’s field equations in general relativity (in addition to the vacuum solutions already discussed). For example, one has the non-vacuum pp-waves (type R3 and R8), the Gödel metric (R10), the Bertotti–Robinson metrics (R7) and the Einstein static universe (R13). For each holonomy type the general idea is, first, to determine the holonomy invariant distributions peculiar to that type and identify any covariantly constant or recurrent vector fields, second to link these vector fields to Riem using the infinitesimal holonomy structure, third to use Lemma 5.1 and Theorem 3.1 to find expressions for the 1-form λ and the tensor a and finally to use (5.7) to complete the procedure. It is convenient to break up the holonomy types into certain subcollections for easier handling. All metric and connection statements and index raising are understood to apply to the structures g and ∇ originally given on M . The following preliminary topological lemma, which is a generalised version of a result in [2], is useful in some of the theorems. Lemma 6.1. (i) Let X be a topological space and let A and B be disjoint subsets of X such that A and A∪B are open in X and A ∪B is dense in X. Suppose B = B1 ∪B2, with B1 and B2 disjoint, and intB ⊂ intB1 ∪ B2 where int denotes the interior operator in the manifold topology of M. Then X may be disjointly decomposed as X = A∪ intB1 ∪ intB2 ∪ J where J is the closed subset of X defined by the disjointness of the decomposition and A∪ intB1 ∪ intB2 is open and dense in X (that is, intJ = ∅). (ii) Let X be topological space and let A1, . . . , An be disjoint subsets of X such that A1 together with ∪i=k i=1Ai for k = 2, . . . , n are open subsets of X and such that ∪i=n i=1Ai is (open and) dense in X. Then X may be disjointly decomposed as X = A1 ∪ intA2 ∪ · · · ∪ intAn ∪K where K is the closed subset of X defined by the disjointness of the decomposition and A1 ∪ intA2 ∪ · · · ∪ intAn is open and dense in X (that is, intK = ∅). Proof. (i) Suppose intJ 6= ∅ and let U be a non-empty open subset of X with U ⊂ J . Then U ∩ (A ∪ B) is open and non-empty (since A ∪ B is dense in M) but U is disjoint from 12 G.S. Hall and D.P. Lonie A, intB1 and intB2. Then U ∩ B is open and non-empty and hence so is U ∩ intB. But ∅ 6= U ∩ intB ⊂ U ∩ (intB1 ∪ B2) = U ∩ B2 and so U ∩ B2 contains the non-empty, open subset U ∩ intB of M from which the contradiction that U ∩ intB2 6= ∅ follows. Thus, U = ∅, intJ = ∅ and the result follows. (ii) Suppose U is an non-empty open subset ofX with U ⊂ K. Then U∩(∪i=n i=1Ai) is not empty but U is disjoint from A1, intA2, . . . , intAn. It follows that if the open set U∩(A1∪A2)(= U∩A2) is non-empty, then U ∩A2 is non-empty and open and gives the contradiction that U ∩ intA2 is (open and) non-empty. Thus U is disjoint from A2. Continuing this sequence one finally gets the contradiction that U is disjoint from each Ai and hence from the open dense set ∪i=n i=1Ai. Thus U = ∅ and the result follows. � Theorem 6.1. Let (M, g) and (M, g′) be space-times with (M, g) non-flat. Suppose that (M, g) is of holonomy type R2, R3 or R4 and that ∇ and ∇′ are projectively related. Then ∇ = ∇′ on M . Proof. Suppose first that (M, g) has holonomy type R2 and, by the non-flat condition, let U be the open dense subset of M on which Riem is non-zero. Then, for m ∈ U (see Section 4), there exists a connected and simply connected open neighbourhood V ⊂ U of m and two orthogonal smooth unit spacelike vector fields X and Y on V , spanning a holonomy invariant distribution at each point of V and which are covariantly constant on V . The Ricci identity then reveals that Ra bcdX d = Ra bcdY d = 0 on V and so Riem takes the curvature class D form (3.3) where F is a smooth simple timelike bivector field on V whose blade is orthogonal to the 2-spaces X(m) ∧ Y (m) at each m ∈ V . Then at each m ∈ V one may construct a null tetrad l, n, x, y, based on the holonomy invariant subspaces, so that x = X(m) and y = Y (m) and then RabcdG cd = 0 is satisfied by at m for G = l ∧ x, l ∧ y, n ∧ x, n ∧ y and x ∧ y. It follows that the conditions of Lemma 5.1 are satisfied on (some possibly reduced version of) V and hence that the conclusions (a), (b) and (c) of this lemma hold. Then part (c) of Lemma 5.1 and Theorem 3.1(i) show that, on V , aab = φgab + µXaXb + νYaYb + ρ(XaYb + YaXb) (6.1) for functions φ, µ, ν and ρ. Also the smoothness of the functions aabX aXb (= φ+ µ), aabY aY b (= φ+ ν), aabX aY b (= ρ) and gabaab (= 4φ+ µ+ ν) reveal the smoothness of the functions φ, µ, ν and ρ on V . Now one substitutes (6.1) into (5.7) to get gabφ,c +XaXbµ,c + YaYbν,c + (XaYb + YaXb)ρ,c = gacλb + gbcλa. (6.2) A contraction of (6.2) successively with laxb and nayb at any m ∈ V shows that λax a = λal a = λay a = λan a = 0. Thus the 1-form λ is zero on V . It follows from (5.6) that the 1-form ψ is identically zero on V and hence, from (5.1) that ∇ = ∇′ on V , hence on U and thus on M . This completes the proof for holonomy type R2 and the proofs for the holonomy types R3 and R4 are similar; for type R3 one has covariantly constant, orthogonal, null and spacelike vector fields L and Y , respectively, on V and the holonomy invariant distributions can be used construct vector fields N and X on (a possibly reduced) V such that L,N,X and Y give a null tetrad at each point of V from which the proof follows in a similar way to that of the R2 type. For type R4 one has covariantly constant null vector fields L and N on V which one can choose to satisfy gabL aN b = 1. Again one easily achieves ∇ = ∇′ on M . � Theorem 6.2. Let (M, g) and (M, g′) be space-times with (M, g) non-flat and of holonomy type R7. Suppose that ∇ and ∇′ are projectively related. Then ∇ = ∇′ on M . Proof. Let U be the open dense subset of M on which Riem does not vanish. If m ∈ U there exists an open, connected and simply connected neighbourhood V ⊂ U and null properly Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 13 recurrent vector fields L and N scaled so that LaNa = 1 on V (noting that this scaling will not affect their recurrence property). Then ∇L = L⊗ P and ∇N = −N ⊗ P for some smooth 1-form field P on V . The simple bivector field F ≡ L ∧ N , (Fab ≡ 2L[aNb]), then satisfies ∇F = 0 and its (simple) dual bivector field also satisfies ∇ ∗ F = 0. The blades of F and ∗ F span, respectively, the timelike and spacelike holonomy distributions on V and reducing V , if necessary, one may write ∗ F ab = 2Xa[Yb] for smooth unit orthogonal spacelike vector fields X and Y on V and which satisfy XaXa;b = Y aYa;b = 0. The condition ∇ ∗ F = 0 on V shows that ∇X = Y ⊗Q and ∇Y = −X⊗Q on V for some smooth 1-form Q on V . At each m ∈ V , L(m), N(m), X(m) and Y (m) give a null tetrad l, n, x, y at m. Now at each m ∈ U , Table 1 and a consideration of the infinitesimal holonomy structure of (M, g) show that the curvature class of Riem at m is either D or B, taking the form (3.2) if it is B and (3.3) with F either timelike or spacelike if it is D. Let B, Ds and Dt denote the subsets of U consisting of those points at which the curvature class is, respectively, B, D with F spacelike or D with F timelike. Then one may decompose M , disjointly, as M = B∪Ds ∪Dt ∪J = B∪ intDs ∪ intDt ∪K where, by the “rank” theorem (see e.g. [2]), B is open in M and J and K are closed subsets of M defined by the disjointness of the decomposition and J satisfies intJ=∅. If intDs 6= ∅ let m ∈ intDs and m ∈ V ⊂ intDs with V as above. Then (3.3) holds on V with F = X ∧ Y and α smooth on V . It is then clear that, at m, l ∧ x, l ∧ y, n ∧ x, n ∧ y and l ∧ n are in ker f̃ and so, from Lemma 5.1(ii)(a), TmM is an eigenspace of ∇λ and so ∇λ = cg on V . Then Lemma 5.1(ii)(c) and Theorem 3.1(i) show that aab = φgab + µLaLb + νNaNb + ρ(LaNb +NaLb) (6.3) for functions φ, µ, ν, and ρ on V which are smooth since aabL aLb, aabN aN b, aabL aN b and gabaab are. Noting that L(aNb) is covariantly constant on V , a substitution of (6.3) into (5.7) gives, on V , gabφ,c + LaLbµ,c + 2µLaLbPc +NaNbν,c − 2νNaNbPc + (LaNb +NaLb)ρ,c (6.4) = gacλb + gbcλa. Successive contractions of (6.4) with LaXb and NaY b then show that λaL a = λaN a = λaX a = λaY a = 0 on V . Thus λ vanishes on V and so on intDs. If intDt 6= ∅ a similar argument using (3.3) with F = L ∧ N can be used to show that λ vanishes on intDt. If B 6= ∅, Lemma 5.1(ii)(b) shows that λ vanishes on B since no non-trivial solutions for λ of the equation displayed there exist at points of curvature class B. So one has achieved the situation that M = B ∪ D ∪ J with int J = ∅ and B and B ∪ D open in M (again by an application of the “rank” theorem) and with λ vanishing on the open subset B ∪ intDs ∪ intDt. That this open subset is a dense subset of M follows from Lemma 6.1(i) by the following argument. Since D ≡ Ds ∪Dt then, from elementary topology, intDs ∪ intDt ⊂ intD. That the reverse inclusion is true in this case follows from noting that it is trivially true if intD = ∅ whilst if intD 6= ∅ one writes equation (3.3) for Riem on some open neighbourhood of any point m in the submanifold intD with α and F smooth (as one can) and then defines the smooth map θ from this neighbourhood to R by θ(m) = (F abFab)(m) (so that θ(m) > 0 if m ∈ Ds and θ(m) < 0 if m ∈ Dt). It is easily seen now that m is either in intDs or intDt. Thus intD = intDs ∪ intDt and the conditions of Lemma 6.1(i) are satisfied. So λ vanishes on an open dense subset of M and hence on M . It follows that ∇ = ∇′ on M . � Theorem 6.3. Let (M, g) and (M, g′) be space-times with (M, g) non-flat and of holonomy type R10, R11 or R13 and with Riem of curvature class C at each point of an open, dense subset U of M . Suppose that ∇ and ∇′ are projectively related. Then ∇ = ∇′ on M . 14 G.S. Hall and D.P. Lonie Proof. Suppose that (M, g) is of holonomy type R10 (the arguments for types R11 and R13 are similar). For any m ∈ U there exists an open, connected and simply connected neighbourhood V ⊂ U of m and a unit spacelike covariantly constant vector field X on V . Then one may choose an orthonormal tetrad u, x, y, z at m such that X(m) = x and, from the Ricci identity on X, Ra bcdX d = 0 on V , and so x∧ u, x∧ y and x∧ z are in ker f̃ at m. As before, one sees that the conditions and conclusions of Lemma 5.1 hold at each point of V and so Theorem 3.1(ii) gives aab = φgab + µXaXb (6.5) on V for functions φ and µ which are smooth since aabX aXb and gabaab are. On substituting (6.5) into (5.7) one finds gabφ,c +XaXbµ,c = gacλb + gbcλa (6.6) and successive contractions of (6.6) with xayb and uazb show that λ vanishes at m, hence on V and so on M . It follows that ∇ = ∇′ on M . � Theorem 6.4. Let (M, g) and (M, g′) be space-times with (M, g) non-flat and of holonomy type R6, R8 or R12. Suppose that ∇ and ∇′ are projectively related. Then ∇ = ∇′ on M . Proof. In each case let U be the open dense subset of M on which Riem does not vanish. Then any m ∈ U admits an open, connected and simply connected neighbourhood V ⊂ U on which are defined orthogonal vector fields L, X and Y , with L null and either recurrent or covariantly constant (depending on the holonomy type) and X and Y unit spacelike vector fields and which together span a 3-dimensional null holonomy invariant distribution on V . By reducing V , if necessary, one may assume the existence of a null vector field N which along with L, X and Y span a null tetrad at each point of V . Now for holonomy type R6 one may take L recurrent and Y covariantly constant. The curvature class at m is either C or D and if it is D, equation (3.3) holds with F = l ∧ x or F = l ∧ n + al ∧ x (a ∈ R). Let the subset of points of M at which the first of these class D possibilities holds be denoted by Dn (null) and at which the second holds by Dnn (non-null). Then let D ≡ Dn∪Dnn. Let the subset of points of M at which at which the curvature class is C be denoted by C. Then C is open in M and U = C ∪D. M admits the disjoint decomposition M = C ∪Dn ∪Dnn ∪ J where J is the closed subset of M determined by the disjointness of the decomposition and intJ = ∅. It is clear that, whatever the curvature type, the conditions and conclusions of Lemma 5.1 hold. If C 6= ∅ and m ∈ C then one may arrange in the previous paragraph that m ∈ V ⊂ U and that, on V , aab = φgab + µYaYb (6.7) for functions φ and µ. The proof now proceeds as for Theorem 6.3 and one obtains ∇ = ∇′ on C. If intDn 6= ∅ then with V as above and m ∈ V ⊂ intDn, Lemma 5.1 and Theorem 3.1(i) give, on V , aab = φgab + µLaLb + νYaYb + ρ(LaYb + YaLb) (6.8) for functions φ, µ, ν and ρ which are smooth since aabX aXb, aabY aY b, aabN aN b and aabN aY b are. On substituting (6.8) into (5.7) and contracting, successively, with XaY b, LaLb and NaXb one again finds that λ = 0 on V and hence on intDn and so ∇ = ∇′ on intDn. (This argument includes within it the curvature class C case above; one simply sets µ and ρ to zero in (6.8).) If intDnn 6= ∅ and m ∈ V ⊂ intDnn with V as above, Riem takes the form (3.3) with F chosen as F = L∧N + σL∧X for a smooth function σ, on V . Now define smooth vector fields Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 15 X ′ = X−σL and N ′ = N +σX− σ2 2 L on V , noting that L, N ′, X ′ and Y still give rise to a null tetrad in the obvious way at each point of V and that now F = L∧N ′ on V . Again Lemma 5.1 applies and Theorem 3.1(i) gives aab = φgab + µX ′ aX ′ b + νYaYb + ρ(X ′ aYb + YaX ′ b), (6.9) where, as before, φ, µ, ν and ρ are smooth functions on V . On substituting (6.9) into (5.7) (and noting that, since L is recurrent, LaX ′ a;b = 0) successive contractions with N ′aN ′b, LaX ′b and LaY b again reveal that ∇ = ∇′ on intDnn. So ∇ = ∇′ on W̃ ≡ C ∪ intDn ∪ intDnn. That the open set W̃ is dense in M follows from Lemma 6.1(i). To see this let K ≡ M \ W̃ and let W be a non-empty open subset of K. Then W is disjoint from C, intDn and intDnn but the open subset W ∩ (C ∪D) = W ∩ U 6= ∅ since U is dense in M . It follows that W ∩D and hence W ∩ intD are open and non-empty. Now, as in the previous theorem, consider the map arising from the bivector F in (3.3) on the (non-empty) open submanifold intD (and note that for points in Dn (where F null) its value is zero whereas for points in Dnn its value is negative). A consideration of the open subset of intD given by the inverse image, under f , of the negative reals shows that intD ⊂ intDnn∪Dn. The result now follows from Lemma 6.1(i) and λ vanishes on W̃ and hence on M . It follows that ∇ = ∇′ and this completes the proof when the holonomy type is R6. If (M, g) has holonomy type R8 and if m ∈ U the curvature class of Riem at m is either D (equation (3.3) with F null) or C. A neighbourhood V and vector fields L, N , X and Y , can be established, as in the above paragraphs, with L covariantly constant. Lemma 1 again applies and Theorem 3.1, parts (i) and (ii), then lead to a straightforward proof that λ = 0 on (and in an obvious notation) the open subset C and on intD. (For class D, useful contractions of (5.7) are with LaLb, LaXb, LaY b, Y aY b and LaN b and in the class C case, with LaXb, LaY b and XaN b.) That the open subset C ∪ intD is dense in M then follows from Lemma 6.1(ii) since C ∪D is open and dense in M . Thus λ = 0 and ∇ = ∇′ on M . If (M, g) has holonomy type R12 then for m ∈ U one may choose a neighbourhood V of m and vector fields L, N , X and Y as above with the null vector field L recurrent. Table 1 shows that l∧x, l∧y and l∧n+ω−1x∧y lie in ker f̃ and so Lemma 5.1 again applies. If m ∈ U the possible curvature classes for Riem at m are D with F null in (3.3) (since the only linear combinations of the members of the bivector algebra for type R12 which are simple are null), C (with Ra bcdl d = 0 at m) or A. One then decomposes M ′, in an obvious notation, as M ′ = A ∪ intC ∪ intD ∪ J (recalling from Section 3 that A is an open subset of M). The conclusion (b) of Lemma 5.1(ii) then immediately shows that λ = 0 on A and procedures similar to those already given reveal that λ = 0 on intC and intD (for the latter contract (5.7) with LaLb, LaXb, LaY b, Y aY b and NaLb). Since A, A ∪C and A ∪C ∪D are open in M , with the latter dense, Lemma 6.1(ii) completes the proof that λ = 0 and ∇ = ∇′ on M . � In summary, if (M, g) is non-flat and if, in addition, it is either of holonomy type R2, R3, R4, R6, R7, R8 or R12 or of holonomy type R10, R11 or R13 and of curvature class C over some open dense subset of M then if g′ is another metric on M projectively related to g, ∇ = ∇′ on M . One can say a little more here because in each of the above holonomy types (including the curvature class C clause applied to the types R10, R11 and R13) the condition that ∇ and ∇′ are projectively related leads to ∇ = ∇′ and hence to the condition that the holonomy groups of (M, g) and (M, g′) are the same. It is then easy to write down a simple relation between the metrics g and g′ [15, 2]. For example, suppose either that M is simply connected or that one is working over a connected and simply connected open region V of M (on which the holonomy type is the same as that of M). Then if this common holonomy type is R2, R3 or R4, one gets a relation of the form g′ab = φgab + µpapb + νqaqb + ρ(paqb + qapb), (6.10) 16 G.S. Hall and D.P. Lonie where φ, µ, ν and ρ are constants (subject only to a non-degeneracy condition and signature requirements) and where p and q are vector fields on M (or V ) spanning the 2-dimensional vector space of covariantly constant vector fields. For holonomy types R6 and R8 or types R10, R11 or R13 (with the above clauses attached) one gets g′ab = φgab + µkakb, (6.11) where φ and µ are constants (again subject to non-degeneracy and signature requirements) and k is a vector field on M (or V ) spanning the vector space of covariantly constant vector fields. For holonomy type R7 one gets g′ab = φgab + µ(lanb + nalb), (6.12) where again φ and µ are constants (again subject to non-degeneracy and signature requirements) and l and n are null, properly recurrent vector fields on M (or V ). For holonomy type R12 one gets g′ab = φgab with φ constant. [It is remarked here that for holonomy types R10, R11 or R13, even without the curvature class C clause, if it is given that ∇ = ∇′ then (6.11) follows for constant φ and µ. But, in this latter case, projective relatedness without this clause does not force the condition ∇ = ∇′ as will be seen later.] It is clear from these results that if g and g′ are projectively related metrics they may have different signatures. 7 Projective structure and holonomy II In this section the holonomy types (and special cases of holonomy types) omitted in the last section will be discussed. Theorem 6.3 dealt in detail with the projective relatedness problem for the (non-flat) holonomy types R10, R11 and R13 when the curvature rank of the map f in (3.1) (and which, for each of these types, is necessarily at most 3) is equal to 2 or 3 (curvature class C) at each point of an open, dense subset of M . One can now consider the situation for these holonomy types when the curvature rank is 1 (curvature class D) at each point of some open, dense subset U of M . For any of these holonomy types define subsets M1 ≡ {m ∈M : λ vanishes on some open neighbourhood of m} and M2 ≡ {m ∈ M : λ does not vanish on any open neighbourhood of m} = M \M1. Then one has the disjoint decomposition of M given by M = M1 ∪M2 with M1 open in M . Let J = M \ U so that J is closed in M and int J = ∅ (since U is open and dense in M). Next, if Dnn is the subset of points of M at which the bivector F in (3.3) is either spacelike or timelike and Dn the subset of points where it is null then U = Dnn∪Dn with Dnn an open subset ofM (by a consideration of the map associated with the smooth function FabF ab on U and described earlier). One then gets a disjoint decomposition M = M1 ∪ (M2 ∩Dnn) ∪ int (M2 ∩Dn) ∪ K. It will be seen, as the proof progresses, that in all cases M2 ∩Dnn is also open in M and hence K, which is defined by the disjointness of the decomposition, is closed in M . It then follows that, with the openness of M2 ∩Dnn assumed, if W is a non-empty open subset of K, W ∩M1 = W ∩ (M2 ∩Dnn) = W ∩ int (M2 ∩Dn) = ∅. But the non-empty open set W ∩ U = W ∩ (Dn ∪ Dnn) = W ∩ Dn (since W ⊂ K and K ∩Dnn = ∅)= W ∩ (M2 ∩Dn). Hence W ∩ int (M2 ∩Dn) 6= ∅ and this contradiction shows that intK = ∅. Thus, considering the open dense subset M1∪ (M2∩Dnn)∪ int (M2∩Dn) of M one sees that since λ = 0 on M1 the interesting parts of M in this context are the open subsets M2 ∩Dnn and int (M2 ∩Dn) (and, in fact, Dn = ∅ in the R13 case). In fact, it will turn out that, in all cases, int (M2 ∩Dn) = ∅. So let (M, g) be a (non-flat) space-time of holonomy type R10, R11 or R13 whose curvature is of class D on some open dense subset U of M . Thus (3.3) holds on U (and Riem vanishes on M \ U). Let m ∈ U and let V ⊂ U be a connected, simply connected, open neighbourhood of m on which is defined a metric g′, (locally) projectively related to g on V , together with Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 17 an associated pair (a, λ). The conditions imposed on (M, g) together with (3.3) show that the conditions and hence the conclusions of Lemma 5.1 hold on U and hence on V . Thus ∇λ = cg holds on V for some constant c. Now suppose that the holonomy type of (M, g) is R11. Then V can be chosen so that it admits a covariantly constant, null vector field l, so that, from the Ricci identity, Ra bcdl d = 0. Also, from the curvature class D condition, V (after a possible reduction, if necessary) admits another (smooth) nowhere-zero vector field p such that l(m′) and p(m′) are independent members of Tm′M for each m′ ∈ V and such that Ra bcdp d = 0 holds on V . Condition (b) of Lemma 5.1 and the curvature class D condition then show that, on V , λa = σla + ρpa (7.1) and so, since λa;b = cgab, one gets ρpa;b + paρ,b + laσ,b = cgab. (7.2) Also, part (c) of Lemma 5.1 and Theorem 3.1(i) give aab = φgab + βlalb + γpapb + δ(lapb + palb) (7.3) for smooth functions φ, β, γ and δ on V . Now there exists a nowhere-null, nowhere-zero vector field q on (a possibly reduced) V which is orthogonal to l and p and is such that qaqa is constant (and hence qaqa;b = 0) on V . Then from (7.3), aabq b = φqa on V . On covariantly differentiating this last equation and using (5.7) after a contraction with qa one finds that φ,a = 0 and so φ is constant on V (and non-zero since a is non-degenerate). Now the simple bivector F in (3.3) satisfies Fabl b = 0 and so is either null or spacelike at m ∈ U , that is, either m ∈ Dn or m ∈ Dnn. Recalling the decomposition of M given above assume int(M2 ∩ Dn) is not empty and let m ∈ V ⊂ int(M2 ∩ Dn) for (some possibly reduced) open neighbourhood V as before. Then p may be chosen a unit spacelike vector field on V orthogonal to l on V and (7.3) shows that aabl b = φla which, after differentiation and making use of (5.7) and the constancy of φ and l immediately shows that λ vanishes on V . This contradicts the fact that m ∈M2. Thus, int(M2 ∩Dn) is empty. If F is spacelike at m, that is, m ∈ Dnn, and assuming that M2 ∩ Dnn is not empty let m ∈ (M2 ∩Dnn) and choose V such that m ∈ V ⊂ Dnn (since Dnn is open in M). Then one may choose p so that p is null and lapa = 1 on V . Then (7.2) on contraction first with la and then with pa (using papa;b = 0 since p is null on V ) yields cla = ρ,a, cpa = σ,a (7.4) and then (7.2) becomes ρpa;b = cTab, (Tab ≡ gab − lapb − palb). (7.5) Next, differentiate (7.3) and use (7.1) and (5.7) to get gac(σlb + ρpb) + gbc(σla + ρpa) (7.6) = lalbβ,c + papbγ,c + γpa;cpb + γpapb;c + (lapb + palb)δ,c + δ(lapb;c + pa;clb). On contracting this successively with lalb, papb and lapb (using lapa;b = 0) one gets 2ρla = γ,a, 2σpa = β,a, σla + ρpa = δ,a. (7.7) If one finally contracts (7.6) first with la and then with pa and use is made of (7.7) one easily finds γpa;b = ρTab, δpa;b = σTab. (7.8) 18 G.S. Hall and D.P. Lonie The condition m ∈ M2 implies that pa;b(m) 6= 0 because, otherwise, (7.5) shows that c = 0 and then (7.4) reveals that ρ and σ are constant on V . Then (7.8) shows that ρ and σ vanish at m and hence on V . Finally, (7.1) implies that λ vanishes on V and a contradiction follows. It thus follows that pa;b does not vanish at m and hence on some open neighbourhood of m (taken to be V ). Further, if λ(m) = 0 then (7.1) shows that σ(m) = ρ(m) = 0 and so, from (7.5), c = 0. Then (7.4) reveals that σ and ρ are constant and hence zero on V and hence, from (7.1), the contradiction that λ ≡ 0 on V . Thus λ(m) 6= 0 and so one may assume that V is chosen (after a possible reduction) so that λ and pa;b are nowhere zero on V . But this implies that V ⊂ M2 and hence that V ⊂ (M2∩Dnn). From this it follows that M2∩Dnn is open in M (as remarked above). Continuing with this case, if pa;b is not a multiple of Tab at some m′ ∈ V then (7.8) shows, since T (m′) 6= 0, that ρ(m′) = σ(m′) = 0 and then (7.1) gives the contradiction that λ(m′) = 0. So pa;b is a non-zero multiple of Tab at each point of V . It follows that the covector fields associated with both l and p are closed 1-forms on V and hence are gradients on (a possibly reduced) V , say of the (smooth) functions u and v, respectively. In addition, l and p are easily checked to have zero Lie bracket and so span an integrable distribution (again on a possibly reduced V ). It can then be shown by using standard techniques that if one chooses x and y coordinates on that part of the 2-dimensional submanifold u = v = 0 contained in (a possibly reduced) V and chooses the other coordinates as parameters along the integral curves of l and p (with x and y constant along these curves) the parameter coordinates may be taken as u and v and l = ∂/∂v and p = ∂/∂u. Then with the coordinates u, v, x, y the metric g takes the form ds2 = 2dudv + gαβdx αdxβ (gαβ = Tαβ), (7.9) where Greek indices take the values 3 and 4 and the second (bracketed) equation in (7.5) has been used. Now suppose that the constant c 6= 0. Then (7.4) shows that one may use the translational freedom in the coordinate u to arrange that ρ = cu. If c = 0 then the first equation in (7.5) together with the non-vanishing of pa;b on V implies ρ = 0 on V and the second equation in (7.4) shows that σ is a non-zero constant on V . Then (7.7) and (7.8) reveal that δ is a linear function of u which is nowhere zero on V . A translation of the u coordinate then ensures that δ = σu (and u is nowhere zero) on V . Thus whether c is zero or not, one can achieve the result that upa;b = Tab. Denoting the Christoffel symbols arising from ∇ by Γ and noting that since l = ∂/∂v is a Killing vector field on V , the functions gαβ(= Tαβ) are independent of v, one calculates that −uΓ2 αβ = gαβ . This, together with upa;b = Tab and an integration then leads to gαβ = u2hαβ for functions hαβ which are independent of u and v. Thus, on V , the original metric is ds2 = 2dudv + u2hαβ ( x3, x4 ) dxαdxβ. (7.10) The next step is, recalling the coordinate scalings above, to write ρ = cu and, from (7.4), σ = cv+e1 and substitute into (7.7) to get β = cv2+2e1v+e2, γ = cu2+e3 and δ = e1u+cuv+e4 for constants e1, e2, e3 and e4. Then (7.5) and (7.8) act as consistency checks on these constants and give e3 = e4 = 0. The equation (7.7) now gives the Sinyukov tensor in (7.3) as aab = φ ( gab + (cv2 + 2e1v + e2)lalb + cu2papb + (e1u+ cuv)(lapb + palb) ) . (7.11) This equation can then be inverted following the techniques described in Section 5. First, with the above values for ρ and σ one finds λ from (7.1) and, from (7.11), ψ can be calculated from the equation ψa = −a−1 ab λ b given in Section 5. Then the potential, χ, of the 1-form ψ can be found. Finally the required metric is then given, on V , by g′ab = e2χaab. This calculation was done using Maple and results in ψ = dχ, χ = 1 2 lnF, F = κ4 [ 1 + 2cuv + 2e1u+ (e21 − ce2)u2 ]−1 (7.12) Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 19 and for the metric g′, projectively related to g, ds′2 = κFg − κ−3F 2 [ (cv2 + 2e1v + e2)du2 + cu2dv2 + 2u(cv + e1 + (e21 − ce2)u)dudv ] (7.13) for a positive constant κ = φ−1. It is noted here that if λa(m) is proportional to l(m) then it follows from (7.1) that ρ(m) = 0 and from (7.5) that c = 0. Then (7.4) reveals that ρ = 0 and that σ is constant on V and so λa is a constant multiple of la = gabl b on V . Conversely, since ∇λ = cg, this last condition implies that c = 0. Thus the special case c = 0 corresponds to the condition that λa is a (non-zero) constant multiple of la on V and then (7.11), (7.12) and (7.13) simplify. For this special case it follows from (7.12) that ψa = r(u)la on V for some (smooth) function r on V . Thus, for any metric g′, projectively related to g on V and with Levi-Civita connection ∇′, one finds from (5.1) that la\|b = −2r(u)lalb, where a stroke denotes a ∇′-covariant derivative. Then the nowhere-zero vector field l′ ≡ e ∫ 2rl on V is null with respect to g′ (from (7.13) with c = 0) and covariantly constant with respect to ∇′. It is, in fact, true that (V, g′) (that is, ∇′) is also of holonomy type R11. This follows from Table 1 since (V, g′) admits a nowhere zero null covariantly constant vector field and so must be of holonomy type R3, R4, R8 or R11 and the first three of these are ruled out because they would, from Theorems 6.1 and 6.4, force ∇ = ∇′. [That ∇′ cannot be flat over some non-empty open subset V ′ of V follows since g′ would then be of constant curvature and hence so would g [16], contradicting the curvature class D condition on (V, g).] The space-time (V, g′) is also of curvature class D on V since, otherwise, it would be of curvature class C over some non-empty open subset V ′ of V . In this case, Theorem 6.3 would show that λ = 0 on V ′ and a contradiction follows. Now let (M, g) be a space-time of holonomy type R10 or R13 which is (non-flat and) of curvature class D, that is the curvature rank equals 1 over some open dense subset U of M and Riem vanishes on M \ U . Let g′ be a metric on some connected open neighbourhood V ⊂ U of m ∈ U which is projectively related to g on V . On V , Riem takes the form (3.3) for some nowhere zero bivector F on V . If the holonomy type is R13 this bivector F is necessarily spacelike and so the subset Dn in the above decomposition of M is empty. Then V may be chosen to admit a covariantly constant, unit timelike vector field u (so that, from the Ricci identity, Ra bcdu d = 0 on V ) and a unit (smooth) spacelike vector field w which is orthogonal to u on V and also satisfies Ra bcdw d = 0 on V . On V one has the useful relations uaua;b = wawa;b = uawa;b = 0. If, however, the holonomy type of (M, g) is R10, V may be chosen so as to admit a covariantly constant unit spacelike vector field (which for later convenience will also be labelled u) and which satisfies Ra bcdu d = 0 and another nowhere-zero vector field (similarly labelled w) which is orthogonal to u and satisfies Ra bcdw d = 0 on V but whose nature (spacelike, timelike or null) may vary. It follows that in the R10 case, each of Dn and Dnn may be non-empty. These remarks permit the totality of possibilities for the types R10 and R13, with one exception, to be taken together (the exceptional case being for type R10 and the subset Dn). The proof is, in fact, very similar to that already given for the holonomy type R11 and so it will be described only briefly. For m ∈ U choose an open, connected and simply connected neighbourhood V of m. Then one has, from Lemma 5.1, λa;b = cgab for some constant c and so, as before, λa = σua + ρwa, ρwa;b + waρ,b + uaσ,b = cgab (7.14) and aab = φgab + βuaub + γwawb + δ(uawb + waub) (7.15) 20 G.S. Hall and D.P. Lonie for functions σ, ρ, φ, β, γ and δ on V . Whichever subcase is chosen an argument similar to that given in the holonomy R11 case leads to the conclusion that φ is constant. Now suppose the holonomy type is R10, that int (M2∩Dn) is non-empty and with V as usual choose m ∈ V ⊂ int (M2∩Dn). On V , u is unit spacelike and covariantly constant and w is null. Then the second of (7.14), on contraction with wa and ua, gives c = 0, σ is constant on V and (ρw)a;b = 0. Thus λ is covariantly constant on V and ρ(m) = 0. [If ρ(m) 6= 0 then ρ would not vanish over some neighbourhood of m and on that neighbourhood ρw would be a nowhere-zero, nowhere-null covariantly constant vector field on V with ρw(m) and u(m) independent members of TmM at each m ∈ V . This would reduce the holonomy type in this neighbourhood to R3 and then, from Theorem 6.1, one finds that λ vanishes on this neighbourhood of m contradicting the fact that m ∈ M2.] Since λ (and u) are covariantly constant on V with λ proportional to u at m it follows that λ is proportional to u on V and hence, from (7.14), that ρ is zero on V . Finally a substitution of (7.15) into (5.7) and a contraction with uawb gives σ = 0 on V and hence, from (7.14) that λ ≡ 0 on V . This contradiction to the statement m ∈ M2 shows that int(M2 ∩Dn) = ∅. The other cases can be handled together. One assumes that M2 ∩ Dnn is non-empty and chooses m ∈ (M2∩Dnn) and the usual neighbourhood V such that m ∈ V ⊂ Dnn. To deal with both cases simultaneously one writes, for the vector fields u and w on V , uaua = ε1, wawa = ε2 (and uawa = 0) with ε1 and ε2 equal to either ±1. Thus the R13 type requires ε1 = −1 and ε2 = 1 whilst the R10 type requires ε1 = 1 and ε2 = ±1. Equations (7.14) and (7.15) still hold and contractions of the second in (7.14) with ua and wa give ε1σ,a = cua, ε2ρ,a = cwa, ρwa;b = cTab (Tab = gab − ε1uaub − ε2wawb). (7.16) Then substituting (7.15) into (5.7) and contracting successively with uaub, wawb and uawb and then with ua and wa gives 2ε1σua = β,a, 2ε2ρwa = γ,a, ε2ρua + ε1σwa = ε1ε2δ,a (7.17) and γwa;b = ρTab, δwa;b = σTab. (7.18) As before one can now show, since λ ∈ M2, that λ and wa;b are nowhere zero on V and hence V ⊂ M2 ∩ Dnn (so that M2 ∩ Dnn is open in M). Also, on V , wa;b is a non-zero multiple of Tab. Thus, V may be chosen so that the 1-forms associated with u and w through g are global gradients on V , say u = ε1dt and w = ε2dz for functions t and z on V . So reducing V , if necessary, to a coordinate domain with coordinates t, z, x3, x4 such that g takes the form ds2 = ε1dt 2 + ε2dz 2 + z2hαβ(x3, x4)dxαdxβ , (7.19) where the functions hαβ are independent of t and z. The equations (7.14) and (7.17) may be integrated to give aab = φ ( gab + (ct2 + 2c2t+ c3)uaub + cz2wawb + (ctz + c2z)(uawb + waub) ) , (7.20) where c2 and c3 are constants. Inverting the pair (a, λ) to obtain the corresponding pair (g′, ψ) leads to ψ = dχ, χ = 1 2 lnF, F = κ4 [ 1 + ε2(c+ ε1(c3c− c22))z 2 + ε1(ct2 + 2c2t+ c3) ]−1 (7.21) Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 21 corresponding to the metric g′ projectively related to g and given by g′ = κFg − κ−3F 2 ( (ct2 + 2c2t+ c3 + ε2(cc3 − c22)z 2)dt2 +(c+ ε1(cc3 − c22))z 2dz2 + 2ε1ε2(ct+ c2)zdtdz ) , (7.22) where κ = φ−1 is a positive constant. In summary, for space-times (M, g) of holonomy types R10, R11 and R13 and which are of curvature class D over some open dense subset U of M , int(M2∩Dn) = ∅ and one may identify an open dense subset of M such that each point m of this subset lies in a neighbourhood where either λ = 0 (when m ∈ M1) or where λ is nowhere zero and in which the situation can be completely resolved (in the open subset M2 ∩ Dnn). It is interesting to remark at this point that more can be said in the cases when Riem is nowhere zero on M and the nature (timelike, spacelike or null) of the bivector F in (3.3) is constant on M . To see this, suppose that such is the case and suppose that F is non-null (and hence always timelike or spacelike on M). Then, in the previous notation, M = M1 ∪M2 = Dnn with M1 open in M . But then the previous work shows that M2 = M2 ∩Dnn is also open in M . Thus, since M is connected, one of M1 and M2 is empty and so either M = M1 or M = M2. It follows that, in this case, either λ = 0 on M or λ is nowhere vanishing on M . In the similar situation, but with F null, M = Dn and the work above shows that λ = 0 on M . Although the class D assumption for these holonomy types was made over an open dense subset of M it is now seen that this is, in fact, a complete solution for holonomy types R10, R11 and R13. For if (M, g) is of any of these holonomy types, then either it is of curvature class C over an open dense subset of M , in which case, ∇ = ∇′ on M , or the curvature class is D over some subset E of M . Now the subset J of M over which Riem vanishes is closed and satisfies intJ = ∅. Also, the subset E ∪ J is closed (by the rank theorem on Riem since M \ (E ∪ J) is exactly the open subset of M on which Riem has curvature rank > 1) and if intE = ∅ it is easily checked that int (E ∪ J) = ∅ and so the curvature class is C over the open dense subset M \ (E ∪ J) of M . Then Theorem 6.3 gives ∇ = ∇′ on M . If intE 6= ∅ let E′ = intE and proceed as above on the (connected) components of E′ with metric g. Where a space-time (M, g) has holonomy type R9 or R14 the situation turns out to be even more complicated. These cases will be discussed in detail elsewhere [25], but, for completeness, non-trivial (that is, λ not identically zero) examples of projective relatedness in such space-times will now be discussed briefly. Consider a spacetime (M, g) and a point m ∈ M for which there exists a coordinate neigh- bourhood U of m with coordinates u, v, x, y (v > 0) such that g is given on U by ds2 = 2dudv + b(u) √ vdu2 + u2ef(x,y) ( dx2 + dy2 ) , (7.23) where b(u) and f(x, y) are arbitrary smooth functions on U . Now the 1-form field l = du may be shown to be recurrent on U , and is covariantly constant on U if and only if b(u) ≡ 0 on U . On any connected open subset of U where b(u) = 0, (7.23) is locally of the form (7.10) and of curvature class D. However, for (7.23) to be of holonomy type R9 or R14, b(u) 6= 0 at least at one point m ∈ M , and hence over some open neighbourhood V of m. Assuming that b(u) is nowhere zero on V the curvature class is A on V and after somewhat lengthy calculations it may be shown that (5.7) may be solved non-trivially to give a = φ ( g + 2ξududv + ξ ( 2v + ub(u) √ v ) du2 ) , (7.24) where φ, ξ ∈ R. Inverting the pair (a, λ) to obtain the corresponding pair (g′, ψ) leads to ψ = dχ, χ = 1 2 lnF, F = κ4 [1 + ξu]−2 (7.25) 22 G.S. Hall and D.P. Lonie with κ = φ−1, corresponding to a metric g′, projectively related to (7.23), and given on V by g′ = κFg − κ−3F 2 [ 2ξu(1 + ξu)dudv + ξ(u(1 + ξu)b(u) √ v + 2v)du2 ] . (7.26) If f(x, y) satisfies ∂2f ∂x2 + ∂2f ∂y2 = 0 everywhere on V then an application of the Ambrose–Singer theorem [14] shows (7.23) has holonomy type R9, and otherwise has holonomy type R14 on V . It is remarked that examples are known of non-trivially projectively related space-times of the Friedmann–Robertson–Walker–Lemaitre (FRWL) type and which are of holonomy type R15 [26]. Other examples of R15 metrics with similar properties will be discussed in [25]. 8 Conclusions This paper studied the question of, given a space-time (M, g), what can be said about those other metrics g′ on M (or some open submanifold of M) which are projectively related to g? The question was inspired from both a geometrical point of view and from a physical stand- point through the principle of equivalence in Einsteins general theory of relativity. If (M, g) is an Einstein space (and this includes the important vacuum metrics of general relativity) the problem is already resolved, as explained in Section 5. This paper then proceeded to extend this by studying the situation through the various possible holonomy groups that are possible for a Lorentz manifold. It was shown that, by using certain algebraic properties of the curvature tensor, much of this problem is tractable and an essentially complete solution was obtained for all the holonomy types except R9, R14 and R15. For these latter types, examples of non-trivially projectively related space-times were shown to exist (and further details will be given elsewhere). Amongst all of these are several well-known, physically important solutions of Einsteins field equations. References [1] Geroch R.P., Spinor structure of space-times in general relativity, J. Math. Phys. 9 (1968), 1739–1744. [2] Hall G.S., Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, Vol. 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. [3] Hall G.S., McIntosh C.G.B., Algebraic determination of the metric from the curvature in general relativity, Internat. J. Theoret. Phys. 22 (1983), 469–476. [4] Hall G.S., Curvature collineations and the determination of the metric from the curvature in general rela- tivity, Gen. Relativity Gravitation 15 (1983), 581–589. [5] McIntosh C.G.B., Halford W.D., Determination of the metric tensor from components of the Riemann tensor, J. Phys. A: Math. Gen. 14 (1981), 2331–2338. [6] Hall G.S., Lonie D.P., On the compatibility of Lorentz metrics with linear connections on four-dimensional manifolds, J. Phys. A: Math. Gen. 39 (2006), 2995–3010, gr-qc/0509067. [7] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. 1, Interscience Publishers, New York, 1963. [8] Schell J.F., Classification of four-dimensional Riemannian spaces, J. Math. Phys. 2 (1961), 202–206. [9] Hall G.S., Lonie D.P., Holonomy groups and spacetimes, Classical Quantum Gravity 17 (2000), 1369–1382, gr-qc/0310076. [10] Hall G.S., Covariantly constant tensors and holonomy structure in general relativity, J. Math. Phys. 32 (1991), 181–187. [11] Besse A., Einstein manifolds, Springer-Verlag, Berlin, 1987. [12] Wu H., On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291–311. [13] Hall G.S., Kay W., Holonomy groups in general relativity, J. Math. Phys. 29 (1988), 428–432. [14] Ambrose W., Singer I.M., A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443. http://arxiv.org/abs/gr-qc/0509067 http://arxiv.org/abs/gr-qc/0310076 Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds 23 [15] Hall G.S., Connections and symmetries in space-times, Gen. Relativity Gravitation 20 (1988), 399–406. [16] Eisenhart L.P., Riemannian geometry, Princeton University Press, Princeton, 1966. [17] Thomas T.Y., Differential invariants of generalised spaces, Cambridge, 1934. [18] Weyl H., Zur Infinitesimalgeometrie: Einordnung der projectiven und der konformen Auffassung, Gött. Nachr. (1921), 99–112. [19] Petrov A.Z., Einstein spaces, Pergamon Press, Oxford – Edinburgh –New York, 1969. [20] Sinyukov N.S., Geodesic mappings of Riemannian spaces, Nauka, Moscow, 1979 (in Russian). [21] Mikes J., Hinterleitner I., Kiosak V.A., On the theory of geodesic mappings of Einstein spaces and their gene- ralizations, in The Albert Einstein Centenary International Conference, Editors J.-M. Alini and A. Fuzfa, AIP Conf. Proc., Vol. 861, American Institute of Physics, 2006, 428–435. [22] Hall G.S., Lonie D.P., The principle of equivalence and projective structure in spacetimes, Classical Quantum Gravity 24 (2007), 3617–3636, gr-qc/0703104. [23] Kiosak V., Matveev V.A., Complete Einstein metrics are geodesically rigid, Comm. Math. Phys. 289 (2009), 383–400, arXiv:0806.3169. [24] Hall G.S., Lonie D.P., Projective equivalence of Einstein spaces in general relativity, Classical Quantum Gravity 26 (2009), 125009, 10 pages. [25] Hall G.S., Lonie D.P., Holonomy and projective structure in space-times, Preprint, University of Aberdeen, 2009. [26] Hall G.S., Lonie D.P., The principle of equivalence and cosmological metrics, J. Math. Phys. 49 (2008), 022502, 13 pages. http://arxiv.org/abs/gr-qc/0703104 http://arxiv.org/abs/0806.3169 1 Introduction 2 Notation and preliminary remarks 3 Curvature structure of space-times 4 Holonomy theory 5 Projective structure 6 Projective structure and holonomy I 7 Projective structure and holonomy II 8 Conclusions References

Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

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