Spacetimes characterized by their scalar curvature invariants
SSpacetimes characterized by their scalarcurvature invariants
Alan Coley , Sigbjørn Hervik , Nicos Pelavas Department of Mathematics and Statistics,Dalhousie University, Halifax, Nova Scotia,Canada B3H 3J5 Faculty of Science and Technology,University of Stavanger,N-4036 Stavanger, Norway aac, [email protected], [email protected]
August 17, 2009
Abstract
In this paper we determine the class of four-dimensional Lorentzianmanifolds that can be completely characterized by the scalar polynomialcurvature invariants constructed from the Riemann tensor and its covari-ant derivatives. We introduce the notion of an I -non-degenerate space-time metric, which implies that the spacetime metric is locally determinedby its curvature invariants. By determining an appropriate set of projec-tion operators from the Riemann tensor and its covariant derivatives, weare able to prove a number of results (both in the algebraically generaland in algebraically special cases) of when a spacetime metric is I -non-degenerate. This enables us to prove our main theorem that a spacetimemetric is either I -non-degenerate or a Kundt metric. Therefore, a metricthat is not characterized by its curvature invariants must be of Kundtform. We then discuss the inverse question of what properties of theunderlying spacetime can be determined from a given a set of scalar poly-nomial invariants, and some partial results are presented. We also discussthe notions of strong and weak non-degeneracy. In matters related to relativity and gravitational physics we are often inter-ested in comparing various spacetime metrics. Often identical metrics (which,of course, would give identical physics) are given in different coordinates andwill therefore be disguising their true equivalence. It is therefore of import tohave an invariant way to distingush spacetime metrics. The perhaps easiestway of distinguishing metrics is to calculate (some of) their scalar polynomialcurvature invariants due to the fact that inequivalent invariants implies inequiv-alent metrics. However, if their scalar polynomial invariants are the same, what1 a r X i v : . [ g r- q c ] A ug A. Coley, S. Hervik and N. Pelavasconclusion can we draw about the (in)equivalence of the metrics? For example,if all such invariants are zero, can we say that the metric is flat? The answerto this question is known to be no, because all so-called VSI metrics have van-ishing scalar invariants. Here, we will address the more general question: if twospacetimes have identical scalar polynomical curvature invariants, what can wesay about these spacetimes? In particular, when do the invariants characterisethe spacetime metric?For a spacetime ( M , g ) with a set of scalar polynomial curvature invariants,there are two conceivable ways in which the metric g can be altered such that theinvariants remain the same. First, the metric can be continuously deformed insuch a way that the invariants remain unchanged. This is what happens for theKundt metrics for which we have free functions which do not affect the curvatureinvariants. Alternatively, a discrete transformation of the metric can leave theinvariants the same. A simple example of when a discrete transformation cangive another metric with the same set of invariants is the pair of metrics:d s = 1 z (cid:0) d x + d y + d z (cid:1) − d τ , (1)d s = 1 z (cid:0) − d x + d y + d z (cid:1) + d τ . (2)One can straight-forwardly check that these metrics have identical invariantsbut are not diffeomorphic (over the reals). These discrete transformations aremore difficult to deal with but the issue will be taken up in a later section.Therefore, first we will consider the continuous metric deformations definedas follows. Definition 1.1.
For a spacetime ( M , g ), a (one-parameter) metric deformation ,ˆ g τ , τ ∈ [0 , (cid:15) ), is a family of smooth metrics on M such that1. ˆ g τ is continuous in τ ,2. ˆ g = g ; and3. ˆ g τ for τ >
0, is not diffeomorphic to g .For any given spacetime ( M , g ) we define the set of all scalar polynomial cur-vature invariants I ≡ {
R, R µν R µν , C µναβ C µναβ , R µναβ ; γ R µναβ ; γ , R µναβ ; γδ R µναβ ; γδ , . . . } . Therefore, we can consider the set of invariants as a function of the metric andits derivatives. However, we are interested in to what extent, or under whatcircumstances, this function has an inverse.
Definition 1.2.
Consider a spacetime ( M , g ) with a set of invariants I . Then,if there does not exist a metric deformation of g having the same set of invariantsas g , then we will call the set of invariants non-degenerate . Furthermore, thespacetime metric g , will be called I -non-degenerate .This implies that for a metric which is I -non-degenerate the invariants char-acterize the spacetime uniquely, at least locally, in the space of (Lorentzian)metrics. This means that these metrics are characterized by their curvatureinvariants and therefore we can distinguish such metrics using their invariants.Since scalar curvature invariants are manifestly diffeomorphism-invariant we canthereby avoid the difficult issue whether a diffeomorphism exists connecting twospacetimes.pacetimes characterized by their scalar curvature invariants 3 Let us first state our main theorems which will be proven in the later sections.The theorems apply to four-dimensional (4D) Lorentzian manifolds. Such space-times are characterized algebraically by their Petrov [1, 2] and Segre [3, 2] typesor, equivalently, in terms of their Ricci, Weyl (and Riemann) types [4, 5, 6]. Thenotation, which essentially follows that of the cited references, is briefly sum-marized in Appendix A. The proofs of these theorems, which are investigatedon a case by case basis in terms of the algebraic type of the curvature tensors,are long and tedious and have therefore been placed in later sections. Once allof the various cases have been explored the theorems follow.Furthermore, let us remark on the technical assumptions made in this paper.The following theorems hold on neighborhoods where the Riemann, Weyl andSegre types do not change. In the algebraically special cases we also need toassume that the algebraic type of the higher-derivative curvature tensors alsodo not change, up to the appropriate order. Most crucial is the definition ofthe curvature operators (see later) and in order for these to be well defined, thealgebraic properties of the curvature tensors need to remain the same over aneighborhood. Henceforth, we will therefore assume that we consider an openneighborhood where the algebraic properties of the curvature tensors do notchange, up to the appropriate order ( ≤ I -non-degenerate metrics. Theorem 2.1 (Algebraically general) . If a spacetime metric is of Ricci type I ,Weyl type I , or Riemann type I/G , the metric is I -non-degenerate. This theorem indicates that the general metric is I -non-degenerate and thusthe metric is determined by its curvature invariants (at least locally, in the senseexplained above). In the above, by Riemann type I/G , we are referring to theexistence of a frame in which components of boost weight +2 vanish for Riemanntype I , and in type G there does not exist a frame in which components withboost weight +2 or -2 vanish, in this case the Weyl and Ricci canonical framesare not aligned. For the algebraically special spacetimes, we need to considercovariant derivatives of the Riemann tensor. Theorem 2.2 (Algebraically special) . If the spacetime metric is algebraicallyspecial, but ∇ R , ∇ (2) R , ∇ (3) R , or ∇ (4) R is of type I or more general, the metricis I -non-degenerate. In terms of the boost weight decomposition, an algebraically special metrichas a Riemann tensor with zero positive boost weight components. In general,type I refers to the vanishing of boost weight components +2 and higher (but notboost weight +1 components). For example, we often use the notation ( ∇ R ) b =0, b ≥ ∇ R of type I (but ( ∇ R ) (cid:54) = 0). The above theorem indicatesthat if by taking covariant derivatives of the Riemann tensor you acquire positiveboost weight components, then the metric is I -non-degenerate. The remainingmetrics which do not acquire a positive boost weight component when takingcovariant derivatives, have a very special structure of their curvature tensors.Indeed, such metrics must be very special metrics: Alternatively, we can assume that the spacetime is real analytic.
A. Coley, S. Hervik and N. Pelavas
Theorem 2.3.
Consider a spacetime metric. Then either,1. the metric is I -non-degenerate; or,2. the metric is contained in the Kundt class. This is a striking result because it tells us that metrics not determined bytheir curvature invariants must be of Kundt form. These Kundt metrics there-fore correspond to degenerate metrics in the sense that many such spacetimescan have identical invariants. The Kundt class is defined by those metrics ad-mitting a null vector (cid:96) that is geodesic, expansion-free, shear-free and twist-free(corresponding to the vanishing of the spin-coefficients κ , σ and ρ ; see alsoAppendix A) (cid:96) β (cid:96) α ; β = 0 (cid:96) ; αα = 0 (cid:96) ( α ; β ) (cid:96) α ; β = 0 (cid:96) [ α ; β ] (cid:96) α ; β = 0 . (3)Any metric in the Kundt class can be written in the following canonical form[7, 4]:d s = 2d u (cid:2) d v + H ( v, u, x k )d u + W i ( v, u, x k )d x i (cid:3) + g ij ( u, x k )d x i d x j . (4)For spacetimes with constant curvature invariants (CSI) Theorem 2.3 hasan important consequence. For CSI metrics, I -non-degenerate implies that thespacetime is curvature homogeneous to all orders; hence, an important corollaryis a proof of the CSI-Kundt conjecture [7]: Corollary 2.4 (CSI spacetimes) . Consider a 4-dimensional spacetime havingall constant curvature invariants (CSI). Then either,1. the spacetime is locally homogeneous; or,2. a subclass of the Kundt spacetimes.
These theorems imply that the Kundt spacetimes play a pivotal role in thequestion of which metrics are I -non-degenerate. Indeed, the Kundt metricsare the only metrics not determined by their curvature invariants (in the senseexplained above).In fact, we can be somewhat more precise since only a subclass of the Kundtspacetimes have these exceptional properties. In the analysis (described be-low) it is found that a Kundt metric is I -non-degenerate if the metric functions W i ( v, u, x k ) in the canonical ( kinematic ) Kundt null frame are non-linear in v (i.e., W i,vv (cid:54) = 0). Hence the exceptional spacetimes are the aligned algebraicallyspecial type- II -Kundt spacetimes or, in short (and consistent with the terminol-ogy of the above theorem) degenerate Kundt spacetimes, in which there exists acommon null frame in which the geodesic, expansion-free, shear-free and twist-free null vector (cid:96) is also the null vector in which all positive boost weight terms ofthe Riemann tensor are zero (i.e., the kinematic Kundt frame and the Riemanntype II aligned null frame are aligned ). We note that the important Kundt-CSIand vanishing scalar invariant (VSI) spacetimes are degenerate Kundt space-times [8, 9, 10, 7].pacetimes characterized by their scalar curvature invariants 5 In order to prove the main theorems we need to introduce some mathematicaltools. These tools, although they are very simple, are extremely useful andpowerful in proving these theorems.A curvature operator, T , is a tensor considered as a (pointwise) linear oper-ator T : V (cid:55)→ V, for some vector space V , constructed from the Riemann tensor, its covariantderivatives, and the curvature invariants.The archetypical example of a curvature operator is obtained by raising oneindex of the Ricci tensor. The Ricci operator is consequently a mapping of thetangent space T p M into itself: R ≡ ( R µν ) : T p M (cid:55)→ T p M . Another example of a curvature operator is the Weyl tensor, considered as anoperator, C ≡ ( C αβµν ), mapping bivectors onto bivectors.For a curvature operator, T , consider an eigenvector v with eigenvalue λ ;i.e., Tv = λ v . If d = dim( V ) and n is the dimension of the spacetime, thenthe eigenvalues of T are GL ( d ) invariant. Since the Lorentz transformations, O (1 , n − ⊂ GL ( d ) on T , the eigenvalues ofa curvature operator is an O (1 , n − -invariant curvature scalar . Therefore,curvature operators naturally provide us with a set of curvature invariants (notnecessarily polynomial invariants) corresponding to the set of distinct eigen-values: { λ A } . Furthermore, the set of eigenvalues are uniquely determined bythe polynomial invariants of T via its characteristic equation. The characteris-tic equation, when solved, gives us the set of eigenvalues, and hence these areconsequently determined by the invariants. We can now define a number of associated curvature operators. For example,for an eigenvector v A so that Tv A = λ A v A , we can construct the annihilatoroperator: P A ≡ ( T − λ A ) . Considering the Jordan block form of T , the eigenvalue λ A corresponds to a setof Jordan blocks. These blocks are of the form: B A = λ A · · · λ A λ A . . . 0... . . . . . . . . . 00 . . . λ A . There might be several such blocks corresponding to an eigenvalue λ A ; however,they are all such that ( B A − λ A ) is nilpotent and hence there exists an n A ∈ N such that P n A A annihilates the whole vector space associated to the eigenvalue λ A . Note that the ’corresponding eigenvalues’ of the operators constructed from the covariantderivatives of the Riemann tensor are also related to scalar curvature invariants built fromcovariant derivatives.
A. Coley, S. Hervik and N. PelavasThis implies that we can define a set of operators (cid:101) ⊥ A with eigenvalues 0 or1 by considering the products (cid:89) B (cid:54) = A P n B B = Λ A (cid:101) ⊥ A , where Λ A = (cid:81) B (cid:54) = A ( λ A − λ B ) n B (cid:54) = 0 (as long as λ B (cid:54) = λ A for all B ). Further-more, we can now define ⊥ A ≡ − (cid:16) − (cid:101) ⊥ A (cid:17) n A where ⊥ A is a curvature projector . The set of all such curvature projectorsobeys: = ⊥ + ⊥ + · · · + ⊥ A + · · · , ⊥ A ⊥ B = δ AB ⊥ A . (5)We can use these curvature projectors to decompose the operator T : T = N + (cid:88) A λ A ⊥ A . (6)The operator N therefore contains all the information not encapsulated in theeigenvalues λ A . From the Jordan form we can see that N is nilpotent; i.e.,there exists an n ∈ N such that N n = . In particular, if N (cid:54) = 0, then N is anegative/positive boost weight operator which can be used to lower/raise theboost weight of a tensor.Considering the Ricci operator, or the Weyl operator, we can show that(where the type refers to either Ricci type or Weyl type): • Type I: N = , λ A (cid:54) = 0. • Type D: N = , λ A (cid:54) = 0. • Type II: N = , λ A (cid:54) = 0. • Type III: N = , λ A = 0. • Type N: N = , λ A = 0. • Type O: N = , λ A = 0.Consider a curvature projector ⊥ : T p M (cid:55)→ T p M . Then, for a Lorentzianspacetime there are 4 categories:1. Timelike: For all v µ ∈ T p M , v ν ( ⊥ ) νµ v µ ≤ v µ ∈ T p M , v ν ( ⊥ ) νµ v µ = 0.3. Spacelike: For all v µ ∈ T p M , v ν ( ⊥ ) νµ v µ ≥ ⊥ A : T p M (cid:55)→ T p M . These projectors can be of any of the aforementionedcategories and we are going to use the Segre-like notation to characterize theset with a comma separating time and space. For example, { , } means wehave 4 projectors: one timelike, and three spacelike. A bracket indicates thatthe image of the projectors are of dimension 2 or higher; e.g., { (1 , } meansthat we have two spacelike operators, and one with a 2 dimensional image. Ifthere is a null projector, we automatically have a second null projector. Givenan NP frame { (cid:96) µ , n µ , m µ , ¯ m µ } , then a null-projector can typically be:( ⊥ ) µν = − (cid:96) µ n ν . Note that ⊥ = ⊥ , but it is not symmetric. Therefore, acting from the leftand right gives two different operators. Indeed, defining( ⊥ ) µν ≡ g να g µβ ( ⊥ ) αβ , we get a second null-projector being orthogonal to ⊥ . The existence of null-projectors enables us to pick out certain null directions; however, note thatthe null-operators, with respect to the aforementioned Newman-Penrose (NP)frame, are of boost weight 0 and so cannot be used to lower/raise the boostweights. In particular, considering the combination ⊥ + ⊥ we see that the ex-istence of null-projectors implies the existence of projectors of type { (1 , } .The existence of curvature projectors is important due to the following result: Theorem 3.1.
Consider a spacetime metric and assume that there exist cur-vature projectors of type { , } , { , } or { , (111) } . Then the spacetimeis I -non-degenerate .Proof. Consider first the case { , } . For any given curvature tensor, R αβ...δ ,we can construct the curvature tensor R [ ij...k ] αβ...δ ≡ R µν...λ ( ⊥ i ) µα ( ⊥ j ) νβ ... ( ⊥ k ) λδ . This enables us to consider the curvature invariant R [ ij...k ] αβ...δ R [ ij...k ] αβ...δ which is, up to a constant factor, the square of the component R ij...k . Thisimplies that it is determined by the invariant (up to a sign) and we get that thespacetime is I -non-degenerate. Consider now the case { , } . We note that in this case we cannotisolate all components of the curvature tensors. However, we can uniquely definetensors r ( A ) IJ...K , I, J, ... = 3 , ⊥ i . The curvature invariantswill now be SO (2)-invariant polynomials in the components of r ( A ) IJ...K . Hence,since SO (2) is compact, the polynomials will separate the SO (2) orbits. Hence,by a similar proof as in [11] we get that the spacetime is I -non-degenerate. Lastly, consider the case { , (111) } . In this case we can define tensors r ( A ) IJ...K , I, J, ... = 2 , , ⊥ i . The curvature invariants will be SO (3)-invariant which is again compact. Hence, using a similar argument as in [11] weget that the spacetime is I -non-degenerate. A. Coley, S. Hervik and N. Pelavas
I /G
Let us consider first the case where the Riemann tensor is of type I or G . Thiscorresponds to the three cases: Ricci type I , Weyl (Petrov) type I , and Ricciand Weyl canonical frames not aligned. We shall consider these in turn. I This case consists of the following Segre types: { , } , { , } { , (111) } , { z ¯ z } , { z ¯ z (11) } . { , } : Here the eigenvalues of the Ricci operator are all distinct and we can diagonalizethe Ricci operator: R = diag( λ , λ , λ , λ ) . It now follows from Theorem 3.1 that the spacetime is I -non-degenerate. Indeed, to determine the spacetime it is sufficient to consider R µν ; α . Choos-ing an orthonormal frame, e i , aligned with the eigendirections of R : R ij ; k = λ i,k g ij + ( λ i − λ j )Γ ijk , where Γ ijk are the connection coefficients, we find that all connection coefficientsmust be determined by the curvature invariants. { , } : This is the special case of above where we have λ = λ . Using Theorem 3.1 the spacetime is I -non-degenerate. { , (111) } : Here we have λ = λ = λ and from Theorem 3.1 we have that the spacetimeis I -non-degenerate. { z ¯ z } and { z ¯ z (11) } : In this case, the Ricci operator has two complex conjugate eigenvalues. We canalways find an orthonormal frame { e i } , so that the Ricci operator takes theform R = a b − b a λ
00 0 0 λ . We can now consider the complex transformation mapping the basis vectors e and e onto the eigenvectors v and v : v = √ ( e + i e ) , v = √ (i e + e ) , (7)with inverse e = √ ( v − i v ) , e = √ ( − i v + v ) . (8)pacetimes characterized by their scalar curvature invariants 9We note that v · v = − v · v = 1, v · v = 0 and so the set { v , v , e , e } can be considered as an orthonormal frame. In this frame the Ricci operatorbecomes diagonal: R = diag( λ , ¯ λ , λ , λ ) . Therefore, we have a set of curvature projectors of the form { , } or { , } and we can use Theorem 3.1. The only difference is that the invariants associatedto the complex frame can now be complex; however, the result still stands.Using the inverse transformation, which induces a transformation between theinvariants from the complex frame to the real frame, we obtain the curvaturecomponents of the real frame. Therefore we can conclude that the spacetime is I -non-degenerate. I (Petrov type I ) For the Weyl tensor any non-trivial isotropy would make it algebraically special.The isotropy group of the Weyl tensor is the subgroup of the Lorentz groupwhose action on the Weyl tensor leaves it invariant; for example a Petrov typeD Weyl tensor has a boost-spin isotropy group. So for the Weyl tensor to be oftype I requires that the isotropy group is trivial. We therefore expect that wewill be able to determine a unique frame using the curvature invariants.We use the bivector formalism and write the Weyl tensor, C αβµν , as anoperator in 6-dimensional bivector space, C = ( C AB ). Using the following indexconvention:[23] ↔ , [31] ↔ , [12] ↔ , [10] ↔ , [20] ↔ , [30] ↔ , a type I Weyl tensor can always be put into the following canonical form [3]: C = a b a b
00 0 a b − b a − b a
00 0 − b a (9)where (cid:80) i a i = (cid:80) i b i = 0 and not all of the a i , b i are zero.First we note that the eigenvalues of C are a i ± i b i . As explained above, a i and b i are uniquely determined by the zeroth order Weyl invariants. Theeigenbivectors are F A = δ Ai ± i δ A i . We can therefore construct annihilatoroperators, ( C − λ ), and projection operators as before (the only difference is that C is 6-dimensional). The eigenbivectors correspond to (complex) antisymmetrictensors. For example, consider the eigenbivector with eigenvalue a + i b : F = 12 F µν ω µ ∧ ω ν = ω ∧ ω − i ω ∧ ω . Hence, from this we can construct an operator P = ( F µα ¯ F αν ) = − − . (10)0 A. Coley, S. Hervik and N. PelavasFor the other eigenbivectors we then get (analogously): P = diag(1 , − , , − , P = diag(1 , − , − , . Thus the linear set { , P i } span all diagonal matrices; in particular, we canconstruct the projection operators: ⊥ = ( + P + P + P ) , ⊥ = ( + P − P − P ) , ⊥ = ( − P + P − P ) , ⊥ = ( − P − P + P ) . It is clear that we will get 3 operators, P i , as long as the 3 sets of complexeigenvalues, λ i = a i + i b i , are all different. Since (cid:80) i λ i = 0, this can only failwhen: λ = λ , λ = − λ ,λ = λ = λ = 0 . The first of these is actually Weyl type D , while the latter is Weyl type O ;hence, these are excluded by assumption.Therefore, we can conclude that as long as the Weyl type is I (and notsimpler), we can define 4 projection operators of type { , } . Therefore, fromTheorem 3.1, the spacetime is I -non-degenerate .At this stage we wish to remark on a certain subtlety in the choice of eigen-vectors. From the Weyl tensor we can actually only determine the product F µν F αβ . Therefore, we can only construct the “square” P ⊗ P . So in orderto get the operator P there is an ambiguity in the choice of sign. Regard-ing the question of I -non-degeneracy as defined above this has no consequence;however, it may have an effect on discrete changes to the metric. This sign am-biguity results in a permutation of the axes; essentially, we don’t know whichaxis corresponds to time. We will get back to this issue later but note that thisphenomenon will recur in several cases below. Consider now the case where both the Ricci tensor and the Weyl tensor arealgebraically special but where there does not exist a null-frame such that boththe Ricci tensor and the Weyl tensor has only non-positive boost weights.First, assume the Weyl type is D and choose the Weyl canonical frame.For Weyl type D the Weyl operator is of the form of eq. (9) with λ = λ , λ = − λ . This immediately implies we have projection operators oftype { (1 , } .In the Weyl canonical frame, the Ricci tensor must have both positive and negative boost weight components (or else there would exist a frame where theyare aligned). Now, by symmetry, we can consider three cases for the Ricci tensor(see Appendix A for ( R ) b notation):1. ( R ) +2 (cid:54) = 0, ( R ) − (cid:54) = 0: Here, we use the (1 , { (cid:96), n } frame): (cid:101) R = (cid:20) a bc a (cid:21) , bc (cid:54) = 0 . (11)pacetimes characterized by their scalar curvature invariants 11This gives two distinct eigenvalues λ = a ± √ bc , and hence, two additionalprojection operators. This case therefore reduces to the case { , } or { z ¯ z (11) } presented earlier. This spacetime is therefore I -non-degenerate.
2. ( R ) +2 = 0, ( R ) +1 (cid:54) = 0, ( R ) − (cid:54) = 0: For this case we note that the square R µα R αν necessarily has boost weight +2 components. Therefore, usingeither R µα R αν or R µα R αν + qR µν , where q is a parameter, we can usethe results of the previous paragraph. This case is consequently I -non-degenerate .3. ( R ) ± = 0, ( R ) +1 (cid:54) = 0, ( R ) − (cid:54) = 0: Here we consider R µα R αν whichnecessarily has non-zero boost weight − I -non-degenerate .Assume now Weyl type N and choose the Weyl canonical frame such that C = ( C ) − . In this frame, either ( R ) +1 or ( R ) +2 is non-zero. If ( R ) +2 iszero, we replace the Ricci tensor R µν with the square R µα R αν in what follows.Therefore, assume that ( R ) +2 (cid:54) = 0. Consider now the operator( T µν ) ≡ C µανβ R αβ Under the above assumptions, this operator can be used to get projectors oftype { (1 , } . Indeed, these projectors are aligned with the Weyl canonicalframe. We can now use one of the spacelike projectors, ⊥ (say), to constructthe symmetric operator: ( ˆ T µν ) ≡ C µανβ ( ⊥ ) βα + qR µν , where q is a parameter. We can use this operator to construct the remainingprojection operators so that we have a set { , } . This case is therefore I -non-degenerate. For Weyl type II we can decompose the Weyl tensor: C = N + (cid:88) A λ A ⊥ A , where the operator N is a “Weyl” operator of type N while the piece (cid:80) A λ A ⊥ A is a “Weyl” operator of type D . By assumption, the Ricci tensor is not alignedwith the Weyl canonical frame; therefore, using the above results, this case isalso I -non-degenerate .Lastly, for Weyl type III , we can consider the square C which is a Weyloperator of type N . The above results imply that this case is I -non-degenerate . Therefore, we have shown that:
If a 4-dimensional spacetime ( M , g ) is eitherRicci type I , Weyl type I or Riemann type I/G , then it is I -non-degenerate. For the algebraically special cases the Riemann tensor itself does not give enoughinformation to provide us with all the required projection operators. Indeed,2 A. Coley, S. Hervik and N. Pelavasin the algebraically special cases it is also necessary to calculate the covariantderivatives. The strategy is as follows: we will consider the two cases of Weyltype D and N in detail. The second Bianchi identity will not be imposed atthis time because we aim to use these results on more general tensors with thesame symmetries, not necessarily the Weyl tensor itself. Weyl type II and III will now follow from these computations and Weyl type O will be treated last.We should also point out that for any symmetric tensor S µν we can alwaysconstruct a Weyl-like tensor with the same symmetries as the Weyl tensor. If S µν is the trace-free Ricci tensor, the corresponding Weyl-like tensor is the so-called Pleba´nski tensor. Explicitly, given the trace-free part of S µν , denotedˆ S µν , the Pleba´nski tensor is given by W αβµν ≡ ˆ S [ α [ µ ˆ S β ] ν ] + δ [ α [ µ ˆ S ν ] γ ˆ S β ] γ − δ [ α [ µ δ β ] ν ] ˆ S γ(cid:15) ˆ S γ(cid:15) . (12)Therefore, to any symmetric tensor there is an associated “Pleba´nski” tensor.Henceforth we are going to use the NP-formalism where we introduce a nullframe { (cid:96), n, m, ¯ m } . (We will use the notation of [2]; also see Appendix A).In order to get the desired results we introduce the canonical frames for thevarious algebraic types. For the Weyl tensor, C , this means that we expressits components in terms of the Weyl scalars Ψ i . Then using the NP-connectioncoefficients, we can express the covariant derivative ∇ C in terms of Ψ i andthe connection coefficients. At this stage it is useful not to assume anythingabout the connection ∇ (i.e., the tensor C need not be the Weyl tensor of theconnection). The advantage of this is that we can utilise the full formalismof projection operators without worrying about the compatibility of the Weyltensor and the connection. Furthermore, the results obtained here will thereforebe more general than what is indicated. These expressions are then utilised toobtain the required results for the curvature tensor. Another important thingto note is that when taking covariant derivatives, some of the components haveterms which are partial derivatives of Ψ i , while other terms are algebraic inΨ i and Γ ijk . These algebraic terms are most useful simply because they givealgebraic relations rather than differential ones. D We choose the canonical frame for which Ψ (cid:54) = 0. From the Weyl operator C wecan construct projectors of type { (1 , } where the (1 , (cid:96) − n -plane, while the (11)-projector projects onto the m − ¯ m -plane.In the following let us use the indices a, b, .. for projections onto the (cid:96) − n planeand the indices i, j, ... for projections onto the m − ¯ m plane.Calculating ∇ C we get the boost weight decomposition ∇ C = ( ∇ C ) +2 + ( ∇ C ) +1 + ( ∇ C ) + ( ∇ C ) − + ( ∇ C ) − . The key observation is that the positive boost weight components vanish if andonly if (cid:96) µ ∇ µ Ψ = 0 and κ = σ = ρ = 0. Therefore, the idea is to define theappropriate operators so that we can isolate the necessary components.Consider the (projected) tensor: T iab ≡ C jij ( a ; b ) pacetimes characterized by their scalar curvature invariants 13This tensor has the following structure, T iab = v i n a n b + t i ( (cid:96) a n b + n a (cid:96) b ) + w i (cid:96) a (cid:96) b , where v i ≡
3( ¯Ψ ¯ κm i + Ψ κ ¯ m i ) , (13) t i ≡ − ( π Ψ − ¯ τ ¯Ψ ) m i − (¯ π ¯Ψ − τ Ψ ) ¯ m i , (14) w i ≡ − νm i + ¯Ψ ¯ ν ¯ m i ) . (15)(16)Furthermore, define the trace-free tensor (cid:98) T iab ≡ T iab +(1 / (cid:96) a n b + n a (cid:96) b ) T ci c ,and then the tensor S abcd = (cid:98) T iab (cid:98) T icd . This tensor can be considered as an operator S = ( S AB ) mapping symmetrictrace-free tensors onto symmetric trace-free tensors. For simplicity, let us alsoconsider the trace-free part of S abcd so that (cid:98) S abcd = v i v i n a n b n c n d + w i w i (cid:96) a (cid:96) b (cid:96) c (cid:96) d . Consider the trace-free tensor M ab = xn a n b + y(cid:96) a (cid:96) b . The operator (cid:98) S has eigen-values λ = ±| v || w | ,
0. Therefore, as long as both v i and w j are non-zero,there are three distinct eigenvalues. Assuming λ (cid:54) = 0, M ab is an eigentensorif x = | v | and y = | w | . In this case we can consider the curvature projectors(up to scaling), M ab M cd . The eigentensor M ab can again be considered as anoperator M = ( M ab ) mapping vectors onto vectors. The eigenvalues of M are λ = ± i | v || w | ; hence, this reduces to the case of two complex eigenvalues.We note that v i v i = 0 if and only if κ = 0. Furthermore, if either of | v | or | w | is non-zero we can assume, by using the discrete symmetry defined later byeq. (26), that w i w i (cid:54) = 0. Therefore, κ (cid:54) = 0 (so that | v | (cid:54) = 0 also) implies that thespacetime is I -non-degenerate .Therefore, assume κ = 0 and consider the symmetric tensor Q ab = C ijka ; l C ijkb ; l . The trace-free part of this tensor is (cid:98) Q ab ∝ | Ψ | (cid:0) | σ | + | ρ | (cid:1) n a n b + | Ψ | (cid:0) | λ | + | µ | (cid:1) (cid:96) a (cid:96) b . If (cid:0) | σ | + | ρ | (cid:1) (cid:0) | λ | + | µ | (cid:1) (cid:54) = 0, then this tensor is of type I . So from the Riccitype I analysis, this would imply that the spacetime is I -non-degenerate .Let us next consider the non-aligned case where κ = 0, λ = µ = 0 and | σ | + | ρ | (cid:54) = 0. We can now consider the mixed tensor: (cid:98) Q ab + (cid:98) S abcd (cid:98) Q cd . This tensor is of type I if ( w i w i ) (cid:0) | σ | + | ρ | (cid:1) (cid:54) = 0 and consequently, the space-time is I -non-degenerate .Assume now that w i = 0, κ = 0, for which we still have an unused discretesymmetry (eq.(26)). If (cid:0) | σ | + | ρ | (cid:1) (cid:0) | λ | + | µ | (cid:1) = 0 we can therefore assumethat ρ = σ = 0. This spacetime is thus Kundt .4 A. Coley, S. Hervik and N. PelavasLastly, consider the case when w i w i (cid:54) = 0, κ = ρ = σ = 0. This automaticallyimplies that the spacetime is Kundt .Let us also consider the differential (cid:96) µ ∇ µ Ψ which in general (not assumingthe Bianchi identities are satisfied) also contributes to ( ∇ C ) +1 . We note thatthe Weyl invariant I , for a Weyl type D tensor, is given by I = 3Ψ . Therefore,we can consider the curvature tensor defined by the gradient ∇ µ I = 6Ψ Ψ ,µ .We can now use the (1 , (cid:96) − n -plane: x a ≡ ∇ a I .1. If x a is either time-like or space-like (and consequently (cid:96) µ ∇ µ Ψ (cid:54) = 0), wecan construct another curvature projector (by considering the operator x a x b ) so that we have a set { , } . Therefore, this case is I -non-degenerate .2. If x a is null or zero, then either n a x a = 0 or (cid:96) a x a = 0. If (cid:96) a x a = 0 then (cid:96) µ ∇ µ Ψ = 0 and does not contribute to positive boost weight components.Assume therefore that (cid:96) a x a (cid:54) = 0, which implies that x a ∝ n a . If ν = λ = µ = 0, we can use the discrete symmetry eq.(26) so that (cid:96) µ ∇ µ Ψ = 0.If any of ν , λ or µ is non-zero, then ( ∇ C ) b< is non-zero. Hence, by con-tracting with x a , we can straight-forwardly construct another projectionoperator so that we get a set { , } . Therefore, this case is I -non-degenerate . D : A Weyl type D spacetime is either I -non-degenerate or Kundt. Moreover, fora Weyl type D spacetime, if ∇ C is of type I or more general, then it is I -non-degenerate. II The Weyl type II tensor can be decomposed as C = N + (cid:88) A λ A ⊥ A . By using the annihilator operators and the projection operators we can, upto scaling, isolate each term in this decomposition. Each term can thus beconsidered a curvature operator in its own right.In particular, by considering only the curvature tensor (cid:80) A λ A ⊥ A , this tensoris identical to a Weyl type D tensor. We can therefore use these results. Inaddition to these results we do have an additional boost weight -2 operator N . This breaks the discrete symmetry present in the Weyl type D tensor andtherefore restricts the choice even more. However, with minor modifications weobtain: a Weyl type II spacetime is either I -non-degenerate or Kundt. We also note that for a Weyl type II spacetime, the Weyl invariant I = 3Ψ as for type D . Therefore, using a similar argument, a Weyl type II spacetime,if ∇ C is of type I or more general, then it is I -non-degenerate. pacetimes characterized by their scalar curvature invariants 15 III
For the Weyl type
III case we get no non-trivial curvature operators from theWeyl tensor itself. The first non-trivial projection operators appears at firstcovariant derivative; however, in order to delineate this case completely we needto consider second covariant derivatives.We note that for the type
III
Weyl operator, C (cid:54) = 0 and is of type N . Theproof for this case is therefore contained in the Weyl type N case consideredbelow. N Consider first the tensor T µν = ∇ γ C αβγµ ∇ δ C αβδν , whose boost weight 0 components are of the form (it has no positive boostweight components) ( T µν ) ∝ ¯ κ ¯Ψ ¯ m µ ¯ m ν + κ Ψ m µ m ν . Therefore, if κ (cid:54) = 0, we can construct curvature operators of type { (1 , } . Thecurvature operator T µν gives rise to a “Pleba´nski” tensor of type D . Therefore,by considering second covariant derivatives, it follows from the Weyl type D analysis that if κ (cid:54) = 0, the spacetime is I -non-degenerate .Henceforth, assume that κ = 0 (and therefore we have no projectors fromfirst derivatives). Consider (cid:3) C µναβ , which has the same symmetries as the Weyltensor itself. This tensor has no positive boost weight components. Consideringthe boost weight 0 components, we note that (cid:3) C µναβ is of type II if and onlyif ρσ (cid:54) = 0. Therefore, if ρσ (cid:54) = 0 we can use the Weyl type II analysis, andcalculate ∇ (cid:3) C ; hence, this spacetime is I -non-degenerate .Therefore, consider the case where either σ or ρ are zero. Define W µν(cid:15)η ≡ C αβγδ ;( µν ) C αβγδ ;( (cid:15)η ) . To get a projection operator we note that the boost weight 0 components of W µαβγ W ναβγ (it has no positive boost weight components) is of the form (cid:0) W µαβγ W ναβγ (cid:1) ∝ | Ψ | | ρ or σ | ( ¯ m µ m ν + m µ ¯ m ν )Therefore, if either ρ or σ are non-zero, we can use this operator and we get(at least) two curvature projectors ⊥ and ⊥ of type { (1 , } . This meansthat we can construct a Weyl-like tensor of type D . Hence, we can use the type D results. Therefore, by considering third derivatives of the curvature tensors,if σ or ρ is non-zero, then the spacetime is I -non-degenerate .The remaining case, for which κ = ρ = σ = 0, is a Kundt spacetime.
III or N : Therefore, we can conclude that a Weyl type
III or N spacetime is either I -non-degenerate or Kundt.6 A. Coley, S. Hervik and N. Pelavas Using the trace-free Ricci tensor, we can construct the Pleba´nski tensor, whichis a Weyl-like tensor. The corresponding algebraic classification of the Pleba´nskitensor is called the Pleba´nski-Petrov (PP) classification. For the various alge-braically special PP types we have the following Segre types: • PP type II : { } , • PP type D : { (1 , } , { (1 , } , { } , { z ¯ z (11) } , { , }• PP type
III : { }• PP type N : { (31) } , { (21)1 }• PP type O : { (1 , } , { , (111) } , { (211) } , { (1 , } .Now, since the Pleba´nski tensor is a tensor with the same symmetries as theWeyl tensor, we can use the previous results for the PP types II , D , III and N . There is consequently only the Weyl (Petrov) type O and PP type O caseleft to consider. O , PP-type O Let us consider the remaining Segre types assuming Weyl (Petrov) type O andPP type O . { (1 , } Using the Bianchi identities we get several differential constraints on the spincoefficients. For this Segre type we have that Φ (cid:54) = 0, so the Bianchi identitiesimmediately imply κ + ¯ κ = ν + ¯ ν = 0. In addition, we get the followingrestrictions: ρ + ¯ σ = s (real) , µ + ¯ λ = m (real) . Furthermore, after some manipulation of the remaining Bianchi identities, weget (cid:15) − ¯ (cid:15) = γ − ¯ γ = τ + ¯ τ + π + ¯ π = 0 , and DR = 24 s Φ , ∆ R = − m Φ . We now split the analysis into 3 cases, according to whether ∇ µ R is timelike,spacelike or null.If ∇ µ R is timelike, we immediately have that this spacetime is I -non-degenerate since we can use ( ⊥ ) µν = ( ∇ µ R )( ∇ ν R ) as a timelike operator,and hence we obtain operators of type { , } .If ∇ µ R is spacelike, we can always use the remaining freedom to choose DR = ∆ R = 0. This implies that m = s = 0. Furthermore, δR − ¯ δR (cid:54) = 0which means we have an additional spacelike projection operator. Therefore,we have a set { (1 , } , which can be used to give a “PP-type” D tensor.Hence, using second covariant derivatives, we find that this is either Kundt or I -non-degenerate.Lastly, ∇ µ R is null. If ∇ µ R is zero, from the Bianchi identities we findthat this is a symmetric space, and hence, is actually locally homogeneous (andpacetimes characterized by their scalar curvature invariants 17Kundt). If ∇ µ R is null, we consider δ Φ − ¯ δ Φ . If this is non-zero we get anadditional projection operator and thus a set { (1 , } . This would thereforegive a “PP-type” D and hence, by considering second derivatives, this is eitherKundt or I -non-degenerate. Therefore, let us assume δ Φ − ¯ δ Φ = 0. TheBianchi identities now imply that α − ¯ β = 0. Using the symmetric operator R αβ ; ν R αβ ; µ we get that this is either of types { } , { (1 , } or { (211) } .The first two of these give a type D “Pleba´nski” tensor which means, by con-sidering second derivatives, they are either I -non-degenerate or Kundt. For thelast case, { (211) } , we can combine with the Ricci operator to break the symme-try down to type { (21)1 } which gives a “Pleba´nski” of type N . Therefore, byconsidering third and fourth derivatives, we get that this is I -non-degenerateor Kundt. { , (111) } This is actually Ricci type I and is therefore I -non-degenerate. { (211) } Choosing a frame where Φ is a constant we get, after using the Bianchi iden-tities (and some manipulation), κ = σ = 0. We then calculate the secondderivatives and compute the operator R αβ ;( γµ ) R αβ ;( γν ) , which gives operatorsof type { (1 , } , assuming ρ (cid:54) = 0. This again gives “PP-type” D tensor andhence, by calculating third derivatives, this is I -non-degenerate or Kundt. { (1 , } This is the maximally symmetric case and is therefore Kundt.In addition to Weyl-type I , Ricci-type I or Riemann-type I/G , we haveshown that there are I -non-degenerate metrics with algebraically special curva-ture types and further conditions on the spin-coefficients. These are summarizedin Tables (1) and (2). We have addressed the question of what is the class of Lorentzian manifolds thatcan be completely characterized by the scalar polynomial invariants constructedfrom the Riemann tensor and its covariant derivatives. Let us now consider the’inverse’ question: given a set of scalar polynomial invariants, what can we sayabout the underlying spacetime? In practice, it is somewhat tedious and alengthy ordeal to determine the spacetime from the set of invariants. However,in most circumstances we only need some partial results or we are dealing withspecial cases. Let us discuss how to determine, from the invariants, whether thespacetime is I -non-degenerate.We remind the reader that the zeroth order Weyl invariants are I and J ,and if all Weyl invariants up to order k vanish, we will denote this by VSI Wk . Note that this is the only case in which we have needed to utilize fourth derivatives; itis possible, by explicitly calculating the components of R µν ; αβγ , that we need only considerthird derivatives, but we have not done this here. P-type Conditions
I —D or II κ (cid:54) = 0 κ = 0 , ( | σ | + | ρ | )( | λ | + | µ | ) (cid:54) = 0 κ = λ = µ = 0 , | σ | + | ρ | (cid:54) = 0N or III κ (cid:54) = 0 κ = 0 ; σ = 0 or ρ = 0 (not both) Table 1: In each Petrov-type we list conditions on the spin-coefficients yieldingdistinct subcases for which the metric is I -non-degenerate. Proposition 6.1. If J (cid:54) = I , then the spacetime is I -non-degenerate. This follows easily from the fact that if 27 J (cid:54) = I then the spacetime is ofWeyl (Petrov) type I.If 27 J = I (cid:54) = 0 (Weyl type II or D), we need to go to higher order invari-ants in order to check whether it is I -non-degenerate or not. Ideally, we wouldlike to have a set of syzygies which gives the appropriate condition for this to bethe case. Such a complete set is not known. However, we have found two suchsyzygies which gives a sufficient condition for I -non-degeneracy. A number ofinvariants of ∇ C were constructed with degrees ranging from 2 to 4 (see Ap-pendix C for details). Imposing the minimal number of conditions required forthe normal form of a ∇ C -type II (boost weight +3,+2,+1 vanish) or D (onlyboost weight 0 is nonzero) results in a degree 8 syzygy, S = 0, and a degree16 syzygy, S = 0, amongst our invariants. Therefore if S (cid:54) = 0 or S (cid:54) = 0 then ∇ C is not of type II or D . Next, we showed that using the normal form of a ∇ C -type G (all components nonzero) or H (boost weight +3 vanish) or I (boostweight +3, +2 vanish) then S (cid:54) = 0 and S (cid:54) = 0. It is important to note that thisimplication refers only to the general types of G , H and I and there is no con-sideration of a secondary alignment type or any further algebraic specializationwithin these types. Indeed, it is possible that there is an algebraically specialsubcase, for example of a ∇ C -type I, that results in S = S = 0. A strongerstatement relating invariants of ∇ C to its algebraic type may be achieved byconsidering a different basis of invariants and a finer algebraic classification of ∇ C . Initially, one would attempt to construct a set of pure ∇ C invariants thatwas complete within each ∇ C algebraic type G , H , I and II , including specialsubcases. We have excluded type D since such a set of invariants is equivalent totype II , and also types III , N or O since these invariants vanish. By complete-ness of the set, an algebraic specialization would result in a dependence amongstinvariants and hence syzygies arise characterizing the algebraically special type.We now have the following invariant characterizations of I -non-degeneracy. Proposition 6.2. If J = I , but S (cid:54) = 0 or S (cid:54) = 0 , then the spacetime is pacetimes characterized by their scalar curvature invariants 19 Segre type of R µν Conditions { , (111) } — { (211) } Φ const., κ = σ = 0 , ρ (cid:54) = 0 (3rd deriv.) { (1 , } ∇ µ R timelike ∇ µ R spacelike : DR = ∆ R = κ + ¯ κ = ρ + ¯ σ = µ + ¯ λ = 0(2nd deriv. ; at least one of κ, σ or ρ is nonzero) ∇ µ R null : ∇ µ R = 0 (symmetric space) ∇ µ R (cid:54) = 0 : δ Φ − ¯ δ Φ (cid:54) = 0 (2nd deriv.) δ Φ − ¯ δ Φ = α − ¯ β = 0(2nd or 3rd and 4th deriv.) Table 2: Within P-type O and PP-type O we list the Segre types that contain I -non-degenerate metrics. The n th derivative conditions indicate that higherorder constraints exist on the spin-coefficients arising from n th order curvatureoperators. These higher-order constraints provide sufficient conditions for themetric to be I -non-degenerate. In all cases, at least one of κ , σ or ρ is nonzero. I -non-degenerate. The remaining cases are when both I and J are zero, and hence, the space-time is VSI W : Proposition 6.3.
Assume a spacetime is VSI W . Then:1. If it is not VSI W , it is I -non-degenerate.2. If it is VSI W , but not VSI W , it is I -non-degenerate. To prove the final result below, we shall assume for simplicity that the space-time is Einstein, so that R µν = λg µν . We therefore only have to consider theRicci scalar (= 4 λ ) and the Weyl invariants. If this is not the case, then wewould need to include the Ricci and mixed invariants. This can be done in astraight-forward manner. A summary of these results is given in Figure 1. Proposition 6.4.
Assume a spacetime is Einstein. Then:3. If it is VSI W , then it is Kundt. From the above results we have conditions on the scalar invariants (in termsof the Weyl tensor and its covariant derivatives) to determine whether the space-time is I -non-degenerate. Consequently, we have a number of conditions in0 A. Coley, S. Hervik and N. PelavasFigure 1: Using invariants in terms of the Weyl tensor and its covariant deriva-tives to determine whether the spacetime is I -non-degenerate.terms of scalar invariants that can be used to determine when a spacetime isnot I -non-degenerate and hence an aligned algebraically special type- II (or degenerate ) Kundt spacetime.Let us further consider to what extent the class of degenerate Kundt space-times can be characterized by their scalar curvature invariants. Clearly suchspacetimes are algebraically special and of type II (or more special) and hence27 J = I . If I = J = 0, then if the spacetime is of Weyl type N , then I = I = 0 if and only if κ = ρ = σ = 0 from the results in [8] (the def-initions of the invariants I and I are given therein). Similar results followfor Weyl type III spacetimes (in terms of the invariants ˜ I and ˜ I ) and in theconformally flat (but non-vacuum) case (in terms of similar invariants I and I constructed from the Ricci tensor) [8]. If 27 J = I (cid:54) = 0 (Weyl types II and D ): essentially if κ = ρ = σ (cid:54) = 0, we can construct positive boost weight termsin the derivatives of the curvature and determine an appropriate set of scalarpacetimes characterized by their scalar curvature invariants 21curvature invariants. For example, consider the positive boost weight terms ofthe first covariant derivative of the Riemann tensor, ∇ ( Riem ). If the spacetimeis I -non-degenerate, then each component of ∇ ( Riem ) is related to a scalar cur-vature invariant. In this case, in principle we can solve (for the positive boostweight components of ∇ ( Riem )) to uniquely determine κ, ρ, σ in terms of scalarinvariants, and we can therefore find necessary conditions for the spacetime tobe degenerate Kundt (there are two cases to consider, corresponding to whetherΨ + Φ is zero or non-zero). We note that even if the invariants exist inprinciple, it may not be possible to construct them in practice. I -non-degenerate Until now we have only considered I -non-degeneracy in terms of a local defor-mation of the metric. It is also of interest to know whether a I -non-degeneratemetric is unique under a discrete transformation. We shall call a spacetime suchthat the set of invariants uniquely specifies the metric strongly I -non-degenerate .Similarly, we shall call a spacetime such that the set of invariants only definesa unique metric up to discrete transformations weakly I -non-degenerate .Let us revisit the examples given by eqs. (1) and (2) in the Introduction.These two examples are both of Weyl type O , but they are of Segre type { , (111) } and { (1 , } . Hence, the eigenvalues of the Ricci operator is thesame but we cannot, from the invariants alone, determine which eigenvalue isassociated with the timelike direction and which is associated with the spacelikedirection. This is linked to the fact that the map where we swap time with aspace direction is not a Lorentz transformation. Note that permuting any twoaxes in the Riemannian-signature case is an O ( n ) transformation, while permut-ing time and space in the Lorentzian case is not an O (1 , n −
1) transformation.Therefore, there is no distinction between weakly and strongly I -non-degeneratein the Riemannian case.In most cases we do actually have a frame in which we know which directionis time. However, if we are only handed a set of invariants we would not havesuch a frame and, a priori, we would not know which eigenvalue is associatedwith time. We also note that the ambiguity in choosing a projection operator incertain cases is linked to the same problem; we do not necessarily know whicheigenvalue is associated with time.Therefore, the question of which I -non-degenerate metrics are strongly I -non-degenerate is linked to the question of when the time direction can beuniquely specified from the set of invariants.Consider an invariant I . Then we can consider the gradient, v µ ≡ ∇ µ I ,which is a curvature “vector”. Assume that the metric is I -non-degenerate,in which case we always have a timelike projection operator, ⊥ . Therefore,we can consider ⊥ v . Now, if ⊥ v (cid:54) = 0 then clearly it is timelike and theinvariant ( ⊥ v ) µ ( ⊥ v ) µ <
0. Therefore, we could uniquely specify time, be-cause ⊥ v would give us the time direction. So if there exists an invariant I for which ( ⊥ ) µν ∇ ν I is timelike (and non-zero), this spacetime is strongly I -non-degenerate.A similar conclusion is reached if we have three spacelike projection operatorsand all of these have similar non-zero gradients. To be more precise:2 A. Coley, S. Hervik and N. Pelavas Proposition 7.1.
Consider a (weakly) I -non-degenerate spacetime. Then, ifeither:1. there exists an invariant I = v µ v µ , where v µ is a curvature 1-tensor, suchthat I < ; or,2. there exist curvature 1-tensors v µ , u µ and w µ such that the invariants I = v µ v µ > , I = u µ u µ > , I = w µ w µ > , and I = v α u β w γ g [ αµ g βν g γ ] λ v µ u ν w λ (cid:54) = 0; then the spacetime is strongly I -non-degenerate.Proof. In case (1) we can construct a timelike projection operator, and theresult follows. In case (2) there exist three spacelike projection operators, andthe condition that I (cid:54) = 0 ensures that these are linearly independent. Hence,the timelike vector is orthogonal to these three and the result follows.Therefore, the only spacetimes that are weakly I -non-degenerate but notstrongly I -non-degenerate must have a timelike and a spacelike derivative whichannihilate all invariants. If the spacetime is weakly I -non-degenerate, but notstrongly I -non-degenerate, there must consequently exist a timelike vector, ξ ,and a spacelike vector, ξ , for which ξ ( I ) = ξ ( I ) = 0 , for all scalar invariants I . If [ ξ , ξ ] = ξ , it also follows that ξ ( I ) = 0.Therefore, there will be a set of vectors, { ξ i } , closed under commutation (conse-quently, the Jacobi identity will also be satisfied), which annihilates all curvatureinvariants. This has several consequences. First, this set will span a timelike(sub)manifold of dimension 2, 3 or 4. We can therefore locally introduce normalcoordinates, so that the invariants only depend on the normal coordinates; i.e., I = I ( x, y ) (dim 2), I = I ( x ) (dim 3) or I = constant (dim 4, and the spacetimeis a CSI spacetime). Second, by the assumption that this spacetime is weakly I -non-degenerate, and the fact that these invariants only depend on the coordi-nates ( x, y ), there exists an orthonormal frame such that all components of thecurvature tensors only depend on the normal coordinates ( x, y ) [12, 2].This indicates that these vectors that annihilate all invariants have a specialgeometric meaning. First, let us consider an arbitrary curvature tensor of rank( n, n ), R α ...α n β ...β n , being a sum, tensor products and contractions of theRiemann tensor and its covariant derivatives. Since this tensor has as manycovariant as contravariant indices, we can interpret this as a curvature operator, R ≡ R α ...α n β ...β n e α ⊗· · ·⊗ e α n ⊗ ω β ⊗· · ·⊗ ω β n , mapping rank n contravarianttensors into rank n contravariant tensors. Let us denote T End as the tensoralgebra of all such curvature operators. It is clear that all polynomial curvatureinvariants can be considered as complete contractions of operators in T End . Theorem 7.2.
Consider a spacetime which is (weakly) I -non-degenerate, anda vector field ξ . Then the following conditions are locally equivalent:1. ξ ( I ) = 0 for all curvature invariants I . pacetimes characterized by their scalar curvature invariants 23
2. The Lie derivative of any curvature operator R ∈ T End with respect to ξ ,vanishes; i.e., £ ξ R = 0 . Proof. (1) ⇒ (2): Assume that ξ ( I ) = 0 for all curvature invariants I . Considerthe 1-parameter group of diffeomorphisms, φ t , associated with the vector field ξ . Then ξ ( I ) = £ ξ ( I ) = lim t → t [ I − φ ∗ t ( I )] = 0 . Assuming the conditions hold over a neighborhood U , this can be integrated andwe get, at a point p ∈ U , I ( p ) = I ( φ − t ( p )). Hence, along the integral curvesthe value of the invariants do not change. Consider now the Lie derivative of anarbitrary curvature operator R = R α ...α n β ...β n e α ⊗ · · · ⊗ e α n ⊗ ω β ⊗ · · · ⊗ ω β n (e.g., see [12]): £ ξ R = lim t → t (cid:104) R − ˆ R t (cid:105) , where ˆ R t is the φ t -transformed tensor defined by:ˆ R t = φ t ( R ) = φ ∗ t ( R α ...α n β ...β n )( φ t ∗ e α ) ⊗· · ·⊗ ( φ t ∗ e α n ) ⊗ ( φ ∗ t ω β ) ⊗· · ·⊗ ( φ ∗ t ω β n ) . The action of φ t preserves the form and symmetries of a tensor. Thus thetransformed tensor ˆ R t will be a curvature tensor of the same kind as R . Thecurvature invariants at p will be I ( p ) for R and I ( φ − t ( p )) for ˆ R t . From the above,these invariants are the same and, from the assumption of I -non-degeneracy, theinvariants characterise the spacetime, which means that there exists a frame suchthat the components of the curvature tensors do not change along φ − t ( p ). Thisframe essentially is the eigenvalue frame of the curvature tensors. In particular,the projection operators define this frame.If v is an eigenvector of R , then Rv − λ v = 0 , ⇒ ˆ R t ˆ v t − λ ˆ v t = 0 , where hatted quantities are transformed under φ t . Eigenvectors are thereforetransformed onto eigenvectors of ˆ R t . Using the fact that there exists a frameso that φ ∗ t ( R α ...α n β ...β n ) = R α ...α n β ...β n , means that the components remainthe same in this frame. For a symmetric operator the eigenvectors are orthogonaland we can introduce a basis of orthonormal eigenvectors { e I } with duals { ω I } .Consider now a symmetric projection operator, ⊥ , written in the eigenvectorbasis: ⊥ = δ AB e A ⊗ ω B , ˆ ⊥ t = δ AB ˆ e A ⊗ ˆ ω B , where the indices run over a subset of eigenvectors with the same eigenvalue,and the hatted basis is the transformed basis. From the above discussion wesee that the eigenspaces are φ t -invariant, and hence there is a transformationmatrix M AB such that ˆ e A = M ˜ AA e ˜ A , and ˆ ω B = ( M − ) B ˜ B ω ˜ B . Consequently, ⊥ = ˆ ⊥ t , and the curvature projection operators are φ t -invariant. Therefore, since all R ∈ T End can be expanded in terms of these projection operators and the4 A. Coley, S. Hervik and N. Pelavascurvature invariants (since it is I -non-degenerate), we have that R = ˆ R t and (2)follows. (2) ⇒ (1): This follows trivially from the observation that ξ ( I ) = £ ξ ( I )and the properties of the Lie derivative. Corollary 7.3.
If there exists a non-zero vector field, ξ , fulfilling£ ξ R = 0 , for all curvature operators R ∈ T End , then the spacetime possesses a Killingvector field, ˆ ξ .Proof. This follows from the equivalence principle [2]. The Cartan scalars arerelated to the components of the Riemann tensor and its derivatives, and alongthe integral curves of ξ we can use φ t at any given point p . We want to comparethe tensors at p and q ≡ φ t ( p ). Consider an arbitrary even-ranked curvaturetensor R . By raising or lowering indices appropriately, we get an operator R .Since the Lie derivative of R along ξ vanishes, there is a frame such that R q and R p has identical components. Therefore, by raising and lowering the indicesappropriately, the components of R q and R p are also the same. The Cartaninvariants of R q and R p are therefore the same. For a curvature tensor, R , ofodd rank we consider R ⊗ R , which is of even rank, and use the fact that φ t iscontinuous in t . Therefore, there exists a frame such that all the componentsof any curvature tensor are identical at p and q . The equivalence principle nowimplies that φ t , for any given t , is an isometry; hence, there must exist a Killingvector field ˆ ξ which generates an isometry ˆ φ ˆ t such that ˆ φ ˆ t ( p ) = φ t ( p ).Note that in most cases ˆ ξ and ξ are the same. However, in some very specialcases with additional symmetries they need not be (although locally they are ofthe same causality; e.g., they are both timelike or both spacelike). For example,for flat space the curvature vanishes identically; hence, £ ξ R = 0 for all ξ andany curvature tensor R , although not all ξ are Killing vectors. However, inthese special cases there will always exist at least two Killing vectors.Therefore, to conclude: Corollary 7.4.
If a spacetime is weakly I -non-degenerate but not strongly I -non-degenerate, then it possesses locally (at least) one timelike Killing vectorand one spacelike Killing vector. In this paper we have addressed the question of what is the class of Lorentzianmanifolds that can be completely characterized by the scalar polynomial in-variants constructed from the Riemann tensor and its covariant derivatives. Inthe Riemannian case the manifold is always locally characterized by the scalarpolynomial invariants and, therefore, all of the Cartan invariants are relatedto the scalar curvature invariants [2]. We have generalized these results to theLorentzian case.We have introduced the important notion of I -non-degenerate spacetimemetrics. In order to prove the main theorems, which is done on a case-by-casepacetimes characterized by their scalar curvature invariants 25(depending on the algebraic type) using a boost weight decomposition, we haveintroduced an appropriate set of curvature operators and curvature projectors.In the (algebraically) general case we have shown that if a 4D spacetime is eitherRicci type I , Weyl type I or Riemann type I/G , then it is I -non-degenerate,which implies that the spacetime metric is determined by its curvature invariants(at least locally, in the sense explained above).For the algebraically special cases the Riemann tensor itself does not giveenough information to provide us with all the required projection operators, andit is also necessary to consider the covariant derivatives. In terms of the boostweight decomposition, for an algebraically special metric (which has a Riemanntensor with zero positive boost weight components) which is not Kundt, bytaking covariant derivatives of the Riemann tensor positive boost weight com-ponents are acquired and a set of higher derivative projection operators areobtained. Consequently, we found that if the spacetime metric is algebraicallyspecial, but ∇ R , ∇ (2) R , ∇ (3) R or ∇ (4) R is of type I or more general, the metricis I -non-degenerate.The remaining metrics which do not acquire a positive boost weight compo-nent when taking covariant derivatives have a very special curvature structure.Indeed, in our main theorem we proved that a spacetime metric is either I -non-degenerate or the metric is a Kundt metric. This is very striking result becauseit implies that a metric that is not determined by its scalar curvature invariantsmust be of Kundt form. The Kundt metrics which are not I -non-degeneratetherefore correspond to degenerate metrics in the sense that many such metricscan have identical scalar invariants. This exceptional property of the the degen-erate Kundt metrics essentially follows from the fact that they do not define aunique timelike curvature operator.The results in the case of Petrov type I spacetimes in 4D follow from theabove theorems. Although these results were not previously known, some partialresults for 4D Weyl (Petrov) type I spacetimes, which are consistent with theabove analysis, can be deduced from previous work. This is discussed in thenext section (also see Appendix D).Therefore, if a spacetime is I -non-degenerate and the algebraic type is ex-plicitly known (using, for example, the Pleba´nski notion for the Segre type inwhich commas are used to distinguish between timelike and spacelike eigenvec-tors and their associated eigenvalues, as is common in general relativity), thespacetime can be completely classified in terms of its scalar curvature invariants.There are a number of important consequences of the results obtained. Acorollary of the main theorem applied to spacetimes with constant curvatureinvariants (CSI) is a proof of the CSI-Kundt conjecture in 4D [13]. In futurework we will study CSI spacetimes in more detail [14].We then considered the inverse question: given a set of scalar polynomialinvariants, what can we say about the underlying spacetime? In 4D we canpartially characterize the Petrov type in terms of scalar curvature invariants. Inmost circumstances we only need some partial results or necessary conditions.For example, we found that if 27 J (cid:54) = I , or if 27 J = I but the invariants S (cid:54) = 0 or S (cid:54) = 0, then the spacetime is I -non-degenerate. Some results werethen presented in the remaining cases when both I and J are zero, and hencethe spacetime is VSI W .We also discussed whether a I -non-degenerate metric is unique under a dis-crete transformation. We introduced the notion strong and weak non-degeneracy.6 A. Coley, S. Hervik and N. PelavasWe provided a necessary criterion to determine spacetimes that are weakly I -non-degenerate but not strongly I -non-degenerate .Having determined when a spacetime is completely characterized by itsscalar curvature invariants, it is also of interest to determine the minimal setof such invariants needed for this classification. For example, in 4D there areresults on determining the Riemann tensor in terms of zeroth order scalar cur-vature invariants (and determining a minimal set of such invariants) [15]. It isalso of interest to study when a spacetime can be explicitly constructed fromscalar curvature invariants.This work is also of importance to the equivalence problem of characterizingLorentzian spacetimes (in terms of their Cartan scalars) [2]. Clearly, by knowingwhich spacetimes can be characterized by their scalar curvature invariants alone,the computations of the invariants (i.e., simple polynomial scalar invariants)is much more straightforward and can be done algorithmically (i.e., the fullcomplexity of the equivalence method is not necessary). On the other hand, theCartan equivalence method also contains, at least in principle, the conditionsunder which the classification is complete (although in practice carrying outthe classification for the more general spacetimes is difficult, if not impossible).Therefore, in a sense, the full machinery of the Cartan equivalence method isonly necessary for the classification of the degenerate Kundt spacetimes (whichwe shall address in future work).Let us briefly discuss this further in the context of two simple examples,which also serve to illustrate the results of the main theorem:1. The Schwarzschild vacuum type D spacetime is an example of an I -non-degenerate spacetime. In the canonical coordinate form of the metric asgiven in [16], the two scalar polynomial invariants C ≡ C abcd C abcd =48 m r − and ( ∇ C ) ≡ C abcd ; e C abcd ; e = 720( r − m ) m r − are function-ally independent and can be used to solve for r and m , and all of the al-gebraically independent Cartan scalars Ψ , ∇ Ψ (cid:48) , ∇ Ψ (cid:48) , and ∇ Ψ (cid:48) are consequently related to the polynomial curvature invariants C and( ∇ C ) [16]. In particular, Ψ = − mr − , ∇ Ψ (cid:48) = 12 m r − − mr − , sothat 48(Ψ ) = C and 120(Ψ )( ∇ Ψ (cid:48) ) = − ( ∇ C ) . We note that thesecond derivative Cartan scalars have the following boost weights: ∇ Ψ (cid:48) is +2, ∇ Ψ (cid:48) is -2 and ∇ Ψ (cid:48) is 0.2. A spatially homogeneous vacuum plane wave, which is a special subcase ofa Petrov type N vacuum spacetime admitting a covariantly constant nullvector, belongs to the class of vanishing scalar invariant (VSI) spacetimes[8] and is consequently an example of a degenerate Kundt spacetime. Sinceit is a VSI spacetime, all scalar polynomial invariants are zero. However,distinct VSI spacetimes give rise to a distinct set of Cartan scalars [2] (e.g.,in flat space all of the Cartan scalars are zero). A spatially homogeneousvacuum plane wave has two non-trivial Cartan scalars, ∇ Ψ (cid:48) and ∇ Ψ (cid:48) . We have addressed the question of what is the class of Lorentzian manifolds thatcan be completely characterized by the scalar polynomial invariants constructedfrom the Riemann tensor and its covariant derivatives. In particular, we provedpacetimes characterized by their scalar curvature invariants 27the result that this is true in the case of Petrov type I spacetimes in 4D. Thisresult was not previously known. However, some partial results for 4D Weyl(Petrov) type I spacetimes are known, which are consistent with the aboveanalysis. Let us review these results.Essentially, in the case of Petrov (Weyl) type I, there exists a unique frame sothat all components of the Riemann tensor are related to curvature invariants.Indeed, in general there are four different curvature invariants (e.g., correspond-ing to the complex invariants I and J ), so that all invariants (which depend on4 coordinates) are functionally dependent on these four invariants. Problemsarise in degenerate cases and cases with symmetries. It is also known that allPetrov type I spacetimes are completely backsolvable [15].Let us consider the Petrov type I case in more detail. From [17, 3] (alsosee Appendix D) it follows that if a 4D spacetime is of Petrov type I it can beclassified according to its rank and it is either:1. curvature class A (and the holonomy group is general and of type R ),2. curvature class C (and of holonomy type R or R , with restricted Segretype).Now, suppose the components of the Riemann tensor R a bcd are given in acoordinate domain U with metric g . In case (1), where the curvature class isof type A, for any other metric g (cid:48) with the same components R a bcd it followsthat g (cid:48) ab = αg ab (where α is a constant); i.e., the metric is determined up toa constant conformal factor and the connection is uniquely determined. Thisimplies that all higher order covariant derivatives of the Riemann tensor arecompletely determined; i.e., given R a bcd , all of the components of the covariantderivatives are determined and we only need classify the Riemann tensor itself.(Note that all of the scalar polynomial curvature invariants are then determined,at least up to an overall constant factor).We can then pass to the frame formalism and determine the frame compo-nents of the Riemann tensor (to do this we need the metric to determine theorthogonality of the frame vectors and hence construct the frame; since g isspecified up to an overall constant conformal factor, orthogonality is unique).The Petrov type I case is completely backsolvable [15] and hence the framecomponents are completely determined by the zeroth order scalar invariants.Therefore, it follows that the spacetime is completely characterized by its scalarcurvature invariants in this case.Let us now consider case (2), where the curvature class is C . Again, let ussuppose that the R a bcd are given in U with metric g . If g (cid:48) is any other metricwith the same R a bcd , it follows that g (cid:48) ab = αg ab + βk a k b (where α and β are constants). The equation R a bcd k d = 0 , (17)has a unique non-trivial solution for k ∈ T m M . Note that R a bcd k a = 0 impliesthat I k e = 0 and hence I = 0, where I is the Euler density: I ≡ [ R abcd R abcd − R ab R ab + R ] . R a bcd ; e k a (cid:54) = 0, then β = 0 and the metric is determined up to a constantconformal factor (and the holonomy type is R ). This is similar to the firstcase discussed above, but now some information on the covariant derivative ofthe Riemann tensor is necessary (to ensure R a bcd ; e k a (cid:54) = 0). Hence, first ordercurvature invariants are needed for the classification of the spacetime. Since R a bcd ; e k a = 0 implies that I k e = 0, where I ≡ [ R abcd ; e R abcd ; e − R ab ; c R ab ; c + R ,a R ,a ] , it follows that the invariant I (cid:54) = 0 implies that R a bcd ; e k a (cid:54) = 0 in this case.If R a bcd ; e k a = 0, then R a bcd k a ; e = 0, and since eqn. (17) has a uniquesolution, k a is recurrent. If k a is null, the spacetime is algebraically special, andsince we assume that the Petrov type is I, this is not possible. Hence, k a is (a)timelike (TL) or (b) spacelike (SL) and is, in fact, covariant constant (CC).In case (2a), the spacetime admits a TL CC vector field k a . The holonomyis R , with a TL holonomy invariant subspace which is non-degenerately re-ducible, and M is consequently locally (1 + 3) decomposable (and static). Thereexist local coordinates (with k = ∂∂t ) such that the metric is given by − dt + g αβ ( x γ ) dx α dx β ( α = 1 , ,
3) (18)where g αβ is independent of t . The metric is unique up to an overall constantscaling and a time translation t → λt , where λ = 1 + β/α (reflecting thenon-uniqueness of the TL CC vector up to a constant scaling λ ). All of thenon-trivial components of the Riemann tensor and its covariant derivatives areconstructed from the 3D positive definite metric g αβ , and can be classified bythe corresponding 3D Riemann curvature invariants. In this case (and case (2 b ))there is an ignorable coordinate and all invariants are functions of 3 independentfunctions; R a bcd ; e must be used to uniquely fix the frame, and hence we needinformation from the first order scalar invariants.In case (2b), the spacetime admits a SL CC vector field k a . The holonomy is R , there exists a holonomy invariant SL vector k a which is non-degeneratelyreducible, and M is this locally (3+1) decomposable. Choosing local coordinatesin which the SL CC vector k = ∂∂x , the metric is given by dx + g αβ dx α dx β ( α = 0 , ,
3) (19)and g αβ is independent of x . The metric is unique up to an overall constantconformal factor and a space translation x → λx ( λ = 1 + β/α ). Classificationnow reduces to the classification of the class of 3D Lorentzian spacetimes withLorentzian metric g αβ (the subclass such that (2) is of Petrov type I ). Wecan now iterate the procedure for 3D Lorentzian spacetimes (such that (2) isPetrov type I ). In the degenerate cases in which additional KV are admitted,we will be led to the locally homogeneous case, and hence the 4D Petrov typeI locally homogeneous spacetimes (which are characterized by their constantscalar invariants). Indeed, in 3D the Riemann tensor is completely determinedby the Ricci tensor. There always exists a frame in which the components ofthe Ricci tensor are constants [7] and so in this case the 4D spacetime is Petrovtype I and CH (curvature homogeneous [18]), and hence generically locallyhomogeneous.pacetimes characterized by their scalar curvature invariants 29 Acknowledgments
We would like to thank Robert Milson for useful comments and questions onour manuscript, and to Lode Wylleman for pointing out a mistake. This workwas supported by the Natural Sciences and Engineering Research Council ofCanada.
A Notation
Throughout we have used a Newman-Penrose (NP) tetrad given by e a = { (cid:96), n, m, m } with inner product η ab = −
10 0 − (20)and directional derivatives defined by D = (cid:96) µ ∇ µ , ∆ = n µ ∇ µ , δ = m µ ∇ µ . (21)Associated with an NP tetrad are the following definitions for the connectioncoefficients that appear frequently above κ = m µ D(cid:96) µ , σ = m µ δ(cid:96) µ , ρ = m µ δ(cid:96) µ (22)with the remaining ones being similarly defined. Given the frame components R abcd = R αβγδ e αa e βb e γc e δd , we have the definitions for the Weyl scalarsΨ = − C , Ψ = − C , Ψ = − C , Ψ = − C , Ψ = − C and the Ricci scalarsΦ = R , Φ = R , Φ = R Φ = ( R + R ) , Φ = R , Φ = R . (23)Given a covariant tensor T with respect to an NP tetrad (or null frame), theeffect of a boost (cid:96) (cid:55)→ e λ (cid:96) , n (cid:55)→ e − λ n allows T to be decomposed according toits boost weight T = (cid:88) b ( T ) b (24)where ( T ) b denotes the boost weight b components of T . An algebraic classifi-cation of tensors T has been developed [6, 5] which is based on the existence ofcertain normal forms of (24) through successive application of null rotations andspin-boost. In the special case where T is the Weyl tensor in four dimensions,this classification reduces to the well-known Petrov classification. However, theboost weight decomposition can be used in the classification of any tensor T inarbitrary dimensions. As an application, a Riemann tensor of type G has thefollowing decomposition R = ( R ) +2 + ( R ) +1 + ( R ) + ( R ) − + ( R ) − (25)0 A. Coley, S. Hervik and N. Pelavasin every null frame. A Riemann tensor is algebraically special if there exists aframe in which certain boost weight components can be transformed to zero,these are summarized in Table 3.A useful discrete symmetry is the following (orientation-preserving) Lorentztransformation: (cid:96) ↔ n, m ↔ ¯ m, (26)which interchanges the boost weights, ( T ) b ↔ ( T ) − b , and makes the replace-ments ( κ, σ, ρ, τ, (cid:15), β ) ↔ − ( ν, λ, µ, π, γ, α ) . (27) Riemann type Conditions
G —I ( R ) +2 = 0II ( R ) +2 = ( R ) +1 = 0III ( R ) +2 = ( R ) +1 = ( R ) = 0N ( R ) +2 = ( R ) +1 = ( R ) = ( R ) − = 0D ( R ) +2 = ( R ) +1 = ( R ) − = ( R ) − = 0O all vanish (Minkowski space) Table 3: The relation between Riemann types and the vanishing of boostweight components. For example, ( R ) +2 corresponds to the frame components R , R , R . B Some special operators
Consider the case where we have a tensor S µναβ , where S µναβ = S ( µν )( αβ ) = S αβµν , This tensor can be considered as an operator: S = ( S µναβ ) : V (cid:55)→ V, where V is the vector space of symmetric 2-tensors M µν .Therefore, we can consider the eigentensors of this map in the standardmanner. We can construct a set of projectors ⊥ A projecting onto each corre-sponding eigenspace. Assume that ⊥ is of rank 1 (as an operator). If M µν isthe corresponding (normalized) eigenvector, this means that( ⊥ ) µναβ = M µν M αβ . We can now consider the eigenvectors of M ≡ M µν . We are actually not consid-ering the operator M itself, but rather ⊥ . However, ⊥ can also be consideredas an operator: P : N µν (cid:55)→ M µα M νβ N αβ pacetimes characterized by their scalar curvature invariants 31Assume that v µ and w ν are eigenvectors of M with eigenvalues λ v and λ w ,respectively. Then, if N µν = v µ w ν , M µα M νβ N αβ = λ v λ w N µν , and is therefore an eigenvector of P with eigenvalue λ = λ v λ w . Clearly, v µ w ν has the same eigenvalue as w µ v ν , so we will not be able to distinguish theseusing projection operators. Furthermore, if λ v = ± λ w , then v µ v ν has the sameeigenvalue as w µ w ν .The above construction is useful in several cases. An example that recurs isthe case where M has two one-dimensional eigenspaces spanned by v µ and w µ ,say. Assume also that λ v = − λ w . Then, P has two projection operators:( P ) µναβ ∝ v µ w ν v α w β + w µ v ν w α v β , (28)( P ) µναβ ∝ v µ v ν v α v β + w µ w ν w α w β , (29)We see that this is somewhat unfortunate because in spite of the fact that M sees the difference between the vectors v µ and w ν , P does not. This isrelated to the fact that for some spacetimes there exists a discrete symmetrywhich interchanges two spacetimes with identical curvature invariants. Herethis manifests itself in that we cannot actually determine which eigenvectorcorrespond to which eigenvalue. C Algebraically special ∇ C The relationship between the invariants of the Weyl tensor and the Petrov typeis well known; however, this is not the case for the covariant derivative of theWeyl tensor. A similar analysis for ∇ C would require an algebraic classificationbased on its boost weight decomposition, and a complete set of its first orderinvariants. We do not attempt to solve this general problem but rather providesome relations relevant to our paper. Restricting attention to four dimensionswe define the following tensors T fghabe = C abcd ; e C cdfg ; h (30) T habe ijk = T fghabe C fgij ; k (31) , T habe ij = T fghabe C fgij (32) T h lmnabe k = T habe ijk C ijlm ; n (33)where the number above the tensor refers to the degree in ∇ C or C . All ofthese tensors are constructed purely from ∇ C with the exception of , T (which2 A. Coley, S. Hervik and N. Pelavasinvolves C ). Next, we consider the following first order invariants w , = T abeabe w , = T eababe w , = T e a ha e h (34) w , = , T eababe w , = , T beaabe w , = , T e b aa e b (35) w , = T e abkabe k w , = T heababe h w , = T h abeabe h (36) w , = T h ebaabe h w , = T abemabe m w , = T a ebmabe m (37) w , = T maebabe m (38)in which w n,i denotes the ith invariant of degree n in ∇ C or C . Since the alignedframes of ∇ C and C need not be the same, the w ,i are mixed invariants andthe remaining invariants are pure ∇ C invariants. For ∇ C and C algebraicallygeneral (type G ) we obtain the following syzygies w , + 2 w , − w , = 0 , w , + 2 w , + 2 w , = 0 , w , − w , = 0which are the result of identities, symmetries and dimensionally dependent rela-tions [19]. In subsequent calculations we always impose these syzygies so thatour set reduces to ten invariants. Now consider ∇ C of algebraically special type,which is obtained by setting the minimal number of appropriate boost weightcomponents to vanish. We obtain the following results:1. If ∇ C is type II or D (i.e., boost weight +3 , +2 , +1components vanish) then the syzygies S = 0 and S = 0 hold.2. If ∇ C is type G or type H (i.e., boost weight +3 vanish), or type I (i.e.,boost weight +3 , +2 vanish) then, in general, S (cid:54) = 0 and S (cid:54) = 0.The second statement refers to the most general types of G , H or I where nofurther algebraically special subcases are taken into account. Below are theexpressions for S and S . Note that S is linear in w , , and when S = 0, weuse this syzygy in the derivation of S ; hence these two invariant expressionsare generally independent. In type II or D we can regard S = 0 as expressingthe dependency of w , in terms of the other invariants of S . In S each ofthe w ,i appear quadratically whereas each of w ,i appear quartically thereforeone of these invariants is dependent with respect to the other invariants in S .Since these syzygies are of degree 8 and 16, and the invariants considered hereare of maximum degree 4, one would expect S and S to attain a simplerform if expressed in terms of higher degree invariants. These calculations were Thanks to Jose M. Martin-Garcia for pointing this out to us. w , does not appear in S . pacetimes characterized by their scalar curvature invariants 33performed with the aid of GRTensorII [20]. S = − w , w , w , − w , w , w , + 15872 w , w , w , +7424 w , w , w , − w , w , w , + 1216 w , w , +640 w , w , + (21504 w , + 2304 w , − w , − w , w , − w , + 2304 w , − w , + 768 w , ) w , − w , w , − w , w , − w , w , − w , w , − w , w , − w , w , + 224 w , w , + 7760 w , w , − w , w , − w , w , + 15232 w , w , + 2680 w , w , + 56320 w , w , +512 w , w , + 6400 w , w , − w , w , − w , w , − w , w , + 180 w , + 6464 w , + 58832 w , + 94208 w , − w , S = − w , w , w , w , + 2223360 w , w , − w , w , − w , w , − w , w , + 55490641920 w , w , − w , w , − w , w , + 148414464000 w , w , + 568104468480 w , w , − w , w , − w , w , + 3288600 w , w , − w , w , +68296366080 w , w , + 2340126720 w , w , + 3975480 w , w , + 305528832000 w , w , +187499520 w , w , + 3116666880 w , w , − w , w , + 12399045120 w , w , − w , w , − w , w , + 2817964800 w , w , + 2603059200 w , w , − w , w , + 22063680 w , w , + 62340480 w , w , + 41428800 w , w , − w , w , − w , w , + 1134028800 w , w , − w , w , +31371264000 w , w , − w , w , − w , w , + 71660160 w , w , +310187520 w , w , + 1551254400 w , w , − w , w , + 1095120 w , +231211008000 w , + 32785562880 w , + 496125 w , + 2437632000 w , − w , w , w , w , + 41724149760 w , w , w , w , − w , w , w , w , +8616960 w , w , w , w , + 109117440 w , w , w , w , + 61908480 w , w , w , w , − w , w , w , w , + 1940244480 w , w , w , w , + 2091409920 w , w , w , w , +414351360 w , w , w , w , + 52652160 w , w , w , w , + 106024320 w , w , w , w , +50016960 w , w , w , w , + 643184640 w , w , w , w , + 201784320 w , w , w , w , − w , w , w , w , + 1132830720 w , w , w , w , − w , w , w , w , − w , w , w , w , − w , w , w , w , + 1610933760 w , w , w , w , +195978240 w , w , w , w , − w , w , w , w , + 2208890880 w , w , w , w , − w , w , w , w , + 26542080000 w , w , w , w , − w , w , w , w , +3734138880 w , w , w , w , + 60439633920 w , w , w , w , + 19318947840 w , w , w , w , +18244730880 w , w , w , w , − w , w , w , w , + 28718530560 w , w , w , w , − w , w , w , w , + 3924910080 w , w , w , w , + 9899274240 w , w , w , w , +6060810240 w , w , w , w , + 304588800 w , w , w , w , + 1175224320 w , w , w , w , − w , w , w , w , − w , w , w , w , + 2413071360 w , w , w , w , +13086720 w , w , w , w , + 5733089280 w , w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , + 78222827520 w , w , w , − w , w , w , + 824785920 w , w , w , + 426528000 w , w , w , − w , w , w , − w , w , w , + 152928000 w , w , w , +90594754560 w , w , w , − w , w , w , − w , w , w , +2974187520 w , w , w , − w , w , w , − w , w , w , +91186560 w , w , w , + 277770240 w , w , w , + 59454259200 w , w , w , +202020480 w , w , w , + 38804520960 w , w , w , − w , w , w , +78681600 w , w , w , − w , w , w , − w , w , w , pacetimes characterized by their scalar curvature invariants 35 − w , w , w , + 396922429440 w , w , w , − w , w , w , +456929280 w , w , w , + 664450560 w , w , w , + 3913482240 w , w , w , +495043200 w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , + 44401582080 w , w , w , + 9839646720 w , w , w , − w , w , w , − w , w , w , + 205920 w , w , w , − w , w , w , − w , w , w , + 226496839680 w , w , w , − w , w , w , + 3500236800 w , w , w , − w , w , w , − w , w , w , + 1986324480 w , w , w , − w , w , w , +10468362240 w , w , w , − w , w , w , − w , w , w , +20755906560 w , w , w , + 101836800 w , w , w , + 237242880 w , w , w , +321085440 w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , − w , w , w , + 260743680 w , w , w , + 68417280 w , w , w , +65399685120 w , w , w , + 6782976000 w , w , w , w , w , − w , w , w , − w , w , w , + 4393267200 w , w , w , − w , w , w , − w , w , w , w , w , + 780894720 w , w , w , w , w , − w , w , w , w , w , + 14745600 w , w , w , w , w , − w , w , w , w , w , + 1975910400 w , w , w , w , w , − w , w , w , w , w , − w , w , w , w , w , − w , w , w , w , w , D Curvature
Let M be a 4-dimensional smooth connected Hausdorff manifold admitting aglobal smooth Lorentz metric h with associated curvature tensor R . It will beconvenient to describe a simple algebraic classification of R according to itsrank (relative to h ). This classification is easily described geometrically and isa pointwise classification [3].A skew-symmetric tensor F of type (0 ,
2) or (2 ,
0) at m ∈ T m M is calleda bivector . If F ( (cid:54) = 0) is such a bivector, the rank of any of its (component)matrices is either two or four. In the former case, one may write (e.g. in the(2 ,
0) case) F ab = 2 r [ a s b ] for r, s ∈ T m M (or alternatively, F = r ∧ s ) and F iscalled simple , with the 2-dimensional subspace (2-space) of T m M spanned by r, s referred to as the blade of F . In the latter case, F is called non-simple .The metric h ( m ) converts T m M into a Lorentz inner product space and thusit makes sense to refer to vectors in T m M and covectors in the cotangent space T ∗ m M to M at m (using h ( m ) to give a unique isomorphism T m M ↔ T ∗ m M ,that is, to raise and lower tensor indices) as being timelike , spacelike , null or orthogonal , using the signature ( − , + , + , +). The same applies to 1-dimensionalsubspaces ( directions ) and 2- and 3-dimensional subspaces of T m M or T ∗ m M . A simple bivector at m is then called timelike (respectively, spacelike or null ) if6 A. Coley, S. Hervik and N. Pelavasits blade at m is a timelike (respectively a spacelike or null) 2-space at m . A non-simple bivector F at m may be shown to uniquely determine an orthogonalpair of 2-spaces at m , one spacelike and one timelike, and which are referred toas the canonical pair of blades of F . A tetrad ( l, n, x, y ) of members of T m M iscalled a null tetrad at m if the only non-vanishing inner products between itsmembers at m are h ( l, n ) = h ( x, x ) = h ( y, y ) = 1. Thus l and n are null. D.1 Classification
Define a linear map f from the 6-dimensional vector space of type (2 ,
0) bivectorsat m into the vector space of type (1 ,
1) tensors at m by f : F ab → R abcd F cd .The condition (2) shows that if a tensor T is in the range of f then h ae T eb + h be T ea = 0 ( ⇒ T ab = − T ba , T ab = h ae T eb ) (39)and so T can be regarded as a member of the matrix representation of the Liealgebra of the pseudo-orthogonal (Lorentz) group of h ( m ). Using f one candivide the curvature tensor R ( m ) into five classes. Class A This is the most general curvature class and the curvature will be saidto be of (curvature) class A at m ∈ M if it is not in any of the classes B , C , D or O below. Class B The curvature tensor is said to be of (curvature) class B at m ∈ M ifthe range of f is 2-dimensional and consists of all linear combinations oftype (1 ,
1) tensors F and G where F ab = x a y b − y a x b and G ab = l a n b − n a l b with l, n, x, y a null tetrad at m . The curvature tensor at m can then bewritten as R abcd ≡ h ae R ebcd = α F ab F cd − β G ab G cd (40)where α, β ∈ R , α (cid:54) = 0 (cid:54) = β . Class C The curvature tensor is said to be of (curvature) class C at m ∈ M if the range of f is 2- or 3-dimensional and if there exists 0 (cid:54) = k ∈ T m M such that each of the type (1 ,
1) tensors in the range of f contains k in itskernel (i.e. each of their matrix representations F satisfies F ab k b = 0). Class D The curvature tensor is said to be of (curvature) class D at m ∈ M ifthe range of f is 1-dimensional. It follows that the curvature componentssatisfy R abcd = λF ab F cd at m (0 (cid:54) = λ ∈ R ) for some bivector F at m whichthen satisfies F a [ b F cd ] = 0 and is thus simple. Class O The curvature tensor is said to be of (curvature) class O at m ∈ M ifit vanishes at m .The following results are useful [3]:1. For the classes A and B there does not exist 0 (cid:54) = k ∈ T m M such that F ab k b = 0 for every F in the range of f .2. For class A , the range of f has dimension at least two and if this dimensionis four or more the class is necessarily A .pacetimes characterized by their scalar curvature invariants 373. The vector k in the definition of class C is unique up to a scaling.4. For the classes A and B there does not exist 0 (cid:54) = k ∈ T m M such that R abcd k d = 0, whereas this equation has exactly one independent solutionfor class C and two for class D .5. The five classes A , B , C , D and O are mutually exclusive and exhaustivefor the curvature tensor at m . If the curvature class is the same at each m ∈ M then M will be said to be of that class. D.2 Properties
Suppose that the components of the Riemann tensor R abcd are given in a coordi-nate domain U with metric h . Suppose that h (cid:48) is another metric with the samecomponents R abcd . It follows from [17] that:Class A h (cid:48) ab = αh ab (42 a )Class B h (cid:48) ab = αh ab + 2 βl ( a n b ) = ( α + β ) h ab − β ( x a x b + y a y b ) (42 b )Class C h (cid:48) ab = αh ab + βk a k b (42 c )Class D h (cid:48) ab = αh ab + βr a r b + γs a s b + 2 δr ( a s b ) (42 d )where α, β, γ, δ ∈ R .Note that h ab ; c = h ab w c for some smooth 1-form w on the open subset A .Using condition (2) above and the Ricci identity we get h ab ;[ cd ] = 0, whichimplies h ab w [ c ; d ] = 0; thus w [ c ; d ] = 0 and so w a is locally a gradient. Hence, foreach m ∈ A , there is an open neighborhood W of m on which w a = w ,c for somesmooth function w . Then on W , g ab = e − w h ab satisfies g ab ; c = 0. Further, if g (cid:48) is any other local metric defined on some neighborhood W (cid:48) of m and compatiblewith Γ then g (cid:48) satisfies condition (2) on W (cid:48) and hence, on W ∩ W (cid:48) , g (cid:48) = φg forsome positive smooth function φ . From this and the result g (cid:48) ab ; c = 0 it followsthat g (cid:48) is a constant multiple of g on W ∩ W (cid:48) . References [1] A.Z. Petrov,
Einstein spaces (Pergamon, 1969)[2] H. Stephani, D. Kramer, M. A. H. MacCallum, C. A. Hoenselaers, E. Herlt2003
Exact solutions of Einstein’s field equations, second edition (Cam-bridge University Press; Cambridge).[3] G S Hall, 2004,
Symmetries and curvature structure in general Relativity (World Science, Singapore).[4] A. Coley, 2008, Class. Quant. Grav. , 033001.[5] R. Milson, A. Coley, V. Pravda and A. Pravdova, 2005, Int. J. Geom. Meth.Mod. Phys. , 41.[6] A. Coley, R. Milson, V. Pravda and A. Pravdova, 2004, Class. Quant. Grav. , L35.[7] A. Coley, S. Hervik and N. Pelavas, 2006, Class. Quant. Grav. , 3053.8 A. Coley, S. Hervik and N. Pelavas[8] V. Pravda, A. Pravdov´a, A. Coley and R. Milson, 2002, Class. Quant.Grav. , 6213.[9] A. Coley, R. Milson, V. Pravda and A. Pravdova, 2004, Class. Quant. Grav. , 5519.[10] A. Coley, A. Fuster, S. Hervik, N. Pelavas, 2006, Class. Quant. Grav. ,7431[11] F. Pr¨ufer, F. Tricerri and L. Vanhecke, 1996, Trans. American Math. Soc., , 4643.[12] S Kobayashi and K. Nomizu, 1963, Foundations of Differential Geometry,Volume 1 (Interscience Publishers).[13] A. Coley, S. Hervik and N. Pelavas, 2008, Class. Quantum Grav. ,025008.[14] A. Coley, S. Hervik and N. Pelavas, 2009, Class.Quant.Grav. , 1474; J. Carminati,E. Zakhary, and R. G. McLenaghan, 2002, J. Math. Phys. , 492; J.Carminati and E. Zakhary, 2002, J. Math. Phys. , 4020[16] F. M. Paiva, M. J. Reboucas and M. A. H. MacCallum, 1993, Class. Quant.Grav. , 1165.[17] G S Hall and W Kay, 1988, J. Math. Phys. , 428[18] R. Milson and N. Pelavas, 2008, Class. Quantum Grav.586-590, arXiv:0802.1274 [cs.SC][20] This is a package which runs within Maple. It is entirely distinct frompackages distributed with Maple and must be obtained independently. TheGRTensorII software and documentation is distributed freely on the World-Wide-Web from the address