aa r X i v : . [ m a t h - ph ] A p r Pseudo-Riemannian VSI spaces II
Sigbjørn Hervik
Faculty of Science and Technology,University of Stavanger,N-4036 Stavanger, Norway [email protected]
February 15, 2018
Abstract
In this paper we consider pseudo-Riemannian spaces of arbitrary sig-nature for which all of the polynomial curvature invariants vanish (VSIspaces). Using an algebraic classification of pseudo-Riemannian spacesin terms of the boost-weight decomposition we first show more generallythat a space which is not characterised by its invariants must possess the S G -property. As a corollary, we then show that a VSI space must possessthe N G -property (these results are the analogues of the alignment theo-rem, including corollaries, for Lorentzian spacetimes). As an applicationwe classify all 4D neutral VSI spaces and show that these belong to oneof two classes: (1) those that possess a geodesic, expansion-free, shear-free, and twist-free null-congruence (Kundt metrics), or (2) those thatpossess an invariant null plane (Walker metrics). By explicit constructionwe show that the latter class contains a set of VSI metrics which have notpreviously been considered in the literature. In this paper we will consider an arbitrary-dimensional pseudo-Riemannianspace of signature ( k, k + m ). We will investigate when such a space has adegenerate curvature structure; in particular, we shall determine criteria forwhen a space, or tensor, has all vanishing polynomial curvature invariants (VSIspace). Recall that a polynomial curvature invariant is defined as the polyno-mial invariants of the components of the curvature tensors. Previously, the VSIspaces for Lorentzian metrics have been studied [1] and it was shown that thesecomprise a subclass of the degenerate Kundt metrics [2]. Here, we will see thatKundt-like metrics also play a similar role for pseudo-Riemannian VSI metricsof arbitrary signature, however, we will see that another class of metrics arisesin the pseudo-Riemannian case, namely the Walker metrics [3]. In order toobtain these results we will utilise invariant theory to obtain important prop-erties of the structure of tensors having degenerate invariants. In particular,tensors not characterised by their invariants will be shown to possess the S G -property, while in the VSI case they necessarily must possess the N G -property.1 S. HervikWe will use this fact to construct a new set of 4-dimensional Walker metricswith vanishing curvature invariants of neutral signature.Walker metrics are metrics possessing a invariant null-plane and have beenstudied in various contexts [3, 4]. Here we will show that they also play a role inthe classification of VSI metrics. Indeed, we will give a new class of VSI metricswhich has not been considered before. These metrics are related to a biggerclass of Walker metrics with a degenerate curvature structure. The curvaturestructure of these metrics are distinct from the Kundt metrics known from theLorentzian case. One of the consequences of this district feature is that we needto consider invariants containing up to four derivatives. Indeed, interestingly,there is a family of Walker metrics which is VSI , but not VSI : the perhapssimplest member of this family is ds = 2 du ( dv + V du ) + 2 dU ( dV + av dU ) , (1)where a is a constant. This peculiar property of being VSI but not VSI hasno analogue in the Lorentzian case .First we will review some of the techniques used in this paper. Then we willprovide with the general result for tensors (or spaces) not being characterisedby its invariants. This result is the analogue of the alignment theorem in theLorentzian-signature case [5]. Then, as a corollary, we will state the importantVSI case. We will then use this VSI result to consider the 4-dimensional neutralcase in detail. Let us first review the boost weight classification, originally used to study de-generate metrics in Lorentzian geometry [6], in the pseudo-Riemannian case[7]. We will assume the manifold is of dimension (2 k + m ) and of signature( k, k + m ). We first introduce a suitable (real) null-frame such that the metriccan be written as:d s = 2 (cid:16) ℓ n + · · · + ℓ I n I + · · · + ℓ k n k (cid:17) + δ ij m i m j , (2)where the indices i = 1 , . . . , m .Let us consider the k independent boosts which forms an abelian subgroupof the group SO ( k, k + m ):( ℓ , n ) ( e λ ℓ , e − λ n )( ℓ , n ) ( e λ ℓ , e − λ n )...( ℓ k , n k ) ( e λ k ℓ k , e − λ k n k ) . (3)This action will be considered pointwise at the manifold.For a tensor T , we can then consider the boost weights of this tensor, b ∈ Z k ,as follows. If we consider the components of T with respect to the above-mentioned null-frame then if a component T µ ...µ n transforms as: T µ ...µ n e ( b λ + b λ + ... + b k λ k ) T µ ...µ n , In the Lorentzian case, VSI implies VSI [1], while VSI Kundt implies VSI [2]. seudo-Riemannian VSI metrics II 3then we will say the component T µ ...µ n is of boost weight b ≡ ( b , b , ..., b k ).We can now decompose a tensor into boost weights; in particular, T = X b ∈ Z k ( T ) b , where ( T ) b means the projection onto the components of boost weight b . Theprojections ( T ) b are the eigentensors of a set of commuting operators (the in-finitesimal generators of the boosts) with integer eigenvalues. For example, atensor P = A ℓ I n J m i m j with I = J and A is some scalar, has boost weight b = ( b , ..., b k ) where b I = − b J = 1, other b i = 0. Indeed, writing out atotally covariant tensor T using the basis in (2), the boost weight is given by b = ( b I ) where b I = n I ) − ℓ I ).By considering tensor products, the boost weights obey the following additiverule: ( T ⊗ S ) b = X ˜ b +ˆ b = b ( T ) ˜ b ⊗ ( S ) ˆ b . (4)We also note that the metric g is of boost weight 0, i.e., g = ( g ) ; hence, raisingand lowering indices of a tensor do not change the boost weights. i - and N-properties Let us consider a tensor, T , and list a few conditions that the tensor componentsmay fulfill [7, 8]: Definition 1.1.
We define the following conditions:B1) ( T ) b = 0, for all b = ( b , b , b , ..., b k ), b > T ) b = 0, for all b = (0 , b , b , ..., b k ), b > T ) b = 0, for all b = (0 , , b , ..., b k ), b > k ) ( T ) b = 0, for all b = (0 , , ..., , b k ), b k > Definition 1.2.
We will say that a tensor T possesses the S -property if andonly if there exists a null frame such that condition B1) above is satisfied.Furthermore, we say that T possesses the S i -property if and only if there existsa null frame such that conditions B1)-B i ) above are satisfied. Definition 1.3.
We will say that a tensor T possesses the N -property if andonly if there exists a null frame such that conditions B1)-B k ) in definition 1.1are satisfied, and ( T ) b = 0 , for b = (0 , , ..., , . Let us also recall the following result [7, 8]:
Proposition 1.4.
For tensor products we have:1. Let T and S possess the S i - and S j -property, respectively. Assuming, withno loss of generality, that i ≤ j , then T ⊗ S possesses the S i -property. S. Hervik
2. Let T and S possess the S i - and N -property, respectively. Then T ⊗ S possesses the S i -property. If i = k , then T ⊗ S possesses the N -property.3. Let T and S both possess the N -property. Then T ⊗ S , and any contractionthereof, possesses the N -property. We extend this and define a set of related conditions which will prove usefulto us. Consider a tensor, T , that does not necessarily meet any of the conditionsabove. However, since the boost weights b ∈ Z k ⊂ R k , we can consider a linear GL ( k ) transformation, G : Z k Γ, where Γ is a lattice in R k . Now, if thereexists a G such that the transformed boost weights, G b , satisfy (some) of theconditions in Def.1.1, we will say, correspondingly, that T possesses the S Gi -property. Similarly, for the N G -property.If we have two tensors T and S both possessing the S Gi -property, with thesame G , then when we take the tensor product:( T ⊗ S ) G b = X G ˆ b + G ˜ b = G b ( T ) G ˆ b ⊗ ( S ) G ˜ b . Therefore, the tensor product will also possess the S Gi -property, with the same G . This will be useful later when considering degenerate tensors and metricswith degenerate curvature tensors. Note also that the S Gi -property reduces tothe S i -property for G = I (the identity). Another useful concept is the question when a tensor/space-time is “charac-terised by its invariants”. Henceforth, by invariants we will always mean the polynomial invariants . Such have been discussed in several papers both in theLorentzian case, as well as in the more general case [9, 10].We will now recall some of the definitions and concepts from invariant theory,see e.g., [11, 12, 13]. For a tensor T , we define the action of the semi-simplegroup G = O ( k, k + m ) on the components of T as follows. For simplicity,assume that the components of T have been lowered: T a ...a p . We form the N -tuple consisting of the components of T as X = [ T a a ...a p ] ∈ R N . Theaction corresponds to a frame rotation and explicitly, if we consider the matrix g = ( M ab ) ∈ O ( k, k + m ), acting as a frame rotation gω = { M a e a , ..., M an e a } ,the frame rotation induces an action on X through the tensor structure of thecomponents: g ( X ) = h M b a ...M b p a p T b ...b p i . The (real) orbit O ( X ) is now defined by: O ( X ) ≡ { g ( X ) ∈ R N (cid:12)(cid:12) g ∈ O ( k, k + m ) } ⊂ R N . We can then extend this definition to a direct sum of vectors, T = T (1) ⊕ ... ⊕ T ( q ) .The action g ( X ) on the components are then extended through the standarddirect-sum representaion of the group G acting on the direct sum of tensors.In the case of a pseudo-Riemannian space, T is a direct sum of the curvaturetensors, T = Riem ⊕ ∇ Riem ⊕ ∇∇
Riem ⊕ ... ⊕ ∇ ( K ) Riemup to some sufficiently high order K .seudo-Riemannian VSI metrics II 5 Definition 1.5.
A tensor T (or pseudo-Riemannian space) is characterised byits invariants if and only if the corresponding orbit O ( X ) is topologically closedin R N with respect to the standard Euclidean topology.The motivation for this definition is given in [5] – essentially, the set of closedorbits: C = {O ( X ) ⊂ V (cid:12)(cid:12) O ( X ) closed } , is parameterised by the invariants, possibly up to a complex rotation (indeed,the complexified orbits are parameterised uniquely, the real orbits intersect thesea finite number of times). For more on these issues we would refer the reader to [11, 12, 13, 5].
A tensor, T , satisfying the S Gi -property or N G -property is not generically de-termined by its invariants in the sense that there may be another tensor, T ′ ,with precisely the same invariants. The S Gi -property thus implies a certain degeneracy in the tensor.Indeed: Theorem 2.1.
A tensor T is not characterised by its invariants if and only ifit possesses (at least) the S G -property.Proof. Assume that T is not charaacterised by its invariants; i.e., the corre-sponding orbit is not closed. Using the results of Richardson-Slodowy [12],there then exists a X ∈ B , where B is the vector subspace of the Lie al-gebra so ( k, k + m ) consisting of symmetric matrices (so that so ( k, k + m ) = B ⊕ K , where K is the Lie algebra of the maximal compact subgroup), such thatexp( τ X )( T ) → p . We note that the maximal compact subgroup of SO ( k, m + k )is K ∼ = SO ( k ) × SO ( m + k ), which we represent as g = ( g , g ) ∈ SO ( k ) × SO ( m + k ). The X can be represented as: X = (cid:20) k SS t m + k (cid:21) , (5)where S is an k × ( k + m ) matrix. The transformation, g − X g induces a trans-formation of S according to g − Sg , ( g , g ) ∈ SO ( k ) × SO ( m + k ). Thus by thesingular value decomposition we can always find a g ∈ K such that S is diagonal: S = diag ( λ , ..., λ k ). This therefore corresponds to a pure boost; specifically,by applying a null-frame the X will be represented the boost given in eq.(3).Henceforth, let us represent λ = ( λ , .., λ k ) as a vector. Then if the tensor T isdecomposed using the corresponding boost-weight components relative to thenull-frame; i.e., T = P b ( T ) b , we can write:exp( τ X )( T ) b = exp( τ b · λ )( T ) b (6)where b · λ = P ni =1 b i λ i . In the limit τ → ∞ , exp( τ X )( T ) has to approach p which is finite: hence, if ( T ) b = 0 we get the requirement b · λ ≤
0. In In [12] they denote this set as
V//G . S. Hervikparticular, exp( τ b · λ )( T ) b → ( T ) b , b · λ = 0 , exp( τ b · λ )( T ) b → , b · λ < , (7)all other ( T ) b must be zero:( T ) b = 0 , b · λ > . (8)Using a G ∈ O ( k ) transformation in boost-weight space we can align λ withthe first basis vector so that G λ = | λ | (1 , , , ... T fulfills the S G -property. For the VSI spaces we now get an important corollary:
Theorem 2.2.
For a tensor T in pseudo-Riemannian space the following isequivalent:1. T has only vanishing polynomial invariants (VSI).2. Any operator constructed from T (by rasing/lowering indices, contrac-tions, and tensor products) is nilpotent.3. T possesses the N G -property.Proof. The proof of 1 ⇔ ⇒ ⇒ T = 0 has a closedorbit, and since the complex orbit consists of only the zero element, the zero-tensor must be the unique tensor which has closed (real) orbits. Thus, we canchoose the limit in the proof to be p = 0. This implies that eq.(8) turns intothe stronger requirement: ( T ) b = 0 , b · λ ≥ . (9)By the same transformation matrix G , we write G λ = | λ | (1 , , , ...,
0) and the N G -property follows. The even-dimensional case with signature ( k, k ) (i.e., m = 0) is called the neutral case. Let us consider the 4D neutral case which is of particular interest (see,e.g., [14, 15]); in particular, we will use the above theorem to find all neutral VSIspaces of dimension 4. Such spaces has been studied before, however, only spacessatisfying the N -property were investigated. Although it was noted that the N G -property was sufficient for VSI this possibility was not investigated in detail.Indeed, we will show that there are VSI spaces satisfying the N G -property, butseudo-Riemannian VSI metrics II 7not the N -property thus establishing a new class of VSI spacetimes. We alsoderive all such metrics and show that they are all Walker metrics possessing aninvariant null-plane.In 4D neutral signature we thus get two classes of metric, the Kundt metricsand the Walker metrics. These will be reviewed in what follows. We will alsoutilise the work of Law [4] where all the spin-coefficients of 4D neutral space wereinvestigated. Using Law’s notation, we adopt the sligthly modified null-frame( ℓ , n , m , ˜ m ) ≡ ( ℓ , n , ℓ , − n ) so that metric (2) can be written: ds = 2 ℓn − m ˜ m . (10)In the neutral case, this frame is purely real. With respect to such a frame, Lawdefined the spin-coefficients which we will use in proving the main theorem.In [4] Law writes the spin-coefficients in terms of κ, ρ, σ, τ, ǫ, α, β, γ ,and their tilded (˜ κ, ˜ ρ, ... .), primed ( κ ′ , ρ ′ , ... .), and primed-tilded (˜ κ ′ , ˜ ρ ′ , ... .)counterparts. All these spin-coefficients are real. For example, the covariantderivatives of the frame-vector ℓ a can be written as: ℓ b ∇ b ℓ a = ( ǫ + ˜ ǫ ) ℓ a + ˜ κm a + κ ˜ m a , ˜ m b ∇ b ℓ a = ( α + ˜ β ) ℓ a + ˜ σm a + ρ ˜ m a ,m b ∇ b ℓ a = (˜ α + β ) ℓ a + ˜ ρm a + σ ˜ m a ,n b ∇ b ℓ a = ( γ + ˜ γ ) ℓ a + ˜ τ m a + τ ˜ m a , (11) ... etc.We refer to [4], in particular, eqs.(2.10-2.11) therein, for details. Here, we will consider the 4D neutral spaces which possesses an invariant null-plane. Such metrics are known as
Walker metrics .Consider two orthogonal null-vectors ℓ and m . These span an invariantnull-plane iff ∇ a ( ℓ ∧ m ) = k a ( ℓ ∧ m ) , (12)for a vector k a . Using [4] this immediately implies the vanishing of certainspin-coefficients : κ = ρ = σ = τ = 0 . Indeed, one can see that the vanishing of these spin-coefficients imply the exis-tence of an invariant null-plane (hence, it is a Walker metric).Furthermore, Walker [3] showed that the requirement of an invariant 2-dimensional null plane implies that the (Walker) metric can be written in thecanonical form:d s = 2d u (d v + A d u + C d U ) + 2d U (d V + B d U ) , (13)where A , B and C are functions that may depend on all of the coordinates.In particular, this implies that we can choose a frame such that [4] κ = ρ = σ = τ = ǫ = β = 0 , α ′ = γ ′ = ρ ′ = τ ′ = 0 , (14)˜ κ = ˜ ρ = ˜ α = ˜ ǫ = 0 , ˜ β ′ = ˜ γ ′ = ˜ σ ′ = ˜ τ ′ = 0 . (15)We note that ˜ σ needs not be zero, and hence, these Walker metrics need not beKundt spacetimes (see below). S. Hervik In the Lorentzian case the Kundt metrics play an important role for degeneratemetrics, and VSI metrics in particular [1]. Their pseudo-Riemannian analoguesalso play an important role for pseudo-Riemannian spaces of arbitary signature[8, 15].We define the pseudo-Riemannian Kundt metrics in a similar fashion, namely:
Definition 3.1.
A pseudo-Riemannian Kundt metric is a metric which pos-sesses a non-zero null vector ℓ which is geodesic, expansion-free, twist-free andshear-free.This implies that, in terms of the spin-coefficients defined in [4], that a spaceis Kundt if and only if there exists a frame such that:˜ κ = κ = ˜ ρ = ρ = ˜ σ = σ = 0 . (16)Therefore, we will consider metrics of the form (which is equivalent to theabove definition)d s = 2d u (cid:2) d v + H ( v, u, x C )d u + W A ( v, u, x C )d x A (cid:3) + g AB ( u, x C )d x A d x B (17)(here, the indices A, B range over the null-indices I = 2 ,
3. The metric (17)possesses a null vector field ℓ obeying ℓ µ ; ν = L ℓ µ ℓ ν + L i ℓ ( µ m iν ) + ˜ L i ℓ ( µ ˜ m iν ) , and consequently it is geodesic, non-expanding, shear-free and non-twisting.Since this is a pseudo-Riemannian space of signature (2 , s = g AB ( u, x C )d x A d x B , will be of signature (1 , Let us now state an important result regarding the deterimination of all 4D VSImetrics.
Theorem 3.2.
A 4D Neutral VSI metric is of one (or both) of the followingtypes:1. A Walker metric possessing an invariant 2-dimensional null-plane.2. A Kundt metric.
In order to prove this theorem one needs to consider theorem 2.1 and con-sider the covariant derivatives ∇ ( N ) (Riemann). We will prove the theorem usingtwo different methods, one is the more indirect method using the one-parameterfamily of boosts B τ = e τ X , the other is the direct method by explicitly com-puting the covariant derivatives. These two illustrate two conseptually differentmethods and both provide us with separate information about the underlyingstructure of these spaces. For example, while the first is a more ’elegant’ proof,the second gives some information of how many derivatives are necessary andprovides with more details about the various special cases. If, in addition L i = ˜ L i = 0, the vector ℓ µ is also recurrent (hence, Walker), and if L i = ˜ L i = L = 0, then ℓ µ is covariantly constant. seudo-Riemannian VSI metrics II 9 I. The boost method
Let us employ the frame which is aligned with thefamily of boosts B τ = exp( τ X ) providing us with the limit in Theorem 2.1. Thisis a pointwise action but consider a point p and assume this is regular implyingthat there exists a neighbourhood U such that the algebraic structure of thespace does not change over U . Consider now a compact K ⊂ U neighbourhoodof p . The boost B τ acts pointwise, however, since K is compact, we can assumethat the B τ does not depend on the point in K . Thus, with respect to theadapted frame, the boost will be constant over K : ℓ e − τλ ℓ , n e τλ n , m e − τλ m , ˜ m e τλ ˜ m Note that such a boost will transform the curvature tensors at p as follows:exp( τ X )( T ) b = exp( τ b · λ )( T ) b . (18)Now, in relation to the ǫ -property [10], we have that this boost manifests thelimit: X = ˜ X + N. Furthermore, since K is compact, || N || will have a maximum, N max , over K sothat || N || ≤ N max ; consequently, || X − ˜ X || ≤ N max . In the VSI case, e X = 0, so that X = N and the ǫ -property implies the compo-nents can be arbitrary close to flat space.Consider now the action of the boost B τ . The vector N is a direct sum oftensorial objects implying that, since it must be of type III, or simpler, thatthere is an a > || B τ ( N ) || ≤ e − aτ || N || ≤ e − aτ N max . We can assume that the neighbourhood U is a coordinate patch and map U into R with p at the origin. Then we can assume that the compact neighbourhood K ⊂ R . We now consider the X = N as a set of differential equations on U asfollows:Express the components of the Riemann tensor (relative to the adaptedframe) in terms of the spin-coefficients Γ µαβ in the standard way: R µαβν = ∂ ν (Γ µαβ ) − ∂ β (Γ µαν ) + (Γ ⋆ Γ) µαβν , (19)where Γ ⋆ Γ indicates the quadratic terms in the spin-coefficients. Similarly, thecovariant derivatives, can also be expressed using the spin-coefficients: ∇ R = ∇ R ( ∂∂ Γ , ∂ Γ , Γ) , ∇∇ R = ∇∇ R ( ∂∂∂ Γ , ∂∂ Γ , ∂ Γ , Γ) , etc.We thus replace the left-hand side of X = N with a PDE: Pde [Γ] =
N . (20)The relation between the frame ∂ α and Γ are given via:[ ∂ α , ∂ β ] = − (Γ µαβ − Γ µβα ) ∂ µ . (21) In the sense of [16]; i.e., the number of independent Cartan invariants do not change at p . U in terms of the functions Γ µαβ . We can now considerthe “boosted” set of equations Pde [ b Γ] = B τ ( N ) (22)over U . This gives us a one-parameter family of equations. Since B τ ( N ) can bemade arbitrary small, this can be seen as a pertubation of a PDE describing flatspace. Let us now consider the Cartan equivalence problem [17] which will giveus a more direct perturbation. Let us make sure we consider sufficient number ofderivatives in X to satify the Cartan bound. Consider the point p . For every τ there is an inverse boost so that the B τ ( N ) is mapped onto X = N . Consideringthe boost that leaves the point p fixed, then the equivalence principle impliesthat there exists a diffeomorphsim φ τ that maps K onto φ τ ( K ), leaving p fixed,and induces (through φ ∗ τ ) the boost B τ acting on the tangent space at p . Thediffeomorphism does not necessarily map K into itself. Consider an increasingsequence τ n such that τ n → ∞ , and define K n = φ τ n ( K ), which is compact. Inparticular, K n is closed and p ∈ K n . This implies further that p ∈ K ∩ ( T n K n )(and closed).Note that the set K ∩ ( T n K n ) may not be a neighbourhood, indeed, in manycases it may be a single line. Thus, the limiting procedure may result in a merepointwise result at p causing the functions b Γ to not necessarily have the rightfunctional dependence in the limit τ → ∞ over K . Thus in the limit we shouldonly consider the value of Γ restricted to the set K ∩ ( T n K n ). On the otherhand, for τ n finite, the result applies to a neighbourhood.It is thus more appropriate to consider the following perturbed PDE: Pde [ b Γ] = B τ ( φ ∗ τ ( N )) (23)where φ ∗ ( N ) should be thought of as acting on the components of N as func-tions; i.e., if N a...b is a component, then φ ∗ ( N a...b ) = N a...b ◦ φ [17].Assuming we are considering a certain metric g , we know that there existsa set of equations to this PDE. In particular, there is a continuous family ofsolutions b Γ( τ ) which solves eq.(22). Moreover, over the compact region K n ,since this is a perturbed PDF implies that it satisfies a Cauchy property, namely,there exists an increasing sequence τ n → ∞ , such that for any ǫ >
0, there existsan M such that: n, m ≥ M ⇒ || b Γ( τ n ) − b Γ( τ m ) || < ǫ. (24)The diffeomorphsm φ τ acts as follows on the connection [16, 17]: if Ω is theconnection form, then ˜ φ ∗ τ Ω = b Ω , where ˜ φ τ is the induced transformation on theframe bundle and b Ω is the transformed connection, we get over U : b Γ µαβ = ( M − ) µν h M γα φ ∗ t (Γ νγδ ) + M γα,δ i M δβ . Furthermore, since p = φ τ ( p ), we have Γ µγδ = φ ∗ τ (Γ νγδ ) at p . Moreover, inthe aforementioned frame, we have M ,µ = − M ,µ , M ,µ = − M ,µ while allother components of M γα,δ are zero.Eq. (24) implies that the connection coefficients can be chosen to be arbi-trary close to flat space. Component-wise we have | b Γ αβγ ( τ n ) − b Γ αβγ ( τ m ) | < ǫ .seudo-Riemannian VSI metrics II 11Since some of the components of the connection transforms as tensor compo-nents under the boost, if the component has boost-weight b , we get: | b Γ αβγ ( τ n ) − b Γ αβγ ( τ m ) | = | exp[ b · λ τ n ]Γ αβγ − exp[ b · λ τ m ]Γ αβγ | = exp[ b · λ τ n ] (cid:12)(cid:12) Γ αβγ − exp[ b · λ ( τ m − τ n )]Γ αβγ (cid:12)(cid:12) < ǫ. (25)If we fix m , then it it is clear that: b · λ ≤ , or Γ αβγ = 0 for b · λ > . This is valid for an arbitrary point p ∈ U ; hence it is valid everywhere in theneighbourhood.We can now consider the connection coefficients that transform tensorially,and consider the various cases. By a simple geometric argument, we get:1. ˜ κ = κ = ˜ ρ = ρ = ˜ σ = σ = 0, and hence Kundt; or
2. ˜ κ = ˜ ρ = ˜ σ = ˜ τ = 0, and hence, a Walker space possessing an invariantnull 2-plane. II. The direct method
Before we embark on the direct method let us remindourselves of some useful identities and formulae. The covariant derivative of atensor T has the formal structure: ∇ T = ∂T − X Γ ⋆ T, (26)where the ∂T indicates the partial derivative piece, and the Γ ⋆ T indicates thealgebraic piece where Γ are the spin-coefficients. Furthermore, also useful arethe 2nd Bianchi identity and the generalised Ricci identity: R ab ( cd ; e ) = 0 , (27)[ ∇ a , ∇ b ] T c ...c k = k X i =1 T c ...d...c k R dc i ab , (28)which enable us to permute covariant derivatives up to algebraic terms. Wenote that all the algebraic terms are of lower order in derivatives of T .Assuming that T fulfills the N G -property, there are therefore two potentialways the covariant derivative ∇ T of the tensor can violate the N G -property;namely, through the components of the partial derivatives propagating the com-ponents of T across the b · λ = 0 line in boost-weight space, and the algebraicterms. At every level of covariant derivatives we can thus first permute thederivatives as much as possible, and the impose the necessary conditions onthe remaining components. Thus we ensure that the N G -property is valid atevery lower derivative so that when using the Ricci identity it does not involve N G -property breaking terms through the algebraic piece.Let us first split the Riemann tensor into its irreducible parts R , S ab , W + abcd and W − abcd . For a VSI space R = 0 so the trace-free Ricci tensor, S ab is equalto the Ricci tensor S ab = R ab .2 S. HervikThen consider a non-zero Ricci tensor. By considering R ac R cb or higherpowers if necessary, we can assume the Ricci tensor is of the form (bracketsmean symmetrisation): R = a ℓℓ + b ( ℓ ˜ m ) + c ˜ m ˜ m . (29)We need to compute the derivatives ∇ ( k ) R ab . The various cases depend on thecomponents a, b and c and let us consider these in turn. ac = 0 . Here, we can boost so that a and c are both constants. Computingfirst ∇ a R , some of the components are proportional to:( − ,
2) : a ˜ σ, (0 ,
1) : a ˜ κ, (1 ,
0) : c ˜ σ, (2 , −
1) : c ˜ κ ;consequently, by the N G -property, ˜ κ = ˜ σ = 0. Computing ∇ b ∇ a R we getsimilarly ˜ ρ = ˜ τ = 0. Thus this is a Walker space. ab = 0 , c = 0 . Here, we can boost so that a and b are both constants. Con-sidering the 1st derivative, ∇ a R , we get (among others) the components:( − ,
2) : a ˜ σ, (0 ,
1) : a ˜ κ, (0 ,
1) : b ˜ σ, (1 ,
0) : b ˜ κ, (0 , −
1) : b ˜ ρ ;hence, there are two possibilities ˜ κ = ˜ ρ = 0, or ˜ κ = ˜ σ = 0. By computing ∇ b ∇ a R , we quickly get ˜ ρ = 0. Thus we need to consider the two cases ˜ σ = 0,and ˜ σ = 0.From the 2nd derivative, and the Law’s eq. (3.4) in [4], we get the conditions:˜ σρ = ˜ σσ = ˜ σκ = ˜ τ κ = τ ˜ σ + ρ ˜ τ = 0 . (30)If ˜ σ = 0, then ρ = κ = σ = τ = 0, and consequently Walker.Assume then ˜ σ = 0. If ˜ τ = 0, then the space is again Walker. Left toconsider is therefore ˜ τ = 0 and ˜ κ = ˜ σ = ˜ ρ = 0. From the equations above, wethus get κ = ρ = 0 also. If σ = 0, then the space is Kundt. We need thus tocheck if σ = 0. By computing ∇ (3) R and ∇ (4) R we get numerous constraintsfrom the requiring the N G -property. Most of these are the same as the Bianchiidentity. Imposing these and some algebraic conditions on the spin-coefficients,we get the following b.w. (0 , R = 12˜ τ σb. By the N G -property this component has to vanish which is contradictory to theassumptions given above. Hence, the space has to be either Walker or Kundt. b = 0 , a = c = 0 Here, we notice that there is a discrete symmetry which flipsboost-weight space with respect to the line b − b = 0. Using this symmetry,the case here essentially reduces to the case ab = 0 above. Thus also here the N G -property implies Walker or Kundt.seudo-Riemannian VSI metrics II 13 a = 0 , b = c = 0 . Lastly we need to consider the case when only a is non-zero.First we look at ∇ (2) R . Using the symmetry ( b , b ) ( b , − b ) we get theconditions: ˜ κ = ˜ σ = ˜ ρ = ρ = 0 . (31)In addition the vanishing of the (0 ,
0) components implies κ ˜ τ = 0. If ˜ τ = 0,then the space is Walker. Assume thus ˜ τ = 0, implying κ = 0.In addition, the Bianchi identities need to be fulfilled. Imposing these andcomputing the symmetric 2-tensor (cid:3) R ab , we note that this is of the followingform: (cid:3) R = A ℓℓ + B ( ℓ ˜ m ) + C ˜ ℓm . (32)If B or C is non-zero, then the previous computations implies that, by consid-ering possibly 4 more derivatives, that its Walker or Kundt. The requirements B = C = 0 impose additional conditions on the spin-coefficients. Eventually,after possibly 4 more derivatives, also this implies its Walker or Kundt. The Weyl tensor
Let us now consider the self-dual (or anti-self-dual byorientation reversion) Weyl tensor. This needs to be of type III, N, or O, see[7]. If it is of type III, then ( W + ) as a bivector operator, is of type N. Considerthus the case of type N. By discrete symmetries, we can thus assume that (in ashort hand notation): W + = φ ( ℓ ∧ m )( ℓ ∧ m ) . (33)We note that the discrete symmetry that act on boost-weight space as ( b , b ) ( − b , − b ), leaves W + invariant. By computing the second covariant derivative, ∇ b ∇ a W + , we pick out the following components (including their boost-weights): n a n b ( ˜ m ∧ m )( ˜ m ∧ m ) : ∝ κ , (2 , m a ˜ m b ( ˜ m ∧ m )( ℓ ∧ n ) : ∝ σ , (0 , − m a m b ( ˜ m ∧ m )( ˜ m ∧ m ) : ∝ ρ , (0 , N G -property of W + and ∇ (2) W + , and using the remaining discretesymmetry we thus get the two cases: κ = σ = 0 , or κ = ρ = 0 . Consider first κ = σ = 0. Computing ∇ d ∇ c ∇ b ∇ a W + , in particular thecomponent m a m b m c m d ( ˜ m ∧ n )( ˜ m ∧ n ) ∝ ρ of boost weight (2 , κ = σ = ρ = 0.Hence, we are left with κ = ρ = 0, while σ need not be zero. Assume thusthat σ = 0. Using the 2nd derivative once again, but this time the components:˜ m a n b ( m ∧ n )( ℓ ∧ m ) ∝ ˜ κσ, (1 , m a ˜ m b ( m ∧ n )( ℓ ∧ m ) ∝ ˜ ρσ, (0 ,
0) (34)thus, ˜ κ = ˜ ρ = 0. From Law’s eq.(3.4a) in [4], it now implies that ˜ σσ = 0; hence,˜ σ = 0.Thus we are in the situation where we are of one of the following cases:4 S. Hervik1. κ = ˜ κ = ρ = ˜ ρ = ˜ σ = 0, σ = 0.2. κ = ρ = σ = 0.It is important here that we keep track of the components of the lower deriva-tives.Consider next the first case where σ = 0. Then using the 4th derivative, weget the component: ˜ m a m b ℓ c ˜ m d ( ˜ m ∧ n )( ˜ m ∧ m ) ∝ σ ˜ τ , of boost weight (0,0); consequently, ˜ τ = 0 and thus all the tilded variables˜ κ = ˜ ρ = ˜ σ = ˜ τ = 0, and this is thus a Walker space.We are left to consider the second case where κ = ρ = σ = 0. If τ = 0, wehave a Walker space. Assume thus that τ = 0. By computing the 4th derivative,we notice that one of the components,˜ m a ˜ m b ℓ c ℓ d ( ℓ ∧ n )( ℓ ∧ n ) ∝ τ . This component has boost-weight (-2,-2) and has the same boost-weight as W + under the exchange of tilded spin-coefficinents with non-tilded ones. Aftera lengthy computation, sometimes needing to go to 8th order, we get that˜ κ = ˜ σ = ˜ ρ = 0 (analogously as above). Thus, implying that this is a Kundtspace.If W + but W − = 0, then we can consider the discrete symmetry whichinterchanges tilded spin-coefficients with non-tilded ones: ˜ x ↔ x , where x is thespin coefficients. Then an identical computation as above implies that the spaceis either Walker with an invariant null 2-plane, or Kundt. The theorem followsthen from these considerations.Although the argument involves 8th derivatives, it is suspected that thenumber of derivatives needed is less that this. In particular, no examples ofspaces which are VSI k but not VSI k +1 are known for k >
3. The example eq.(1)is VSI but not VSI , however, this is a Walker metric which is a restrictedclass. This example, and an explanation of how this example can be extendedto other similar examples, will be given later. However, a question still remains:Are there examples of non-Walker metrics which are VSI k but not VSI k +1 for k > Using d s = 2 ( ℓn − m ˜ m ) . (35)We will consider the pseudo-Riemannian Kundt case for which the transversespace is 2-dimensional. Requiring the N -property, this must be flat space (see,[8, 15]). Therefore, we can write: − m ˜ m = 2d U d U = − d T + d X . seudo-Riemannian VSI metrics II 15There are two classes of 4D Neutral Kundt VSI metrics, they can be written[8, 15]: d s = 2d u (d v + H d u + W µ d x µ ) + 2d U d V, (36)where: Null case: W µ d x µ = vW (1) U ( u, U )d U + W (0) U ( u, U, V )d U + W (0) V ( u, U, V )d V,H = vH (1) ( u, U, V ) + H (0) ( u, U, V ) , (37) Spacelike/timelike case: W µ d x µ = vW (1) d X + W (0) T ( u, T, X )d T + W (0) X ( u, T, X )d X,H = v (cid:16) W (1) (cid:17) + vH (1) ( u, T, X ) + H (0) ( u, T, X ) , (38)and W (1) = − ǫX , where ǫ = 0 , . (39)We note that these possess an invariant null-line if W (1) = 0, and a 2-dimensional invariant null-plane if W (0) V = 0 for the null case . This class of metrics provides us with a new set of VSI metrics which have notbeen considered before. This is due to the fact that these VSI metrics doesnot in general possess the N -property, but rather the weaker requirement of the N G -property.Using the following Walker form,d s = 2d u (d v + A d u + C d U ) + 2d U (d V + B d U ) , (40)the result can be summarised in the following theorem: Theorem 3.3.
Consider the metric (40) where A = vA ( u, U ) + V A ( u, U ) + A ( u, U ) ,B = V B ( u, v, U ) + B ( u, v, U ) C = C ( u, v, U ) + V C ( u, U ) + C ( u, U ) . (41) Then the following holds:1. The metric is a VSI space. If A ∂ B ∂v = 0 , or A ∂ C ∂v = 0 , then it is not VSI . In order for the spacelike/timelike case to possess an invariant null 2-plane, it needs to bea special case of the null case.
2. If B = vB ( u, U ) + B ( u, U ) C = v C ( u, U ) + vC ( u, U ) + C ( u, U ) , (42) then it is a VSI space. If in addition, A ∂ B ∂v = 0 , then it is not VSI .3. If eq. (42) holds and, in addition: B = v B ( u, U ) + v B ( u, U ) + vB ( u, U ) + B ( u, U ) (43) then the space is VSI. The proof is this result is partly by direct computation of the curvaturetensors and requiring N G -property. Let us indicate how the proof goes and inthe process we elude to how these can be generalised.Starting with the Walker form eq.(40) we can compute the Riemann tensor.We notice that the metric gives Riemann components in the lower triangularpart of boost-weight space. Let us for short use notation such that the basisone-forms are { ω , ω , ω , ω } = { d u, d v + A d u + C d U, d U, d V + B d U } (44)Then a component of a tensor would have the boost-weight as follows (indicesdownstairs): ( b , b ) = ( − , − R , say, will have boost-weight ( − , N G -property are:(2 , −
2) : R = − B ,vv , (45)(1 , −
1) : R = − C ,vv , R = − B ,vV (46)(0 ,
0) : R = − A ,vv , R = − C ,vV R = − B ,V V (47)( − ,
1) : R = A ,vV , R = C ,V V , (48)( − ,
2) : R = − A ,V V (49)while ( R ) ( b ,b ) = 0 for b + b >
0. Thus the Riemann tensor automaticallysatisfies the S G -property. In order for it to satisfy the N G -property we can setthe components (2 , − , −
1) and (0 ,
0) to zero. Solving these equations givesthe functional dependencies as given in (41). This is thus a VSI space. Indeed,by direct computation we note that ∇ (Riemann) satisfies the N G -property also,hence, it is in addition VSI .Assume then that (41) is satisfied. Regarding the ∇ (2) (Riemann) we notethat this does not necessarily satisfy the N G -property (thus not VSI ). One suchnonvanishing scalar is R abcd ; ef R abcd ; ef . However, component-wise, the criticalcomponents are: R = A ( B ) ,vv , R = 12 A (2( B ) ,vv − ( C ) ,vvv ) . seudo-Riemannian VSI metrics II 17These give rise to the conditions mentioned and equating these to zero gives thesolutions (42). Satisfying eq.(42) will now give the N G -property and thus VSI ;indeed, VSI by direct computation.Thus assume (41) and (42) are satisfied. Computing R abcd ; efgh R abcd ; efgh weget: R abcd ; efgh R abcd ; efgh = 576[( B ) ,vvvv ] A . Hence, it is not VSI if A ( B ) ,vvvv = 0. Requiring that ( B ) ,vvvv = 0, givesthe solution in (43) and by inspection ∇ (4) (Riemann) satisfies the N G -property.This is sufficient for the metric to be VSI.We note that this proof also provides us with examples of metrics being VSI but not VSI . For example, if (41) and (42) are satisfied, but A , ( B ) ,vvvv = 0,then it is VSI but not VSI . The example given in the introduction, eq.(1) isperhaps the simplest member of this family.Similarly, metrics being VSI but not VSI can be found analogously; as asimple set of examples of metrics of this kind: ds = 2 du ( dv + V du + bv dU ) + 2 dU ( dV + aV v dU ) , (50)where a and b are constants, or functions depending on ( u, U ), not both beingzero. In this paper we have studied pseudo-Riemannian metrics with degenerate cur-vature structure in the sense that they are not characterized by their polynomialcurvature invariants. In particular, we related these to the S G -property. Specif-ically, we have three main results:1. In a pseudo-Riemannian space of arbitrary dimension and signature, aspace (tensor) not characterized by its polynomial invariants possessesthe S G -property.2. In the special case where the invariants vanish, the space (tensor) mustpossess the N G -property.3. In 4D neutral signature, a VSI space is either Kundt or a Walker space.Indeed, in the latter case we constructed a new family of Walker VSI spaces.This shows that in the pseudo-Riemannian case these Walker metrics can pro-vide new examples of metrics not being characterized by their invariants. In-deed, using the ideas given in this paper examples of VSI Walker metrics can begiven in any signature ( k, k + m ) where k ≥
2. As an example, the following isa neutral VSI Walker metric (with a 3D invariant null-space) in six dimensions: ds = 2 du ( dv + V du ) + 2 dU ( dV + V dU ) + 2 d U ( d V + v d U ) . In future work, pseudo-Riemannian VSI metrics will be studied further andthe ultimate aim is a full classification of VSI metrics in any dimension andsignature.8 S. Hervik
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