We prove a generalisation of the $\epsilon$-property, namely that for any
dimension and signature, a metric which is not characterised by its polynomial
scalar curvature invariants, there is a frame such that the components of the
curvature tensors can be arbitrary close to a certain "background". This
"background" is defined by its curvature tensors: it is characterised by its
curvature tensors and has the same polynomial curvature invariants as the
original metric.Comment: 6 page

Cartan E

Coley A

Eberlein P

Hervik S

Milson R

Procesi C

Sigbjørn Hervik

Tolley A J

English

arXiv

Crossref

Classical and Quantum Gravity

All metrics have curvature tensors characterized by its invariants as a limit: the ϵ-property

This is an author-created, un-copyedited version of an article accepted for publication in Classical and quantum gravity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at doi:10.1088/0264-9381/28/15/157001.We prove a generalisation of the ϵ-property, namely that for any dimension and signature, a metric which is not characterised by its polynomial
scalar curvature invariants, there is a frame such that the components of
the curvature tensors can be arbitrary close to a certain “background”.
This “background” is defined by its curvature tensors: it is characterised
by its curvature tensors and has the same polynomial curvature invariants
as the original metric

Hervik, Sigbjørn

UiS Brage

All metrics have curvature tensors characterised by its invariants as a limit: the ϵ-property

HSN Open Archive

We prove a generalisation of the ϵ-property, namely that for any dimension and signature, a metric which is not characterised by its polynomial
scalar curvature invariants, there is a frame such that the components of
the curvature tensors can be arbitrary close to a certain “background”.
This “background” is defined by its curvature tensors: it is characterised
by its curvature tensors and has the same polynomial curvature invariants
as the original metric

NORA - Norwegian Open Research Archives

arXiv:1107.3210v1  [gr-qc]  16 Jul 2011All metrics have curvature tensors characterisedby its invariants as a limit: the ǫ-property.Sigbjørn HervikFaculty of Science and Technology,University of Stavanger,N-4036 Stavanger, Norwaysigbjorn.hervik@uis.noJuly 19, 2011AbstractWe prove a generalisation of the ǫ-property, namely that for any dimen-sion and signature, a metric which is not characterised by its polynomialscalar curvature invariants, there is a frame such that the components ofthe curvature tensors can be arbitrary close to a certain “background”.This “background” is defined by its curvature tensors: it is characterisedby its curvature tensors and has the same polynomial curvature invariantsas the original metric.1 IntroductionFor Lorentzian spacetimes having all vanishing polynomial curvature invariants(VSI spaces) it has been proven that the ǫ-property holds [1]. The ǫ-propertyimplies the components of the curvature tensors can, by chosing a suitable frame,be arbitrarily small. Clearly, this property can only hold for VSI spaces as theinvariants are all zero. In the general case however, in particular for degeneratemetrics, we will here prove a similar property but at the cost of having toreplace the ǫ-property with respect to a “background”. This is remincent of thefact that any matrix M can be split, using the Jordan decomposition, into adiagonalisable matrix, D, and a nilpotent matrix, N :M = D +N.The components of the nilpotent matrix can, by a change of basis, be made assmall as possible. Also note that the matrices M and D have the same polyno-mial invariants: tr(Mn) = tr(Dn). Therefore, since the eigenvalues (=diagonalcomponents) of D, are determined by the polynomial invariants tr(Dn) we cansay that D is “characterised by its invariants”. For a tensor this concept can bedefined in a similar way [2, 3, 5, 4].In order for us to state the corresponding result for curvature tensors, we willreview some results from invariant theory and define the appropriate concepts12 S. Hervikwhich we need. Furthermore, we will consider the polynomials invariants andso in what follows ’invariants’ is to be understood as ’polynomial invariants ’.The idea is to consider a group G acting on a vector space V . In our case wewill be consider a real G and a real vector space V . However, it is advantageousto review the complex case with a complex group GC acting on a complex vectorspace V C. Then for a vector X ∈ V C we can define the orbit of X under theaction of GC as follows:OC(X) ≡ {g(X) ∈ V C∣∣ g ∈ GC} ⊂ V CThen ([2], p555-6):Theorem 1.1. If GC is a linearly reductive group acting on an affine varietyV C, then the ring of invariants is finitely generated. Moveover, the quotientV C/GC parameterises the closed orbits of the GC-action on V C and the invari-ants separate closed orbits.Here the term closed refers to topologically closed with respect to the stan-dard vector space topology and henceforth, closed will mean topologically closed.This implies that given two distinct closed orbits A1 and A2, then there is aninvariant with value 1 on A1 and 0 on A2. This enables us to define the set oforbits:CC = {OC(X) ⊂ V C∣∣ OC(X) closed.} (1)Based on the above theorem we can thus say that the invariants separate ele-ments of CC and hence we will say that an element of CC is characterised by itsinvariants.In our case we will consider the real case where we have the Lorentz group,O(1, n−1) which is a real semisimple group. For real semisimple groups acting ona real vector space we do not have the same uniqueness result as for the complexcase [3]. However, by complexification, [G]C = GC we have [O(1, n − 1)]C =O(n,C), and complexification of the real vectorspace V we get V C ∼= V + iV .The complexification thus lends itself to the above theorem, and consequentlywe will define characterised by its invariants as follows. For a tensor, T , arotation of a frame naturally defines a group action on the components of thetensor. Then:Definition 1.2. Consider a (real) tensor, T ∈ V , or a direct sum of tensors,then if the orbit of the components of T under the complexified Lorentz groupGC is an element of CC, i.e., OC(T ) ∈ CC, then we will say that T is characterisedby its invariants.As the invariants parameterise the set CC and since the group action defines anequivalence relation between elements in the same orbit this definition makessense.Let us similarly define the real orbits:O(X) ≡ {g(X) ∈ V∣∣ g ∈ G} ⊂ VHow do these results translate to the real case? The real orbit O(T ), is a realsection of the complex orbit OC(T ). However, there might be more than onesuch real section having the same complex orbit. Using the results of [3], thesereal closed orbits are disjoint, moreover:The epsilon property 3Theorem 1.3. O(T ) is closed in V ⇔ OC(T ) is closed in V C.Thus the question of whether T is characterised by its invariants is thusequivalent to whether O(T ) is closed in V . Thus we can define similarly:C = {O(X) ⊂ V ∣∣ O(X) closed.}, (2)hence, we have that T is characterised by its invariants iff O(T ) ∈ C.However, as pointed out, there might be other closed real orbits O(T˜ ) havingthe same invariants as O(T ) (in line with the comments in [6, 5]). An exampleof this is the pair of metrics:ds21 = −dt2 +1x2(dx2 + dy2 + dz2),ds22 = dτ2 +1x2(dx2 + dy2 − dζ2) , (3)The curvature tensors1 of these metrics lie in separate orbits O(T ), but in thesame complex orbit OC(T ).2 The ǫ-propertyWe introduce a basis of orthonomal (or null) vectors ω = {e1, ..., en} for ann-dimensional pseudo-Riemannian space. We express the components of thecurvature tensors up to kth derivatives, with respect to the frame ω, in termsof the vector Xω = [Rabcd, Rabcd;e, ..., Rabcd;e1...ek ]. At a point p on the manifoldthis vector can be considered as Xω ∈ Rm, for some m. We will also use thestandard Euclidean norm ||X || in Rm.We note that there is no requirement that the components of Rabcd etc.need to be independent (nor is the order significant). One way this can bethought of is that we have a metric from which we compute the Riemann tensorswith respect to an orthonormal frame ω. We when construct the m-tuple Xωfrom these components. These components may be dependent (which wouldbe the case if we just naively write down all components of Riemann withoutconsidering the symmetries). If the metric is of signature (q, n − q), then theaction of the corresponding orthogonal group O(q, n − q) will preserve thesesymmetries and thus this has no consequence for our result.The aim of this Note is to prove that (assuming C∞ and that the algebraicstructure does not change over a neighbourhood2):Theorem 2.1 (ǫ-property). Consider a spacetime (M, g) of any dimension(and signature) and a fixed number of derivatives, k, of the Riemann tensor.Then locally either:1. The curvature tensors Xω are characterised by its invariants; or,1Both of these metrics are symmetric and conformally flat, so the only non-zero curvaturetensor is the Ricci tensor.2This is a technical assumption implying that the Segre/Petrov or Ricci/Weyl types cannotchange over the neighbourhood. In some cases this assumption can be relaxed, for examplefor real analytic metrics, or if we are only interested in a pointwise ǫ-property, however wewill not consider this here.4 S. Hervik2. For any ǫ > 0, there exists a suitable frame and an X˜ω ∈ Rm such thatthe components Xω = X˜ω +Nω, where ||Nω|| < ǫ and X˜ω is characterisedby its invariants. Moreover, the polynomial invariants of X˜ω are the sameas those of Xω.We will prove this theorem using the orbits of Xω under frame rotations.However, in the 4 dimensional Lorentzian spacetime there is a much simplerproof following along the same lines as in the VSI case.Proof: 4D Lorentzian case. Here we can use the results of [6, 7] which impliesthat either the spacetime is I-non-degenerate, hence characterised by its in-variants, or degenerate Kundt. For a degenerate Kundt the vector Xω canbe written as Xω = (Xω)0 + (negative boost weight terms). Thus we can setX˜ω = (Xω)0 (which is characterised by its invariants) and Nω is the negativeboost weight part. By applying a boost we can thus get the components ofNω as small as possible, in particular ||Nω|| < ǫ. Clearly also the polynomialinvariants of X˜ω are the same as those of Xω.It is believed that a similar mechanism can be used in abitrary dimensionsand signatures (for Lorentzian case, see [4]). However, in order to give a proof inthe general case we will use a different method using the space of orbits of Xω.In order to do this we will review some known results from invariant theory.A frame-rotation in n-dimensions (arbitrary signature) at a point P ∈ Mis an action of the group g ∈ O(q, n − q). A rotation of the frame gω ={Ma1ea, ...,Manea}, induces an action of g on Xω through the tensor structureof the components:g(Xω) =[M b1a1...Mbk1ak1R(1)b1...bk1,M b1a1...Mbk2ak2R(2)b1...bk2, ...,M b1a1...MbkNakNR(N)b1...bkN].Consider now the orbit of a point X ∈ Rm:O(X) = {g(X)|g ∈ O(q, n− q)} ⊂ Rm.Clearly, if X and Y are in the same orbit, then they would be equivalent ascurvature tensors because they are separated by a mere rotation of frame. Theequivalence problem is then reduced to a question of classifying the variousorbits.In the complex case, where we consider the complexification of O(q, n−q) 7→[O(q, n − q)]C = O(n,C) and Rm 7→ Cm, the orbits that are characterised bytheir invariants are the closed orbits (those that are elements of CC). In ourcase we do not consider orbits under the the complex group O(n,C), rather oneof the real forms O(q, n− q) for which we need to consider, as discussed in theintroduction, the set of real orbits C.Assume thus that O(X) ∈ C; i.e., that O(X) is closed and from the previousdiscussion that O(X) is characterised by its invariants. Indeed this implies thatthere exists a X˜ ∈ O(X) which is minimal [3]. Recall that a vector X˜ ∈ Rmis minimal iff ||g(X˜)|| ≥ ||X˜ || for all g ∈ G where is || · || the O(q) × O(n− q)-invariant Euclidean norm on Rm. There may be several minimal vectors of eachorbit but the minimal vectors are in some sense the vectors with the “smallest”norm with respect to the Euclidean metric on Rm. The existence of minimalvectors are not essential for this discussion, however, it is useful to note thatThe epsilon property 5the limit in the theorem can always be chosen to be a minimal vector [3]. Inparticular, the existence of a minimal X˜ in the orbit of X is equivalent to sayingthat O(X) is closed and hence that X is characterised by its invariants [3].We are now ready to complete the proof:Proof: General case. Assume that Xω has a closed orbit; i.e., O(Xω) is closed.Due to the existence of a minimal X˜ in its orbit it is characterised by its invari-ants by the above argument.Assume thus that O(Xω) is not closed. Then by the results of [3], there isa unique closed orbit in the closure of O(Xω). In particular, there would be aminimal X˜ω in O(Xω). Since X˜ω is in the closure of O(Xω), there would exist,for any ǫ > 0 a sequence xn ∈ O(Xω) and an integer N so that ||xn − X˜ω|| < ǫfor all n > N . Thus since the sequence, xn, is in the orbit of Xω, we can choosea frame such that xn = Xω for an n > N ; hence, there exists a frame such that||Xω − X˜ω|| < ǫ.As regards to the polynomial invariants, Ii, we note that these are continuousas functions Ii : Rm 7→ R. Since the invariants are constants over the orbitO(Xω), they must also, due to continuity, be constants over the closure O(Xω).Consequently, X˜ω and Xω have the same polynomial invariants. This completesthe proof.Let us also make some final remarks on this result.Firstly, there is no guarantee that the minimal vector X˜ω corresponds tothe curvature tensors of an actual metric but there are important cases whereit does. For example, in the VSI case X˜ω = 0, and for sufficiently large k, thiswould be flat space. For other spacetimes, for example, CSI spacetimes there isa frame such that X˜ω is constant over the neighbourhood.As an example, consider the Weyl (Petrov) type III vacuum solution ofRobinson-Trautman [8]:ds2 = −dt2 + t2dx2 + t 25(e−√65xdy +√52t45dx)2+ t65 e4√65xdz2.This has all vanishing zeroth order invariants (VSI0); hence, including only theRiemann tensor, X(0) = [Rabcd], we have X˜(0) = 0. The first order invari-ants, however, are non-zero. In particular, ∇(Riem) is of type II, so definingX(1) = [Rabcd, Rabcd;e], then both X˜(1) and N (1) are non-zero. However, con-sider the second derivatives then X(2) = [Rabcd, ..., Rabcd;e1e2 ] is characterisedby its invariants. Thus this is an example where the ǫ-property depends on thenumber of derviatives considered.Also note that the theorem is not a “uniform” ǫ-property in the sense thatwe do need to truncate Xω so that it contains only a finite number of derivativesof the Riemann tensor. Thus the validity of an ǫ-property for an infinite numberof derivatives is still elusive and remains an open question.3We would also point out that the actual limit arises from the existence ofa boost: by boosting the frame appropriately, this limit can be achieved (see[4] for the Lorentzian case). Physically, this implies that an observer in, for3On the same token, Cartan [9] showed that, as far as the equivalence problem is concerned,it is sufficient to know the covariant derivatives ∇(q)(Riem) up to a certain order q (see also[10] which considers 4D Lorentzian manifolds).6 S. Hervikexample, an AdS-gyraton [11] can experience the space arbitrarily close to AdSspace. Interestingly, similar spacetimes are of particular interest when it comesto supersymmetry [12] and the ǫ-property thus puts these solutions and theirphysical significance in new light.This Note also gives a set of new ideas and techniques to study the relation-ship between the polynomial invariants and the curvature tensors. In particular,these ideas can also be used to study more closely the alignment of degeneratetensors [4].References[1] N. Pelavas, A. Coley, R. Milson, V. Pravda, A. Pravdova, J.Math.Phys. 46:063501, 2005.[2] C. Procesi, Lie Groups: An approach through Invariants and Representa-tions, Springer, 2007.[3] P. Eberlein, M. Jablonski, Contemp. Math. 491: 283, 2009.[4] S. Hervik, The alignment theorem, CQG submitted.[5] S. Hervik and A. Coley, Class.Quant.Grav. 27: 095014, 2010.[6] A. Coley, S. Hervik and N. Pelvas, Class.Quant.Grav. 26: 025013, 2009.[7] A. Coley, S. Hervik, G. Papadopoulos, N. Pelavas, Class.Quant.Grav. 26:105016, 2009[8] I. Robinson and A. Trautman, Proc. Roy. Soc. London, A265: 463, 1962.[9] E. Cartan, Lec¸ons sur la Ge´ome´trie des Espaces de Riemann (Paris:Gauthier-Villars) , 1946[10] R. Milson, N. Pelavas Class. Quant. Grav. 25: 012001, 2008[11] V. Frolov and A. Zelnikov, Phys.Rev. D72: 104005, 2005.[12] J. Gauntlett et al., Phys. Rev. D 74: 106007, 2006.A. Coley, A. Fuster, and S. Hervik, Int.J.Mod.Phys.A24:1119-1133,2009.A. J. Tolley, C. P. Burgess, C. de Rham and D. Hoover, JHEP 0807: 075,2008.

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All metrics have curvature tensors characterised by its invariants as a limit: the \epsilon-property

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