On the largest scales the universe appears to be almost perfectly homogeneous and isotropic adhering to the cosmological principle. On smaller scales inhomogeneities and anisotropies become increasingly prominent reflecting the origin emergence and formation of structure in the Universe and its cosmological impact. Also a range of tensions between various cosmological observations may suggest it to be necessary to explore the consequences of such deviations from the ideal uniform universe. In this study we restrict this to an investigation of anisotropies on the nature of the Universe. This motivates a more thorough understanding of the manifestation of anisotropy in cosmological applications. When letting go of the assumption of isotropy spacetime metrics become homogeneous and completely anisotropic. As such the Lie algebras of Killing vector fields will be 3D and fit into the socalled 9part Bianchi classification. This work strives to given a suitable 3D Lie algebra of vector fields xiaa123 reconstruct the basis for a metric on which this Lie algebra is Killing. Through finding a determining equation for the frame invariant under xia and using the method of characteristics to solve it expressions for said invariant frame in terms of the xia are obtained. This leads to general equations for the invariant frame in terms of the xia organized by Bianchi class. Some examples demonstrating this method are worked out.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Mathematical Physics']
Cosmic Anisotropy and Bianchi Characterization Killing vector fields and the implied finding of their metric frame
{'Cosmic Anisotropy and Bianchi Characterization: Killing vector fields and the implied finding of their metric frame': 'Robbert W. Scholtens, 1, 2, ∗ Marcello Seri, 2, † Holger Waalkens, 2, ‡ and Rien van de Weygaert 1, §  \n1  \nKapteyn Astronomical Institute, University of Groningen, Groningen, The Netherlands 2 Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands (Dated: September 13, 2024)  \nOn the largest scales the universe appears to be almost perfectly homogeneous and isotropic, adhering to the cosmological principle. On smaller scales inhomogeneities and anisotropies become increasingly prominent, reflecting the origin, emergence and formation of structure in the Universe and its cosmological impact. Also, a range of tensions between various cosmological observations may suggest it to be necessary to explore the consequences of such deviations from the ideal uniform universe. In this study, we restrict this to an investigation of anisotropies on the nature of the Universe. This motivates a more thorough understanding of the manifestation of anisotropy in cosmological applications. When letting go of the assumption of isotropy, spacetime metrics become homogeneous and completely anisotropic. As such, the Lie algebras of Killing vector fields will be 3D, and fit into the so-called 9-part Bianchi classification. This work strives to, given a suitable 3D Lie algebra of vector fields { ξ a } a =1 , 2 , 3 , reconstruct the basis for a metric on which this Lie algebra is Killing. Through finding a determining equation for the frame invariant under { ξ a } and using the method of characteristics to solve it, expressions for said invariant frame in terms of the { ξ a } are obtained. This leads to general equations for the invariant frame in terms of the { ξ a } , organized by Bianchi class. Some examples demonstrating this method are worked out.', 'I. INTRODUCTION': "The present study centers on an investigation of the mathematical structure and of anisotropic homogeneous  \nBianchi metrics for the geometry of the universe on a global scale.  \nThe current cosmological worldview is based on the central tenet of the cosmological principle . According to the cosmological principle the universe on the largest scales is spatially homogeneous and isotropic . In other words, it is uniform, the same everywhere and looking the same in every direction. This constraint on the nature of our universe mathematically implies a strong constraint on the geometry. The curvature is restricted to only three possible options, specified by the RobertsonWalker (RW) metric,  \nds 2 = -dt 2 + a ( t ) 2 d x 2 , (1)  \nwhere d x 2 is the line element either of Euclidean (flat) space, or unit curvature hyperbolic or spherical space, and a ( t ) the scale factor, normalized to a ( t = now) = 1 .  \nThe fundamental importance of the highly symmetric RW geometry is its ability to simplify the complex, nonlinear system of Einstein's field equations of General Relativity. Instead of a nonlinear system of 10 equations, cosmologists can work with just two independent equations of motion: the Friedmann-Lemaˆıtre-RobertsonWalker (FLRW) equations [see e.g. 1-8]. These equations, which describe the universe's evolution in terms of the scale factor a , have been instrumental in probing the depths and history of the universe from its primordial beginnings 13.8 Gyrs ago up to the present epoch. Only with the help of the RW metric and FLRW equations the incessantly growing body of cosmological observations can be turned into understandable physical insights and interpretations.  \nWhile an overwhelming body of observational evidence has shown that the cosmological principle represents an  \namazingly accurate description of the global universe, at least asymptotically, we still need to establish how close the universe fulfils these conditions and what the scale of homogeneity actually is. First and foremost evidence is that of the amazing isotropy of the Cosmic Microwave Background, at a level of ≈ 10 -5 [9-11]. Strong indications also exist for the universe to be close to homogeneous on large scales. Most notably this is indicated by the scaling of the angular clustering of galaxies as a function of depth of the galaxy survey: the angular two-point correlation function of galaxies is seen to scale almost perfectly with the survey depth as expected for a spatial galaxy distribution that assumes uniformity on large scales [12, 13]. Nonetheless, it has not yet been unequivocally established beyond what scale the universe may be considered to be homogeneous and isotropic. General contention is that the universe may be considered homogeneous and isotropic beyond a scale of a few to several hundreds of Megaparsec [see e.g. 14].  \nAn increasing number of observations indicate that although cosmological principle may be a very accurate description of physical reality, it may also be in need of either a more precise formulation or, at worst, a reconsideration. Even within the context of the observed universe, it is more accurate to consider the cosmological principle as a statistical principle. Indeed, since the COBE CMB maps, we know that even on the largest scales the mass distribution in the universe is marked by a stochastic spatial pattern of density variations (be it of a minute amplitude). In fact, we know that structure in the universe arose through gravitational growth of these primordial perturbations, and that these were imprinted on all scales. Since COBE revealed the map of the large scale angular structure of the cosmic microwave background temperature, we know that at the time of recombination there were structures much larger than the particle horizon at the time. By implication, this is still true at the current cosmic epoch, so that the visible Universe presents us with a minute deviation from the global cosmological average. Hence, it is more precise to state that the statistical properties of the corresponding inhomogeneous mass distribution obey the cosmological principle and do not vary with direction and location, but that the particular realization within our visible universe may not be perfectly uniform [see e.g. 2].  \nAt a more fundamental level, various studies forwarded observational indications for deviations from pure isotropy or homogeneity, and hence may have serious implications for the validity of FLRW-based cosmologies. An intriguing indication for possible intrinsic anomalous anisotropies is the finding by Copi et al. [15, 16] that the low multipole components of the cosmic microwave background anisotropies are mutually aligned, and aligned with our motion through the universe (i.e. the dipole motion). The existence of this Axis of Evil would seriously question the validity of the isotropy of our universe, and hence the cosmological principle. At an even more critical level, Sarkar and collaborators have highlighted some  \nserious problems in the regular interpretation of cosmological observations within the context of the standard FLRW cosmology. One aspect concerns the fact that the dipole in the CMB is fully ascribed to the gravitational attraction by local inhomogeneities, and hence can be interpreted as a dipole effect due to our Galaxy moving within the restframe blackbody cavity of the cosmic microwave background radiation. However, recently Secrest et al. [17] found that the dipole with respect to the high redshift sample of quasars does not coincide with the CMB dipole, which it would be expected to do when our Galaxy's motion is solely a local effect. It may imply the existence of an anomalously large dipole that is not of local origin and directly calls into question the validity of the assumed isotropy of the universe. This finding is augmented by a potentially even more disruptive result that questions the reality of dark energy and instead may indicate a bulk flow on a cosmological scale. Colin et al. [18] inferred from a sample of 740 Supernovae Ia that the cosmological acceleration parameter is direction dependent, which may imply that we are embedded in a deep bulk flow that conflicts with the assumption of an isotropic universe.  \nFollowing the indications for the reality of possible deviations from the cosmological principle, the incentive for the present study is to explore cosmologies that do not entail uniformity. With the ultimate goal to extend this to the geometric structure of fully inhomogeneous and anisotropic cosmologies, we here first explore the implications for cosmologies that are anisotropic, yet homogeneous. No longer involving the symmetry of isotropy, there is a substantial extension of possible metrics with respect to the 3 types of highly symmetric RW metrics. This consideration gives rise to the class of so-called Bianchi models , derived from the classification scheme for three-dimensional Lie algebras [19]. These models, first introduced by Bianchi in 1898 [20], provide a framework for studying universes that are perfectly homogeneous yet have varying degrees of anisotropy.  \nIn this work, we investigate the mathematical basis of the Bianchi models from the perspective of the infinitesimal isometries, or Killing vector fields (KVFs), belonging to the metrics involved. Namely, requirements on homogeneity and isotropy of a metric manifest themselves mathematically as such infinitesimal isometries. Denoting by g the relevant metric, a KVF ξ is characterized by the following equation:  \nL ξ g = 0 , or equivalently, ξ µ ; ν + ξ ν ; µ = 0 , (2)  \nwhere L indicates Lie derivative and semicolon notation is used for covariant derivative. For instance, if g is taken as a RW metric (1), one will find exactly six solutions for ξ to this equation: three characterizing homogeneity, and three characterizing full isotropy. Moreover, due to the property of Lie derivatives that L [ ξ,η ] = [ L ξ , L η ]-so that the commutator of KVFs is again a KVF-we see that KVFs form a Lie algebra of vector fields (Killing Lie  \nalgebra; KLA). It is known that a metric in dimension n may allow for up to n ( n +1) / 2 KVFs [21, § 17.1], and certainly does not need to reach this bound.  \nRetracting the requirement of isotropy, and replacing it by that of anisotropy while retaining full homogeneity, substantially extends the pool of potential metrics with respect to the three RW type metrics. Instead, we obtain the so-called Bianchi models, named after the classification scheme for 3D Lie algebras [19]. This is because the KVFs effecting homogeneity will be three in number, and hence the KLA will be three-dimensional. An example metric of the so-called Bianchi I type is given by  \nds 2 = -dt 2 + a ( t ) 2 dx 2 + b ( t ) 2 dy 2 + c ( t ) 2 dz 2 , (3)  \nfor a ( t ), b ( t ), and c ( t ) mutually different-a generalization of the flat RW metric, allowing for different length scales in the different directions. This metric is anisotropic, as the 'rotation generating' vector fields, e.g. x∂ y -y∂ x , fail to be Killing for all t . (One could make the KVFs dependent on time, but since we wish for our symmetries to be purely spatial, this is outside of the scope of this work.)  \nBianchi models, with their inherent anisotropy, have been investigated extensively in the literature-see [22] for a review. The hopes are that, due to their inherent anisotropy, they could (partially) explain some of the anisotropies that are currently observed, such as that of the CMB. Particular focus has been placed on Bianchi types I, V, VII, and IX, as these provide 'natural' anisotropic extensions of the flat, hyperbolic, and spherical FLRW models, respectively, and so would allow for the FLRW model of cosmology to remain somewhat standing. A calculation detailing contributions of Bianchi-type perturbations to the CMB was performed in [23]. A Bianchi type VII contribution was tested against the Planck data, though it was ruled out [24].  \nSince a Bianchi model is defined by the structure of its metric's KLA, our approach shall be to build the models starting from this structure. That is to say, given a suitable 'candidate' 3D Lie algebra of vector fields, we wish to find a (frame basis for a) metric on which those vector fields will be Killing. This stands in contrast with the usual approach, where a priori a metric is chosen whose KLA is of a certain Bianchi type. Such metrics are tabulated, e.g. [25, § 6.4] and [26, § 8.2]. In some sense, we aim to reverse the traditional treatment: instead of being given a metric and finding its KVFs through the Killing equation (2), we declare that we want a given Lie algebra to be Killing, and then find a metric on which it is so.  \nBy linking the KLA structure with the metric in this manner, we establish a more robust methodology for studying Bianchi metrics. This approach ideally eliminates the need for any specific choice of KLA within a given Bianchi type, thereby elminating the frame dependence of the metric.  \nThe key observation is to consider vector fields that commute with each proposed KVF-these are known as invariant . Namely, it can be shown that the Lie deriva-  \ntive w.r.t. a candidate KVF of the dual of an invariant vector field, vanishes. As such, if three invariant vector fields which form a frame are found (invariant frame; IF), then the dual to this IF can be used as a basis for constructing a metric; the candidate KLA will then be a bona fide KLA for that metric.  \nOur approach in this article is as follows. We first recall mathematical details pertaining to homogeneity and isotropy. We then introduce the Bianchi classification of Lie algebras, and define what we mean by a Bianchi model. In the subsequent section we implement the approach as outlined above, by explicitly calculating the invariant vector fields belonging to a candidate KLA. The result is a tabulation of invariant frames in terms of the candidate KLAs, ordered by Bianchi type, and dependent only on the particular KLA chosen. We then conclude the article by exemplifying some of the constructions, and discussing further potential research questions.", 'A. General symmetry considerations': "Definition II.1 (Isometry) . Let ( M , g ) a (pseudo)Riemannian manifold. We say that a diffeomorphism ϕ from M to itself is an isometry if the pullback of the metric by ϕ leaves it invariant, that is, ϕ ∗ g = g . We denote the resulting collection of isometries of a given space with known metric as Isom( M ).  \nThe set of all isometries forms a group, acting on the manifold. In fact, by the Myers-Steenrod theorem, this group is a Lie group. As such, we may also consider the infinitesimal version of the isometries, i.e. the corresponding elements in the Lie algebra. These are captured by vector fields, which are known as Killing vector fields (KVFs). A KVF ξ has the (defining) property that  \nL ξ g = 0 , (4)  \nwhere L denotes the Lie derivative. Since all KVFs belonging to a certain metric form a Lie algebra under commutation of vector fields, we may speak of a metric's Killing Lie algebra (KLA).  \nIn a local coordinate system { x i } , we can write an explicit equation to solve for a KVF's components. In said coordinate system, expand the KVF as ξ = ξ i ∂ i , where ∂ i indicates the derivative in the x i -direction, and then apply this to the equation  \nL ξ [ g ( ∂ i , ∂ j ) ] = ( L ξ g ) ( ∂ i , ∂ j ) + g ( L ξ ∂ i , ∂ j ) + g ( ∂ i , L ξ ∂ j ) , (5)  \nfrom which we can deduce that  \nξ k g ij,k = -ξ k ,i g kj -ξ k ,j g ik , (6)  \nwhere comma notation was used to indicate partial differentiation ( ,a ≡ ∂ a ). Given the metric, this defines  \na collection of PDEs to solve for the ξ i . If instead covariant differentiation is used, the lhs vanishes and we are left with the famous equation ξ i ; j + ξ j ; i = 0 defining KVFs.  \nIn general a metric may allow for multiple KVFs; the maximum allowed is n ( n +1) / 2, where n is the dimension of the manifold [21, § 17.1]. This also means that the dimension r of the isometry group is bounded by r ≤ n ( n +1) / 2. While on a general manifold it is possible that no KVF is allowed for a metric, this is never the case in the class of spacetime manifolds considered in this work: we wish to investigate homogeneous spacetimes, and as we shall see below, this means they have KVFs.  \nOnce we have our infinitesimal descriptions of isometries, we can quantify the orbit of a starting point under these infinitesimal isometries. To this end, we need the following definition.  \nDefinition II.2 (Transitivity) . A (sub)group of isometries G ≤ Isom( M ) is said to act transitively on some submanifold S if G ( s ) = S [27, § 5] for all s ∈ S . In this case S is the orbit of the isometries, and also referred to as surfaces of transitivity. The dimension of the transitivity subgroup dim S we denote by s .  \nEffectively, through transitivity we infer an equivalence relation on M of which points can be reached via an isometry. However, there may be 'redundancy' in the surface of transitivity; we may reach individual points via more than one isometry. We say that G acts simply transitively on the surface of transitivity if there is precisely one isometry that connects pairs of points. Or, equivalently, when the infinitesimal generators of isometries are linearly independent as vector fields [25, § 6.2]. Otherwise, we call G multiply transitive .  \nWe also need to touch on the concept of isotropy . The isotropy subgroup (of isometries) at some p ∈ M consists of those isometries which leave the point p unchanged:  \nStab( p ) := { ϕ ∈ Isom( M ) : ϕ ( p ) = p } . (7)  \nThe symbol Stab comes from the alternative name for the isotropy subgroup, being the stabilizer . Directly, this implies that the infinitesimal generators of those isometries, i.e. their associated KVFs, vanish at p .  \nLemma II.3. The dimension of the isotropy subgroup dimStab ≡ q is constant over a surface of transitivity.  \nProof. Suppose a and b lie in the same surface of transitivity, and let ϕ be an isometry so that ϕ ( a ) = b . We notice that we can send an isotropy transformation at a to one at b by noticing  \nψ ∈ Stab( b ) = ⇒ ϕ -1 · ψ · ϕ ∈ Stab( a ) = ⇒ ψ · ϕ ∈ ϕ · Stab( a ) = ⇒ Stab( b ) · ϕ ⊆ ϕ · Stab( a ) (8)  \nBy interchanging the roles of a and b , the reverse inclusion can also be established, so that we have equality. Hence, Stab( a ) ∼ = Stab( b ), and so they have the same dimension.  \nOne also says that the surface of transitivity is completely anisotropic if the isotropy subgroup is trivial (and so generated by the zero vector field).  \nFollowing the above Lemma, we recognize that dimStab( S ) is a well-defined entity for a surface of transitivity S . We can then make an important conclusion about the structure of symmetries, by noticing the following: we have that  \ndimIsom( S ) = dim Stab( S ) + dim S (9)  \n[27, § 5.1]; or, r = q + s . Here, Isom( S ) indicates the group of isometries defined on the surface of transitivity. That this is true can be seen heuristically by considering that KVFs (which generate isometries) either vanish at some point on S or they do not. If no KVF vanishes anywhere on S , then evidently dim Isom( S ) = dim S ; if not, then dim Stab( S ) is the maximum number of KVFs that simultaneously vanish at some p ∈ S . Then S must be of dimension dim Isom( S ) -dimStab( S ), as that is how many KVFs at said point still generate the surface of transitivity.", 'B. Bianchi models': "We now narrow our symmetry considerations down to the case of interest, namely that of a 4-dimensional pseudo-Riemannian manifold ( M , g ).  \nDefinition II.4 (Spatial homogeneity) . We say that if ( M , g ) permits a transitive KLA of dimension 3, then ( M , g ) is spatially homogeneous . We may also shorten this to homogeneous if it is clear from context that spatial homogeneity is meant.  \nA surface of transitivity generated by such a KLA is called a spatial section .  \nIt is then clear that we should consider the spatial sections as 'space,' with the remaining dimension fulfilling the role of 'time.' This gives rise to the split M = R × S , a consequence of the assumption of global hyperbolicity that is often present in mathematical treatises of general relativity (e.g. [28, Thm. 5.44]), and implicitly made in the study of cosmology. Using the symmetry language built up in the previous section, we can then articulate precisely what is meant with a Bianchi model.  \nDefinition II.5 (Bianchi model) . A Bianchi model is a spatially homogeneous spacetime, with a KLA of dimension precisely 3.  \nRemark II.6 (Differing nomenclature) . One may also adhere to the definition that a Bianchi model is a spacetime which is spatially homogenous, without the dimensionality restriction. Under this definition, locally rotationally symmetric (LRS) Bianchi models (which have a KLA of dimension 4) and the FLRW models (dimension 6) will be special cases of Bianchi models, instead of being completely separated. Since for our purposes we  \nwish to restrict to three-dimensional KLAs, we make the additional dimensionality requirement.  \nIn view of equation (9), it follows immediately that a Bianchi model is completely anisotropic. Furthermore, the vector fields which compose the KLA form a frame for the tangent space T p S for all p ∈ S : indeed, if somewhere they did not form a frame, then locally the spatial section would be of a dimension smaller than 3, which is not possible due to (9).  \nRemark II.7 (Different choices q , s ) . Effectively in the definition of Bianchi models we prescribed dimensions for the isotropy subgroup and surfaces of transitivity. Since mathematically there is nothing special about these choices, others can be considered as well. For instance, [27, § 5.2] presents a table outlining the valid choices of dimensions, and the names of the resulting models.  \nNow that we have defined what a Bianchi model is, we may introduce a further level of detail by invoking the Lie algebraic structure of the KLA. Since for Bianchi models the dimension of the KLA is 3, a Bianchi model is effectively characterized by a 3-dimensional Lie algebrawe shall write its basis as { ξ a } a =1 , 2 , 3 . Note that here the subscripts do not indicate coordinate dependence, but are simply the labels of the vector fields.  \nThere are infinitely many Lie algebras of dimension 3, but if we construct equivalence classes through GL(3 , R ) invariance of the Lie algebra's basis, it turns out that there are only nine distinct classes of Lie algebras. One could classify these in various ways that one sees fit, and depending on the intended use. In the astrophysics literature it is common to consider the Bianchi(-Behr) classification of Lie algebras. This is given in Table I, with additional mathematical information [22, 29]. The structure constants are determined from the table's entries as  \n     [ ξ 1 , ξ 2 ] = n 3 ξ 3 -aξ 2 [ ξ 2 , ξ 3 ] = n 1 ξ 1 [ ξ 3 , ξ 1 ] = n 2 ξ 2 + aξ 3 . (10)  \nThat this indeed yields a valid classification (i.e. every Lie algebra belongs to precisely one of these types by a suitable GL(3) transformation of its basis) is shown in [19]. The classes are thus composed of Lie algebras which, when their basis is suitably chosen, have the prototypical structure constants.  \nWe are then able to articulate what we mean when we consider metrics of Bianchi type n .  \nDefinition II.8 (Bianchi type n metric) . A spacetime model ( M , g ) (and in particular its metric) is of Bianchi type n if it is  \n- 1. spatially homogeneous,\n- 2. its KLA has dimension precisely 3, and\n- 3. the KLA is of Bianchi type n , i.e. it permits a basis with structure constants as found through Table I and equation (10), for Bianchi type n .  \nTABLE I: Table of Bianchi types, together with further mathematical detail. Here, i refers to the 1-dimensional Lie algebra, and l (2) to the irreducible 2-dimensional Lie algebra.  \nWe call the metric g of such a Bianchi model a Bianchi type n metric.  \nFor further details, see [21, § 2.7.3 & § 17.1.3] and [27, § 5].", 'III. FROM PRE-KLA TO METRIC': "In the preceding section, we have assumed that there was a spacetime model featuring a metric, for which we would then find the appropriate Killing vector fields (KVFs), for instance through use of the Killing equation (4). Now, we wish to turn this scenario around: for some given 3D Lie algebra of smooth, non-vanishing, transitive vector fields { ξ a } a =1 , 2 , 3 on a 3D manifold Σ 3 , which we interpret as a spatial section, we desire to find a metric so that these vector fields are Killing with respect to it. That is to say, knowing the isometries that we want, we wish to find a metric on which they are indeed realized as isometries. Let us call a Lie algebra with the above characterisation a pre-KLA.  \nIn order to cook up a suitable metric, we need i) a collection of vector fields, { X i } i =1 , 2 , 3 , and ii) a collection of one-forms { e i } i =1 , 2 , 3 , both on Σ 3 , which satisfy the following relations:  \n[ ξ a , X i ] = 0 and e i ( X j ) = δ i j . (11)  \nIn this case, the collection { X i } is known as an invariant frame (IF), and { e i } is its dual (or the dual frame , DF). In the mathematical literature, what we have termed an IF is known as a left-invariant frame [30, § 8]. This terminology stems from the frame { X i } being invariant under the left action of the ξ a , emblemized by their commutator vanishing as in (11).  \nWe note that the first condition implies that the invariant frame is also a Lie algebra. Assume there are functions D k ij on Σ 3 such that [ X i , X j ] = D k ij X k . Then, we can calculate  \nL ξ a ( D k ij X k ) = ( ξ a D k ij ) X k (12a)  \nand  \nL ξ a [ X i , X j ] = [ ξ a , [ X i , X j ]] = 0 , (12b)  \ndue to the Jacobi identity. Hence, we conclude that ξ a D k ij = 0 for all a , everywhere, so that the functions D k ij must in fact be constants. It follows that the IF is a Lie algebra.  \nThe two properties (11) are sufficient to construct a tensor product which vanishes under Lie differentiation by a pre-KLA, which is shown in the following proposition.  \nProposition III.1. Let { ξ a } a pre-KLA, and { X i } and { e i } an IF and DF, respectively. Furthermore, let γ = γ ij e i ⊗ e j for γ ij arbitrary constants. Then, L ξ a γ = 0 .  \nProof. Let us first show that L ξ a e i = 0. To see this, observe that  \n0 = L ξ a ( e i ( X j )) = ( L ξ a e i ) ( X j ) + e i ( L ξ a X j ) = ( L ξ a e i ) ( X j ) , (13)  \nwhere we used that [ ξ a , X j ] = 0. So we conclude that L ξ a e i = 0 for all i and a . We can then calculate the Lie derivative of γ :  \nL ξ a γ = ( ξ a γ ij ) e i ⊗ e j + γ ij ( L ξ a e i ) ⊗ e j + γ ij e i ⊗ ( L ξ a e j ) , (14)  \nand see that it vanishes: the latter two terms due to our previous observation, and the first since γ ij is constant. Finally, since ξ a was chosen arbitrarily, this holds for all a .  \nAlthough the tensor product γ introduced above is certainly not a metric, it could be made into one by symmetrizing the tensor product and demanding that the constants γ ij form a positive, symmetric matrix. This then results in a constant metric on the spatial slices Σ 3 . However, for our purposes, we wish to obtain a metric for spacetime, and so we somehow need to involve time as well.  \nThis is done by 'adding a time axis,' and creating a spacetime manifold R × Σ 3 (cf. the global hyperbolicity assumption). We furthermore introduce the vector field X 0 along the added dimension, and its corresponding dual e 0 .  \nThe requirements we place on e 0 is that { e µ } µ =0 , 1 , 2 , 3 and its dual { X µ } µ =0 , 1 , 2 , 3 satisfy i) the properties as outlined in (11), but with the Roman indices replaced by Greek ones:  \n[ ξ a , X µ ] = 0 and e µ ( X ν ) = δ µ ν ; (15)  \nand ii) the relation [ X 0 , X i ] = 0 for all i . From this latter condition, we see that essentially X 0 = ∂ 0 is the only choice up to a coefficient, which we could absorb into the definition of t .  \nAs such, we can construct a spacetime metric on which a pre-KLA is indeed Killing: set g = g µν ( t ) e µ e ν for some appropriate functions g µν ( t ). The { ξ a } do not see the dependence on t in the metric components, and hence the reasoning for the proof of Proposition III.1 is valid to show L ξ a g = 0 for all a .  \nSince commutativity of the Killing and IF vector fields persists if we multiply the latter with functions of time, a linear combination of IF vector fields with functions in time-say Y µ ( t ) := y ν µ ( t ) X ν , for functions y ν µ ( t )-will still form an IF. This freedom corresponds to the freedom to 'line up' the spatial slices along the t -axis in the product M = R × Σ 3 [25, § 6.4].  \nRemark III.2 (Orthonormal synchronous form) . For any metric g = g µν ( t ) e µ e ν , we can also consider a redefinition of the dual basis into ˜ e µ ( t ) := b µ ν ( t ) e ν , for b having the property that η µν b µ α b ν β = g αβ , where η µν = diag( -1 , 1 , 1 , 1) is the Minkowski metric. In that case, g = η µν ˜ e µ ˜ e ν , and so the metric becomes the Minkowski one. This form is called the orthonormal synchronous form of a Bianchi model [21].  \nAs many equations greatly simplify/trivialize with constant metric components, this form may be useful to utilize if one is willing to make the tradeoff for more complicated (time evolutions of) basis vectors.  \nSo, let us summarize what we have found up to now. Given some pre-KLA { ξ a } , if we find  \n1. some vector fields { X µ } satisfying [ ξ a , X µ ] = [ X 0 , X µ ] = 0, and  \n- 2. the covector fields { e µ } dual to these,  \nthen we can find a spacetime metric, depending only on time t , so that the { ξ a } are Killing on it. It rests us to construct these vector and covector fields, i.e. to detail the steps 1 and 2 above.  \nIn fact, it is known that the second step can be made in a unique way, and its computation is straightforward. Since the first step is nontrivial, however, that is what we shall detail in the following subsection.", 'A. To frame a Killing': "To simplify matters, let us make the choice X 0 ≡ ∂ 0 , so that the problem reduces to needing to find only { X i } for i = 1 , 2 , 3.  \nLet us note that since a pre-KLA { ξ a } forms a basis for the tangent space at each point in Σ 3 (by spatial homogeneity), we may decompose any vector field in its basis. Utilizing this for our proposed invariant basis { X i } , and  \ntaking inspiration from [31], we make the calculation  \n0 ! = [ ξ a , X i ] = [ ξ a , X b i ξ b ] = ( ξ a X b i ) ξ b + X b i [ ξ a , ξ b ] = ( ξ a X b i + X c i C b ac ) ξ b , (16)  \nwhere the C b ac are the structure constants of the Lie algebra spanned by { ξ a } . Thus, the X i satisfy the linear differential equations,  \nξ a X b i = -C b ac X c i . (17)  \nfor all a and b . We emphasize that the above equations are actually three sets of equations, one for each i . That is to say, one for each of three vector fields { X i } which together form an IF. In order to make headway in solving these differential equations, we employ the method of characteristics [32, § 3.2]. Assume that we parametrize a path in (a local coordinate) ( x 1 , x 2 , x 3 )-space with parameter s , so that the tangent to this path is given by F ( x ) = F j ( x ) ∂ j :  \ndx j ds = F j , (18)  \nwhere x = x ( s ). That is, we follow the flow of F from a given starting point. Then, along this flow, we can examine how the components of X i evolve:  \ndX b i ds = ∂X b i ∂x j dx j ds = F j ∂ j X b i ⋆ = F a ξ a X b i = -C b ac F a X c i , (19)  \nwhere in ⋆ we used that F j ∂ j = F a ξ a for suitable components F a ( x ).  \nObserve now that if we supply an initial condition, then along this flow from this initial condition, our problem will boil down to an initial value problem (IVP), and standard results in the theory of ODEs will give us existence and uniqueness of the evolution of (the components of) X i along said flow.  \nThe initial condition we pick is that at some p ∈ Σ 3 , we have X i | p := δ a i ξ a | p . The reason for this is that, if this is the case, we have the following useful Proposition at our disposal.  \nProposition III.3. Let { ξ a } a pre-KLA on Σ 3 , and { X i } its IF. Let C c ab denote the structure constants of the pre-KLA, and D k ij those of the IF. Assume moreover that there exists a point p ∈ Σ 3 such that X i | p = δ a i ξ a | p . Then we have D k ij = -C k ij .  \nConsequently, the Lie algebra spanned by { X i } is of the same Bianchi type as that spanned by { ξ a } .  \nProof. We adopt the proof from [25, § 6.3]. Since we assumed homogeneity, we have  \nD k ij X k = [ X i , X j ] = [ X a i ξ a , X b j ξ b ] = ✘ ✘ ✘ ✘ ✘ ✘ X a i ( ξ a X b j ) ξ b -X b j ( ξ b X a i ) ξ a + ✘ ✘ ✘ ✘ ✘✘ X a i X b j C c ab ξ c = X a j X c i C b ac ξ b , (20)  \nwhere the cancellation follows upon applying (17). As such, we may derive the equality  \nD k ij X b k = X a j X c i C b ac . (21)  \nThe premise implies that at p , X a i | p = δ a i . If we consider the equality at point p , we find the relation D k ij = C k ji = -C k ij . This completes the first part of the proof.  \nTo see the second point, simply consider the isomorphism x ↦→ -x for x in the Lie algebra spanned by { X i } . Under this isomorphism, the structure constants are sent to their negative, and so will be the same as those for the Lie algebra spanned by { ξ a } . Hence, they must have the same Bianchi type.  \nWithout loss of generality, we can choose p , the point of coincidence in the above Proposition, to be the origin in a local coordinate space. We may then formulate an IVP for what we shall name the IF:  \n- 1. along the flow of F , parametrized by s , evolve X b i as:  \ndX b i ds = -C b ac F a X c i ; (22a)  \n2. initial condition at p ∈ Σ 3 :  \nX i | p = δ a i ξ a | p . (22b)  \nIn the especially simple case F a = δ a b , for b ∈ { 1 , 2 , 3 } , we get the flow along the corresponding KVF-this flow is known as the characteristic . Hence, we can also reformulate our IVP (22) as being along the characteristic curves:  \n- 1'. the characteristic of ξ a , parametrized by s :  \ndx j ds = ξ j a ; (23a)  \n2'. evolve the components of X i along this characteristic as:  \ndX b i ds = -C b ac X c i ; (23b)  \n3'. initial condition at p ∈ Σ 3 :  \nX i | p = δ a i ξ a | p . (23c)  \nNote that since we assumed that our vector fields were transitive, we know that we can reach any point p ∗ ∈ Σ 3 from p by following certain characteristics for a certain amount of distance. If the number of characteristics required is more than one, we see that our path from p to p ∗ will be only piecewise smooth.  \nThe IVP (23), for each i and a , can be regarded as a matrix differential equation: if we define ⃗ X i := ( X 1 i X 2 i X 3 i ) ⊺ , then  \nd ds ⃗ X i = -C a ⃗ X i = ⇒ ⃗ X i ( s ) = exp( -C a s ) ⃗ X i, 0 , (24)  \nfor the matrix C a with entries ( C a ) b c = C b ac . Note that due to the skew symmetry of the structure constants, we will have that the a th column in C a will be the zero column, so that the matrix exponential will look like, e.g. for a = 1,  \nC 1 =   0 ∗ ∗ 0 ∗ ∗ 0 ∗ ∗   = ⇒ exp( -C 1 s ) =   1 ∗ ∗ 0 ∗ ∗ 0 ∗ ∗   . (25)  \nRemark III.4. The matrices C a are also the images of the adjoint representation of the pre-KLA spanned by { ξ a } , when expressed in the { ξ a } -basis.  \nConsidering the relative simplicity of exponentiating constant matrices, the formulation (23) suggests a good way to find an IF when presented with a pre-KLA { ξ a } . Namely, if we want to know the IF at some point p ∗ ∈ Σ 3 , p ∗ = p , we make the following steps:  \n̸  \n- 1. Find a path consisting of at most three segments, where each segment\n- (a) follows a characteristic, say ξ a ,\n- (b) for a suitable length s a ,  \nso that the end point of the full path is p ∗ .  \n- 2. For each segment, calculate the matrix exp( -C a s a ) (no sum).\n- 3. Multiply ⃗ X i | p with the first, and then second and third previously calculated matrices (if applicable), corresponding to the order of the matrices.  \nThe result is, for each p ∗ , the 'vector of vectors' ⃗ X i | p ∗ , which when read in the { ξ a } -basis, gives vector fields of the IF.  \nRemark III.5. We may also consider a different flavor of the IVP (23). Since the pre-KLA { ξ a } is homogeneous, there should exist an inverse tensor A = A ( x ) so that A a e ξ f a = δ f e for ξ a expressed in a local coordinate neighborhood around p as ξ a = ξ e a ∂ e . Here { ∂ e } are the coordinate vector fields. If we then contract equation (17) with A a e , we obtain  \n∂ e X b i = -A a e C b ac X c i . (26)  \nWith the identical justifcation as the one leading up to (22), we can write down what the evolution of (the components of) X i is along some curve parametrized by s whose tangent curve is F :  \n- 1. along the flow of F , parametrized by s , evolve X b i as:  \ndX b i ds = -C b ac F e A a e X c i ; (27a)  \n- 2. initial condition at p ∈ Σ 3 :  \nX i | p = δ a i ξ a | p (27b)  \n(note that F = F e ∂ e ). This formulation makes it very simple to find a path connecting the starting point p to some other (nearby) p ∗ ∈ Σ 3 : simply follow the flows of ∂ 1 , ∂ 2 , and ∂ 3 for the required length. The tradeoff is in the complexity of the integrand: this now depends on the (generally nontrivial) functions A a e , so that one has to resort to complex integration methods.", 'B. The question of uniqueness': "So far, then, given a pre-KLA { ξ a } , we can find the components of the IF (that is, an invariant frame { X i } with a point p ∈ Σ 3 so that X i | p = δ a i ξ a | p ) in the basis of the pre-KLA at any given point within our spatial section (recall that the spatial section is by definition all those points that can be reached via the isometries generated by the KVFs): we have existence for the IF. Now then rises the question of whether the so found IF is unique to this pre-KLA.  \nThis question is not related to the unique solvability of the IVP along the characteristics: standard ODE results tell us that uniqueness is guaranteed in this case. The issue arises when one chooses to use a path not strictly composed of characteristics in order to reach a point at which the IF is desired.  \nThe following theorem tells us that uniqueness is indeed the case.  \nTheorem III.6. For a given pre-KLA { ξ a } , the corresponding IF { X i } with X i | p = δ a i ξ a | p at some p ∈ Σ 3 , is unique.  \nProof. Suppose there are two such IFs, { A i } and { B i } , with p ∈ Σ 3 such that A i | p = B i | p = δ a i ξ a | p . Then their difference, D i := A i -B i , satisfies  \n[ ξ a , D i ] = 0 = ⇒ ξ a ( D b i ) = -C b ac D c i , (28)  \nby the same reasoning as we used above to write equation (17) (note D i is decomposed in the { ξ a } -basis). So, we can create an IVP for the D i along the same lines we did above: let F = F a ξ a a the tangent to some curve parametrized by s , so then  \n- 1. along the flow of F , parametrized by s , evolve D b i as:  \ndD b i ds = -C b ac F a D c i ; (29a)  \n2. initial condition at p :  \nD i | p = 0 . (29b)  \nRewrite the evolution equation into a matrix system: ⃗ D i := ( D 1 i D 2 i D 3 i ) ⊺ , so that we can write  \nd ds ⃗ D i = -F a C a ⃗ D i = ⇒ ⃗ D i ( s ) = P exp ( -∫ s 0 F a C a ds ' ) ⃗ D i, 0 , (30)  \nwith C a defined as in (24) and P exp denoting the pathordered exponential. This latter is necessary, as the integrand depends on position (namely, F a = F a ( x ( s )) generally).  \nObserve that if we choose to start from the point p , then ⃗ D i, 0 = ⃗ D i | p = 0, so that ⃗ D i ( s ) = 0 for all s -more importantly, this conclusion holds for any vector field F . Hence, the vector field D i is well-defined as a function of the spatial section Σ 3 , and moreover this function returns the zero vector everywhere: D i | q = 0 for all q ∈ Σ 3 . Therefore A i = B i everywhere, completing the proof.  \nCorollary III.7. Let { X i } be the IF for the pre-KLA { ξ a } with X i | p = ξ i | p , p ∈ Σ 3 . Suppose that { Y i } is a collection of vector fields on Σ 3 such that [ ξ a , Y i ] = 0 for all a and i , and Y i | p = y a i ξ a | p for constants y a i forming a GL(3) matrix. Then Y i = y j i X j .  \nProof. Define ˜ Y i := ( y -1 ) j i Y j , and observe that  \n[ ξ a , ˜ Y i ] = 0 and ˜ Y i | p = δ a i ξ a | p . (31)  \nBy the above Theorem, the only collection of vector fields which has this property is the IF { X i } , so ˜ Y i = X i .", 'C. Change of the invariant frame': "In the previous two subsections we have established that, given a pre-KLA { ξ a } on Σ 3 , we have existence and uniqueness of the accompanying IF. That is to say, we have that there exist vector fields { X i } so that [ ξ a , X i ] = 0 for all a and i , that X i | p = δ a i ξ a | p for some p ∈ Σ 3 , and this collection of vector fields is unique.  \nWe would now like to understand how the IF changes if we apply some manner of transformation to the original { ξ a } . After all, it would hardly be efficient if for each possible pre-KLA { ξ a } we would need to calculate the IF all over again, even if it is clear that there exist a relation to some other pre-KLA for which the IF was known already. Ideally, we want to export a relation between the pre-KLAs to one between their corresponding IFs.  \nTo this, non-exhaustive effect, we have the following Propositions.  \nProposition III.8. Let { ξ a } a pre-KLA on M with IF { X i } , and { ξ ' a } one on N . Suppose that ϕ : ξ a ↦→ ξ ' a for a = 1 , 2 , 3 . If ϕ = Φ ∗ for some diffeomorphism Φ : M→ N , then the IF belonging to { ξ ' a } is given by { ϕ ( X i ) } .  \nProof. It is known that for a diffeomorphism Φ : M→ N and vector fields A,B ∈ X ( M ), we have that [Φ ∗ A, Φ ∗ B ] = Φ ∗ [ A,B ]-see for instance [30, Ch. 8]. Applying this to our case, observe  \n[ ξ ' a , ϕ ( X i )] = [Φ ∗ ξ a , Φ ∗ X i ] = Φ ∗ [ ξ a , X i ] = 0 , (32)  \nestablishing commutativity. Now let p ∈ M be a point so that X i | p = δ a i ξ a | p . Then, we have  \nΦ ∗ X i | Φ( p ) f = X i | p ( f · Φ) = δ a i ξ a | p ( f · Φ) = δ a i Φ ∗ ξ a | Φ( p ) f, (33)  \nfor any function f ∈ C ∞ ( N ). Conclude therefore that ϕ ( X i ) | Φ( p ) = δ a i ξ ' a | Φ( p ) , and so that { ϕ ( X i ) } constitutes the IF for { ξ a } .  \nProposition III.9. Under a GL(3) transformation ϕ of the pre-KLA { ξ a } , ξ a ↦→ ξ ' s = ϕ a s ξ a , we have that the components of the corresponding IF change as X a i ↦→ X ' t s = ϕ i s X a i ( ϕ -1 ) t a .  \nProof. Since the KVFs and the IF commute with each other, so does any constant linear combination of them. As such, we see that for the GL(3) transformation ϕ , we have that  \n[ ϕ a s ξ a , ϕ i s X i ] = 0 and ϕ i s X i | p = ϕ a s ξ a | p , (34)  \nfor p ∈ Σ 3 such that X i | p = δ a i ξ a | p . Hence, { ϕ i s X i } is the IF for the transformed pre-KLA. The components of the IF vector fields in the new basis are given by  \nϕ i s X i = ϕ i s X a i ξ a = ϕ i s X a i ( ϕ -1 ) t a ϕ b t ξ b = ϕ i s X a i ( ϕ -1 ) t a ϕ t . (35)  \nIf we write X ' s = ϕ i s , then from the above equation we conclude that X ' t s = ϕ i s X a i ( ϕ -1 ) t a , which completes the proof.  \nFrom this second Proposition, we make an important observation. Namely, GL(3) actions acting on a pre-KLA conserve Bianchi type of said pre-KLA. As such, once we know what the Bianchi class of our pre-KLA is, we also know that there will exist a certain GL(3) to bring it into its tabulated, standard form. In finding IFs for pre-KLAs, we can thus assume that the pre-KLA is in standard form; if a given pre-KLA were not, we would perform a suitable GL(3) transformation to make it so, find the IF, and then 'untransform' utilizing Proposition III.9.  \nThough Proposition III.9 tells us that any pre-KLA can be transformed into standard Bianchi form, this does not imply that any two pre-KLAs of the same Bianchi type can be transformed into each other. To illustrate, consider the following example of two Bianchi II preKLAs (even with identical structure constants C 1 23 = 1 and rest zero), in some local coordinate frame:  \n     ξ 1 = ∂ 2 ξ 2 = ∂ 3 ξ 3 = ∂ 1 + x 3 ∂ 2 and      ξ 1 = ∂ 1 ξ 2 = ∂ 2 -1 2 x 3 ∂ 1 ξ 3 = ∂ 3 + 1 2 x 2 ∂ 1 . (36)  \n(The former is type II in Ryan & Shepley [25, § 6.4], and the latter is the Heisenberg algebra.) Evidently one cannot perform a GL(3) transformation from one to the other, and it can be verified that they are not diffeomorphic. Hence, these two cannot be related to each other by the previous two Propositions. We shall discuss this case further in the next section.", 'D. List of invariant frames': 'From subsection III C we learned that if we know the IF to some pre-KLA, we can also calculate the IF belonging to a GL(3)-transformed pre-KLA, and from III B that any found IF-with a given initial condition-is unique to the pre-KLA. Combining these two observations, we see that it has value to find expressions for the IFs belonging to pre-KLAs in the various Bianchi classes. The methodology from subsection III A allows us to do this, given only the structure constants of a pre-KLA.  \nThis has been done here, with the Table II as its result. For each class, one considers a pre-KLA { ξ a } such that they possess the structure constants as indicated (with any unmentioned structure constants equal to zero). 1 The general form of the IF is then given, with relations to be satisfied by the functions p ( x ), q ( x ), and r ( x ). Initial conditions for these functions are such that, at some reference point (e.g. a coordinate origin x = 0), X i = δ a i ξ a .  \nRemark III.10. Since the relation [ ξ a , X i ] is symmetric, we could also reverse the roles that ξ a and X i are playing: take the ξ a as the invariant frame, and X i as the Killing vectors. In this way, the table allows us to find KVFs belonging to a metric without having to employ the Killing equation. If a given metric can be written in the form ds 2 = -dt 2 + γ ij ( t ) e i e j , and constants C i jk found so that the 1-forms e i satisfy  \nde i = -1 2 C i jk e j ∧ e k , (37)  \nthen inputting ξ i as the dual of e i in the table will yield the X i as the non-vanishing KVFs.', 'IV. EXAMPLES': 'In the preceding section we developed a methodology for calculating the IF belonging to a given pre-KLA. The result was the listing in § III D, which gives the IF in terms of the pre-KLA that was given, with some functions that need to be determined and that are pre-KLA specific. This was done so that a general structure may be observed, without reference to any particular choice of pre-KLA, say, in terms of coordinates.  \nHowever, in order to gain some intuition into Bianchi models, the symmetries that may arise, and practical application of this result, it is evidently useful to consider some specific examples. This is what we shall do in this section.', 'A. Twice Bianchi V': "Let us work out two examples of the Bianchi V type. This is the anisotropic 'generalization' of the open FLRW model, and so one may already expect that somehow a hyperbolic nature to these models will arise.  \nModel 1: Consider the pre-KLA  \nξ 1 = ∂ 2 , ξ 2 = ∂ 3 , ξ 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (38)  \n(this is type V in [25, § 6.4]). Its structure constants are C 1 13 = C 2 23 = 1, and the rest is zero. This is precisely type V in the above list, and so no additional GL(3) transformation is required. The solution for the functions p , q , and r are found to be:  \np ( x ) = x 2 , q ( x ) = x 3 , r ( x ) = x 1 , (39)  \nso that the IF, and corresponding dual frame, is  \n     X 1 = e x 1 ∂ 2 X 2 = e x 1 ∂ 3 X 3 = ∂ 1 = ⇒      e 1 = e -x 1 dx 2 e 2 = e -x 1 dx 3 e 3 = dx 1 . (40)  \nThese are thus the basis components for our spacetime metric. Upon setting e 0 ≡ dt , we can then find the general spacetime metric to be  \nds 2 = g µν ( t ) e µ e ν , (41)  \nwhere the coefficients g µν ( t ) are functions of time t and satisfy the usual assumptions. Making the particular choice of diagonal spacetime metric, setting g 00 ≡ -1, and writing out the coordinate components, we obtain the form  \nds 2 = -dt 2 + g 11 ( t ) 2 ( dx 1 ) 2 + e -2 x 1 ( g 22 ( t ) 2 ( dx 2 ) 2 + g 33 ( t ) 2 ( dx 3 ) 2 ) . (42)  \nThe functions g ii ( t ), i = 1 , 2 , 3 (no sum) are essentially arbitrary for the purposes of ensuring this metric is Bianchi V (indeed, has the KLA defined by (38)).  \nModel 2: We now pick as pre-KLA  \nξ 1 = e x 1 ∂ 2 , ξ 2 = e x 1 ∂ 3 , ξ 3 = ∂ 1 . (43)  \n(This is exactly the IF in the above construction!) Its structure constants are C 1 13 = C 2 23 = -1; this does not occur as a standard one in the above list, and so we need a GL(3) transformation. The relevant transformation is ξ a ↦→ ξ ' a = -ξ a , for which then C ' 1 13 = C ' 2 23 = 1, and this is again exactly type V in the above list. We can then solve for the functions p , q , and r (remembering that we are doing this for ξ ' a ),  \np ( x ) = -x 2 e -x 1 , q ( x ) = -x 3 e -x 1 , r ( x ) = -x 1 , (44)  \n̸  \nTABLE II: General forms of the IFs { X i } belonging to pre-KLAs { ξ a } of the various Bianchi types. The functions p = p ( x ), q = q ( x ), and r = r ( x ) must be solved for any particular choice of pre-KLA, to satisfy the indicated relations. In Bianchi VII h , ∆ := 1 2 √ 4 -h 2 .  \nand so construct the IF and corresponding dual frame:  \n     X ' 1 = ∂ 2 X ' 2 = ∂ 3 X ' 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (45)  \n⋆ = ⇒      X 1 = ∂ 2 X 2 = ∂ 3 X 3 = ∂ 1 + x 2 ∂ 2 + x 3 ∂ 3 (46)  \n= ⇒      e 1 = -x 2 dx 1 + dx 2 e 2 = -x 3 dx 1 + dx 3 e 3 = dx 1 . (47)  \nThe step ⋆ boiled down to a relabeling, as a scaling does not affect the components of the IF (cf. Proposition III.9). Now that we have the dual basis, we can construct a spacetime metric along the same lines as above. Once more assuming a diagonal spacetime metric (in the basis { e µ } ), g 00 ≡ -1, and writing the spacetime metric in coordinates, we find  \nds 2 = -dt 2 + g 33 ( t ) 2 ( dx 1 ) 2 + g 11 ( t ) 2 ( dx 2 -x 2 dx 1 ) 2 + g 22 ( t ) 2 ( dx 3 -x 3 dx 1 ) 2 (48)  \nHow do we interpret the metrics resulting from both these models? Ignoring the temporal components, the spatial metrics of these models are  \ndℓ 2 1 = a 2 1 ( dx 1 ) 2 +( b 1 e -x 1 ) 2 ( dx 2 ) 2 +( c 1 e -x 1 ) 2 ( dx 3 ) 2 (49a)  \nand  \ndℓ 2 2 = a 2 2 ( dx 1 ) 2 + b 2 2 ( dx 2 -x 2 dx 1 ) 2 + c 2 2 ( dx 3 -x 3 dx 1 ) 2 , (49b)  \nfor constants a 1 etc. The spheres of radius R around a point x ∗ , S R ( x ∗ ), for the former case are ellipsoids, whilst for the latter the shape is more difficult to categorize. Note that these S R ( x ∗ ) are also the shapes of the lightcones in the relevant spacetime: as such, they are also the shape of a 'flash of light' emitted from x ∗ . Some representative S R ( x ∗ ) for both cases are shown in Figures 1 and 2.  \nIt must also be noted that these shape seem to have some rotational symmetry still, which would beleaguer the assumption of anisotropy. However, we recall that these metrics only hold for a specific time, namely such a time at which the metric coefficients become the constants a 1 etc. Evolving the full spacetime metric in time (with the metric coefficients following Einstein's equation), these S R ( x ∗ ) will start to evolve as well. The only way the apparent rotational symmetry might persist is if, in general, the KVFs were dependent on time, which we demanded they not be.", 'B. Twice Bianchi II': 'In the above subsection § III C we touched upon two forms for Bianchi II pre-KLAs:  \n     ξ RS 1 = ∂ 2 ξ RS 2 = ∂ 3 ξ RS 3 = ∂ 1 + x 3 ∂ 2 &      ξ H 1 = ∂ 1 ξ H 2 = ∂ 2 -1 2 x 3 ∂ 1 ξ H 3 = ∂ 3 + 1 2 x 2 ∂ 1 , (50)  \nwith RS and H denoting Ryan & Shepley and Heisenberg, respectively. Both have the tabulated structure constants C 1 23 = 1 belonging to type II. As both algebras are written in the { ∂ i } -basis we can find a linear transformation mapping one to the other: ϕ : ξ RS a ↦→ ξ H a . In the { ∂ i } -basis, it is expressed as  \nϕ =   1 2 x 2 -x 3 1 -1 2 x 3 0 0 1 1 0 0   . (51)  \nThis is not a Jacobian matrix, and so we cannot use Proposition III.8 to our advantage; the methodology will be calculate the IFs separately, using the Table II. So we do this:  \n     X RS 1 = ∂ 2 X RS 2 = x 1 ∂ 2 + ∂ 3 X RS 3 = ∂ 1 &      X H 1 = ∂ 1 X H 2 = ∂ 2 + 1 2 x 3 ∂ 1 X H 3 = ∂ 3 -1 2 x 2 ∂ 1 . (52)  \n(In fact, if ϕ were the Jacobian matrix of a diffeomorphism, then due to Proposition III.8 it would have to map ϕ : X RS a ↦→ X H a ; we can verify by direct calculation that this does not occur.) The dual frames follow:  \n     e RS 1 = dx 2 -x 1 dx 3 e RS 2 = dx 3 e RS 3 = dx 1 (53a)  \nand  \n     e H 1 = dx 1 + 1 2 ( x 2 dx 3 -x 3 dx 2 ) e H 2 = dx 2 e H 3 = dx 3 . (53b)  \nAssuming a diagonal metric (in the { e µ } -bases), the spatial parts of the metrics when written out in coordinates become  \ndℓ 2 RS = a 2 RS ( dx 1 ) 2 + b 2 RS ( dx 3 ) 2 + c 2 RS ( dx 2 -x 1 dx 3 ) 2 (54a)  \nand  \ndℓ 2 H = a 2 H ( dx 1 + 1 2 ( x 2 dx 3 -x 3 dx 2 )) 2 + b 2 H ( dx 2 ) 2 + c 2 H ( dx 3 ) 2 , (54b)  \nfor constants a RS etc. Examples of various S R ( x ∗ ) are presented in Figures 3 and 4 for both metrics.  \nFIG. 1: The sphere of radius R about x ∗ , S R ( x ∗ ) at various x ∗ , for the spatial metric (49a); R = 5, a 1 = b 1 = c 1 = 1. Since S R ( x ∗ ) does not depend on x 2 nor x 3 , only x 1 is fixed in the subfigures. In this particular case, they are all ellipsoids. Note also the different scales used for the different subfigures.  \n<!-- image -->  \nFIG. 2: The sphere of radius R about x ∗ , S R ( x ∗ ) at various x ∗ , for the spatial metric (49b); R = 5, a 2 = b 2 = c 2 = 1. Since S R ( x ∗ ) does not depend on x 1 , only x 2 and x 3 are fixed in the subfigures.  \n<!-- image -->  \nFIG. 3: The sphere of radius R about x ∗ , S R ( x ∗ ), at various x ∗ , for the spatial metric (54a); R = 7, a RS = b RS = c RS = 1. Since S R ( x ∗ ) does not depend on x 2 nor x 3 , only x 1 is fixed in the subfigures.  \n<!-- image -->  \nFIG. 4: The sphere of radius R about x ∗ , S R ( x ∗ ), at various x ∗ , for the spatial metric (54b); R = 7, a H = b H = c H = 1. Since S R ( x ∗ ) does not depend on x 1 , only x 2 and x 3 are fixed in the subfigures.  \n<!-- image -->', 'C. Separated form of homogeneous metrics': 'The existence of the IF when given a pre-KLA not only allows us to formulate metrics on which the pre-KLA is realized as a KLA, but also to rewrite given homogeneous metrics into a standard form, wherein the components depend only on time. This is evidently useful for GR-related purposes, as then the Einstein equation will reduce to coupled ODEs, instead of PDEs, simplifying the solving procedure.  \nTheorem IV.1. Let g be a spatially homogeneous metric. Then there exists a frame of 1-forms { e µ } and coefficients g µν ( t ) such that  \ng = g µν ( t ) e µ e ν . (55)  \nProof. Let { ξ i } i =1 , 2 , 3 be those KVFs of g which are purely spatial and form a frame everywhere, which exist due to spatial homogeneity. Since the KLA spanned by { ξ i } is trivially a pre-KLA, we can find the IF { X i } by the results of Section III. Set X 0 ≡ ∂ 0 = ∂ t , and then observe that we have  \nξ i [ g ( X µ , X ν )] = ( L ξ i g )( X µ , X ν ) + g ([ ξ i , X µ ] , X ν ) + g ( X µ , [ ξ i , X ν ]) = 0 , (56)  \n( µ, ν = 0 , 1 , 2 , 3), where the first term vanishes due to ξ i being a KVF, and the second and third term due to the IF commuting with KVFs. (That [ ξ i , X 0 ] = 0 follows from ξ i being purely spatial.) Since the KVFs form a frame, this means that the functions g ( X µ , X ν ) are constant in space, and hence are dependent only on time: g ( X µ , X ν ) = g ( X µ , X ν )( t ).  \nGiven the IF { X i } , we can uniquely find its dual frame of 1-forms { e i } . Moreover, with the added vector field X 0 = ∂ t ↔ e 0 = dt , we still maintain the desired dual frame relations e µ ( X ν ) = δ µ ν . When we then write the metric in said basis,  \ng = g ( X µ , X ν ) e µ e ν , (57)  \nwe see that we have exactly recovered the form of the metric we wanted to show exists.  \nWe emphasize that the proof is constructive: due to Table II, once the KVFs are determined, the IF follows after having solved for the functions p , q , and r from that table. Furthermore, the proof shows that any spatially homogeneous metric can be written into a form in which the temporal and spatial dependencies-generally e µ = e µ ( x )-are separated. As such, we dub this to be the separated form of the spatially homogeneous metric.', 'V. SUMMARY AND CONCLUSIONS': "According to the standard cosmological worldview, based on the Friedman-Lemaˆıtre-RobertsonWalker (FLRW) equations as its foundation, the universe on the largest scales adheres to the cosmological  \nprinciple. It means that the universe is perfectly uniform, isotropic and homogeneous, on scales in the order of the Hubble radius. Yet, we do know that in physical reality the universe contains a rich infrastructure, reflected in the existence of density and velocity deviations whose amplitude becomes more prominent towards smaller scales. It is these deviations which are essential for the formation and evolution of all structure and objects in our cosmos. A range of recent observational studies have started to question the validity of the assumption that globally the universe is very close to perfectly isotropic and homogeneous, while the deviations represent a minor perturbation to the overall FLRW universe.  \nOf fundamental significance for assessing the consequences of a universe that does not obey the cosmological principle is to establish its geometric foundation. In this paper we illustrate the implications for the spacetime metric once we abandon the assumption of isotropy. This entails a study of the metric structure implied by the specific symmetry properties of the cosmological model considered. In the present investigation this involves cosmologies with a perfect homogeneity, yet with a distinctly anisotropic angular structure. For this configuration, the corresponding homogeneous and fully anisotropic spacetime metric fits into the so-called 9-part Bianchi classification [see 21].  \nThe intention of the current study is to investigate the mathematical basis of the Bianchi models from the perspective of the infinitesimal isometries, or Killing vector fields (KVFs), belonging to the metrics involved. Requirements on homogeneity and isotropy of a metric manifest themselves mathematically as such infinitesimal isometries. The resulting homogeneous, completely anisotropic metric has precisely three KVFs, which together form a Lie algebra. As such, the resulting Lie algebra of KVFs (Killing Lie algebra; KLA) belongs to one of Bianchi's classes I-IX.  \nThe approach of the current study is that of a reversal of the process. Given a suitable 3D Lie algebra of vector fields { ξ a } a =1 , 2 , 3 (termed a pre-KLA ), the program is to construct a metric on which those vector fields would be Killing. We proceeded by showing that the basis of the metric should be the frame dual to the invariant frame (IF), i.e. that frame { X i } i =1 , 2 , 3 so that [ ξ a , X i ] = 0 for all a and i . Expanding the X i in the ξ -basis, and substituting this into the commutator relation above, we infer a differential equation that is solved by the metric of characteristics. This yields an immediate expression for the X i in terms of ξ a , albeit involving some functions that need to be determined.  \nSeveral results are obtained and proven that expand the applicability of our method. Specifically, this concerns the situation in which an initial condition is provided and for which the IF is unique to the pre-KLA. We demonstrate that when given a pre-KLA that is not in canonical Bianchi form, the following procedure should be followed: perform a GL(3) operation to bring it to a  \ncanonical form, construct the IF, and then operate with the inverse GL(3) to get the IF belonging to the original pre-KLA. To complete our inventory, we calculate and tabulate for the different Bianchi classes the above relations between pre-KLA and IF. The results are tabulated in Table II.  \nThe relevance of the presented formalism and results lies in the generalization of the implications of Bianchi type metrics. It aims to establish a notation rooted solely in the Lie algebra structure of the KVFs, independent of any specific coordinate formulation. This approach opens the road for further research, particularly in constructing tensors of cosmological interest-such as the Ricci and shear tensors-for the various Bianchi type cosmologies. 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