Spatially isotropic homogeneous spacetimes

We classify simply-connected homogeneous ($D+1$)-dimensional spacetimes for kinematical and aristotelian Lie groups with $D$-dimensional space isotropy for all $D\geq 0$. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for $D=1,2$. These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.


Motivation and contextualisation.
The laws of physics are to a good approximation invariant under spatial rotations, spatio-temporal translations and inertial transformations (boosts). This leads to space and time homogeneity and space isotropy, but does not determine their precise realisation completely. This freedom is for example evident from the way boosts act on space and time (e.g., compare the galilean and Poincaré boosts) and through the existence of curvature (e.g., compare Minkowski and de Sitter spacetimes). This leads to the fundamental question [1]: What are the possible kinematical symmetries of space and time?
overview section will focus on those spacetimes which exist in all dimensions; although we will mention at the end how these results are modified in low dimension. Our starting points are the de Sitter spacetimes. These are lorentzian spacetimes which are locally isometric to quadric hypersurfaces in pseudo-euclidean spaces. Concretely, de Sitter and anti de Sitter spacetimes in D + 1 dimensions with radius of curvature R are locally isometric, respectively, to the quadrics and (2) More precisely, the de Sitter spacetimes are the simply-connected universal covers of these quadrics. Taking the limit R → ∞ is equivalent to the zero curvature limit in which we recover Minkowski spacetime: the real affine space A D+1 with a metric which, when expressed relative to affine coordinates, is given by where we have introduced the speed of light c. We may now take the non-relativistic limit (on the cometric) in which c → ∞ or the ultra-relativistic limit in which c → 0. In the former case we arrive at the galilean spacetime, whereas in the latter, we arrive at the carrollian spacetime [44,39]. These spacetimes are no longer lorentzian: the (co-)metric becomes degenerate in the limit, leading to a galilean and a carrollian structure, respectively. This is not to say that on the underlying manifold of such spacetimes one could not define a lorentzian metric, but simply that any such metric would not be invariant under the kinematical symmetries of the spacetime, in the way that the Minkowski metric is Poincaré invariant.
Geometrically, the ultra-and non-relativistic limits can be understood in terms of what they do to the light cone present in the tangent space at any point in Minkowski spacetime, as depicted in Figure 1, where we see that the light cone collapses to a timelike line or a spacelike hyperplane in the ultraand non-relativistic limits, respectively. Since the tangent spaces in a lorentzian manifold are lorentzian vector spaces containing their own light cones, we can consider these limits not just for Minkowski spacetime, but for any lorentzian manifold. In particular, we can do this with the de Sitter spacetimes. The non-relativistic limits of the de Sitter spacetimes are the galilean (anti) de Sitter spacetimes (also known as the Newton-Hooke or non-relativistic cosmological spacetimes) [1,45,46,47], whereas the ultra-relativistic limits are the carrollian (anti) de Sitter spacetimes (also known as para-euclidean and para-Minkowski spacetimes) [1]. As in the case of Minkowski spacetimes, the limiting spacetimes are no longer lorentzian, but have galilean and carrollian structures, respectively.
. Effect on light cone of ultra-(left) and non-relativistic (right) limits.
Just like Minkowski spacetime is the zero-curvature limit of the de Sitter spacetimes, the galilean (resp. carrollian) spacetime can be obtained as a zero-curvature limit of the galilean (resp. carrollian) (anti) de Sitter spacetimes. These spacetimes are not lorentzian and thus, in contrast to the Minkowski case, the curvature being taken to zero is not the Riemann curvature of a (non-existing) Levi-Civita connection. Indeed, as we will see, these spacetimes are homogeneous spaces of kinematical Lie groups and as homogeneous spaces they are reductive and symmetric and hence in possession of a canonical torsion-free invariant connection. It is that connection whose curvature is being sent to zero.
The galilean and carrollian symmetric spaces can also arise as limits of the riemannian symmetric spaces. The physical interpretation of the riemannian analogues of the non-and ultra-relativistic limits of euclidean space to galilean and carrollian spacetimes, respectively, is not so clear. There is no longer a light cone and hence no longer a privileged timelike direction. Nevertheless we may choose any direction (all are equivalent, since the riemannian symmetric spaces are isotropic) and rescale the metric along that direction or along the perpendicular plane, and in this way arrive at the galilean and carrollian spacetimes. Neither the clock one-form in the galilean spacetime nor the invariant vector field in the carrollian spacetime are actually induced in the limit, so they have neither a minkowskian nor a euclidean preferred interpretation.
The resulting picture (incomplete at this stage) is summarised in Figure 2, where M, C and G stand for Minkowski, carrollian and galilean spacetimes, respectively; E, H and S for euclidean space, hyperbolic space and the round sphere, respectively; (A)dS for (anti) de Sitter spacetimes; and (A)dSG and (A)dSC for the galilean and carrollian (anti) de Sitter spacetimes, respectively. Diagonal arrows are flat limits, whereas horizontal and vertical arrows are, respectively, ultra-and non-relativistic limits. Notice that whereas the analogue of the ultra-relativistic limit of hyperbolic space is carrollian AdS, the analogue of the non-relativistic limit is galilean dS, and vice versa for the round sphere. One of the main results in this paper is to complete this picture to the one illustrated by Figure 3 (and also Figures 4 and 5 for lower dimension) which includes the (simply-connected) homogeneous spacetimes of all the kinematical Lie groups, with the exception of the riemannian maximally symmetric spaces, whose inclusion might obscure more than enlighten. Much of this paper is devoted to explaining that picture and describing how to arrive at it, but for now let us describe briefly its salient features: • The galilean de Sitter spacetime dSG is actually the unique symmetric point in a one-parameter family dSG γ , with γ ∈ [−1, 1], of reductive homogeneous spaces, distinguished by the torsion of the canonical connection, which vanishes at the symmetric point γ = −1. • Similarly, the galilean anti de Sitter spacetime AdSG is the unique symmetric point in a oneparameter family AdSG χ , with χ 0, of reductive homogeneous spaces, distinguished by the torsion of the canonical connection, which vanishes at the symmetric point χ = 0. Moreover, AdSG ∞ := lim χ→∞ AdSG χ = dSG 1 .
• There is a non-reductive homogeneous spacetime LC of SO(D + 1, 1) with an invariant carrollian structure admitting a limit to the carrollian spacetime. This will be shown to be isomorphic to the (future) light cone in Minkowski spacetime in one dimension higher, hence the notation. • There are several aristotelian homogeneous spacetimes, which are spacetimes without boosts: the static affine spacetime S to which all other spacetimes have limits, -a torsional aristotelian spacetime TS corresponding to the group manifold of a non-abelian solvable Lie group, and the Einstein static universe R × S D and its hyperbolic version R × H D , which do not arise from kinematical groups for D = 3. In addition, although not depicted in Figure 3, there are also the riemannian maximally symmetric spaces (sphere S D+1 , hyperbolic H D+1 and euclidean E D+1 ), whose rôle as spacetimes, due to their compact "boosts", is questionable.
When D = 2, Figure 3 is slightly modified to Figure 4. From that figure we see that all that happens now is that there is a new aristotelian spacetime (A24) and a new two-parameter family (S12 γ,χ ) of galilean spacetimes interpolating between the torsional galilean (anti) de Sitter spacetimes. There are limits from every spacetime in that family to the galilean spacetime. This figure also omits the riemannian maximally symmetric spaces.
Finally, when D = 1, Figure 3 is also modified. Now there are accidental pairwise isomorphisms between some of the symmetric spacetimes due to the possibility of redefining what we mean by space . Three-dimensional homogeneous spacetimes and their limits. and time. In addition there are new two-dimensional spacetimes with no discernible structure: two spacetimes (S17 and S18) and two continua (S19 χ and S20 χ ). The resulting picture is depicted in Figure 5.
1.3. Organisation of the paper. The rest of this paper is organised as follows. Section 2 contains the basic definitions and a summary of the main results in the paper. Readers who are pressed for time should perhaps read that section and then skip to Section 6. In Section 2.1 we define the main objects of interest: kinematical Lie algebras, their homogeneous spacetimes and their infinitesimal description in terms of Lie pairs, relegating to Appendix B a more careful treatment including the proof that (geometrically realisable, effective) Lie pairs are in one-to-one correspondence with . Two-dimensional homogeneous spacetimes and their limits simply-connected homogeneous spaces. In Section 2.2 we summarise the main results and list the resulting isomorphism classes of simply-connected kinematical and aristotelian homogeneous spacetimes in Tables 1 and 2. Sections 3 and 4 contains the details leading up to Table 1. (Table 2 is the result of the classification of aristotelian Lie algebras in Appendix A.) In Section 3 we classify the isomorphism classes of kinematical Lie pairs. This is achieved by going one by one through the isomorphism classes of kinematical Lie algebras and determining for each one the possible Lie pairs up to isomorphism. We do this in turn for generic kinematical Lie algebras in D 3 in Section 3.1, kinematical Lie algebras unique to D = 3 in Section 3.2, kinematical Lie algebras in D = 2 in 3.3 and finally those in D = 1 in Section 3.4. The classification of kinematical Lie algebras are summarised in Tables 4, 6, 8 and 11 and that of their corresponding Lie pairs are summarised in Tables 5,7,9 and 12. Some Lie pairs in D 2 can be seen to be the low-dimensional avatars of some Lie pairs which exist for all D. Tables 10 and 13 describe this correspondence. In Section 4 we select from the Lie pairs obtained in Section 3 those which are effective and then show, using a variety of methods, that all effective Lie pairs are geometrically realisable. The end result of this section and one of the main results of this work is the list of simply-connected homogeneous kinematical spacetimes in Table 1.
The classification of homogeneous spacetimes provides the "objects" in Figure 3. The "arrows" between these objects is provided by limits. In Section 5 we flesh out Figures 3,4 and 5 by exhibiting the limits relating different spacetimes in Tables 1 and 2. Section 6 starts the geometric study of the homogeneous spacetimes in Tables 1 and 2. In Section 6.1 we review some basic notions like reductivity, symmetry, the linear isotropy representation and the invariant structures of interest: lorentzian, riemannian, galilean, carrollian and aristotelian. In Section 6.2 we briefly review the notion of an invariant (affine) connection on a homogeneous space and explore how properties (or absence of) canonical invariant connections help us characterise the spacetimes. In Section 6.3 we list the resulting spacetimes dividing them into classes depending on the whether they are non-reductive, flat symmetric, non-flat symmetric and reductive torsional. Table 16 summarises these results for the homogeneous spacetimes in Tables 1 and 2.
Finally, Section 7 we offer some conclusions and point to further work. In addition, there are two appendices in the paper. In Appendix A we classify aristotelian Lie algebras and in Appendix B we prove that our infinitesimal approach actually classifies simply-connected homogeneous spacetimes.

Reader's guide.
It is our intention that this paper should be useful to the community, but realise that there is a risk that the results are hard to extract from the details of how we arrived at them. We have therefore tried to write the paper in a way that a reader who is happy to believe the classification can reach it quickly without having to wade through the details of how we arrived at it. The main information content of this paper is contained in Sections 2 and 6 and a reader who is pressed for time should perhaps concentrate on those two sections at a first reading. In particular, Tables 1 and 2 contain the  list of simply-connected homogeneous kinematical and aristotelian spacetimes and Table 16 lists their basic geometrical properties: whether they are reductive/symmetric/affine, whether they admit parity and/or time reversal symmetry, the type of invariant structure that they possess (if any): lorentzian, riemannian, galilean, carrollian or aristotelian, and, for the reductive examples, whether the canonical invariant connection is flat and/or torsion-free. In addition, Figures 3, 4 and 5 illustrate limits between the spacetimes and it too contains useful information. Busy readers should be able to read the introduction and Section 2 and then skip to Section 6 where we discuss some geometrical properties of the homogeneous spacetimes.
Many of the tables and figures contain hyperlinks to ease navigation. Let us explain here how to use them. The spacetimes in Figures 3, 4 and 5 as well as our summary table of the properties, Table 16, are hyperlinked to Tables 1 and 2 which provide the explicit kinematical and aristotelian spacetimes. Every row in these tables starts with a label corresponding to one of the kinematical spacetimes. These labels are hyperlinked to the corresponding label in Table 14. That table lists for each spacetime the corresponding effective Lie pairs. These Lie pairs are themselves hyperlinked to the relevant tables of Lie pairs: Table 5 for D 3, Table 9 for D = 2 and Table 12 for D = 1. Those tables also contain the information of the isomorphism class of kinematical Lie algebra associated to that Lie pair and the label of the Lie algebra is hyperlinked to the relevant tables of kinematical Lie algebras: Table 4 for D 3, Table 8 for D = 2 and Table 11 for D = 1. In addition, Table 15 shows which homogeneous kinematical spacetimes are associated with which kinematical Lie algebra.
For example, if we click on LC in Figure 3 we are taken to Table 1. Clicking on S16 in Table 1, we are taken to Table 14 where we see that it can be described by several Lie pairs, depending on dimension: LP17 (D 3), LP45 (D = 2) or LP79 (D = 1). Clicking, say, on LP17, we are taken to Table 5, where we see that it comes from Lie algebra LA11 and clicking on LA11 we are taken to Table 4, where we see that it corresponds to so(D + 1, 1) with the Lie brackets given there.

K
The well-known spacetimes in Figure 2 are all symmetric homogeneous spaces of kinematical Lie groups. In this section we will review this description.
2.1. Basic definitions. Before we properly define the main objects of interest, let us motivate our definitions and summarise the philosophy of the construction. By definition, a kinematical Lie algebra has the same dimension as the Lie algebra of isometries of a maximally symmetric riemannian/lorentzian spacetime, and it consists of spatial rotations and additional generators we call boosts and (space and time) translations. However before we specify how the kinematical Lie algebra acts on the spacetime, we cannot make a precise identification of which generators are boosts and/or translations. This is the reason to refine the discussion from the kinematical Lie algebras to the homogeneous spacetimes. The spacetimes are constructed in such a way that the stabiliser of any point contains the rotations about that point. This spatial isotropy implies that all invariant tensors are isotropic (i.e., so(D) rotationally invariant). The vectorial generators in the stabiliser are interpreted as boosts, whereas the additional generators are interpreted as translations. The resulting spacetime is by construction homogeneous, which roughly means that every point of the manifold looks like any other point. We now provide the precise details.

Definition 1.
A kinematical Lie algebra (with D-dimensional space isotropy) is a real Lie algebra k satisfying the following two properties: (1) k contains a Lie subalgebra r ∼ = so(D), the Lie algebra of rotations of D-dimensional euclidean space; and (2) k decomposes as k = r ⊕ 2V ⊕ S as a representation of r, where 2V are two copies of the Ddimensional (vector) irreducible representation of so(D) and S is the one-dimensional (scalar) trivial representation of so(D).
By a kinematical Lie group we mean any Lie group whose Lie algebra is a kinematical Lie algebra.
It follows from this definition that we can describe a kinematical Lie algebra explicitly in terms of a basis J ab = −J ba for the rotational subalgebra r, V (i) a with i = 1, 2 for the two copies of V and H for S.
The definition implies that the Lie brackets of k in this basis include the following: and any other Lie brackets are subject only to the Jacobi identity, which implies, in particular, equivariance under r. It is convenient to relabel V (1) a as B a and V (2) a as P a , a notation reminiscent of the boosts and translations in kinematics; although it must be stressed that there is no a priori geometrical interpretation of these generators: they only acquire such an interpretation when we realise them in a spacetime on which the kinematical Lie group acts.
Kinematical Lie algebras have been classified up to Lie algebra isomorphism [1,2,3,4,5]. These classifications are the starting point to the classification of homogeneous spacetimes and hence they will be briefly recalled at the start of Sections 3.1, 3.2, 3.3 and 3.4 and contained in Tables 4,6,8 and 11. In this paper we obtain a classification of simply-connected homogeneous spacetimes of kinematical Lie groups (up to isomorphism). Not every homogeneous space of a kinematical Lie group is a spacetime, so this requires a definition. Notice that, in particular, a homogeneous spacetime of a kinematical Lie group with D-dimensional space isotropy is (D + 1)-dimensional.

Definition 2.
By a homogeneous kinematical spacetime we mean a homogeneous space M of a kinematical Lie group K, satisfying the following properties: (1) M is a connected smooth manifold, and (2) K acts transitively and locally effectively 2 on M with stabiliser H, where (3) H is a closed subgroup of K whose Lie algebra h contains a rotational subalgebra r ∼ = so(D) and decomposes as h = r ⊕ V as an adjoint r-module, where V is an irreducible D-dimensional vector representation of so(D). It follows that M is K-equivariantly diffeomorphic to K/H. Within the confines of this paper, and because we will need to refer to them often, we will say that H and h are admissible.
It may be convenient to keep an example in mind, so let us consider the Poincaré Lie group, whose Lie algebra is defined relative to the basis (J ab , B a , P a , H) by the nonzero Lie brackets This is obtained from the more familiar expression by decomposing J µν = (J ab , B a := J 0a ) and P µ = (P a , H := P 0 ), where η 00 = −1 and η ab = δ ab . Let us take K to be the Poincaré Lie group and H the Lorentz subgroup; that is, the (admissible) subgroup generated by the Lie subalgebra spanned by (J ab , B a ). Then K/H is diffeomorphic to Minkowski spacetime, as is well known.
Perhaps not so well known is the fact that the Poincaré group admits a second homogeneous spacetime. If we now let H ′ denote the (again, admissible) subgroup generated by the Lie subalgebra h ′ spanned by (J ab , P a ), then K/H ′ is diffeomorphic to the carrollian anti de Sitter spacetime (AdSC) (also known as para-Minkowski spacetime), as we will see below. To more easily distinguish between these two homogeneous spacetimes of the Poincaré group, it is convenient to change basis in the Poincaré Lie algebra in such a way that the admissible Lie subalgebra h ′ is also spanned by (J ab , B a ). Doing so we arrive at what is often termed the para-Poincaré Lie algebra, with nonzero Lie brackets Of course, this Lie algebra is isomorphic to the Poincaré Lie algebra, but not in a way which fixes the admissible subalgebra. We see from the above Lie brackets that translations no longer commute, signalling that this spacetime is not flat; although, as we will see, the non-flat connection, which is the canonical Poincaré-invariant connection on this symmetric homogeneous space, is not a metric connection since AdSC does not admit a Poincaré-invariant metric. If it did, by dimension, it would have be maximally symmetric and hence isometric to Minkowski spacetime, where translations do commute.
As we explain in Appendix B, the classification of isomorphism classes of simply-connected homogeneous kinematical spacetimes can be arrived at infinitesimally, by classifying isomorphism classes of (geometrically realisable, effective) kinematical Lie pairs. This too requires a definition. Definition 3. A (kinematical) Lie pair is a pair (k, h) consisting of a kinematical Lie algebra k and an admissible subalgebra h. Two Lie pairs (k 1 , h 1 ) and (k 2 , h 2 ) are isomorphic if there is a Lie algebra isomorphism ϕ : k 1 → k 2 with ϕ(h 1 ) = h 2 . A Lie pair (k, h) is effective if h does not contain any nonzero ideals of k. It is said to be (geometrically) realisable if there exists a connected Lie group K ′ with Lie algebra k ′ and a closed Lie subgroup H ′ with Lie algebra h ′ with (k ′ , h ′ ) isomorphic to (k, h). The homogeneous space K ′ /H ′ is said to be a geometric realisation of (k, h).
The relationship between Lie pairs and homogeneous spaces extends the relationship between Lie algebras and Lie groups. Recall that associated with every finite-dimensional Lie algebra k there exists a unique (up to isomorphism) simply-connected Lie group whose Lie algebra is isomorphic to k. A similar correspondence exists between homogeneous spaces and Lie pairs, except that we need to restrict to Lie pairs which are geometrically realisable (existence) and effective (uniqueness). This is explained in detail in Appendix B.
In this paper we will classify isomorphism classes of geometrically realisable, effective Lie pairs (k, h), where k is a kinematical Lie algebra and h an admissible Lie subalgebra. As just explained, this is equivalent to classifying isomorphism classes of simply-connected homogeneous kinematical spacetimes. We will actually exploit the classification of kinematical Lie algebras up to isomorphism and in this way fix a kinematical Lie algebra k in each isomorphism class and classify isomorphism classes of (geometrically realisable, effective) Lie pairs (k, h) with h an admissible subalgebra, with two such Lie pairs (k, h 1 ) and (k, h 2 ) declared to be isomorphic if there is an automorphism of k sending h 1 to h 2 .
2.2. Summary of main results. The classification is described in Sections 3 and 4, but we think it might be helpful to the reader to collect here already the results. In the tables below we use an abbreviated notation in which we do not write the so(D) indices explicitly. We write J, B, P and H for the generators of the kinematical Lie algebra k and write the kinematical Lie brackets of (4) as For D = 2, any other brackets can be reconstructed unambiguously from the abbreviated expression since there is only one way to reintroduce indices in an so(D)-equivariant fashion; that is, using only the so(D)-invariant tensors δ ab and ǫ a1···aD on the right hand side of the brackets. For example, In D = 3 we may also have brackets of the form where we employ Einstein's summation convention. Similarly, for D = 2, ǫ ab is rotationally invariant and can appear in Lie brackets. So we will write, e.g., et cetera. Table 1 summarises all the hard work in Sections 3 and 4 and lists all the inequivalent geometrically realisable effective Lie pairs (k, h) with k a kinematical Lie algebra and h an admissible Lie subalgebra. A basis has been chosen in such a way that h is spanned by (J ab , B a ). In this way, Lie pairs are uniquely characterised by specifying the Lie brackets of k in this basis. The kinematical Lie brackets (8) are common to all kinematical Lie algebras, so that we need only specify those Lie brackets which do not involve the rotations. The table is divided into five sections, separated by horizontal rules. From top to bottom, the first four correspond to the lorentzian, riemannian, galilean and carrollian spacetimes. The final section corresponds to "exotic" two-dimensional spacetimes admitting no such structures. Two remarks are in order about this table and both concern the case of D = 1. Since there are no rotations in D = 1, in any row where J appears, we are tacitly assuming that we set J = 0 if D = 1. Also, some of the D 1 spacetimes become accidentally pairwise isomorphic in D = 1: namely, carrollian and galilean, de Sitter and anti de Sitter, carrollian dS and galilean AdS, and carrollian AdS and galilean dS. This explains why we write D 2 for carrollian, de Sitter, and carrollian (anti) de Sitter. In this way no two rows are isomorphic and hence every row in the table specifies a unique isomorphism class of simplyconnected homogeneous kinematical spacetime. Perhaps it bears repeating that, as mentioned already in the introduction, the galilean (A)dS spacetimes are often called the Newton-Hooke spacetimes.

T
. Simply-connected homogeneous (D + 1)-dimensional kinematical spacetimes Geometrical properties of the spacetimes are provided in Table 16. Table 2 lists the Lie pairs corresponding to the aristotelian spacetimes. Many of them arise from noneffective Lie pairs (k, h) with k a kinematical Lie algebra. If such a pair is not effective, it means that h contains a nonzero ideal of k. For k a kinematical Lie algebra, this cannot be other than the span b of the (B a ), assuming they do form an ideal. When this is the case, we may quotient both k and h by the ideal b and arrive at an effective (by construction) Lie pair (k/b, h/b) = (a, r), where a = k/b is an aristotelian Lie algebra (see Appendix A) and r = h/b is a rotational subalgebra of a.
To fix ideas, let us consider the example of the galilean algebra g. This is the kinematical Lie algebra with nonzero Lie brackets [H, B] = −P in addition to those in (8). As we will see, there are two isomorphism classes of Lie pairs associated to g. If we choose bases for g so that in both cases the admissible subalgebra is the span of (J ab , B a ), then the two Lie pairs are described by the following Lie brackets for g in addition to those in (8): • [H, B] = −P, which is the standard galilean spacetime, and The latter Lie pair is not effective because h contains the ideal b spanned by the (B a ). If we quotient this Lie pair by b, which boils down to discarding any B a in the Lie brackets, we see that now [H, P] = 0 and we arrive at the static aristotelian Lie algebra. Aristotelian Lie pairs are always geometrically realisable because the rotational subalgebra r generates a compact subgroup and compact subgroups are always closed. Furthermore, since r is fixed, every aristotelian Lie algebra gives rise to a unique aristotelian Lie pair, so that the classification in Appendix A is also a classification of simply-connected aristotelian spacetimes up to isomorphism.  Table 1, whereas Section 5 contains the details leading up to Figures 3, 4 and 5. The busy reader may wish to skip them at a first reading and go directly to Section 6 where we explore some of the geometrical properties of these spacetimes, culminating in Table 16.

C L
This section, which can be skipped at a first reading, contains the details of the classification of kinematical Lie pairs (k, h) up to isomorphism, where k is a kinematical Lie algebra and h is an admissible Lie algebra. At this stage we will not worry about whether the resulting Lie pairs are effective and/or geometrically realisable. The results of this section are collected in Tables 5, 7, 9 and 12, which will be the starting point for the analysis in Section 4, where we extract from those tables the effective Lie pairs which admit a geometric realisation to arrive at Table 1.
We will simplify the analysis by making use of the classification of kinematical Lie algebras, which we briefly review. Kinematical Lie algebras (with D-dimensional space isotropy) have been classified up to Lie algebra isomorphism. In D = 0 there is a unique one-dimensional Lie algebra, whereas in D = 1 there are no rotations, so any three-dimensional Lie algebra is kinematical. These were classified by Bianchi [48] (see [49] for an English translation) in the context of his classification of three-dimensional homogeneous spaces; although here they will play the rôle of symmetries of two-dimensional homogeneous spaces. The other classic case is D = 3, where the kinematical Lie algebras were classified by Bacry and Nuyts [2] refining earlier work of Bacry and Lévy-Leblond [1]. The remaining cases D > 3 and D = 2 are recent classifications. Following earlier work on the galilean algebra [50], the D = 3 classification was recovered recently in [3] using methods of deformation theory. These methods were then extended to arrive at the classifications for D > 3 [4] and D = 2 [5]. These classifications are recalled in this paper and are contained in Tables 4, 6, 8 and 11.
The classification of kinematical Lie algebras allows us to fix a kinematical Lie algebra k and then consider Lie pairs (k, h), with two such pairs (k, h 1 ) and (k, h 2 ) being isomorphic if there is an automorphism of k taking h 1 to h 2 . This suggests the following methodology, which is how we will proceed in this paper.
For each kinematical Lie algebra k in the classification, we will determine the admissible subalgebras h and, if necessary, the action of the automorphism group A = Aut(k) on them. We will then pick one representative from each A-orbit. Finally, we will change basis (if needed) for k so that h is always spanned by J ab and B a . The resulting Lie pair is then described uniquely by specifying the Lie brackets of k in this basis. Furthermore, whenever possible, we choose a basis in which the Lie pair is manifestly reductive.
It follows from the classification of kinematical Lie algebras that there are kinematical Lie algebras which exist for all D 1, but there are additional Lie algebras for D 3, due to accidents in small dimension: namely, the existence of a rotationally invariant vector product in D = 3, a rotationally invariant symplectic structure in D = 2, and the absence of rotations in D = 1. As explained in [5], the case D = 2 is special and it is convenient to work with the complexified Lie algebra. The analysis in that case does not embed easily into the general discussion of D 3 and therefore we will have to do it separately. Similarly the case D = 1 is special in that any three-dimensional Lie algebra is kinematical and any onedimensional subalgebra is admissible. We also treat this case separately. These considerations suggest first treating the generic case (restricted to D 3), which we do in Section 3.1. Then in Section 3.2 we consider the additional kinematical Lie algebras which are unique to D = 3. In Section 3.3 we determine the Lie pairs associated to the kinematical Lie algebras in D = 2. As we will see, there are two kinematical Lie algebras unique to D = 2 which do not admit any Lie pairs. Then in Section 3.4 we consider the Lie pairs associated to the three-dimensional Lie algebras.

Lie pairs for D
3. Let us consider those kinematical Lie algebras which have analogs for all D 3. (They also have analogs for D < 3, but we will discuss them again separately in that context.) These are listed in Table 4, which is borrowed from [4, Table 17] with some small modification. We only list those nonzero Lie brackets in addition to (8), which are understood as given. The numbering in this and other tables is somewhat arbitrary, but might help in referring to those Lie algebras which are otherwise nameless. Although we are aware that referring to some of these kinematical Lie algebras by name presupposes a geometrical interpretation of the generators which may or may not be the right interpretation depending on the actual homogeneous space under consideration, we do feel that it helps to orient the reader if we point out to which named Lie algebras (when the name exists) the Lie algebras in Table 4 are isomorphic. In this spirit, let us also mention in this and other tables the isomorphism type of the Lie algebra, when known. To this end let us introduce in Table 3 some notation for the named Lie algebras appearing in this section.
In discussing the automorphisms of kinematical Lie algebras we will fix the rotational subalgebra r and concentrate on automorphisms which are the identity on r and in this way focus only on their action on the non-rotational generators. Since automorphisms must commute with the action of the for some a, b, c, d, ∆ ∈ R and ∆(ad − bc) = 0. In other words, the automorphism group of k will be a subgroup of GL(2, R) × R =0 .
3.1.1. Lie pairs associated to Lie algebra #1. Kinematical Lie algebra #1 in Table 4 is isomorphic to the static Lie algebra, with all brackets zero except for those in equation (8). The automorphism group is the full GL(2, R) × R =0 . An admissible Lie subalgebra is spanned by the rotations and the D vectors αB a + βP a , for some α, β ∈ R not both zero. We will abbreviate the span of these vectors as αB + βP, but let us not forget that we mean a D-dimensional subspace. Under an automorphism, αB + βP → (aα + bβ)B + (cα + dβ)P, so that all that happens is that the vector (α, β) ∈ R 2 is transformed under GL(2, R) according to the defining representation. Given any nonzero vector (α, β) ∈ R 2 , there is a change of basis which sends it to the elementary vector (1, 0), so that up to the action of GL(2, R), we can take the Lie subalgebra h to be spanned by the rotations and B.

3.1.2.
Lie pairs associated to Lie algebra #2. Kinematical Lie algebra #2 in Table 4 is isomorphic to the galilean Lie algebra. The automorphism group A is easily determined to be Any vectorial subspace W of the form αB + βP defines a subalgebra, since B and P commute. Under an automorphism, If α = 0, we can choose a = α −1 and c = −a 2 ∆β and hence bring (α, β) → (1, 0), so that W is the span of B.
3 In D = 3 one might conceive of adding multiples of the rotation generators J to the transformation properties of B and P, but one can show that in automorphisms of k no such terms appear.
If α = 0, then β = 0 and we can choose a∆ = β −1 so that (0, β) → (0, 1). This means that now W is the span of P. We change basis in the Lie algebra so that W is the span of B. In the new basis, the galilean algebra has the additional bracket (16) This is often known as the para-galilean algebra, but it is of course isomorphic to the galilean algebra.
3.1.3. Lie pairs associated to Lie algebra #3 γ . The automorphism group of Lie algebra #3 γ in Table 4 is determined to be Any vectorial subspace W of the form αB + βP defines a subalgebra, since B and P commute. Under an automorphism, We have three cases to consider, depending on whether α = 0, β = 0 or neither are zero. If β = 0, then we can choose a so that aα = 1, bringing (α, 0) → (1, 0). Here W is the span of B.
If α = 0, then similarly W is the span of P. We change basis so that W is again the span of B, which means that the additional Lie brackets are now  Finally, if αβ = 0, then we can choose a, d so that (α, β) → (1, 1) and W is spanned by B + P. We change basis to that W is spanned by B, bringing the additional Lie brackets to the form   Table 4 is determined to be Any vectorial subspace αB + βP is a subalgebra, but under the automorphisms we can always bring it to B.

3.1.5.
Lie pairs associated to Lie algebra #5. The automorphism group of Lie algebra #5 in Table 4 is determined to be Any vectorial subspace αB + βP is a subalgebra. The analysis of the action of automorphisms is very similar to that of algebra #3, except that using automorphisms not in the identity component, we can relate the subspace spanned by B to that spanned by P. Therefore we have two inequivalent Lie pairs, corresponding to the Lie algebra in the original basis: and the one corresponding to 3.1.6. Lie pairs associated to Lie algebra #6. The automorphism group of Lie algebra #6 in Table 4 is determined to be Any vectorial subspace αB + βP is a subalgebra. Using the automorphisms, we may send (α, β) → (aα, cα + aβ). We can distinguish between two cases. If α = 0, then take a = α −1 and c = −α −2 β so arrive at (1, 0), so that W is spanned by B. If α = 0, then W is spanned by P. In the former case, we have the original Lie brackets (26) although we prefer to change basis so that P → −(P + B). In that basis, the brackets are now In the latter case, we change basis so that W is again spanned by B, but then the Lie brackets are now 3.1.7. Lie pairs associated to Lie algebra #7 χ . The automorphism group of Lie algebra #7 χ in Table 4 is determined to be Any vectorial subspace αB + βP is a subalgebra. Using the automorphisms we can bring any (α, β) to (1, 0), so that W is spanned by B. We prefer to change basis in the Lie algebra so that P → −(P + χB). Doing so, the Lie brackets become 3.1.8. Lie pairs associated to Lie algebra #8. The automorphism group of Lie algebra #8 in Table 4 is determined to be (31) Any vectorial subspace αB + βP is a subalgebra. As in the previous case, using only the automorphisms in the identity component we can already bring any (α, β) to (1, 0), so that W is spanned by B. There is thus a unique Lie pair associated to this Lie algebra.
3.1.9. Lie pairs associated to Lie algebra #9. Lie algebra #9 in Table 4 is isomorphic to the Carroll algebra. Its automorphism group is determined to be Any vectorial subspace αB + βP is a subalgebra since However up to the automorphisms we can always bring (α, β) to (1, 0) and hence W is spanned by B.
There is a unique homogeneous Lie pair associated to the Carroll algebra.
3.1.10. Lie pairs associated to Lie algebra #10 ε . Lie algebra #10 −1 is isomorphic to the euclidean Lie algebra, whereas #10 +1 is isomorphic to the Poincaré Lie algebra. The automorphism group is determined to be Any vectorial subspace αB + βP is admissible, since Under the automorphisms, we may bring (α, β) to (±α, cα + dβ). We must distinguish between two cases. If α = 0, we can bring (α, β) to (α, 0) with α > 0, which says that W is the span of B. If α = 0, then W is spanned by P. In the former case, we have the original Lie brackets whereas in the latter case we change basis so that W is again spanned by B, but this changes the Lie brackets to where we have also changed the sign of H in order to keep the [B, P] = H bracket uniform. These Lie algebras are often known as the para-Poincaré and para-euclidean algebras, depending on the sign of ε.
3.1.11. Lie pairs associated to Lie algebra #11. Lie algebra #11 in Table 4 is isomorphic to so(D + 1, 1). The automorphism group consists of two connected components: Every vectorial subspace αB + βP is admissible, since If either α = 0 or β = 0, then we can use the automorphisms to bring (α, β) to (1, 0), which says that W is spanned by B. If αβ = 0, then we can use the automorphisms to bring (α, β) to ( |αβ|, ± |αβ|), depending on whether αβ is positive or negative. This says that W is spanned by B ± P. Changing basis so that W is again spanned by B, and redefining H, we arrive at the following brackets where ε = ±1 according to whether ±αβ > 0.
3.1.12. Lie pairs associated to Lie algebra #12 ε . Lie algebra #12 ε in Table 4 is isomorphic to so(D, 2) for ε = 1 and to so(D+2) for ε = −1. The automorphism group is isomorphic to O(2) and hence has two connected components: However using only the automorphisms connected to the identity, we can rotate (α, β) to ( α 2 + β 2 , 0) and hence W is spanned by B. Therefore each of these kinematical Lie algebras has a unique Lie pair.
3.1.13. Summary. We summarise the above results in Table 5, which lists the equivalence classes of kinematical Lie pairs which exist for all D 3. In this and other similar tables throughout the paper, each row consists of an incremental label "LP#" for the Lie pair for easy reference in the rest of the paper and also a label "LA#" of the isomorphism type of kinematical Lie algebra k to which the Lie pair is associated. The rest of the row contains the Lie brackets of k in a basis where the h is spanned by J ab and B a and perhaps some relevant comments. In some cases we have made changes of basis (leaving alone the subalgebra h) in order to arrive at a more uniform description.

3.2.
Lie pairs unique to D = 3. Table 6 lists those kinematical Lie algebras which are unique to D = 3. It is a sub-table of [3, Table 1], with small modifications. 4 As usual we only list any nonzero Lie brackets in addition to (8).
For the kinematical Lie algebras in this table, the condition on the vectorial subspace W to be admissible is very restrictive and there is no need to worry about the action of the automorphisms. Therefore we have no need to determine the automorphism groups.

3.2.1.
Lie pairs associated to Lie algebra #13 ε . The only linear combinations αB + βP which are admissible are those with β = 0, so that W is already spanned by B. Therefore there is a unique spacetime for each of these kinematical Lie algebras.

3.2.2.
Lie pairs associated to Lie algebra #14. There are two admissible subspaces: the span of B and the span of P. In the first case, the Lie brackets are as shown in the table, whereas in the second case, changing basis so that W is spanned by B, we find    Table 1] the rotational generator R is a vector and is related to J in Table 6 by J ab = −ǫ abc R c .

LP# LA# Nonzero Lie brackets in addition to
3.2.5. Lie pairs associated to Lie algebra #17. There is a unique admissible subspace: the one spanned by P.
Changing basis so that it is spanned by B, we arrive at the Lie bracket 3.2.6. Lie pairs associated to Lie algebra #18. The span of P is the only admissible subspace. Changing basis so that it is spanned by B, we arrive at 3.2.7. Summary. Table 7 lists the Lie pairs associated to the kinematical Lie algebras which are unique to D = 3.

3.3.
Lie pairs for D = 2. Table 8 lists the kinematical Lie algebras with D = 2 in complex form. It is borrowed partially from [5, Table 1], where we explain the rationale for working with the complexified Lie algebras. If k is a kinematical Lie algebra in D = 2, we let its complexification k C be the complex span of H, J, B, P, B, P, where B = B 1 + iB 2 , P = P 1 + iP 2 , B = B 1 − iB 2 and P = P 1 − iP 2 . The standard rotational generator J ab is related to J by J ab = −ǫ ab J, from where we see that on a vectorial generator To see how to translate between the complex and real notations, the bracket Automorphisms are also different than for D > 2 due to the fact that r is one-dimensional. If k is a kinematical Lie algebra in Table 8, then the group of automorphisms is a subgroup of GL(2, C)×GL(2, R).

It consists of pairs
and acting on the basis by (B, P) → (aB + cP, bB + dP) and (This uses the observation that for any of the Lie algebras in Table 8, the only elements in the real span of H and J which act on B and P as rotations have the form J + tH and t can be nonzero only when H is central, which is the case for the static and Carroll algebras and also for the kinematical Lie algebras #29 and #31 in Table 8.) If h is an admissible subalgebra, its complexification h C is the complex span of J+tH, αB+βP,ᾱB+βP, for some α, β ∈ C not both zero and some t ∈ R. (Again this uses the above mentioned observation that the rotational generators are of the form J + tH.) The real subalgebra h is then the real span of J + tH and the real and imaginary parts of αB + βP. We let W denote the two-dimensional real vector space spanned by the real and imaginary parts of αB+βP, which transforms as a two-dimensional real (vector) representation of r. We will, however, abbreviate this by saying W is spanned by αB + βP.

3.3.1.
Lie pairs associated to Lie algebra #19. Any subspace αB + βP is admissible. The automorphism group is the full group As in the case of D 3, we may bring (α, β) to (1, 0) by an automorphism which preserves H and J, so we can take W to be spanned by B without loss of generality. There is thus a unique spacetime associated to this Lie algebra.

3.3.2.
Lie pairs associated to the Lie algebra #20. Any subspace αB + βP is admissible. The automorphism group is now Using the automorphisms, we can bring any nonzero (α, β) ∈ C 2 to either (1, 0) or (0, 1), depending on whether or not α = 0. In the former case, W is spanned by B and the nonzero brackets are unchanged. In the latter case, W is spanned by P. Changing basis so that it is again spanned by B changes the nonzero brackets to [H, P] = −B.

3.3.3.
Lie pairs associated to Lie algebra #21. Again every nonzero subspace αB + βP is admissible. The automorphisms are given by Using automorphisms we can bring any nonzero (α, β) to one of two vectors: (1, 0) or (0, 1), depending on whether or not β = 0. In the former case, W is already spanned by B and the nonzero brackets are unchanged. In the latter case, W is spanned by P, but changing basis so that it is again spanned by B, results in the following nonzero Lie brackets: for ξ ∈ C given by ξ = γ + iχ with γ ∈ [−1, 1] and χ 0. For kinematical Lie algebra #22 γ+iχ , ξ = −1, which corresponds to Lie algebra #23. It is convenient to distinguish three cases: In all cases, any nonzero subspace αB + βP is admissible, but the automorphisms differ. For ξ = 1, the automorphisms are given by and it is clear that we can bring any nonzero (α, β) ∈ C 2 to (1, 0). So without loss of generality we can take W to be spanned by B and the brackets are unchanged: For Re ξ = −1, the automorphisms are given by Any nonzero (α, β) ∈ C 2 may be brought to one of two normal forms: (1, 0) or (1, 1) according to whether one of α or β are zero or neither are zero, respectively. In the first case, W is already spanned by B and there is no need to change basis. In the second case, W is spanned by B + P and after the change of basis, the new brackets are (60) Finally, if ξ = 1 and Re ξ = −1, the automorphisms are Again we can bring any nonzero (α, β) to one of three normal forms In the first case there is no need to change basis. In the second case, W is spanned by P and changing basis results in 3.3.5. Lie pairs associated to Lie algebra #24. Again every nonzero subspace W is admissible. The automorphism group has now two connected components: Via an automorphism, any nonzero (α, β) ∈ C 2 can be brought to either (1, 0) if at least one of α or β is zero, or (1, 1) otherwise. In the former case, W is already spanned by B and the brackets are unchanged, whereas in the latter case W is spanned by B + P and changing basis changes the brackets to The condition αβ ∈ R means that we can choose α and β real and still span the same subspace. It is unnecessary to determine the precise automorphism group. It suffices to remark that it includes the subgroup and that using such an automorphism, we can take any (α, β) ∈ R 2 to (1, 0). There is thus a unique spacetime.
3.3.7. Lie pairs associated to Lie algebra #26 ε . A subspace αB + βP is admissible if and only if αβ is real: As in the previous case, we can take α, β both real without altering their span.
The automorphism group consists of Under automorphisms we can bring every admissible subspace to either the span of the B or the span of the P. In the former case we do not change the brackets, whereas in the latter case we change basis and arrive at 3.3.8. Lie pairs associated to Lie algebra #27. Because of the bracket a subspace αB + βP is admissible if and only if αβ +ᾱβ = 0.
Suppose that αβ = 0. Then either α = 0 or β = 0 and we may apply an automorphism to send such (α, β) to (1, 0). On the other hand, if αβ = it for some real t = 0, we can always choose an automorphism to bring (α, β) to a multiple of (1, ±i). The former case, W is spanned by B and in the latter cases by B±iP. In these latter cases, changing basis so that W is spanned again by B, we find the following brackets: 3.3.9. Lie pairs associated to Lie algebra #28 ε . A subspace αB + βP is admissible if and only if αβ ∈ R.
Such an admissible subspace is therefore always of the form αB + βP with α, β ∈ R. It is not necessary to determine the precise form of the automorphism group, but simply to remark that it contains the following SO(2) subgroup: Any (α, β) ∈ R 2 can be rotated to ( α 2 + β 2 , 0) which corresponds to the subspace spanned by B. Therefore there is a unique spacetime for each sign ε.

No Lie pairs associated to Lie algebras #29 and #30.
There is no admissible subalgebra in these cases, since in one case and in the other whereas the right-hand sides do not span a rotational subalgebra. Therefore there are no spacetimes associated these kinematical Lie algebras.
3.3.11. Lie pairs associated to Lie algebras #31, #32 and #33 ε . We can treat these cases simultaneously, since they only differ in the adjoint action of H. In all cases, the only admissible subspace is the one spanned by P, since and H does not span a rotational subalgebra. Changing basis so that it is spanned by B, changes the brackets to 3.3.12. Summary. In Table 9, we list the Lie pairs associated to the kinematical Lie algebras in D = 2. We have reverted to the real form of the D = 2 Lie algebras. The notation is then very similar to D 3. In particular, the rotational generator J is again J ab as in D 3. This eases the comparison with D 3 and allows us to quickly determine which Lie pairs are unique to D = 2 and which are just the D = 2 avatars of Lie pairs which exist also for D > 2. The main difference between D = 2 and D > 2 is that some brackets feature the ǫ symbol and we have therefore introduced P and B as explained at the start of Section 2.2. We have also changed basis in some cases in order to uniformise the presentation and ease the comparison with D > 2.
The table is divided into two: the ones above the horizontal line are D = 2 avatars of Lie pairs which exist also for D > 2. The precise dictionary is shown in Table 10. The ones below the line exist only for D = 2.
galilean AdS 40 25   [48] as part of his classification of three-dimensional homogeneous manifolds. In this section, however, they will be associated to two-dimensional homogeneous spaces. The Bianchi classification of three-dimensional real Lie algebras is recalled in Table 11. We have omitted Bianchi III because it is isomorphic to Bianchi VI 1 .
Because of dimension and the absence of rotations, every one-dimensional subspace is an admissible subalgebra. Therefore to classify pairs (k, h) we need, for each k in the Bianchi classification, classify the one-dimensional subspaces up to the action of automorphisms. In other words, we must determine the orbits of the action of the automorphism group of k on the projective space Pk of one-dimensional subspaces of k. We will now do this for every Bianchi type in turn. We will choose a basis for k where h is spanned by B and will choose P, H, if possible, so that their span is h-invariant. The resulting Lie pairs are summarised in Table 12 below. Those Lie pairs below the horizontal line correspond to Lie algebras unique to D = 1; although the Lie pair might actually be the 1 + 1 case of a Lie pair which exists for all

D.
One notational remark about the automorphism groups. Since we have chosen a basis (e 1 , e 2 , e 3 ) for k, we can identify the automorphism group A = Aut(k) with a subgroup of GL(3, R) and therefore we will be describing it as a set of matrices.
3.4.1. Bianchi I. Since k is abelian, every invertible linear map is an automorphism. The general linear group acts transitively on the projective space, so we can take h to be spanned by B = e 1 , say.

Bianchi II.
Here k is the Heisenberg algebra, whose automorphisms can be determined to be A nonzero vector x 1 e 1 + x 2 e 2 + x 3 e 3 determines a one-dimensional subspace and hence an admissible subalgebra. Under a typical automorphism, this vector transforms as The line spanned by e 1 is sent to itself, whereas any other line can be transformed to the line spanned by e 2 , say.
In the first case we let B = e 1 , P = e 2 and H = e 3 , arriving at the Lie algebra with nonzero brackets In the second case we let B = e 2 , P = e 1 and H = e 3 , arriving at

Bianchi IV. The automorphism group is now
There are three A-orbits in Pk, labelled by the vectors (1, 0, 0 3.4.5. Bianchi VI 0 . The automorphism group is this case is There are three A-orbits in Pk, labelled by 3.4.6. Bianchi VI χ>0 . The automorphism group in this case is the identity component of the automorphism group of Bianchi VI 0 : There are four A-orbits in Pk labelled by the vectors (0, 0, 1), (1, 0, 0), (0, 1, 0) and (1, 1, 0).
3.4.7. Bianchi VII 0 . The automorphism group has two connected components: (99) Using only the identity component, we can bring any line in Pk to of of two lines: the one spanned by (0, 0, 1) and the one spanned by (1, 0, 0). In 3.4.8. Bianchi VII χ>0 . The automorphism group in this case is the identity component of the automorphism group for Bianchi VII 0 : This was all that was needed to bring every line to one of two lines, spanned by either 3.4.11. Summary. In Table 12 we list the Lie pairs associated to the Bianchi Lie algebras. In that table we have often redefined H and P linearly in order to ease the comparison with the Lie pairs in D 3. This comparison leads to a division of the table into two: the ones above the horizontal line are D = 1 avatars of Lie pairs which exist also for D > 1. Table 13 shows the correspondence between the D = 1 Lie pairs and the D 3 Lie pairs in those cases where there is one. Notice that in D = 1 there are exceptional isomorphisms between Lie pairs which might differ in D 3. For example, exchanging P and H (i.e., re-interpreting what is time and space), we see that the following pairs of spacetimes are isomorphic: de Sitter/anti de Sitter, galilean/carrollian, galilean dS/carrollian AdS and galilean AdS/carrollian dS.
As in previous tables (Tables 5, 7 and 9) each row in Table 12 is an isomorphism class of kinematical Lie pair with sequential label "LP#". Similarly, the label "Bianchi" in Table 12 identifies the Bianchi type of the Lie algebra in Table 11. It bears reminding that the Lie brackets in the tables below are expressed in a basis where the admissible subalgebra h is spanned by B.

C -
In Section 3 we have classified the equivalence classes of Lie pairs (k, h) where k is a kinematical Lie algebra and h is an admissible subalgebra. Our aim is to classify simply-connected homogeneous spacetimes and, as explained in Appendix B, this requires selecting those Lie pairs which are effective (so that h contains no nonzero ideal of k) and also geometrically realisable, so that there exists a kinematical Lie group K ′ and an admissible subgroup H ′ whose Lie pair (k ′ , h ′ ) is isomorphic to (k, h). Let us discuss both of these selection criteria in turn.

Effective Lie pairs.
Recall that we have chosen a basis for the kinematical Lie algebra k such that the subalgebra h is spanned by J ab and B a . This means that the only possible nonzero ideal of k contained in h is the span of the boosts B a . Therefore to check if a Lie pair (k, h) is effective, all we need to do is inspect the Lie brackets [B a , X] for X in the span of (B a , P a , H) and see whether they all lie in the span of the B a (in which case the Lie pair is not effective) or not (in which case it is). Reducing a non-effective Lie pair by the ideal b spanned by the boosts, we arrive at an effective aristotelian Lie pair. Since it turns out that not all aristotelian Lie pairs arise in this way, we treat their classification separately in Appendix A.
4.1.1. Effective Lie pairs for all D 3. Inspecting Table 5 we see that Lie pairs 1, 3, 4 γ , 5 γ , 7, 8 and 11 are not effective, since the span of the B a define an ideal of k. Reducing by that ideal we obtain an aristotelian Lie pair of the ones in Table 2, which are classified in Appendix A: • Lie pairs 1 and 3 reduce to the static aristotelian Lie pair A21; whereas galilean AdS (AdSG = AdSG 0 ) S11 χ 12 χ 50 2  Table 15 is included for convenience. We have found this list useful at times and thus we think it might be useful to other readers. Table 15 lists which simply-connected homogeneous kinematical or aristotelian spacetimes are associated to which kinematical Lie algebras. We do not list aristotelian Lie algebras since there is a one-to-one correspondence between the Lie algebras and the simply-connected homogenous aristotelian spacetimes in that case. The table is separated into three: corresponding to the Lie algebras for

Geometric realisability.
Having selected the effective Lie pairs, we must then select those which are geometrically realisable, so that they correspond to the Lie pair of a homogeneous spacetime. It will turn out that all effective Lie pairs are geometrically realisable. Recall that (k, h) is geometrically realisable if there exists a connected Lie group K, a Lie algebra isomorphism ϕ : Lie(K) → k and a closed Lie subgroup H ⊂ K whose Lie algebra is isomorphic to h under ϕ. There are a number of criteria which can be brought to bear in order to help decide whether a Lie pair admits a geometric realisation.

Riemannian maximally symmetric spaces.
We start by showing that the Lie pairs corresponding to the riemannian symmetric spaces are geometrically realisable; although this is of course well-known.

Criterion 1. Compact subgroups are closed, so if if h generates a compact subgroup, then (k, h) is geometrically realisable.
From this criterion we see that the following Lie pairs are geometrically realisable: • Lie pairs 15 −1 , 44 and 72, which we can identify with euclidean spaces; • Lie pairs 18 +1 , 46, 76, which we can identify with hyperbolic spaces; and • Lie pairs 19 −1 , 48, 77, which we can identify with the round spheres.

A sufficient criterion.
Another useful criterion (sufficient, but by no means necessary) applies to linear Lie algebras; that is, Lie algebras isomorphic to Lie algebras of matrices. By Ado's theorem (see, e.g., [51, Ch. VI]) every kinematical Lie algebra k, being finite-dimensional, has a faithful linear representation and hence is a linear Lie algebra. Exponentiating inside the matrix algebra, we obtain a connected Lie group K with k (or, more precisely, its isomorphic image in the matrix algebra) as its Lie algebra. In particular, if the Lie pair is reductive, so that k = h⊕m as adjoint h-modules, then we may decompose X ∈ k uniquely as X = X h + X m , with X h ∈ h and X m ∈ m. Now, rotations (when present) act reducibly, so [J ab , X] ∈ h means that X m = cH for some c. So we then need to inspect whether [B a , H] ∈ h.

Two-dimensional spacetimes.
To show that D = 1 Lie pairs 63, 65, 68, 71 χ , 73, 74 χ , 79 and 82 χ=1 are geometrically realisable, we may use yet a third criterion for when a one-dimensional subgroup of a matrix group is (not) closed.

Criterion 3. A one-parameter subgroup of a matrix group is not closed if and only if the generating matrix in the Lie algebra is similar to a diagonal matrix with imaginary entries, at least two of which have an irrational ratio
(see, e.g., [52,Pr. 2.7.5]).
All the Bianchi Lie algebras have faithful representations of dimension 2 or 3. So it is simply a matter of calculating the eigenvalues of B in each of these representations to deduce that all these Lie pairs are geometrically realisable. Indeed, let us go back to the description of the Bianchi Lie algebras in terms of the basis (e 1 , e 2 , e 3 ) as in Table 11 and let us write the generic element as X = xe 1 + ye 2 + ze 3 . For each of the cases of interest, we will write down the matrix ρ(X) representing X and the matrix ρ(B) representing B. We will see that in no case does ρ(B) have imaginary eigenvalues.
(63) This is Bianchi II, This matrix is not diagonalisable. (65) This is Bianchi IV, which is not diagonalisable. (68) This is Bianchi VI 0 , but let us consider the general Bianchi VI χ , which is not diagonalisable. (71 χ ) This is again Bianchi VI χ , so that we can reuse the previous calculation. In this case also which is not diagonalisable. (74 χ ) This is again Bianchi VI χ and again B = e 1 + e 2 , with the same matrix non-diagonalisable as above. (73) This is Bianchi VII 0 , but we will treat the general Bianchi VII χ whose matrix representation is which is not diagonalisable. (82 1 ) This is Bianchi VII 1 and B = e 3 , so that the matrix representing it is This matrix is diagonalisable, but the eigenvalues are not imaginary: −1 ± i. (79) This is Bianchi VIII ( ∼ = sl(2, R)) which has a faithful two-dimensional representation Here B = e 1 + e 3 , which has matrix whose eigenvalues are real: ±1.
In summary, after this discussion we are left with the carrollian Lie pairs: 14, 16 ε and 17 in D 3 and 40, 41, 42 and 45 in D = 2.

The carrollian light cone.
It is easy to show that Lie pairs 17 and 45 are geometrically realisable. We can treat them together and, in fact, the argument works also to give an alternative proof of geometric realisability for the D = 1 Lie pair 79. These Lie pairs are all of the form (so (D+1, 1), h) where, as we now show, h is the subalgebra of so(D + 1, 1) corresponding to the stabiliser of a null vector in the lorentzian vector space R D+1,1 . Indeed let e µ = {e a , e + , e − } be a Witt basis for R D+1,1 where the lorentzian inner product η µν := η(e µ , e ν ) is given by η ab = δ ab and η +− = 1. Then so(D + 1, 1) has generators J µν with Lie brackets: Letting µ = (a, +, −) and decomposing J µν into {J ab , J a+ , J a− , J +− }, we obtain the following nonzero brackets:  2) and B a . The key observation is that h is the subalgebra which annihilates the basis vector e + under the usual action: The connected subgroup H of SO(D + 1, 1) generated by h is (the identity component of) the stabiliser of e + and hence it is a closed subgroup. Since H is connected, it is actually a subgroup of SO(D + 1, 1) 0 , the identity component of SO(D+ 1, 1). We conclude that SO(D+ 1, 1) 0 /H is thus a geometric realisation of (so(D + 1, 1), h). Geometrically, it corresponds to the future light cone L + ⊂ M, where M is (D + 2)dimensional Minkowski spacetime. 5 Indeed, as shown in [39] (see also [28]), null hypersurfaces (such as the future light cone in Minkowski spacetime) are carrollian spacetimes. This idea turns out to be very fruitful in order to prove the geometric realisability of the remaining symmetric carrollian Lie pairs, as we will now see.

Symmetric carrollian spacetimes.
Finally, we show that the symmetric carrollian Lie pairs 14 and 16 ε in D 3 and 40, 41 and 42 in D = 2 are geometrically realisable. As mentioned above already, one way to do this is to construct the geometric realisations explicitly by exhibiting them as null hypersurfaces in lorentzian manifolds one dimension higher. This was done originally for the carrollian spacetime C (S13) in [39], who embedded it as a null hypersurface in Minkowski spacetime one dimension higher.
In a similar way we will construct the carrollian (anti) de Sitter spacetimes dSC (S14) and AdSC (S15) as null hypersurfaces in (anti) de Sitter spacetimes one dimension higher. All we need to show is that the Lie pairs describing these null hypersurfaces are the symmetric carrollian Lie pairs 14, 16 ε , 40, 41 and 42.
Introducing a parameter ε = 0, ±1, we define the kinematical Lie algebra k ε by the following Lie brackets in addition to the ones in (8): We shall let h ε denote the admissible subalgebra spanned by J and B. The Lie pair (k ε , h ε ) is isomorphic to 14 and 40 when ε = 0, to 16 +1 and 41 when ε = +1 and to 16 −1 and 42 when ε = −1. We will now exhibit homogeneous manifolds whose Lie pairs are isomorphic to (k ε , h ε ) for each value of ε. We start with ε = 0, which is the construction of flat carrollian space C (S13) in [39]. To this end, let M denote (D + 2)-dimensional Minkowski spacetime with coordinates x µ for µ = 0, 1, . . . , D + 1 (although 5 Strictly speaking this description holds for D 2. For D = 1, the future light cone is not simply-connected and hence the simply-connected geometric realisation of the Lie pair 79 is the universal cover of the future light cone, which we may think of as the submanifold of R 3 consisting of points (r cos θ, r sin θ, θ) with r > 0, projecting to the light cone in R 1,2 , by sending (r cos θ, r sin θ, θ) → (r, r cos θ, r sin θ).
we will let ♮ stand for D + 1) and metric where x ± = 1 √ 2 (x ♮ ± x 0 ). Let C 0 ⊂ M denote the null hypersurface defined by x − = 0. We claim that C 0 is a geometric realisation of the Lie pair (k 0 , h 0 ).
The Poincaré Lie algebra is spanned by the following vector fields on M: where x µ = η µρ x ρ . It is a transitive Lie algebra for M, which means that for every p ∈ M, the values J µν (p) and P µ (p) span the tangent space T p M. The subalgebra of the Poincaré algebra consisting of vector fields which are tangent to C is spanned by The subalgebra spanned by J ab ,P a , B a and H satisfy the Lie brackets of k 0 (the Carroll algebra) and the subalgebra h 0 which vanishes at the point o ∈ C 0 with coordinates x µ = 0 is the span of J ab and B a . By dimension, P a and H span the tangent space T o C 0 and indeed the same is true at any other point of C with different (but isomorphic) stabiliser subalgebra. Hence k 0 is a transitive Lie algebra for C 0 with stabiliser h 0 at o. Therefore C 0 is a geometric realisation of (k 0 , h 0 ). Now let us consider ε = −1 and let M now stand for Minkowski spacetime of dimension D + 3, with coordinates x µ with µ = 0, 1, . . . , D + 2 and metric Let Q − denote the quadric defined by for some (fixed) R > 0. Its universal cover (with the induced metric) is de Sitter spacetime in dimension D + 2. The Lorentz Lie algebra, spanned by with x µ = η µρ x ρ , is a transitive Lie algebra for Q − and isomorphic to so(D + 2, 1). Let N ⊂ M denote the null hypersurface defined by the equation x 0 = x D+2 and let C − = Q − ∩ N . This is defined by the following two equations: which shows that C − is diffeomorphic to R × S D and hence is simply-connected for D 2. We claim that C − is a geometric realisation for the Lie pair (k − , h − ). We will show this by determining the subalgebra of the Lorentz Lie algebra consisting of vector fields tangent to C − , a subalgebra of which, by the same argument as in the previous case, is transitive on C − . We will show that this subalgebra is isomorphic to k − and that the stabiliser at a suitably chosen point o ∈ C − is isomorphic to h − . The following Lorentz generators are tangent to C − : for all i = 1, . . . , D + 1. The subalgebra spanned by J ij and V i is isomorphic to the euclidean algebra in dimension D + 1, with nonzero Lie brackets apart from (8). We see that this Lie algebra is isomorphic to k − . Now let o ∈ C − denote the point with coordinates x a = x 0 = x D+2 = 0 and x ♮ = R. Then the vector fields which vanish at o are the span of J ab and B a , which is isomorphic to h − . Therefore C − is a geometric realisation of (k − , h − ).
Finally, we consider the case ε = +1. Now E is pseudo-euclidean space with signature (D + 1, 2) with coordinates x µ for µ = 0, 1, . . . , D + 2 and metric (132) metric as above. Now we fix R > 0 and let Q + ⊂ E denote the quadric defined by the equation Its universal cover (with the induced metric) is anti de Sitter spacetime in dimension D + 2. The Lie algebra spanned by the vector fields with x µ = η µρ x ρ , is a transitive Lie algebra for Q + and isomorphic to so(D + 1, 2). Let N ⊂ E denote now the null hypersurface with equation x D+1 = x D+2 . The intersection Q + ∩ N is described by the two equations: which has two connected components determined by the sign of x 0 , which is never zero. Let C + denote the component where x 0 > 0. The following vector fields generate the subalgebra of the span of the J µν which are tangent to C + : for a = 1, . . . , D. The subalgebra spanned by J ab , P a , B a and H has, in addition to (8), the following nonzero Lie brackets showing that it is isomorphic to k + . The stabiliser subalgebra at the point o ∈ C + with coordinates x a = x D+1 = x D+2 = 0 and x 0 = R, is the span of J ab and B a , so isomorphic to h + . As before, dimension says that k + is transitive at o ∈ C + with stabiliser h + and also at other points with isomorphic stabilisers. Therefore C + is a geometric realisation of (k + , h + ).
It is worth remarking that in all the homogeneous carrollian spacetimes discussed here, the boost generators can be interpreted as null rotations in the ambient Minkowski or pseudo-euclidean spaces.

L
In the previous two sections we have classified the simply-connected homogeneous spacetimes. This provides the objects in Figure 3, which also contains arrows between the spacetimes. These arrows are explained by limits between spacetimes and in this section we will discuss these limits and in this way explain Figure 3. We will also explain how the picture gets modified in D 2 and explain Figures 4 and 5.
In the infinitesimal description of the homogeneous spacetimes in terms of Lie pairs, most (but not all) limits between homogeneous spacetimes manifest themselves as contractions of the underlying kinematical Lie algebras.

Contractions.
Recall that a (finite-dimensional, real) Lie algebra consists of a vector space V, together with a linear map ϕ : Λ 2 V → V satisfying the Jacobi identity. The Jacobi identity defines an algebraic variety J ⊂ Λ 2 V * ⊗ V, every point of which is a Lie algebra structure on V. The general linear group GL(V) acts on V and hence also on the vector space Λ 2 V * ⊗ V and since the action is tensorial it preserves the variety J . If ϕ ∈ J defines a Lie algebra g and g ∈ GL(V), then g · ϕ ∈ J and, by definition, the Lie algebra it defines is isomorphic to g. Indeed, the GL(V) orbit of ϕ consists of all Lie algebras on V which are isomorphic to g. This orbit may not be closed relative to the induced topology on J . The closure of the orbit may contain Lie algebras which are not isomorphic to g: they are said to be "degenerations" of g. A special class of degenerations are the contractions of g, which are limit points of curves in the GL(V)-orbit of ϕ. More precisely, let t ∈ (0, 1] and let g t be a continuous curve in GL(V) with g 1 the identity. Define ϕ t := g t · ϕ. Explicitly, the Lie bracket [−, −] t associated to ϕ t is given by Then ϕ 1 = ϕ and for every t ∈ (0, 1], ϕ t defines a Lie algebra on V isomorphic to g. By continuity, if the limit ϕ 0 := lim t→0 ϕ t exists, it defines a Lie algebra, but since the linear transformation g 0 := lim t→0 g t of V (even if it exists) need not be invertible, the Lie algebra defined by ϕ 0 need not be isomorphic to g. It is, however, by definition a contraction of g.
We will now explicitly exhibit contractions between kinematical Lie algebras which induce the limits between the (D + 1)-dimensional homogeneous spaces in Figure 3 for D 3. We will also explain other non-contraction limits in that figure, as well as in D 2.

D
3. We will start with the lorentzian and riemannian space forms with nonzero curvature, whose kinematical Lie algebras are the (semi)simple Lie algebras: so(D + 1, 1) for de Sitter spacetime (S2) and hyperbolic space (S6), so(D, 2) for anti de Sitter spacetime (S3) and so(D + 2) for the round sphere (S5).
Let R D+2 have basis e µ = (e 0 , e a , e ♮ ), with a = 1, . . . , D and let η denote the inner product with coefficients η µν = η(e µ , e ν ) given by η ab = δ ab and all other components zero except for η 00 and η ♮♮ . The generators of so(R D+2 , η) are J µν = {J ab , B a := J 0a , P a := J a♮ , H := J 0♮ } subject to the following Lie brackets The first three brackets are the standard kinematical Lie brackets, so we will focus attention on the remaining brackets and we will change to shorthand notation where the so(D) indices are implicit. Let us consider a three-parameter (κ, c, τ) family of linear transformations g κ,c,τ defined on generators by g κ,c,τ · J = J, g κ,c,τ · B = τ c B, g κ,c,τ · P = κ c P and g κ,c,τ · H = τκH.  The flat limit corresponds to taking κ → 0, the non-relativistic limit to c → ∞ and the ultra-relativistic limit to τ → 0. We may take any two of the three limits or, indeed, all limits at once; although whenever we combine a non-relativistic limit with an ultra-relativistic limit we arrive at a non-effective Lie pair reducing to the aristotelian static spacetime (A21), denoted S in Figure 3.
Let us first take the flat limit κ → 0 of the Lie brackets in (141) For τ c = 0, this is either the Poincaré Lie algebra for η 00 = −1 or the euclidean Lie algebra for η 00 = 1. The corresponding Lie pairs are those of Minkowski spacetime (S1) and euclidean space (S4). In Figure 3 only η 00 = −1 is considered and these limits explain the arrows AdS → M and dS → M, but in Figure 2 this also explains the arrows H → E and S → E. We may now take the non-relativistic limit c → ∞ to arrive at the galilean algebra (after rescaling H by −1/(η 00 τ 2 )), or alternatively the ultra-relativistic limit τ → 0 to arrive at the Carroll algebra (after setting c = 1): The corresponding Lie pairs are the galilean (S7) and carrollian (S13) spacetimes. This explains the arrows M → G and M → C in Figure 3 and the arrows E → G and E → C in Figure 2.
Taking now the non-relativistic limit of the Lie brackets in (141), we have For τκ = 0, we obtain the Lie pairs corresponding to the galilean de Sitter spacetime (S8), if η ♮♮ η 00 = −1, or the galilean anti de Sitter spacetime (S10), if η ♮♮ η 00 = 1, thus explaining the arrows dS → dSG and AdS → AdSG in Figure 3 and also the arrows H → dSG and S → AdSG in Figure 2. If we then take the flat limit κ → 0, we obtain the galilean spacetime, thus explaining the arrows dSG → G and AdSG → G. If instead we take the ultra-relativistic limit we obtain a non-effective Lie pair reducing to the aristotelian static spacetime (A21). Finally, let us start by taking the ultra-relativistic limit (τ → 0) in the brackets (141), to arrive at If κ c = 0, we obtain either the carrollian de Sitter spacetime (S14) if η ♮♮ = 1 or the carrollian anti de Sitter spacetime (S15) if η ♮♮ = −1. This explains the arrows dS → dSC and AdS → AdSC in Figure 3 and the arrows H → AdSC and S → dSC in Figure 2. If we now take the flat limit we arrive at the carrollian spacetime, which explains the arrows dSC → C and AdSC → C. If instead we take the non-relativistic limit, we arrive at a non-effective Lie pair reducing to the aristotelian static spacetime (A21).
Of all the arrows to the aristotelian static spacetime, only G → S and C → S are shown explicitly in Figure 3. Taking any two limits in the brackets (141), the resulting Lie algebra does not depend on the order in which we take the limits. This means that the arrows in (141) "commute" and thus, for instance, that the arrow (not shown) dSC → S is to be understood as the composition of the arrows (shown) dSC → C → S. Similarly, the arrows (not shown) AdSC → S, dSG → S and AdSG → S can be understood as compositions of arrows which are shown: AdSC → C → S, dSG → G → S and AdSG → G → S, respectively.
We have so far explained the limits in Figure 3 (or even Figure 2) corresponding to the known symmetric spacetimes and it now remains to explain the limits from the new spacetimes in our classification.

5.2.1.
AdSG χ → G. Let t ∈ (0, 1] and let g t be defined by The new brackets are now Taking the limit t → 0 we recover the Carroll algebra [B, P] = H.

LC → TS.
Let t ∈ (0, 1] and let g t be defined by so that the brackets become The resulting Lie pair is not effective because the span of the B a is an ideal. Quotienting by this ideal gives the aristotelian Lie algebra defined by [H, P] = P, whose associated spacetime is the torsional static spacetime A22.

TS → S.
Let t ∈ (0, 1] and let g t · H = tH and g t · P = P. The new bracket is [H, P] = tP, which vanishes in the limit t → 0.

5.2.6.
A non-contracting limit. Finally, we discuss a limit which does not come from a contraction of Lie algebras. The Lie algebra of AdSG χ depends on a parameter χ 0 and this parameter determines the isomorphism class of the Lie algebra. A natural question is what spacetime corresponds to AdSG χ in the limit χ → ∞. The answer turns out to be that lim χ→∞ AdSG χ = dSG 1 . To see this we start with the Lie algebra corresponding to AdSG χ and we change basis to This is a vector space isomorphism for any χ > 0, but becomes singular in the limit χ → ∞. In this sense this is reminiscent of a contraction, but it is not a contraction since we are changing the isomorphism type of the Lie algebra as we change χ. In the new basis, and now taking χ → ∞ we arrive at which is the Lie algebra corresponding to dSG 1 .

D = 2.
In D = 2 there is an additional two-parameter family of spacetimes not present in D 3: namely, spacetime S12 γ,χ , for γ ∈ [−1, 1) and χ > 0. The Lie brackets are given by (8) where z = γ + iχ ∈ C lies in the infinite vertical strip in the upper-half plane defined by −1 Re z < 1.
Parenthetically, let us mention that for z = −1 + iχ, the complexified Lie pair for spacetime S12 −1,χ is isomorphic to the complexification of the Lie pair for dSG. This can be seen by the following complex change of basis: When γ = 1, this corresponds to AdSG 2/χ . Now let us consider the (singular) limit χ → ∞, so that the Lie brackets become which we recognise as AdSG.
The picture resulting from this discussion is illustrated in Figure 6. This shows that in D = 2, spacetime S12 γ,χ interpolates between the one-dimensional continua of torsional galilean de Sitter and anti de Sitter spacetimes. Figure 4 shows how to insert this figure into Figure 3. . At first one might be surprised at this statement since after all de Sitter space has positive scalar curvature, whereas anti de Sitter space has negative scalar curvature and surely they are geometrically distinguishable. This is perhaps a good place to point out that scalar curvature is not an invariant of a homogeneous space, but rather of the homogeneous space together with the choice of an invariant metric. In (A)dS there is a one-parameter family of invariant metrics, labelled by the radius of curvature, all sharing the same connection and curvature: after all, the Levi-Civita connection and hence the Riemann curvature are homothety invariant. Even the Ricci tensor is homothety invariant and it is only the Ricci scalar, which involves tracing with the metric, that distinguishes between the (A)dS spacetimes with different radii of curvature. In two dimensions, we have the possibility of exchanging space and time, which results in multiplying the metric by −1, which is formally a homothety, so that the Riemann and Ricci tensors are unchanged. It is only when we calculate the Ricci scalar that we see the effect of this homothety: namely, changing the sign.
There are other accidental isomorphisms coming from exchanging space and time, e.g., galilean and carrollian spacetimes are isomorphic. This is clear from Figure 1 since exchanging space and time rotates the light cones by 90 degrees. Similarly, carrollian dS and galilean AdS spacetimes are isomorphic, as are carrollian AdS and galilean dS spacetimes: the result of both changing the sign of the curvature and rotating the light cone by 90 degrees.
In addition to these identifications, there are additional homogeneous spacetimes which are unique to two dimensions: namely, S17, S18, S19 χ>0 and S20 χ>0 and which admit none of the low-rank invariant structures we have been focussing on. We describe two kinds of limits between these exotic spacetimes: limits which manifest themselves infinitesimally as contractions of the relevant Bianchi Lie algebra as well as limits which are not of this type. The contractions between the Bianchi Lie algebras have been determined in [53]. All Lie algebras contract to the abelian Lie algebra (here, Bianchi I), and we will not mention these contractions explicitly. The resulting spacetimes and their limits are depicted in Figure 5. 5.4.1. S17 → S18 and S17 → S7/S13. The Lie algebra associated to S17 is Bianchi IV, which can contract to Bianchi II and Bianchi V [53]. Indeed, both contractions arise from limits of spacetime S17: a limit to galilean/carrollian spacetime and a limit to S18.
The Lie algebra has brackets [H, B] = −P and [B, P] = −H − 2P. This is an extension by B of the abelian Lie algebra spanned by H and P. The action of B is via a non-diagonalisable endomorphism with one eigenvector with eigenvalue −1. This means that we can change basis in the span of H and P so that relative to the new basis H ′ , P ′ , If we now introduce a parameter t ∈ (0, 1] and a one-parameter family g t of invertible endomorphisms defined by g t · H ′ = H ′ , the brackets become so that the limit t → 0 recovers the Lie algebra corresponding to spacetime S18.
If instead we define g t by the brackets become so that the limit t → 0 recovers the Lie algebra corresponding to spacetime S7/S13 after exchanging H ′ and P ′ .
5.4.2. S19 χ → S7/S13. The Lie algebra is Bianchi VI χ for χ > 0, which can contract to Bianchi II. The Lie brackets are [H, B] = (1 − χ)H and [B, P] = (1 + χ)P. Change basis to H ′ = H + P and P ′ = P − H, so that the Lie algebra in this new basis is Define g t , for t ∈ (0, 1], by The brackets become so that the limit t → 0 recovers the Lie algebra corresponding to spacetime S7/S13. 5.4.3. S20 χ → S7/S13. The Lie algebra is Bianchi VII χ for χ > 0, which can contract to Bianchi II. Under the same one-parameter family of invertible transformations g t as in the previous case: the brackets become Taking the limit t → 0 and changing B to −B, recovers the Lie algebra of spacetime S7/S13.

S
In this section we start to study some of the geometrical properties of the homogeneous spacetimes in Tables 1 and 2. We will concentrate on those geometrical properties which are easy to glean from the infinitesimal description in terms of the Lie pair (k, h) and which help distinguish between the different homogeneous spacetimes. In a follow-up paper [54] we will study in more detail the local geometry of these homogeneous spacetimes. In that case M is an abelian group and, if simply-connected, then a vector space. The homogeneous space is then an affine space modelled on M. We will say then that the homogeneous space is affine. (One should not confuse this with the more general notion of an affine symmetric space from [55,56].) In Table 1 we have chosen a basis for k in such a way that, in the reductive examples, h is spanned by J and B and m is the span of P and H. It is clear by inspection that all of the spacetimes in Table 1 are reductive with one exception: the carrollian light cone LC. We will see below that LC does not admit any invariant connections for D 2, providing a separate proof that it is indeed non-reductive. We will also classify the invariant connections for D = 1.

6.1.2.
The linear isotropy representation. The natural geometric objects in a homogeneous space (e.g., metric, connections, curvature, torsion,...) are those which are invariant under the group action. Invariance means, in particular, that their value at any point is invariant under the stabiliser subgroup of that point, which acts on the tangent space at that point via the linear isotropy representation, which we now introduce at the Lie algebraic level.
Even if (k, h) is not reductive, we have a representation of h on k/h, called the linear isotropy representation and denoted λ : h → gl(k/h) and sending X ∈ h to λ X . For every Y ∈ k, let Y denote its image in k/h. Then X ∈ h acts on Y as which is well defined because h is a Lie subalgebra. In the reductive case, m is isomorphic to k/h as a representation of h. In the non-reductive case, we may choose a vector space complement m and then define on it a representation of h by transporting the linear isotropy representation on k/h via the vector space isomorphism m ∼ = k/h. In practice all this means is that when calculating the action of X ∈ h on Y ∈ m, we can compute the Lie bracket [X, Y] in k and set to zero anything on the right-hand side that lies in h.
If the Lie pair (k, h) has a geometric realisation, the vector space k/h is an algebraic model for the tangent space at the origin of any homogeneous space M := K/H with K a Lie group with Lie algebra k and H a Lie subgroup with Lie algebra h. If H is connected-which we can assume with no loss of generality by passing to the universal cover of the homogeneous space, if necessary-there is then a bijective correspondence between h-invariant tensors on k/h and K-invariant tensor fields on M.

Invariant structures.
For the purposes of this paper, we are particularly interested in invariant tensors of low rank; that is, h-invariant tensors in k/h, (k/h) * , S 2 (k/h) and S 2 (k/h) * . A non-degenerate invariant tensor in S 2 (k/h) * gives rise to a K-invariant metric on M, whereas h-invariant tensors in k/h and (k/h) * give rise to an invariant vector field and one-form on M, respectively. We shall say that a homogeneous kinematical spacetime M is lorentzian or riemannian if it admits a K-invariant metric of lorentzian or riemannian signature. This is the case if the associated Lie pair admits an h-invariant nondegenerate tensor in S 2 (k/h) * of the right signature. In cases where rotations are present in h (i.e., D 2), the only possible invariant in k/h must be proportional to H and any invariants in S 2 (k/h) must lie in the subspace spanned by H 2 and P 2 := δ ab P a P b .
Let us introduce a basis (π a , η) for (k/h) * canonically dual to the basis (P a , H) for k/h; that is, Then similarly, for D 2, any invariants in (k/h) * are proportional to η, whereas in S 2 (k/h) * any invariant lies in the span of η 2 and π 2 , where π 2 = δ ab π a π b . If (k, h) is such that the one-form η ∈ (k/h) * and the co-metric P 2 ∈ S 2 (k/h) are h-invariant, we say that M admits an invariant galilean structure. The one-form η is called the absolute clock reflecting that galilean spacetimes are absolute in time. This means, two points of the homogeneous space that are at same time stay that way, irrespective of any galilean kinematical transformation. If H ∈ k/h and π 2 ∈ S 2 (k/h) * are h-invariant we say that M admits an invariant carrollian structure. Analogous to the galilean case, the fundamental vector field H reflects the absolute space character of carrollian spacetimes. Notice that aristotelian spacetimes in Table 2 admit simultaneously a galilean and carrollian structure, since H, η, P 2 , π 2 are all rotationally invariant. It follows that they also admit many invariant lorentzian and riemannian structures. It is easy to determine the existence of these invariants from the data in Table 1. After writing down down the possible rotationally invariant tensors, we only need to check invariance under B. The action of B is induced by duality from its action via the linear isotropy representation on g/h: with the brackets being those of k. In practice, we can determine this from the tables, by computing the brackets in k and simply dropping any B or J from the right-hand side. The only possible invariants in k/h are proportional to H, which is invariant provided that [B, H] ∈ h. Dually, the only possible invariants in (k/h) * are proportional to η, which is invariant provided that there is no X ∈ k such that H appears in [B, X].

Parity and time reversal.
We define a parity transformation on a kinematical Lie pair (k, h) to be an automorphism of k which changes the sign of B and P and leaves H and J inert. Similarly, we define a time reversal transformation to be an automorphism of k which changes the sign of B and H, but leaves P and J inert. For aristotelian Lie pairs (a, r), only P changes sign under a parity transformation and only H changes sign under a time reversal transformation. The combination of a parity and time-reversal transformations is then an automorphism of k (or a) which changes simultaneously the signs of P and H and leaves other generators inert.
It follows from equivariance under rotations, that for D = 1, 3, every kinematical Lie algebra possesses a parity transformation. It is only in D = 1 and D = 3 where we can have Lie brackets which violate parity: in D = 3 because we have a vector product which is only invariant under the orientation preserving orthogonal transformations, and in D = 1 because there are no rotations. We saw that there are no effective Lie pairs for D = 3 whose kinematical Lie algebra involves the vector product, so that it is only in D = 1 where we can expect to have effective Lie pairs without parity (or time reversal, since in D = 1 what is time and space is a matter of convention) symmetry.
Any spacetime whose canonical connection (see Section 6.2 below) has torsion cannot be invariant under PT , hence it cannot be invariant under both P and T . Since, as explained above, parity is guaranteed for D = 1, it is time reversal invariance which fails for torsional geometries.
It is simply a matter of inspecting the brackets in Tables 1 and 2 to determine whether the corresponding spacetimes possess parity and/or time reversal invariance. The results are summarised in Table 16. 6.2. Invariant connections, curvature and torsion. As a final geometric property, we discuss the existence of invariant affine connections and their curvature and torsion. Let (k, h) be an effective Lie pair associated to a homogeneous space. By an invariant (affine) connection on (k, h) we shall mean a linear map Λ : k → gl(k/h), denoted X → Λ X , satisfying the following two properties: (1) Λ| h = λ, the linear isotropy representation, and (2) Λ is h-equivariant. Notice that Λ is not generally a Lie algebra homomorphism; although its restriction to h is. In fact, as we will see, the obstruction of Λ being a Lie algebra homomorphism is the curvature of the connection.
The equivariance condition for Λ says that for all X, Y ∈ k and Z ∈ h, which is the h-invariance of Λ as an element in the space of linear maps Hom(k, gl(k/h)). Notice that if either X or Y are in h, then this is automatically satisfied: it is clear if Y ∈ h, since then Y = 0, but then also if X ∈ h, Λ X = λ X and the invariance condition reads which holds because λ is a representation. So the only nontrivial condition comes from X, Y ∈ h. Now choose m such that k = h ⊕ m as vector spaces. In the reductive case, we can choose m such that [h, m] ⊂ m and hence m is an h-module. But even in the non-reductive case, m ∼ = k/h as a vector space and there is a unique way to give m the structure of an h-module so that this is also an isomorphism of h-modules. Let us assume we have done that. Now let Λ be an invariant affine connection. Since the restriction of Λ to h is fixed, Λ is determined by its component mapping m → gl(k/h) and, as argued above, it is only this component that is involved in the invariance condition. In the reductive case, this component is an h-equivariant linear map m → gl(m) or, equivalently, an h-equivariant bilinear map m × m → m, called the Nomizu map. In a reductive homogeneous space, we may always take the Nomizu map to be zero and in this way arrive at a canonical invariant connection (termed "of the second kind" in [55]). All invariant connections are then classified by their Nomizu maps.
In the non-reductive case, it can very well be the case that there are no invariant connections. Turning this around, if a given homogeneous space does not admit any invariant connections, it cannot be reductive.
Given an invariant affine connection, its torsion is given for all X, Y ∈ k, One checks that this only depends on the images X, Y of X, Y in k/h, so it defines an h-equivariant skewsymmetric bilinear map Θ : k/h × k/h → k/h. The curvature Ω is given, for all X, Y ∈ k and Z ∈ k/h, by from where we see that it measures the failure of Λ : k → gl(k/h) to be a Lie algebra homomorphism.
The h-equivariance of Λ guarantees that this expression only depends on X, Y and hence it defines an h-equivariant skewsymmetric bilinear map Ω : k/h × k/h → gl(k/h). When the curvature vanishes we say that the canonical connection is flat.
In the reductive case, the torsion and curvature of an invariant affine connection are given in terms of its Nomizu map α : m × m → m by the following expressions for all X, Y, Z ∈ m, where These expressions simplify for the canonical connection with zero Nomizu map: and If the space is symmetric, so that [X, Y] ∈ h for X, Y ∈ m, then we see that the canonical connection is torsion-free. If, on the contrary, the canonical connection is flat, then [X, Y] ∈ m for all X, Y ∈ m and hence m is a Lie subalgebra of k. In this case we can identify the homogeneous space (assumed simplyconnected) with the group manifold of the simply-connected Lie group with Lie algebra m. If the canonical connection is both torsion-free and flat, then the Lie algebra m is abelian and hence the homogeneous space (if simply-connected) is the group manifold of a simply-connected abelian Lie group. A simplyconnected abelian Lie group is a vector space and hence the homogeneous space in that case is an affine space.
The holonomy of the canonical connection on a reductive homogeneous space M = K/H can be calculated in general. Indeed, by a result of Nomizu's [55, §12], the Lie algebra of the holonomy group is isomorphic to the ideal [m, m] h of h acting on m via (the restriction of) the linear isotropy representation. As a corollary, the covariant derivative of a K-invariant tensor field T , say, on M with respect to the canonical connection vanishes: ∇T = 0. If M is simply-connected, then parallel-transporting T along any closed loop leaves it invariant. In particular, the curvature and torsion tensor fields (of any invariant connection, but in particular of the canonical connection), being themselves invariant, are parallel, and so are the tensor fields corresponding to any invariant lorentzian, riemannian, galilean, carrollian or aristotelian structure that M might possess. In particular the connection is compatible with the (co-)metrics. It is thus clear that the galilean (G) and carrollian (C) spacetimes reproduce the standard flat Newton-Cartan and Carroll structures, respectively (cf. [39]).
All spacetimes, except for LC are reductive. It is then easy to inspect Tables 1 and 2 and determine the torsion and curvature of the canonical invariant connection with zero Nomizu map. The results are summarised in Section 6.3 and in Table 16. The existence of invariant connections for the non-reductive spacetime LC has to be studied separately and we do so below. We will see that for D 2 the spacetime admits no invariant connections, whereas for D = 1 it admits a three-parameter family of invariant connections and a unique torsion-free, flat connection. The follow-up paper [54] presents the classification of invariant connections and the calculation of their torsion and curvature for the reductive kinematical and aristotelian spacetimes.
with the tacit understanding that the term proportional to ν is only present if D = 3. Invariance demands, in particular, that which becomes Taking any b = c = a, we arrive at a contradiction. Therefore there are no invariant connections for D 3.
(D = 2). Now the most general rotationally equivariant Λ is given by But as before, equivariance requires, in particular, that which again for any b = c = a results in a contradiction. Therefore there are no invariant connections for D = 2 either. (D = 1). Now there are no rotations, and Λ is a general linear map: Invariance under B says that for all X, Y ∈ {H, P}, Taking (X, Y) = (H, H), (H, P), (P, H), (P, P) in turn we arrive at the following conditions: This results in the following three-parameter family of invariant connections 6.3. Summary of properties of homogeneous spacetimes. We now summarise the properties of the simply-connected homogeneous spacetimes in Tables 1 and 2.
There is precisely one non-reductive spacetime: LC (S16), which we identified with the future light cone in Minkowski spacetime one dimension higher. It is carrollian, but for D > 1 does not admit any invariant connections. We determined above the invariant connections when D = 1.
The remaining spacetimes in Tables 1 and 2 are reductive and we proceed to list them according to the type of reductive structure they possess. 6.3.1. Flat symmetric spacetimes. These are symmetric spacetimes where the canonical connection is flat (Ω = 0). This means that the homogeneous space is a principal homogeneous space for the translations. If simply-connected, then it is an affine space modelled on the vector space of translations.
The notation is such that we interpret Ω as a two-form with values in endomorphisms of the tangent space, which for a homogeneous space localises to a linear map Ω : Λ 2 m → gl(m) and we write it in terms of the image in gl(m) of the linear isotropy representation λ : h → gl(m).
et cetera. Notice that the curvature 2-form of the canonical connection of de Sitter spacetime does not see the radius of curvature. This is because the canonical connection (and hence its curvature) is an invariant of the reductive homogeneous space, whereas the radius of curvature is an additional structure: namely, an invariant lorentzian metric. The same happens with anti de Sitter spacetime, the round sphere and hyperbolic space.
(S3) anti de Sitter spacetime, with curvature Ω(H, P a ) = −λ Ba and Ω(P a , P b ) = −λ Jab , which is formally like for de Sitter spacetime except for an overall sign. (S5) round sphere, with curvature formally identical to that in equation (196), except that of course, the linear isotropy representation λ Ba differs. (S6) hyperbolic space, with curvature formally identical to that in equation (198), except that again the linear isotropy representation λ Ba differs. (S8) galilean de Sitter spacetime, with curvature (S10) galilean anti de Sitter spacetime, with curvature which is again a sign off the one for galilean de Sitter spacetime. (S14) carrollian de Sitter spacetime, whose curvature is formally identical to that in equation (196), with the different action of λ Ba . (S15) carrollian anti de Sitter spacetime, whose curvature is formally identical to that in equation (198), with the different action of λ Ba . (A23) aristotelian: R × S D for ε = −1 and R × H D for ε = 1, with curvature given by 6.3.3. Reductive torsional spacetimes. The canonical connection of these reductive spacetimes has torsion and hence they are not symmetric spaces. If the connection is flat, then the spacetime is actually a principally homogeneous space for a Lie group whose Lie algebra is isomorphic to m, which is a Lie algebra in the flat case.
so that it is never flat, but it is torsion-free if and only if χ = 0, corresponding to the symmetric space (S10). (S12) This is a two-parameter family of three-dimensional galilean spacetimes S12 γ,χ with γ ∈ [−1, 1) and χ > 0. The torsion and curvature are given by Θ(H, P a ) = −(1 + γ)P a + χǫ ab P b and and zero curvature. It is a principally homogeneous space for the simply-connected Heisenberg Lie group with Lie algebra [P a , P b ] = ǫ ab H. Table 16 we summarise the basic properties of the homogeneous kinematical spacetimes in Table 1 and aristotelian spacetimes in Table 2. The first column is simply our label in this paper, the second column specifies the value of D, where the dimension of the spacetime is D + 1. For the columns labeled "R", "S" and "A" we indicate with a when a spacetime is reductive, symmetric and/or affine, respectively. The columns labelled "L", "E", "G" and "C" indicate the kind of invariant structures the spacetime possesses: lorentzian, riemannian ("euclidean"), galilean and carrollian, respectively. Again a indicates that the spacetime possesses that structure. The columns "P", "T" and "PT" indicate whether the spacetime is invariant under parity, time reversal or their combination, respectively, with signalling when they are. The columns "Ω" and "Θ" tell us, respectively, about the curvature and torsion of the canonical invariant connection for the reductive spacetimes (that is, all but S16). A = 0 indicates the presence of curvature and/or torsion. Otherwise the connection is flat and/or torsion-free, respectively. The final column contains any relevant comments, including, when known, the name of the spacetime. The table is divided into six sections. The first four correspond to lorentzian, euclidean, galilean and carrollian spacetimes. The fifth section contains two-dimensional spacetimes with no invariant structure of these kinds. The sixth and last section contains the aristotelian spacetimes.

T
. Properties of simply-connected homogeneous spacetimes This table describes if a spacetime of dimension D + 1 is reductive (R), symmetric (S) or affine (A). A spacetime might exhibit a lorentzian (L), riemannian (E), galilean (G) or carrollian (C) structure, and be invariant under parity (P), time reversal (T) or their combination (PT). Furthermore the canonical connection might be have curvature (Ω) and/or torsion (Θ). For the precise definitions of these properties see Sections 6.1 and 6.2.
In particular, we see how all the spacetimes in Figure 2 are indeed symmetric: with S1 (M), S2 (dS) and S3 (AdS) lorentzian; S4 (E), S5 (S) and S6 (H) riemannian; S7 (G), S8 (dSG) and S10 (AdSG) galilean; and S13 (C), S14 (dSC) and S15 (AdSC) carrollian. It is clear from a dimension count that there can be no other lorentzian or riemannian kinematical spacetimes than the ones in Figure 2. The dimension of the kinematical group associated to a (D + 1)-dimensional homogeneous spacetime is given by (D + 1)(D + 2)/2, which is also the maximal dimension of the isometry group of a (D + 1)-dimensional pseudo-riemannian manifold, so the homogeneous lorentzian and riemannian homogeneous spaces are necessarily maximally symmetric. The perhaps remarkable fact is that for D 2, every homogeneous (kinematical) spacetime which is not lorentzian or riemannian is either galilean, carrollian or aristotelian. The new spacetimes in Figures 3 and 4, not already present in Figure 2, are therefore necessarily galilean, carrollian or aristotelian. We see that the class of galilean spacetimes is particularly rich: admitting two one-dimensional continua of such spacetimes: one, denoted AdSG χ (S11 χ ), which extends the galilean anti de Sitter spacetime AdSG (S10) and a second, denoted dSG γ (S9 γ ), which extends the galilean de Sitter spacetime dSG (S8). The carrollian spacetimes are all realisable as null hypersurfaces in either Minkowski or (anti) de Sitter spacetimes one dimension higher. In particular, the carrollian spacetime LC (S16) can be realised as the future light cone in Minkowski space one dimension higher. There are also several aristotelian spacetimes. The situation in D = 2 is even richer, with a two-dimensional continuum interpolating between the one-dimensional continua present in all D 1. Finally, in D = 1 there are exotic (i.e., without any discernible invariant structures) homogeneous spacetimes, including two one-dimensional continua.

C
In this paper we have classified isomorphism classes of simply-connected homogeneous spacetimes of kinematical and aristotelian Lie groups with D-dimensional space isotropy for all D 0. We have done this by classifying the corresponding infinitesimal algebraic objects, namely (geometrically realisable, effective) Lie pairs. A number of observations follow from the classification.
It follows from our classification (see, e.g., Table 15) that inequivalent spacetimes may have the same transitive kinematical Lie algebra, which might have interesting consequences (e.g., as in AdS/CFT). Conversely, non-isomorphic kinematical Lie algebras may have isomorphic spacetimes. For example, the para-galilean and static kinematical Lie algebras lead to the same homogeneous spacetime: the static aristotelian spacetime (A21).
The classification yields novel (at least to us) spacetimes: particularly, the family of torsional galilean (anti) de Sitter spacetimes (S9 γ and S11 χ ) and the torsional static aristotelian spacetime (A22), as well as the new families of two-and three-dimensional spacetimes: S12 γ,χ , S19 χ and S20 χ . These novel spacetimes can be distinguished from the known ones (see Figure 2) in one of several (equivalent) ways: • they are not symmetric homogeneous spaces; • they are not invariant under both parity and time-reversal (at least for D 2); and • they do not arise as limits of the maximally symmetric riemannian and lorentzian spaces. In particular, this last characterisation allows us to see their existence as a purely non-relativistic prediction. Conversely, one can ask whether there is a relativistic set-up that leads to these spacetimes via limits. None of these characterisations is compelling reason to ignore the novel spacetimes.
We observed that not all aristotelian spacetimes arise from kinematical Lie algebras and this motivated us to present the separate classification of aristotelian Lie algebras in Appendix A.
We also explored some of the geometrical properties of the spacetimes, particularly those which can be easily read from the infinitesimal description: namely the existence of invariant (pseudo-)riemannian, galilean, carrollian and aristotelian structures. In the reductive cases, which are the vast majority, we have paid particular attention to the torsion and curvature of the canonical connection, as this provides an identifiable invariant for the spacetime in question.
The main results are contained in Tables 1, 2 and 16 and their interrelations are conveniently summarised in Figures 3, 4 and 5. In this paper we have restricted ourselves to the classification of the simplyconnected homogeneous spacetimes, without paying very close attention to each of the geometries. This will be remedied in a follow-up paper [54], where we will revisit the classification and investigate the local geometry of the spacetimes.

A
This work started during a visit of SP to Edinburgh and continued during a visit of JMF to Brussels. It is our pleasure to thank the corresponding institutions for hospitality and support. Additional work was done during our participation at the MITP Topical Workshop "Applied Newton-Cartan Geometry" (APPNC2018), held at the Mainz Institute for Theoretical Physics, where we presented a preliminary version of our results. We are grateful to the MITP for their support and hospitality and for providing such a stimulating research atmosphere. We are particularly grateful to Eric Bergshoeff and Niels Obers for the invitation to participate. JMF would like to thank Andrea Santi for the invitation to visit Bologna and the opportunity to talk about this work. SP is grateful to Eric Bergshoeff, Daniel Grumiller, Joaquim Gomis, Marc Henneaux, Axel Kleinschmidt, Victor Lekeu, Javier Matulich, Arash Ranjbar, Jan Rosseel, Jakob Salzer, Friedrich Schöller, and David Tong for useful discussions. The research of JMF is partially supported by the grant ST/L000458/1 "Particle Theory at the Higgs Centre" from the UK Science and Technology Facilities Council. The research of SP is partially supported by the ERC Advanced Grant "High-Spin-Grav" and by FNRS-Belgium (convention FRFC PDR T.1025.14 and convention IISN 4.4503.15).
JMF would like to dedicate this paper to the memory of his friend Andrew Ranicki, the late Professor of Algebraic Surgery at the University of Edinburgh. Andrew was a great practitioner in the art of turning topological/geometrical problems into algebra, which is the philosophy we have tried to follow in arriving at the results described in this paper. A

A. C L
In this appendix we present the classification of aristotelian Lie algebras. In complete analogy with the definition of a kinematical Lie algebra, we have the following.

Definition 4.
A real Lie algebra a is said to be aristotelian (with D-dimensional space isotropy) if it satisfies two properties: (1) a contains a Lie subalgebra r ∼ = so(D), and (2) a decomposes as a = r ⊕ V ⊕ S under r, where now we only have one copy of the vector representation of so(D).
We can choose a basis (J ab , P a , H) for a, relative to which the Lie brackets include the following: and any other Lie brackets are subject only to the Jacobi identity, which implies, in particular, equivariance under r. Every aristotelian Lie algebra a gives rise to a unique aristotelian homogeneous spacetime with effective Lie pair (a, r). Therefore classifying aristotelian Lie algebras up to isomorphism also classifies the aristotelian spacetimes. Aristotelian Lie algebras all share the Lie brackets (207) and are thus distinguished by the [H, P a ] and [P a , P b ] brackets, which are only constrained by the Jacobi identity.
Many aristotelian Lie algebras arise as quotients of kinematical Lie algebras by a vectorial ideal. Indeed, we have seen in Section 4 that kinematical Lie algebras giving rise to non-effective Lie pairs always reduce to an aristotelian Lie algebra after quotienting by the ideal generated by the boosts. However, as we shall see below, not all aristotelian Lie algebras arise in this way.
We now proceed to classify aristotelian Lie algebras, starting with those in D > 3 and making our way down in dimension.
Let D > 3. Equivariance under r ∼ = so(D) forces for some α, β ∈ R. The Jacobi identity says that αβ = 0, giving rise to four isomorphism classes of aristotelian Lie algebras: ( for some α, β, γ ∈ R. The Jacobi identity now says that αβ = αγ = 0. But in D = 3 we can change basis to P a → P ′ a = P a + λǫ abc J bc for some λ ∈ R, apart from an overall scale. Choosing λ = 1 4 γ, we can assume that γ = 0 without loss of generality. In terms of the new basis, we are back to the case D > 3 with the same results: (A1) the static aristotelian Lie algebra (α = 0, β = 1 4 γ 2 ); (A2) [H, P a ] = P a and [P a , P b ] = 0 (α = 0, β = γ = 0); and (A3) [H, P a ] = 0 and [P a , P b ] = εJ ab , with ε = ±1 (α = 0, β = 1 4 γ 2 ). It is only for D = 3 that the aristotelian Lie algebra A3 arises by reduction from a kinematical Lie algebra. Indeed, since J ab can be written as a Lie bracket of translations and since boosts transform nontrivially under rotations, boosts cannot commute with translations in the kinematical Lie algebra. Since the boosts define an ideal, there would have to be a nonzero bracket [B, P] = B, whose existence requires a nontrivial vector product invariant under rotations, and this only exists in D = 3.
Let D = 2. Now r ∼ = so(2) is abelian, so equivariance under r implies where we have defined J via J ab = −ǫ ab J. But now we can change basis to H → H ′ = H − λJ for some λ ∈ R, apart from an overall scale. Choosing λ = δ, we can assume with no loss of generality that δ = 0.
In general, the Jacobi identity says that αβ = αγ = 0. There is now an additional aristotelian Lie algebra: ( In summary, the isomorphism classes of aristotelian Lie algebras with D-dimensional space isotropy are recorded in Table 17. For psychological reasons, in this appendix g will denote a finite-dimensional real Lie algebra and not the galilean Lie algebra as in the bulk of the paper. In this appendix we will prove that the classification of simply-connected homogeneous spaces (up to isomorphism) is equivalent to the classification of isomorphism classes of geometrically realisable, effective Lie pairs. This statement is the analogue in the homogeneous space setting of the well-known fact that associated to every finite-dimensional real Lie algebra g there exists a unique (up to isomorphism) simply-connected Lie group whose Lie algebra is isomorphic to g. Indeed, a Lie group is a principally homogeneous space over itself, so that the group/algebra statement is a special case of the general statement about homogeneous spaces. The crucial difference is that given a Lie pair the corresponding simply-connected homogeneous space need not exist nor be unique, unless we impose additional conditions on the Lie pair: effective (for uniqueness) and geometrically realisable (for existence). It is surprisingly difficult to find this more general statement in the literature, but it is certainly standard and one can piece it together from results in [57].
B.1. Transitive actions of Lie groups. Let M be a connected smooth manifold. By an action of a Lie group G on M, we mean a smooth map α : G × M → M satisfying axioms which are easier to state after we introduce the following notation. If g ∈ G and m ∈ M, we will write α(g, m) as g · m. Then α is an action if, for e ∈ G the identity element, e · m = m for all m ∈ M, and if g 1 · (g 2 · m) = (g 1 g 2 ) · m for all g 1 , g 2 ∈ G and m ∈ M.
Let G act on M and let g be the Lie algebra of G. The action induces a Lie algebra (anti)homomorphism ξ : g → X (M) from g to the Lie algebra of vector fields on M sending every X ∈ g to the fundamental vector field ξ X on M. If m ∈ M, then ξ X (m) is the velocity of the curve exp(tX) · m at t = 0.
Let G act on M and let N = {g ∈ G | g · m = m, ∀m ∈ M} denote the kernel of the action. The action of G on M is said to be effective if N = {e} and it is said to be locally effective if N is a discrete group. This is equivalent to the map ξ : g → X (M) being injective. In any case, N is a normal subgroup of G and the G action on M induces an effective action of G/N . Nevertheless, although assuming that the action is effective seems to represent no loss of generality, we will allow for locally effective actions.
Let G act on M. If given any two points m 1 , m 2 ∈ M, there is some g ∈ G such that m 2 = g · m 1 , we say that the action is transitive. If the action is both transitive and locally effective, then M is said to be a homogeneous space of G. We will assume that G is connected. This represents no loss of generality because if G acts transitively on a connected manifold M, so does the connected component of the identity of G.

Definition 5.
Let M and M ′ be homogeneous spaces of G and G ′ , respectively. We say that M and M ′ are isomorphic if there is a diffeomorphism f : M → M ′ and a Lie group isomorphism Φ : G → G ′ such that f(g · m) = Φ(g) · f(m) for all m ∈ M and g ∈ G.
Let G be a connected Lie group acting transitively on M and let m ∈ M. The set G m = {g ∈ G | g · m = m} is a closed Lie subgroup of G called the stabiliser of m. It need not be a connected subgroup. Pick an "origin" o ∈ M and let H = G o . Then M is diffeomorphic to the space G/H of right H-cosets. The diffeomorphism ι : M → G/H is such that ι(o) = eH and if m = g · o then ι(m) = gH, which is welldefined because H is the stabiliser of o. It follows that ι is G-equivariant: ι(g · m) = gι(m) for all g ∈ G and m ∈ M. In the language of the previous definition, the homogeneous spaces M and G/H of G are isomorphic. We say that G/H is a coset model for M.
If we change the origin, we get a different (but isomorphic) coset model. Indeed, let o ′ ∈ M have stabiliser H ′ . Then if o ′ = g · o, H ′ = gHg −1 and in the language of the above definition, Φ : G → G is the inner automorphism corresponding to conjugation by g and f : G/H → G/H ′ is such that f(kH) = gkg −1 H ′ . B.2. Lie pairs. Every coset space G/H has a corresponding Lie pair (g, h), where g is the Lie algebra of G and h is the Lie algebra of H, and so there is a way to assign a Lie pair to a homogeneous space M of G and a choice of origin. A different choice of origin results in a different Lie pair, but how are they related? Let o, o ′ ∈ M be two choices of origin with stabilisers H and H ′ = gHg −1 , where g · o = o ′ . Then the resulting Lie pairs are (g, h) and (g, h ′ ), where h ′ is the Lie algebra of H ′ . Let Ad g : g → g be the inner automorphism of g induced by conjugation by g in G. Then h ′ = Ad g h, so the Lie pairs are related by an inner automorphism. This motivates the following definition. Definition 6. Two Lie pairs (g 1 , h 1 ) and (g 2 , h 2 ) are said to be isomorphic if there is a Lie algebra isomorphism ϕ : g 1 → g 2 with ϕ(h 1 ) = h 2 .
This notion of isomorphism is stronger than what is needed in order to classify homogeneous spaces, but it is easier to implement algebraically. In this Appendix we will show that it corresponds to classifying homogeneous spaces up to coverings. Equivalently, we will see that to each isomorphism class of (certain) Lie pairs there corresponds a unique simply connected homogeneous space (up to isomorphism). Proof. It is enough to show that any Lie pair associated to M is isomorphic to at least one Lie pair associated to M ′ , since as we have seen above all Lie pairs associated to a homogeneous space are isomorphic (by an inner automorphism). Since M and M ′ are isomorphic homogeneous spaces, we have an isomorphism of Lie groups Φ : G → G ′ and a diffeomorphism f : M → M ′ obeying the equivariance property f(g · m) = Φ(g) · f(m) for all m ∈ M and g ∈ G. We will show that the Lie algebra isomorphism ϕ : g → g ′ induced by Φ is the desired isomorphism between the Lie pairs.
So choose an origin o ∈ M with stabiliser H ⊂ G, leading to the Lie pair (g, h) and let o ′ = f(o) ∈ M ′ have stabiliser H ′ , leading to the Lie pair (g ′ , h ′ ). It follows from the equivariance property that if g ∈ H, then Φ(g) ∈ H ′ : But if g ′ ∈ H ′ , then the unique g ∈ G such that g ′ = Φ(g) lies in H: but since f is one-to-one, g · o = o. Therefore Φ(H) = H ′ and the Lie algebra isomorphism ϕ : g → g ′ induced by Φ sends h isomorphically to h ′ .
It turns out that not all Lie pairs come from homogeneous spaces.

Definition 7.
A Lie pair (g, h) is said to be effective if h does not contain a nonzero ideal of g.
It follows from this definition that if two Lie pairs are isomorphic and one is effective, so is the other. The following lemma justifies the definition. Proof. We start by proving that if (g, h) is not effective, then G does not act locally effectively. If (g, h) is not effective, then there is a nonzero ideal n of g contained in h. Let N be the unique connected subgroup of G generated by n. Since n is an ideal, N is a normal subgroup. We claim that N stabilises every point on M. Since N ⊂ H, it stabilises any point o ∈ M with stabiliser H. Let m ∈ M be any other point and let g ∈ G be such that g · o = m. Then the stabiliser of m is gHg −1 , which contains gN g −1 = N .
Conversely, suppose that G does not act locally effectively, so that the Lie algebra (anti)homomorphism ξ : g → X (M) has nonzero kernel n, which is an ideal of g. Let o ∈ M have stabiliser H. Then h consists of those X ∈ g for which ξ X (o) = 0. But if X ∈ n, ξ X (m) = 0 for all m ∈ M, so in particular, ξ X (o) = 0 and hence X ∈ h. This means that there is an ideal of g contained in h and hence (g, h) is not effective.
In summary, to a homogeneous space of G and a choice of origin, we may assign an effective Lie pair and up to isomorphism the choice of origin is immaterial. We now wish to examine the inverse problem: namely, does every effective Lie pair arise as the Lie pair of a homogeneous space and a choice of origin? B.3. Geometric realisations. It turns out that not every effective Lie pair arises from a homogeneous space. For example, consider g = su(3), the simple Lie algebra of 3 × 3 traceless skewhermitian complex matrices, and let h be the one-dimensional subalgebra spanned by the matrix for some irrational real number α. We claim that there is no Lie group G with Lie algebra isomorphic to g for which the subgroup corresponding to h is closed. Indeed, there are (up to isomorphism) precisely two connected Lie groups with Lie algebra isomorphic to su(3): SU(3) itself and the adjoint group Ad SU(3) ∼ = SU(3)/Z 3 . The one-parameter subgroup of either of these groups generated by X α is not closed. It is enough to see this for the simply-connected group SU(3), since if the subgroup of SU(3)/Z 3 generated by X α were closed, then so would be its pre-image under the covering homomorphism π : SU(3) → SU(3)/Z 3 , which is the subgroup generated by X α in SU(3). So let H α denote the subgroup generated by X α in SU(3). It is clearly contained in the maximal torus of diagonal matrices in SU (3), which is a closed subgroup. So it defines a one-parameter subgroup of the torus with an irrational slope and it's easy to see that the closure of this subgroup is the whole torus. 6 This suggests the following definition.

Definition 8. A Lie pair (g, h)
is geometrically realisable if there is a connected Lie group G with Lie algebra Lie(G) isomorphic to g and a closed Lie subgroup H with Lie algebra Lie(H) isomorphic to h (by restricting the isomorphism Lie(G) ∼ = g). The coset space G/H is then a geometric realisation of (g, h).
It is clear from this definition that if two Lie pairs are isomorphic and one pair admits a geometric realisation then so does the other pair.
B.4. Simply-connected homogeneous spaces. Finally, we are ready to prove the main result of this Appendix. Namely, we show that every geometrically realisable, effective Lie pair admits a unique (up to isomorphism) simply-connected geometric realisation.
Let M := G/H be a geometric realisation of the Lie pair (g, h), where G is connected. Let π : G → G be the universal covering group of G. Since π is surjective, G also acts transitively on M via g · m = π(g) · m, for g ∈ G and m ∈ M. If o ∈ M denotes the identity coset, then its stabiliser in G is H = π −1 H = 6 Since the counterexample here is the irrational slope flow on a torus, one might have wondered why we didn't simply consider the abelian Lie algebra g = u(1) ⊕ u(1) and the subalgebra h spanned by (i, αi), with α irrational. Indeed, the subgroup of U(1) × U(1) generated by h is not closed, but the subgroup generated by h in the universal covering group R 2 is closed, so that the Lie pair (g, h) is geometrically realisable. Proof. It is clear that M is a homogeneous space of G and hence it is the base of a principal H 1 -bundle whose homotopy long exact sequence ends with π 1 ( G) where all maps are group homomorphisms. Since G is connected and simply connected, π 0 ( G) = π 1 ( G) = 0 and since H 1 is connected, π 0 ( H 1 ) = 0, resulting in π 1 ( M) = 0. The map p : M = G/ H 1 → M = G/ H, defined by p(g H 1 ) = g H, is manifestly G-equivariant and moreover it is a covering since it is the projection of a principal bundle with base M and discrete fibre π 0 ( H). The Lie pair associated to M is (Lie( G), Lie( H 1 )), but since π : G → G is a covering homomorphism of Lie groups, the Lie map π * : Lie( G) → g is a Lie algebra isomorphism which restricts to an isomorphism Lie( H 1 ) = Lie( H) → h.
We have shown that every geometrically realisable Lie pair (g, h) has a simply-connected geometric realisation M as above. It turns out that this is unique up to isomorphism.

Lemma 4. M is the unique (up to isomorphism) simply-connected geometric realisation of (g, h).
Proof. Suppose that M ′ is another simply-connected geometric realisation of (g, h). This means that there is a connected Lie group G ′ and a closed subgroup H ′ such that M ′ = G ′ /H ′ and such that the Lie pairs (g ′ , h ′ ) and (g, h) are isomorphic, where g ′ and h ′ are the Lie algebras of G ′ and H ′ , respectively. This also means that (g ′ , h ′ ) is isomorphic to the Lie pair (Lie( G), Lie( H 1 )) of M. Let ϕ : (g ′ , h ′ ) → (Lie( G), Lie( H 1 )) denote this isomorphism. Passing to the universal covering group (if necessary), we may assume without loss of generality that G ′ is simply-connected and, since M ′ = G ′ /H ′ is simply connected, that H ′ is connected. The isomorphism ϕ : g ′ → Lie( G) lifts to a unique Lie group isomorphism Φ : G ′ → G which restricts to an isomorphism H ′ → H 1 and hence induces a unique isomorphism of homogeneous spaces φ : M ′ = G ′ /H ′ → M = G/ H 1 , sending gH ′ → Φ(g) H 1 .
As a corollary of the above lemma, we see that two homogeneous spaces with isomorphic Lie pairs have isomorphic universal covers. Indeed, let M = G/H and M ′ = G ′ /H ′ have isomorphic Lie pairs (g, h) and (g ′ , h ′ ), respectively. Then the universal cover M of M has a Lie pair which is isomorphic to (g, h) and the universal cover M ′ of M ′ has a Lie pair which is isomorphic to (g ′ , h ′ ) and hence also to (g, h). Therefore M and M ′ are simply-connected geometric realisations of (g, h) and by Lemma 4 they are isomorphic as homogeneous spaces.
In summary, we have proved the following.
Theorem 5. Isomorphism classes of geometrically realisable, effective Lie pairs are in one-to-one correspondence with isomorphism classes of simply-connected homogeneous spaces.
We may paraphrase this result as follows. Introduce an equivalence relation between homogeneous spaces by declaring two homogeneous spaces to be equivalent if their universal covers are isomorphic as homogeneous spaces. The isomorphism classes of geometrically realisable, effective Lie pairs are in one-to-one correspondence with equivalence classes of homogeneous spaces.
If we wish to classify homogeneous spaces up to isomorphism and not just up to covering, we would start from the classification of simply-connected homogeneous spaces and then classify their homogeneous quotients. That, however, is beyond the scope of this paper.