Lorentzian and Newtonian spacetimes and their quantum (noncommutative) deformations

Transcription

1 Lorentzian and Newtonian spacetimes and their quantum (noncommutative) deformations FRANCISCO J. HERRANZ University of Burgos, Spain XX-th Edition of Conference on Geometry,Integrability and Quantization June 2-7, 2018, Varna - Bulgaria

2 Scheme 1. The nine two-dimensional Cayley Klein geometries 1.1. Spacetimes as Cayley Klein spaces 1.2. Matrix realization and vector model 2. Metric structure and coordinate systems of the 2D CK spaces of points 2.1. Vector fields 2.2. Laplace/wave-type equations 3. Conformal symmetries 4. N-dimensional CK spaces 4.1. Orthogonal CK algebras 4.2. Symmetrical homogeneous CK spaces 5. Conclusions 2

3 1. The nine two-dimensional Cayley Klein geometries The motion groups of the nine 2D Cayley Klein (CK) geometries can be described in a unified setting by means of two real coefficients κ 1, κ 2 and are collectively denoted SO κ1,κ 2 (3). The generators {P 1, P 2, J 12 } of the corresponding Lie algebras so κ1,κ 2 (3) have Lie commutators: [J 12, P 1 ] = P 2 [J 12, P 2 ] = κ 2 P 1 [P 1, P 2 ] = κ 1 J 12. There is a single Lie algebra Casimir: C = P κ 2 P κ 1 J The CK algebras so κ1,κ 2 (3) can be endowed with a Z 2 Z 2 group of commuting automorphisms generated by: Π (1) : (P 1, P 2, J 12 ) ( P 1, P 2, J 12 ) Π (2) : (P 1, P 2, J 12 ) (P 1, P 2, J 12 ). The two remaining involutions are the composition Π (02) = Π (1) Π (2) and the identity. 3

4 Each involution Π determines a subalgebra of so κ1,κ 2 (3) whose elements are invariant under Π leading to the following Cartan decompositions: so κ1,κ 2 (3) = h (1) p (1), h (1) = J 12 = so κ2 (2), p (1) = P 1, P 2. so κ1,κ 2 (3) = h (2) p (2), h (2) = P 1 = so κ1 (2), p (2) = P 2, J 12. so κ1,κ 2 (3) = h (02) p (02), h (02) = P 2 = so κ1 κ 2 (2), p (02) = P 1, J 12. The elements defining a 2D CK geometry are The plane as the set of points corresponds to the 2D symmetrical homogeneous space S 2 [κ 1 ],κ 2 SO κ1,κ 2 (3)/H (1) SO κ1,κ 2 (3)/SO κ2 (2) H (1) = J 12 SO κ2 (2). The generator J 12 leaves a point O (the origin) invariant, thus J 12 acts as the rotation around O. The involution Π (1) is the reflection around the origin. In this space P 1 and P 2 generate translations which move the origin point in two basic directions. 4

5 The set of lines is identified as the 2D symmetrical homogeneous space S 2 κ 1,[κ 2 ] SO κ 1,κ 2 (3)/H (2) SO κ1,κ 2 (3)/SO κ1 (2) H (2) = P 1 SO κ1 (2). In this space, the generator P 1 leaves invariant the origin line l 1, which is moved in two basic directions by J 12 and P 2. Therefore, within S 2 κ 1,[κ 2 ], P 1 should be interpreted as the generator of rotations around l 1. There is a second set of lines corresponding to the 2D symmetrical homogeneous space SO κ1,κ 2 (3)/H (02) SO κ1,κ 2 (3)/SO κ1 κ 2 (2) H (02) = P 2 SO κ1 κ 2 (2). In this case, P 2 leaves invariant an origin line l 2 in this space while J 12 and P 1 do move l 2. By a two-dimensional CK geometry we will understand the set of three symmetrical homogeneous spaces of points, lines of first-kind and lines of second-kind. The group SO κ1,κ 2 (3) acts transitively on each of these spaces. 5

6 The coefficients κ 1, κ 2 play a twofold role. The space S 2 [κ 1 ],κ 2 has a quadratic metric coming from the Casimir, whose signature corresponds to the matrix diag(1, κ 2 ). This metric is riemannian (definite positive) for κ 2 > 0, lorentzian (indefinite) for κ 2 < 0 and degenerate for κ 2 = 0. This space has a canonical conexion which is compatible with the metric, and has constant curvature equal to κ 1. In the notations S 2 [κ 1 ],κ 2, S 2 κ 1,[κ 2 ] for the spaces, the κ i in square brackets is the constant curvature, and the remaining constant determines the signature. Alternatively, the coefficients κ 1, κ 2 determine the kind of measures of separation amongst points and lines in the Klein sense: The pencil of points on a first-kind line is elliptical/parabolical/hyperbolical according to whether κ 1 is greater than/equal to/lesser than zero. Likewise for the pencil of points on a second-kind line depending on the product κ 1 κ 2. Likewise for the pencil of lines through a point according to κ 2. 6

7 For κ 1 positive/zero/negative the isotropy subgroup H (2) is SO(2)/R/SO(1, 1), and the same happens for H (1) (resp. H (02) ) according to the value of κ 2 (resp. κ 1 κ 2 κ 02 ). Whenever the coefficient κ 1 (resp. κ 2 ) is different from zero, a suitable choice of length unit (resp. angle unit) allows us to reduce it to either +1 or 1. Hence we obtain nine 2D real CK geometries. There exists an automorphism of the whole family, called ordinary duality D, which is given by: D : (P 1, P 2, J 12 ) ( J 12, P 2, P 1 ) D : (κ 1, κ 2 ) (κ 2, κ 1 ). The map D leaves the commutation rules invariant while it interchanges the space of points with the space of first-kind lines, S[κ 2 1 ],κ 2 Sκ 2 1,[κ 2 ], and the corresponding curvatures κ 1 κ 2, preserving the space of second-kind lines. The vanishment of a coefficient κ i corresponds to an Inönü Wigner contraction. The limit κ 1 0 is a local-contraction (around a point), while the limit κ 2 0 is an axial-contraction (around a line). 7

8 Measure of distance Measure Elliptic Parabolic Hyperbolic of angle κ 1 = 1 κ 1 = 0 κ 1 = 1 Elliptic Euclidean Hyperbolic SO(3) ISO(2) SO(2, 1) Elliptic [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 κ 2 = 1 [J 12, P 2 ] = P 1 [J 12, P 2 ] = P 1 [J 12, P 2 ] = P 1 [P 1, P 2 ] = J 12 [P 1, P 2 ] = 0 [P 1, P 2 ] = J 12 C = P2 2 + P1 2 + J12 2 C = P2 2 + P1 2 C = P2 2 + P1 2 J12 2 H (1) = SO(2) H (1) = SO(2) H (1) = SO(2) H (2) = SO(2) H (2) = R H (2) = SO(1, 1) H (02) = SO(2) H (02) = R H (02) = SO(1, 1) Co-Euclidean Galilean Co-Minkowskian Oscillating NH Expanding NH ISO(2) IISO(1) ISO(1, 1) Parabolic [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 κ 2 = 0 [J 12, P 2 ] = 0 [J 12, P 2 ] = 0 [J 12, P 2 ] = 0 [P 1, P 2 ] = J 12 [P 1, P 2 ] = 0 [P 1, P 2 ] = J 12 C = P2 2 + J12 2 C = P2 2 C = P2 2 J12 2 H (1) = R H (1) = R H (1) = R H (2) = SO(2) H (2) = R H (2) = SO(1, 1) H (02) = R H (02) = R H (02) = R Co-Hyperbolic Minkowskian Doubly Hyperbolic Anti-de Sitter De Sitter SO(2, 1) ISO(1, 1) SO(2, 1) Hyperbolic [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 [J 12, P 1 ] = P 2 κ 2 = 1 [J 12, P 2 ] = P 1 [J 12, P 2 ] = P 1 [J 12, P 2 ] = P 1 [P 1, P 2 ] = J 12 [P 1, P 2 ] = 0 [P 1, P 2 ] = J 12 C = P2 2 P1 2 + J12 2 C = P2 2 P1 2 C = P2 2 P1 2 J12 2 H (1) = SO(1, 1) H (1) = SO(1, 1) H (1) = SO(1, 1) H (2) = SO(2) H 8 (2) = R H (2) = SO(1, 1) H (02) = SO(1, 1) H (02) = R H (02) = SO(2)

9 1.1. Spacetimes as Cayley Klein spaces Let H, P and K be the generators of time translations, space translations and boosts, respectively, in the most simple (1 + 1)D homogeneous spacetime. Under the identification P 1 H P 2 P J 12 K the six CK groups with κ 2 0 (second and third rows of table ; NH means Newton Hooke) are the motion groups of (1 + 1)D spacetimes: S 2 [κ 1 ],κ 2 is a (1 + 1)D spacetime, and points in S 2 [κ 1 ],κ 2 are spacetime events; the spacetime curvature equals κ 1 and is related to the usual universe (time) radius τ by κ 1 = ±1/τ 2. The space of first-kind lines Sκ 2 1,[κ 2 ] corresponds to the space of time-like lines. The coefficient κ 2 is the curvature of the space of time-like lines, linked to the relativistic constant c as κ 2 = 1/c 2. Relativistic spacetimes occur for κ 2 < 0 (the signature of the metric is diag(1, 1/c 2 )) and their non-relativistic limits correspond to κ 2 = 0. The space of second-kind lines SO κ1,κ 2 (3)/H (02) is the 2D space of space-like lines. 9

10 1.2. Matrix realization and vector model The following 3D real matrix representation of the CK algebra so κ1,κ 2 (3): 0 κ κ 1 κ P 1 = P 2 = J 12 = 0 0 κ gives rise to a natural realization of the CK group SO κ1,κ 2 (3) as a group of linear transformations in an ambient linear space R 3 = (x 0, x 1, x 2 ) in which SO κ1,κ 2 (3) acts as the group of linear isometries of a bilinear form with matrix: Λ = diag (1, κ 1, κ 1 κ 2 ). Their exponential leads to a representation of the one-parametric subgroups H (2), H (02) and H (1) generated by P 1, P 2 and J 12 as: C κ1 (α) κ 1 S κ1 (α) exp(αp 1 ) = S κ1 (α) C κ1 (α) 0, exp(γj 12 ) = 0 C κ2 (γ) κ 2 S κ2 (γ) S κ2 (γ) C κ2 (γ) C κ1 κ 2 (β) 0 κ 1 κ 2 S κ1 κ 2 (β) exp(βp 2 ) = S κ1 κ 2 (β) 0 C κ1 κ 2 (β) 10

11 where the generalized cosine C κ (x) and sine S κ (x) functions are defined by S κ (x) := C κ (x) := l=0 l=0 ( κ) l x 2l (2l)! = ( κ) l x 2l+1 (2l + 1)! = cos κ x κ > 0 1 κ = 0 cosh κ x κ < 0 1 κ sin κ x κ > 0 x κ = 0 1 κ sinh κ x κ < 0 Two other useful functions are the versed sine V κ (x) and the tangent T κ (x):. V κ (x) := 1 κ (1 C κ(x)) T κ (x) := S κ(x) C κ (x). These curvature-dependent trigonometric functions coincide with the usual circular and hyperbolic ones for κ = 1 and κ = 1, respectively; the case κ = 0 provides the parabolic or Galilean functions: C 0 (x) = 1, S 0 (x) = x, V 0 (x) = x 2 /2. 11

12 The CK group SO κ1,κ 2 (3) can be seen as a group of linear transformations in an ambient space R 3 = (x 0, x 1, x 2 ), acting as the group of isometries of a bilinear form An element X SO κ1,κ 2 (3) satisfies Λ = diag(1, κ 1, κ 1 κ 2 ). X T Λ X = Λ. The action of SO κ1,κ 2 (3) on R 3 is linear but not transitive, since it conserves the quadratic form (x 0 ) 2 + κ 1 (x 1 ) 2 + κ 1 κ 2 (x 2 ) 2. The action becomes transitive if we restrict to the orbit in R 3 of the point O, which is contained in the sphere Σ: Σ (x 0 ) 2 + κ 1 (x 1 ) 2 + κ 1 κ 2 (x 2 ) 2 = 1. This orbit is identified with the CK space S 2 [κ 1 ],κ 2, and (x 0, x 1, x 2 ) are called Weierstrass coordinates; these allow us to obtain a differential realization of the generators as first-order vector fields in R 3 with i = / x i : P 1 = κ 1 x 1 0 x 0 1 P 2 = κ 1 κ 2 x 2 0 x 0 2 J 12 = κ 2 x 2 1 x

13 2. Metric structure and coordinate systems of the 2D CK spaces of points Hereafter we consider the homogeneous space of points S 2 [κ 1 ],κ 2 SO κ1,κ 2 (3)/H (1) SO κ1,κ 2 (3)/SO κ2 (2) H (1) = J 12 SO κ2 (2). Table 1: The nine two-dimensional CK spaces S 2 [κ 1 ],κ 2 = SO κ1,κ 2 (3)/SO κ2 (2). Elliptic: S 2 Euclidean: E 2 Hyperbolic: H 2 S[+],+ 2 = SO(3)/SO(2) S2 [0],+ = ISO(2)/SO(2) S2 [ ],+ = SO(2, 1)/SO(2) Oscillating NH: NH Galilean: G 1+1 Expanding NH: NH 1+1 (Co-Euclidean) (Co-Minkowskian) S[+],0 2 = ISO(2)/ISO(1) S2 [0],0 = IISO(1)/ISO(1) S2 [ ],0 = ISO(1, 1)/ISO(1) Anti-de Sitter: AdS 1+1 Minkowskian: M 1+1 De Sitter: ds 1+1 (Co-Hyperbolic) (Doubly Hyperbolic) S[+], 2 = SO(2, 1)/SO(1, 1) S2 [0], = ISO(1, 1)/SO(1, 1) S2 [ ], = SO(2, 1)/SO(1, 1) 13

14 If both coefficients κ i are different from zero, SO κ1,κ 2 (3) is a simple Lie group, and the space S[κ 2 1 ],κ 2 is endowed with a non-degenerate metric g 0 coming from the non-singular Killing Cartan form in the Lie algebra so κ1,κ 2 (3). At the origin, g 0 is given by: g 0 (P 1, P 1 ) = 2κ 1 g 0 (P 2, P 2 ) = 2κ 1 κ 2 g 0 (P 1, P 2 ) = 0. To cover the cases with κ 1 = 0 where g 0 vanishes identically, we take out a factor 2κ 1 out of g 0, and introduce the space main metric g 1 as 2g 1 := g 0 /κ 1. If κ 2 = 0, g 1 is a degenerate metric and the action of SO κ1,0(3) on S[κ 2 1 ],0 invariant foliation. The restriction of g 1 to each foliation leaf vanishes, but has an g 2 = 1 κ 2 g 1 has a non-vanishing and well defined restriction to each leaf; we call g 2 the subsidiary metric. 14

15 Proposition. The metric structure for a generic space S 2 [κ 1 ],κ 2 is characterized by: A connection which is invariant under SO κ1,κ 2 (3). A hierarchy of two metrics g 1 and g 2 = 1 κ 2 g 1 compatible with. The action of SO κ1,κ 2 (3) on S 2 [κ 1 ],κ 2 is by isometries of both metrics. The main metric g 1 is actually a metric in the true sense and has constant curvature κ 1 and signature diag(+, κ 2 ). If κ 2 0, g 2 is a true metric proportional to g 1. If κ 2 = 0, the subsidiary metric g 2 gives a true metric only in each leaf of the invariant foliation in S[κ 2 1 ],0, whose set of leaves can be parametrized by (x 0 ) 2 + κ 1 (x 1 ) 2 = 1 S[κ 1 1 ] ; g 2 has signature (+). In terms of Weierstrass coordinates in the linear ambient space R 3, the two metrics in S 2 [κ 1 ],κ 2 come from the flat ambient metric ds 2 = (dx 0 ) 2 + κ 1 (dx 1 ) 2 + κ 1 κ 2 (dx 2 ) 2 in the form (ds 2 ) 1 = 1 κ 1 ds 2 (ds 2 ) 2 = 1 κ 2 (ds 2 ) 1. We introduce three coordinate systems of geodesic type. 15

16 Let us consider the origin O (1, 0, 0), two (oriented) geodesics l 1, l 2 which are orthogonal through the origin, and a generic point Q with Weierstrass coordinates x = (x 0, x 1, x 2 ). We have: If x = exp(ap 1 ) exp(yp 2 )O, we call (a, y) the type I geodesic parallel coordinates of Q. If x = exp(bp 2 ) exp(xp 1 )O, we call (x, b) the type II geodesic parallel coordinates of Q. The geodesic polar coordinates of the point Q are (r, φ) if x = exp(φj 12 ) exp(rp 1 )O. l 2 Q 2 b O x r φ a l 2 Q y Q 1 l l 1 l 1 We compute the Weierstrass coordinates x of a generic point Q in the three geodesic 16

17 Q 1 x 0 y O 1 O l 2 Q r 1 l 1 l 2 l 1 φ l r Q 2 S 2 H 2 x r 2 x 2 O 2 x 0 l l 1 Q l 2 l x l 1 y 2 r Q 2 φ Q 1 O x 2 x 1 x 1 (a) (b) coordinate systems. By substitution in the expressions of the metrics in Weierstrass coordinates we find the main and subsidiary metrics in either geodesic coordinates. From them we may compute the conexion symbols Γ i jk. The area element ds in coordinates say u 1, u 2 is det g 1 /κ 2 du 1 du 2. 17

18 Table 2: Weierstrass coordinates, metric, canonical connection and area element for S 2 [κ 1 ],κ 2 given in the three geodesic coordinate systems. Parallel I (a, y) Parallel II (x, b) Polar (r, φ) x 0 = C κ1 (a)c κ1 κ 2 (y) x 0 = C κ1 (x)c κ1 κ 2 (b) x 0 = C κ1 (r) x 1 = S κ1 (a)c κ1 κ 2 (y) x 1 = S κ1 (x) x 1 = S κ1 (r)c κ2 (φ) x 2 = S κ1 κ 2 (y) x 2 = C κ1 (x)s κ1 κ 2 (b) x 2 = S κ1 (r)s κ2 (φ) (ds 2 ) 1 = C 2 κ 1 κ 2 (y)da 2 + κ 2 dy 2 (ds 2 ) 1 = dx 2 + κ 2 C 2 κ 1 (x)db 2 (ds 2 ) 1 = dr 2 + κ 2 S 2 κ 1 (r)dφ 2 (ds 2 ) 2 = dy 2 for a = a 0 (ds 2 ) 2 = C 2 κ 1 (x)db 2 for x = x 0 (ds 2 ) 2 = S 2 κ 1 (r)dφ 2 for r = r 0 Γ y aa = κ 1 S κ1 κ 2 (y)c κ1 κ 2 (y) Γ x bb = κ 1κ 2 S κ1 (x)c κ1 (x) Γ r φφ = κ 2S κ1 (r)c κ1 (r) Γ a ay = κ 1 κ 2 T κ1 κ 2 (y) Γ b bx = κ 1T κ1 (x) Γ φ φr = 1/T κ 1 (r) ds = C κ1 κ 2 (y) da dy ds = C κ1 (x) dx db ds = S κ1 (r) dr dφ 18

19 S 2 = S 2 [+],+ E 2 = S 2 [0],+ H 2 = S 2 [ ],+ x 0 = cos a cos y x 0 = 1 x 0 = cosh a cosh y x 1 = sin a cos y x 1 = a x 1 = sinh a cosh y x 2 = sin y x 2 = y x 2 = sinh y (ds 2 ) 1 = cos 2 y da 2 + dy 2 (ds 2 ) 1 = da 2 + dy 2 (ds 2 ) 1 = cosh 2 y da 2 + dy 2 Γ y aa = sin y cos y Γ y aa = 0 Γ y aa = sinh y cosh y Γ a ay = tan y Γ a ay = 0 Γ a ay = tanh y ds = cos y da dy ds = da dy ds = cosh y da dy NH = S 2 [+1/τ 2 ],0 G 1+1 = S 2 [0],0 NH 1+1 = S 2 [ 1/τ 2 ],0 x 0 = cos(t/τ) x 0 = 1 x 0 = cosh(t/τ) x 1 = τ sin(t/τ) x 1 = t x 1 = τ sinh(t/τ) x 2 = y x 2 = y x 2 = y (ds 2 ) 1 = dt 2 (ds 2 ) 1 = dt 2 (ds 2 ) 1 = dt 2 (ds 2 ) 2 = dy 2 t = t 0 (ds 2 ) 2 = dy 2 t = t 0 (ds 2 ) 2 = dy 2 t = t 0 Γ y tt = 1 τ 2 y Γ t ty = 0 Γ y tt = 0 Γ t ty = 0 Γ y tt = 1 τ 2 y Γ t ty = 0 ds = dt dy ds = dt dy ds = dt dy AdS 1+1 = S 2 [+1/τ 2 ], 1/c 2 M 1+1 = S 2 [0], 1/c 2 ds 1+1 = S 2 [ 1/τ 2 ], 1/c 2 x 0 = cos(t/τ) cosh(y/cτ) x 0 = 1 x 0 = cosh(t/τ) cos(y/cτ) x 1 = τ sin(t/τ) cosh(y/cτ) x 1 = t x 1 = τ sinh(t/τ) cos(y/cτ) x 2 = cτ sinh(y/cτ) x 2 = y x 2 = cτ sin(y/cτ) (ds 2 ) 1 = cosh 2 (y/cτ)dt 2 1 c 2 dy 2 (ds 2 ) 1 = dt 2 1 c 2 dy 2 (ds 2 ) 1 = cos 2 (y/cτ)dt 2 1 c 2 dy 2 Γ y tt = c τ sinh(y/cτ) cosh(y/cτ) Γy tt = 0 Γ y tt = c τ sin(y/cτ) cos(y/cτ) Γ t ty = 1 cτ tanh(y/cτ) Γt ty = 0 Γ t ty = 1 cτ tan(y/cτ) ds = cosh(y/cτ) dt dy ds = dt 19 dy ds = cos(y/cτ) dt dy

20 2.1. Vector fields The differential realization of the generators as first-order vector fields in R 3 P 1 = κ 1 x 1 0 x 0 1 P 2 = κ 1 κ 2 x 2 0 x 0 2 J 12 = κ 2 x 2 1 x 1 2. Geodesic parallel I coordinates (a, y) P 1 = a P 2 = κ 1 κ 2 S κ1 (a)t κ1 κ 2 (y) a C κ1 (a) y J 12 = κ 2 C κ1 (a)t κ1 κ 2 (y) a S κ1 (a) y Geodesic parallel II coordinates (x, b) P 1 = C κ1 κ 2 (b) x κ 1 T κ1 (x)s κ1 κ 2 (b) b P 2 = b J 12 = κ 2 S κ1 κ 2 (b) x T κ1 (x)c κ1 κ 2 (b) b Geodesic polar coordinates (r, φ) P 1 = C κ2 (φ) r + S κ 2 (φ) T κ1 (r) φ J 12 = φ P 2 = κ 2 S κ2 (φ) r C κ 2 (φ) T κ1 (r) φ 20

21 These vectors fields satisfy the Killing equations for the metrics g 1, g 2 of the space S 2 [κ 1 ],κ 2, that is, L X g i = µ X g i = 0, where L X g i is the Lie derivative of g i. Therefore we find an application to Lie systems Laplace/wave-type equations Let us consider a 2D space with coordinates (u 1, u 2 ), a differential operator E = E(u 1, u 2, 1, 2 ) acting on functions Φ(u 1, u 2 ) defined on the space ( i / u i ), and consider the differential equation: EΦ(u 1, u 2 ) = 0. An operator O is a symmetry if O transforms solutions into solutions: E O = Q E or [E, O] = Q E where Q is another operator and Q = Q O. We consider the differential equation obtained by taking as E the Casimir C of the CK algebra so κ1,κ 2 (3) in the space S 2 [κ 1 ],κ 2 : CΦ = 0. 21

22 Recall that C = P κ 2 P κ 1 J 2 12, [C, X] = 0, X {P 1, P 2, J 12 }. In the three geodesic coordinate systems, such an equation turns out to be ( ) κ2 Cκ 2 1 κ 2 (y) 2 a + y 2 κ 1 κ 2 T κ1 κ 2 (y) y Φ(a, y) = 0 ( κ 2 x 2 κ 1 κ 2 T κ1 (x) x + 1 ) Cκ 2 1 (x) 2 b Φ(x, b) = 0 ( κ 2 r 2 + κ 2 T κ1 (r) r + 1 ) Sκ 2 1 (r) 2 φ Φ(r, φ) = 0. 22

23 Table 3: The Laplace Beltrami operator C giving rise to differential Laplace and wave-type equations CΦ = 0 in geodesic parallel I Φ(a, y) Φ(t, y) and polar Φ(r, φ) Φ(r, χ) coordinates for the nine CK spaces (when κ 2 < 0 the angle is denoted as χ and is a rapidity in the kinematical interpretation). so(3) : S 2 = S[+],+ 2 iso(2) : E 2 = S[0],+ 2 so(2, 1) : H 2 = S[ ],+ 2 1 cos 2 y 2 a + y 2 tan y y a 2 + y 2 1 cosh 2 y 2 a + y 2 + tanh y y 1 sin 2 r 2 φ + r tan r 1 r r 2 2 φ + r r 1 r sinh 2 r 2 φ + r tanh r r iso(2) : NH = S 2 [+1/τ 2 ],0 iiso(1) : G 1+1 = S 2 [0],0 iso(1, 1) : NH 1+1 = S 2 [ 1/τ 2 ],0 y 2 y 2 y τ 2 sin 2 (r/τ) 2 χ r 2 2 χ τ 2 sinh 2 (r/τ) 2 χ so(2, 1) : AdS 1+1 = S 2 [+1/τ 2 ], 1/c 2 iso(1, 1) : M 1+1 = S 2 [0], 1/c 2 so(2, 1) : ds 1+1 = S 2 [ 1/τ 2 ], 1/c 2 1 c 2 cosh 2 (y/cτ) 2 t + y 2 + tanh(y/cτ) y cτ 1 τ 2 sin 2 (r/τ) 2 χ 1 1 c 2 2 r c 2 τ tan(r/τ) r 1 c 2 2 t + 2 y 1 r 2 2 χ 1 c 2 2 r 1 c 2 r r 1 c 2 cos 2 (y/cτ) 2 t + y 2 tan(y/cτ) y cτ 1 τ 2 sinh 2 (r/τ) 2 χ 1 1 c 2 2 r c 2 τ tanh(r/τ) r 23

24 The usual 2D Laplace equation in E 2 and the corresponding non-zero curvature Laplace Beltrami versions in the sphere and hyperbolic plane. An equation which does not involve time in the three non-relativistic spacetimes (indeed reducing to a 1D Laplace equation). This agrees with the known absence of a true Galilean invariant wave equation and is the main reason precluding a further development of non-relativistic electromagnetic theories, where only two separate electric and magnetic essentially static limits are allowed. The proper (1 + 1)D wave equation is associated to M 1+1 ; its curvature versions correspond to anti-de Sitter and de Sitter electromagnetisms in both AdS 1+1 and ds

25 3. Conformal symmetries It is possible to enlarge the CK algebra by considering: - A dilation generator D. - Two specific conformal transformations G 1, G 2. Procedure: by imposing cycle-preserving transformations. Geodesic parallel I coordinates (a, y) P 1 = a P 2 = κ 1 κ 2 S κ1 (a)t κ1 κ 2 (y) a C κ1 (a) y J 12 = κ 2 C κ1 (a)t κ1 κ 2 (y) a S κ1 (a) y D = S κ 1 (a) C κ1 κ 2 (y) a C κ1 (a)s κ1 κ 2 (y) y 1 G 1 = C κ1 κ 2 (y) (V κ 1 (a) κ 2 V κ1 κ 2 (y)) a + S κ1 (a)s κ1 κ 2 (y) y G 2 = κ 2 S κ1 (a)t κ1 κ 2 (y) a (V κ1 (a) κ 2 V κ1 κ 2 (y)) y 25

26 The commutation rules and Casimirs of conf κ1,κ 2 read [J 12, P 1 ] = P 2 [J 12, P 2 ] = κ 2 P 1 [P 1, P 2 ] = κ 1 J 12 [J 12, G 1 ] = G 2 [J 12, G 2 ] = κ 2 G 1 [G 1, G 2 ] = 0 [D, P i ] = P i + κ 1 G i [D, G i ] = G i [D, J 12 ] = 0 [P 1, G 1 ] = D [P 2, G 2 ] = κ 2 D [P 1, G 2 ] = J 12 [P 2, G 1 ] = J 12 C 1 = J κ 2 D 2 + κ 2 (P 1 G 1 + G 1 P 1 ) + (P 2 G 2 + G 2 P 2 ) + κ 1 (κ 2 G G 2 2) C 2 = J 12 D + (G 1 P 2 P 1 G 2 ) All spaces in the family S 2 [κ 1 ],κ 2 with the same κ 2 have isomorphic conformal algebras. These are: so(3, 1) ((2 + 1)D de Sitter algebra) as the conformal algebra of the three 2D Riemannian spaces with κ 2 > 0. iso(2, 1) ((2 + 1)D Poincaré) for the (1 + 1)D non-relativistic spacetimes with κ 2 = 0. so(2, 2) ((2 + 1)D anti-de Sitter) for the (1 + 1)D relativistic spacetimes with κ 2 < 0. 26

27 In relation with the usual approach to conformal groups by solving the conformal Killing equations we state the following: Proposition. All the conformal vectors fields X satisfy the conformal Killing equations for the metrics g 1, g 2 of the space S 2 [κ 1 ],κ 2, that is, L X g i = µ X g i, where L X g i is the Lie derivative of g i. In Weierstrass coordinates the conformal factors µ X are given by µ P1 = µ P2 = µ J12 = 0 µ D = 2x 0 µ G1 = 2x 1 µ G2 = 2κ 2 x 2. Hence the conformal vector fields of the CK spaces would allow one the construction of new Lie systems. 27

28 4. N-dimensional CK spaces 4.1. Orthogonal CK algebras Let us consider the real Lie algebra so(n + 1) whose 1 2 N(N + 1) generators J ab (a, b = 0, 1,..., N, a < b) satisfy the non-vanishing Lie brackets given by [J ab, J ac ] = J bc, [J ab, J bc ] = J ac, [J ac, J bc ] = J ab, a < b < c. A grading group Z N 2 of so(n + 1) is spanned by the following N commuting involutive automorphisms Θ (m) (m = 1,..., N): Θ (m) (J ab ) = { Jab, if either m a or b < m; J ab, if a < m b. A large family of contracted real Lie algebras can be obtained from so(n + 1); this depends on 2 N 1 real contraction parameters which includes from the simple pseudo-orthogonal algebras so(p, q) (the B l and D l Cartan series) (when all the contraction parameters are different from zero) up to the Abelian algebra at the opposite case (when all the parameters are equal to zero). 28

29 Properties associated with the simplicity of the algebra are lost at some point beyond the simple algebras in the contraction sequence. There exists a particular subset of contrated Lie algebras which are close to to the simple ones, whose members are called CK or quasi-simple orthogonal algebras: all the CK algebras share, in any dimension, the same rank defined as the number of (functionally independent) Casimir invariants. This orthogonal CK family, here denoted so κ (N + 1), depends on N real contraction coefficients κ = (κ 1,..., κ N ): [J ab, J ac ] = κ ab J bc, [J ab, J bc ] = J ac, [J ac, J bc ] = κ bc J ab, a < b < c, without sum over repeated indices and where the two-index coefficients κ ab are expressed in terms of the N basic ones through κ ab = κ a+1 κ a+2 κ b, a, b = 0, 1,..., N, a < b. Each non-zero real coefficient κ m can be reduced to either +1 or 1 by a rescaling of the Lie generators. There are 3 N CK algebras. 29

30 The case κ m = 0 can be interpreted as an Inönü Wigner contraction, with parameter ε m 0, and defined by the map Γ (m) (J ab ) = { Jab, if either m a or b < m; ε m J ab, if a < m b. Each involution Θ (m) provides a Cartan decomposition as a direct sum of antiinvariant and invariant subspaces, denoted p (m) and h (m), respectively: so κ (N + 1) = p (m) h (m), with the linear sum referring to the linear structure; Lie commutators fulfil: [h (m), h (m) ] h (m), [h (m), p (m) ] p (m), [p (m), p (m) ] h (m), and thus h (m) is always a Lie subalgebra with a direct sum structure: h (m) = so κ1,...,κ m 1 (m) so κm+1,...,κ N (N + 1 m), while the vector subspace p (m) is generally not a subalgebra. 30

31 The Cartan decomposition can be visualized in array form as follows: J 01 J J 0 m 1 J 0m J 0 m+1... J 0N J J 1 m 1 J 1m J 1 m+1... J 1N J m 2 m 1 J m 2 m J m 2 m+1... J m 2 N J m 1 m J m 1 m+1... J m 1 N J m m+1... J m N.... J N 1 N The subspace p (m) is spanned by the m(n + 1 m) generators inside the rectangle; the left and down triangles correspond, in this order, to the subalgebras so κ1,...,κ m 1 (m) and so κm+1,...,κ N (N + 1 m) of h (m). 31

32 When all κ a 0 a, so κ (N + 1) is a (pseudo-)orthogonal algebra so(p, q) (p + q = N + 1) and (p, q) are the number of positive and negative terms in the invariant quadratic form with matrix (1, κ 01, κ 02,..., κ 0N ). When κ 1 = 0 we recover the inhomogeneous algebras with semidirect sum structure so 0,κ2,...,κ N (N + 1) t N so κ2,...,κ N (N) iso(p, q), p + q = N, where the Abelian subalgebra t N is spanned by J 0b ; b = 1,..., N and so κ2,...,κ N (N) preserves the quadratic form with matrix diag(+, κ 12,..., κ 1N ). When κ 1 = κ 2 = 0 we get a twice-inhomogeneous pseudo-orthogonal algebra so 0,0,κ3,...,κ N (N+1) t N (t N 1 so κ3,...,κ N (N 1)) iiso(p, q), p+q = N 1, where the metric of the subalgebra so κ3,...,κ N (N 1) is (1, κ 23, κ 24,..., κ 2N ). When κ a = 0, a / {1, N}, these contracted algebras can be described as t a(n+1 a) (so κ1,...,κ a 1 (p, q) so κa+1,...,κ N (p, q )), p+q = a, p +q = N +1 a. The fully contracted case in the CK family corresponds to setting all κ a = 0. This is the so called flag algebra so 0,...,0 (N + 1) i... iso(1) such that iso(1) R. 32

33 4.2. Symmetrical homogeneous CK spaces If we now consider the CK group SO κ (N + 1) with Lie algebra so κ (N + 1) we find that each Lie subalgebra h (m) generates a subgroup H (m) leading to the homogeneous coset space denoted by: S (m) SO κ (N + 1) /( SO κ1,...,κ m 1 (m) SO κm+1,...,κ N (N + 1 m) ). The dimension of S (m) is that of p (m) : dim(s (m) ) = m(n + 1 m). Then S (m) is a symmetrical homogeneous space and there are N such symmetrical homogeneous spaces S (m) (m = 1,..., N) for each CK group SO κ (N + 1). We define the rank of the CK space S (m) as the number of independent invariants under the action of the CK group for each generic pair of elements in S (m) : rank(s (m) ) = min(m, N + 1 m). The sectional curvature of S (m) turns out to be constant and equal to κ m. 33

34 Table 4: Isotopy subgroup, sectional curvature, dimension and rank of the set of N symmetrical homogeneous spaces S (m) SO κ (N + 1)/H (m). Isotopy subgroup Curv. Dimension Rank H (1) = SO κ2,...,κ N (N) κ 1 N 1 H (2) = SO κ1 (2) SO κ3,...,κ N (N 1) κ 2 2(N 1) 2 H (3) = SO κ1,κ 2 (3) SO κ4,...,κ N (N 2) κ 3 3(N 2) H (m) = SO κ1,...,κ m 1 (m) SO κm+1,...,κ N (N + 1 m) κ m m(n + 1 m) min (m, N + 1 m).... H (N 2) = SO κ1,...,κ N 3 (N 2) SO κn 1,κ N (3) κ N 2 (N 2)3 3 H (N 1) = SO κ1,...,κ N 2 (N 1) SO κn (2) κ N 1 (N 1)2 2 H (N) = SO κ1,...,κ N 1 (N) κ N N 1 34

35 5. Conclusions We have provided a review on CK spaces and their vector fields. These results could be applied to the field of Lie and Lie-Hamilton systems. The case for the vector fields coming from isometries is currently in progress. The case for the vector fields coming from conformal symmetries is devoted for the future. References - Yaglom I M, Rozenfel d B A and Yasinskaya E U 1966 Sov. Math. Surveys Yaglom I M 1979 A Simple Non-Euclidean Geometry and its Physical Basis (New York: Springer) - Ballesteros A, FJH, del Olmo M A and Santander M 1993 J. Phys. A: Math. Gen FJH, Ortega R and Santander M 2000 J. Phys. A: Math. Gen FJH and Santander M 2002 J. Phys. A: Math. Gen Ballesteros A, FJH, Ragnisco O, and Santander M 2008 Int.J. Theor. Phys