Geometry of the homogeneous and isotropic spaces

Transcription

1 Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant to cosmology. The pupose of this note is to summaize the geomety of maximally symmetic spaces, o homogeneous and isotopic spaces. We ae inteested in thee dimensional spaces, but in the following we will study the spaces of an abitay dimension N = 2, 3, 4,.... The thee dimensional cases ae obtained by taking N = 3. 1 Homogeneity and isotopy We ae inteested in an N dimensional space fo which the cosmological pinciple applies. The cosmological pinciple is the assumption that thee is no special point o diection in ou univese. We can fomulate this moe pecisely as follows. If the distances between two abitay points ae kept invaiant unde a map P f(p ), then the map is called an isomety. Given a paticula point in space, isometies can be classified into two: those moving the point and those leaving the point fixed. We call a space homogeneous if given abitay points P and Q, thee exists an isomety mapping P to Q. We call a space isotopic with espect to a point P if thee exist N(N 1) 2 independent isometies fixing the point P. (oughly speaking these ae otations aound P.) We call a space maximally symmetic if it is homogeneous and isotopic. We note that due to the homogeneity a maximally symmetic space is isotopic with espect to any points. Actually a space which is isotopic with espect to any points is maximally symmetic. In the following we take it fo ganted that the homogeneous and isotopic spaces can be classified into thee types: euclidean spaces, sphees, and hypebolic spaces. 1 Since we know eveything about the geomety of euclidean spaces, we will only discuss sphees and hypebolic spaces. In each case we fist conside the simplest two dimensions and then genealize the esults to abitay N dimensions. 1 Fo a poof, see Chapte 13 of S. Weinbeg, Gavitation and Cosmology (Wiley). 1

2 2 Sphees Let x i (i = 1, 2, 3) be thee othogonal coodinates of a thee-dimensional euclidean space. A two-dimensional sphee of adius is defined by the equation x x x 2 3 = 2 (1) In the euclidean space an SO(3) tansfomation is given by x i x i = O ij x j (2) whee the epeated indices j ae summed ove 1 to 3, and O is a thee-by-thee eal othogonal matix of a unit deteminant: O T O = 1 3 (3) We note that the sphee is invaiant unde the SO(3) tansfomation goup, i.e., unde the tansfomations an abitay point on the sphee is mapped to a point on the same sphee. It is easy to see that the SO(3) tansfomations ae isometies of the sphee since dx 2 i = dx i2. Let us conside the infinitesimal tansfomations explicitly. An abitay infinitesimal tansfomation can be witten as O = ɛ T (4) whee the thee eal antisymmetic matices T ae given by T , T , T (5) Consideing the actions of the isometies on the point P (0, 0, 1), we find that T 1,2 move P, and T 3 leaves P invaiant. Hence, T 1,2 give homogeneity, and T 3 gives isotopy to the thee dimensional sphee. x_1 x_3 T_3 P T_2 T_1 Q x_2 Let us now conside the line element on the sphee. We fist conside the familia spheical coodinates: x 1 = sin θ cos φ x 2 = sin θ sin φ x 3 = cos θ (6) 2

3 whee 0 < θ < π and 0 < φ < 2π. The line element is given by 3 (dx i ) 2 = 2 ( (dθ) 2 + sin 2 θ(dφ) 2) (7) i=1 Aound eithe the noth pole N o the south pole S, we can altenatively use x 1,2 as local coodinates. The thid coodinate x 3 is dependent on x 1,2 as x 3 = ± 2 x 2 1 x2 2 (8) depending on whethe the point is in the noth o south hemisphee. The above implies dx 3 = x 1dx 1 + x 2 dx 2 x 3 (9) Hence, the line element on the sphee is given by (dx 1 ) 2 + (dx 2 ) 2 + (x 1dx 1 + x 2 dx 2 ) 2 2 (x x2 2 ) (10) Finally we expess x 1,2 in tems of the pojected adius x x2 2 and the angle φ as Then, the line element is given by x 1 = cos φ x 2 = sin φ x 3 = ± 2 2 (11) (d)2 + 2 (dφ) 2 (12) This is the fom most convenient to cosmology. (See the late section on the obetson-walke metic.) Let us make a final note on the elation between the adial coodinate and the pope adius l on the sphee. The pope adius l is the distance between the oigin (noth pole P in the figue) and a point at. Since we find l = 0 d dl = (13) d = acsin Since the cicumfeence at is given by 2π, we find (cicumfeence) 2π (pope adius) = l = acsin (14) < 1 (15) 3

4 All of the above can be easily genealized to the N-dimensional sphee. It is defined in the N + 1 dimensional euclidean space as x x 2 N+1 = 2 (16) The isometies ae given by the SO(N+1) tansfomations: x x = O x (17) whee O is an N-by-N eal othogonal matix of a unit deteminant. An infinitesimal tansfomation can be witten as O = 1 N+1 + ɛ ij T ij (18) i<j N+1 whee T ij is a eal antisymmetic matix whose only non-vanishing elements ae ( T ij ) ij = ( T ij) ji = 1 (19) Let us conside a point P = (0,..., 0, 1) on the sphee. T ij (i < j N) give N(N 1) 2 independent otations aound P, and T i,n+1 (i N) move P in N independent diections. In the spheical coodinates we obtain x 1 = sin θ sin θ 1 sin θ 2... sin θ N 2 sin θ N 1 x 2 = sin θ sin θ 1 sin θ 2... sin θ N 2 cos θ N 1 x 3 = sin θ sin θ 1 sin θ 2... cos θ N 2... x N = sin θ cos θ 1 x N+1 = cos θ (20) whee 0 < θ < π, 0 < θ 1,...,N 2 < π, 0 < θ N 1 < 2π. The line element on the sphee is given ecusively as (dl N ) 2 = 2 [ (dθ) 2 + sin 2 θ (dl N 1 ) 2] (21) whee dl N 1 is the line element on the unit N 1 sphee. If we use N i=1 (x i) 2, the distance fom the N + 1-th axis, instead of the angle θ, we find x 1 = sin θ 1 sin θ 2... sin θ N 2 sin θ N 1 x 2 = sin θ 1 sin θ 2... sin θ N 2 cos θ N 1 x 3 = sin θ 1 sin θ 2... cos θ N 2... x N = cos θ 1 x N+1 = ± 2 2 (22) 4

5 and the line element is given by 3 Hypebolic spaces (dl N ) 2 = (d)2 + 2 dl 1 2 N 1 2 (23) 2 To constuct a hypebolic suface, we stat fom a thee-dimensional Minkowski space as opposed to a euclidean space. Let x 1,2,3 be the coodinates of the Minkowski space with the metic η ij dx i dx j (dx 1 ) 2 + (dx 2 ) 2 (dx 3 ) 2 (24) Hence, x 3 plays the ole of time. The space-time distance is invaiant unde the Loentz tansfomations SO(2, 1): whee L is a thee-by-thee eal matix satisfying x x = L x (25) L T ηl = η, det L = 1, L 33 > 0 (26) We define a hypebolic suface of adius by x x 2 2 x 2 3 = 2 (27) This defines two sepaate space-like sufaces, and we take the one with point P = (0, 0, ) on. x_3 x_1 P x_2 Since the suface is invaiant unde the SO(2, 1) tansfomations (i.e., Loentz tansfomations in dimensional space-time), the isometies ae given by SO(2, 1). An infinitesimal SO(2, 1) tansfomation is given in the fom L = ɛ T (28) whee T 1 = , T 2 = , T 3 = (29)

6 With espect to P, T 1,2 coespond to the homogeneity, and T 3 to the isotopy of the hypebolic suface. Using x 1,2 as local coodinates, the line element is given by (dx 1 ) 2 + (dx 2 ) 2 (dx 3 ) 2 = (dx 1 ) 2 + (dx 2 ) 2 (x 1dx 1 + x 2 dx 2 ) x x2 2 (30) With x x2 2 as the distance fom the x 3 axis, we find The line element is given by x 1 = cos φ x 2 = sin φ x 3 = (31) (d)2 + 2 (dφ) 2 (32) A final note on the on the elation between the adial coodinate and the pope adius l on the hypebolic suface. We obtain ( ) d l = = sinh = ln (33) 2 2 Hence, we find (cicumfeence) 2π (pope adius) = l = sinh 1 > 1 (34) Now, let us summaize the popeties of N dimensional hypebolic spaces. We define an N dimensional hypebolic space as a space-like subspace in the N + 1 dimensional Minkowski space. Let the metic of the Minkowski space be η ij dx i dx j (dx 1 ) (dx N ) 2 (dx N+1 ) 2 (35) We define a hypebolic space of adius by x x 2 N x 2 N+1 = 2 (36) The isometies of the hypebolic space is given by the SO(N, 1) tansfomations whee L is an N + 1-by-N + 1 eal matix satisfying x x = L x (37) L T ηl = η, detl = 1, L N+1,N+1 > 0 (38) 6

7 The infinitesimal SO(N, 1) tansfomations ae given in the fom L = 1 N+1 + i<j ɛ ij T ij (39) whee T ij (i < j N) ae the eal antisymmetic matices with the only nonvanishing elements (T ij ) ij = (T ij ) ji = 1, and T i,n+1 (i N) ae the eal symmetic matices with (T i,n+1 ) i,n+1 = (T i,n+1 ) N+1,1 = 1. With espect to the special point P = (0,..., 0, 1) on the hypebolic space, T ij coespond to the isotopy, and T i,n+1 to the homogeneity of the space. With x x2 N as the distance fom the x N+1 axis, we find The line element is given by x 1 = sin θ 1... sin θ N 1 x 2 = sin θ 1... cos θ N 1... x N = cos θ 1 x N+1 = (40) (d)2 + 2 (dl N 1 ) 2 (41) whee dl N 1 is the line element on the unit N 1 sphee. 4 obetson-walke metic If we adopt the cosmological pinciple that ou univese is both isotopic and homogeneous, the space must be one of the thee types discussed above: euclidean space, sphee, and hypebolic space. Hence, the space-time metic of the univese must be given by (ds) 2 = c 2 (dt) 2 (dl) 2 (42) whee c is the light velocity, and dl is the line element in space. In the pevious sections, we have found if the space is flat (o euclidean), (d) ( (dθ) 2 + sin 2 θ(dφ) 2) (43) (d)2 + 2 ( (dθ) 2 + sin 2 θ(dφ) 2) (44) 1 2 a(t) 2 if the space is a sphee of adius a(t), and (d)2 + 2 ( (dθ) 2 + sin 2 θ(dφ) 2) (45) a(t) 2 7

8 if the space is a hypebolic space of adius a(t). By escaling by a(t), we can incopoate all the above as [ (d) a(t) K 2 + ( 2 (dθ) 2 + sin 2 θ(dφ) 2) ] (46) whee K = 0, 1, 1 fo euclidean space, sphee, and hypebolic space. 5 De Sitte space In the above we have consideed the maximally symmetic spaces. In this final section we genealize the notion of maximal symmety fom spaces to spacetimes. We wish to constuct dimensional maximally symmetic spacetimes which have 4 tanslational isometies and 6 otational isometies. We can constuct such a space, known as the de Sitte space, as follows. We stat fom a five dimensional Minkowski space with the metic (ds) 2 = η µν dx µ dx ν (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 + (dx 4 ) 2 (dx 0 ) 2 (47) We conside the fou-dimensional subspace M defined by 2 x x x x 2 4 x 2 0 = 2 (48) The isometies of the subspace M is given by the SO(4, 1) tansfomations x x = L x (49) whee L satisfies L T ηl = η, det L = 1, and L 00 > 0. Taking x = x 1,2,3 and x 0 as independent local coodinates, we find (ds) 2 = (d x) 2 (dx 0 ) 2 + ( x d x x 0dx 0 ) x 2 0 x2 (50) The fou coodinates x, x 0 ae not entiely fee, but they ae constained by the condition 2 + x 2 0 x 2 0 (51) We can ewite the above line element using diffeent choices of space-time coodinates. We give two examples hee. The fist example is given by [ x x 0 = cosh x 0 (1 + x + ) ] sinh x 0 ( ) x x = x exp 0 (52) 2 Note the diffeence in the sign of the ight-hand side compaed to the definition of the hypebolic spaces. 8

9 The coodinates x and x 0 ae entiely fee. The line element becomes ( ) 2x (ds) 2 = (dx 0) 2 exp 0 (d x ) 2 (53) Note that the space metic appeas flat so that this becomes a obetson-walke metic with a paticula scale paamete: a(t) = e 2t. The second example is given by [ x 0 = x 0 2 ln x 1 ( ) ] 2 2x 2 exp 0 ( ) x x = x exp 0 = x (54) The coodinates x (and hence x) ae constained by x 2 < 2. Then the line element becomes time independent as (ds) 2 x = (1 ) ( ) 2 2 x d x 2 (dx 0) 2 (d x ) 2 (55) 2 x 2 The metic in this fom was fist applied by de Sitte to a theoy of the steady state univese. 9