CC BY
1
UDC 514.82 + 512.81 + 517.977+517.838+519.46 DOI 10.24147/1812-3996.2024.3.4-13
THE GODEL UNIVERSE AS A LIE GROUP WITH LEFT-INVARIANT LORENTZ METRIC AND THE IWASAWA DECOMPOSITION
V. N. Berestovskii
Dr.Sc. (Phys.-Math.), Professor, e-mail: [email protected]
Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia
Abstract. We discuss models of the Godel Universe as Lie groups with left-invariant Lorentz metric for two simply connected four-dimensional Lie groups, the Iwa-sawa decomposition for semisimple Lie groups, and left-invariant Lorentz metric on rmSL(2, R), following K.-H. Neeb. Also we show that the isometry between two non-isomorphic sub-Riemannian Lie group, constructed by A. Agrachev and D. Bar-ilari, is induced by some Iwasawa decomposition of SL(2, R).
Keywords: Godel Universe, Iwasawa decomposition, left-invariant Lorentz metric, left-invariant sub-Riemannian metric, Lie algebra, Lie group.
1. Introduction
Kurt Godel in paper [1] of 1949 introduced the Lorentz metric (1) of the signature (+, —, —, —) on the space R4. The Godel Universe (space-time) S is a solution of the General Relativity Theory (the Einstein gravitation equations).
In paper [2], the author found timelike and isotropic geodesics of the Godel Universe, considered as the Lie group G = (R, +) x A+(R) x (R, +) with left invariant Lorentz metric. Here A+(R) is connected Lie group of affine transformations on R.
The above left-invariant Lorentz metric on G and behaviour of its geodesics are defined essentially by the corresponding induced left-invariant Lorentz metric ds2 on the subgroup Gs = (R, +) x A+ (R).
Professor Karl-Hermann Neeb wrote to the author that it is possible to realize the Godel Universe otherwise. He sent an electronic version of his joint with Joachim Hilgert book [3], where in section 2.7 "Godel's cosmological model and universal covering of
SL(2, R)" is suggested a left-invariant Lorentz metric on SL(2, R) which is isometric to (Gs, dsl) as stated there.
In this connection, it is useful to mention the paper [4] by A. Agrachev and D. Barilari, where the autors obtained a full classification of left-invariant sub-Riemannian metrics on three-dimensional Lie groups and "explicitly find a sub-Riemannian isometry between nonisomorphic Lie groups SL(2, R) and rmSO(2) x A+(R)" [4].
The existence of such isometry was indicated ealier in [5] by Falbel and Gorodski.
In a message to the author, Professor Neeb explains the mentioned two isometries by a diffeomormism of Lie groups SL(2, R) and SO(2) x A+(R) by means of the Iwasawa decomposition for SL(2, R); let us cite Theorems 6.5.1 and 9.1.3 from [6].
In p. 10.6.4 (i) from [6] are indicated the following isomorphisms of Lie algebras:
sl(2, R) = su(1,1) ^ so(2,1) = sp(1, R).
Consequently, simply connected Lie groups with these Lie algebras are isomorphic.
In Theorem 3 we prove some properties of special left-invariant Lorentz metrics on Lie groups which support the mentioned statement on the isometry of two left-invariant Lorentz metrics from [3]. Also we show in Proposition 3 that the isometry between two non-isomorphic sub-Riemannian Lie group, constructed by A. Agrachev and D. Barilari, is induced by some Iwasawa decomposition of SL(2, R).
The author expresses his gratitudes to Professor Neeb for fruitful discussions and anonymous referee for useful remarks.
2. The Godel Universe as a Lie group with left-invariant Lorentz metric
Godel introduced in [1] his space-time S as R4 with the linear element
/ e2xi \ ds2 = a2 i dx^ + 2eX1 dx0dx2 +—— dx\ — dx\ — dx\ \ , a > 0. (1)
Godel noticed in [1] that it is possible to rewrite this quadratic form in view of
ds2 = a2
2 g2^l (dx0 + eXl dx2) — dx\--— dx2 — dx\
(2)
which shows obvious that its signature is equal everywhere to (+, —, —, —). We shall assume that a =1.
Godel noticed in [1] that on (S, ds2) acts simply transitively a four-dimensional isometry Lie group. It is easy to see that such action could be written as
x0 = x'0 + a, X\ = x[ + b, x2 = x'2e-b + c, x3 = x'3 + d (3)
with arbitrary a,b,c,d G R. This implies that corresponding Lie group G is the simplest simply connected noncommutative four-dimensional Lie group of the view
G = [(R, +) x G2] x (R, +), (4)
where G2 is unique up to isomorphism, necessary isomorphic to R2, two-dimensional noncommutative Lie group. The Lie group G2 is isomorphic to the Lie group A+(R) of preserving orientation affine transformations of the real direct line (R, +).
In case under consideration, identifying the quad (x'0,^,x'2,x'3) with the vector (x'2, x\,x'0,x'3, 1)T, where T is the sign of transposition, the action of the group G on R4 by formula (3) has the view (x2,xi,x0,xs, 1)T = A(xl2,xll,xl0,xl3,1)T, where
/ e-b 0 0 0 c\
0 10 0 b
A =0010 a . (5)
0 0 0 1 d
\ 0 0 0 0 1 /
Under this the equality
/ e-Xl 0 0 0 X2 \
0 1 0 0 xi
0 0 1 0 X0
0 0 0 1 X3
V 0 0 0 0 1
(0, 0, 0, 0, 1)T = (X2,X!,X0,X3, 1)
T
(6)
sets the bijection of the group G onto R4 and the unit of G corresponds to the zero-vector (0,0,0,0) e R4. On base of this, (4) and (1), we can identify (S, ds2) with the Lie group G equipped with left-invariant Lorentz metric. Let
eo
dx0
(0),ex
d
dx\
(0),e2
dx2
(0),es
dx3
(0))
be the basis of the Lie algebra g of the Lie group G at the unit of G, corresponding to coordinates (x0, x\,x2,x3). Then, according to what has been said and (1), the components of the linear element ds2 with respect to this basis are equal to
9oo = 1,go2 = 920 = 1,922 = 2,911 = 1,933 = -1,9íj = 9jí = M = j,j = 2. (7)
According to (6), the Lie subgroup G3 := matrix Lie group
, +) x G2 can be identified with the
/ e-X1 0 0 X2 \
0 1 0 xi
0 0 1 X0
V 0 0 0 1 /
(X0,X1,X2) e R3.
(8)
It is obvious that (S, ds2) = (S0, dsl) x (Si,ds2), where S0
p2x\
R3, Si
R,
d$0 = dx0 + 2eX1 dx0dx2 +
2
dx 2 dx i, ds i — dx 2.
(9)
Also it is clear that we can consider (S0, dsl) as the matrix Lie group (8) with left-invariant Lorentz metric, which according to (7) has components
900 = 1^02 = 920 = 1^22 = 2,9\1 = -1,9ij = 9ji = M = i,j = 2; (10)
with respect to the basis e0,e\,e2 of the Lie algebra 03 of the matrix Lie group (8). In consequence of (6), for the Lie algebra 03 of the Lie group G3,
e0 =
0 0 0 0 / -1 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 1 , ei = 0 0 0 0 , e2 = 0 0 0 0
0 0 0 0 V 0 0 0 0 0 0 0 0
(11)
Then in the Lie algebra g3,
[ei, e2] = eie2 - e2ei = -e2, [e0, ei] = [e0, e2] = 0.
(12)
3. The Iwasawa decompositions of Lie algebras and Lie groups
Let 0 be a semisimple real Lie algebra, a be some Cartan involution of 0, and 0 = k © p be the corresponding Cartan decomposition (k is the Lie subalgebra of g, consisting of fixed points relative to a). Let us denote by a a maximal commutative subspace in p. Then there is the following Iwasawa decomposition of Lie algebra g.
Theorem 1. (4.7.2) in [7]. Let 0 be a semisimple real Lie algebra. Then there exists a direct sum of vector subspaces in g
0 = k © a © n,
(13)
where n is a nilpotent subalgebra in g such that the endomorphism adX is nilpotent for every X G n, and a © n is a solvable subalgebra in g.
As an example, the authors of [7] give the decomposition (13) for 0 = si(n, R). In this case k is the Lie subalgebra of skew-symmetric matrices, a is the Lie subalgebra of diagonal matrices with zero trace, and n is the Lie subalgebra of strictly upper triangular matrices. In particular, for g = sl(2, R) we have
k
0 t -t 0
t 0 0 -t
n
0 t 00
, t G R, (14)
with natural basis
fc
( -1 0 ) , h = ( 0 -01 ) , h = ( 0 0 )
and Lie brackets for this basis
[fc, h] = 2/c - 4/2, [fc, /2] = h, [h,f2] = 2/2.
(15)
(16)
Let K = exp(k), A = exp(a), N = exp(n) be Lie subgroups of the semisimple Lie group G, corresponding to the decomposition (13).
Theorem 2. (Theorem 9.1.3 in [6]) Let G be a connected semisimple real Lie group. Then G = KAN and the mapping
(k, a, n) ^ kan (17)
is the diffeomorphism of manifold K x A x N onto the Lie group G.
Corollary 1. The Lie group G is diffeomorphic to Lie groups K x AN and K x A x N.
Theorem 1 and Theorem 6.1.1 from [6] imply the following
Proposition 1. The sets K, A, N, and AN are connected closed Lie subgroups of the Lie group G, where ) is compact, A is commutative, N is nilpotent, and AN is
solvable. The subgroup K contains the center Z of the Lie group G. In addition, K is compact if and only if the center Z of G is finite; in this case K is a maximal compact subgroup of the Lie group G.
a
The statements above imply the following
Corollary 2. If G = SL(n, R), then K = SO(n), A is the group of all real diagonal (n x n)-matrices with unit determinant, N could be considered as the group of all real upper triangular (n x n)-matrices with units on the main diagonal, and Sol(n) := AN as the group of all real upper triangular (n x n)-matrices with unit determinant.
Corollary 3. The Lie group SL(n, R) is diffeomorphic to Lie groups SO(n) x AN and SO(n) x A x N. As a consequence, SL(2, R) is diffeomorphic to the commutative Lie group SO(2) x A x N.
4. (S0, ds2) as (R, +) x Sol(2) with left-invariant Lorentz metric
This is a preparatory section.
Proposition 2. There exist an isomorphism of the Lie group G3 onto the Lie group (R, +) x Sol(2) and corresponding realization of ( So, d s as the Lie group (R, +) x Sol(2) with left-invariant Lorentz metric.
Proof. Comparing (12) and (16), we see that the linear map p : a © n ^ g2 such that
(i)
= e u p( ¡2) = e2
(18)
is an isomorphism of Lie algebras. Let Sol(2) be the Lie group with the Lie algebra a © n for (14). Then (18) defines isomorphism of Lie groups 4 : Sol(2) ^ G2 :
4
e-s/2 0
0
s/2
)(
(
1 r 0 1
0 0 e-sr \
4
We can consider
1 r 0 1
)(
0 10 s
0 0 10
\ 0 0 0 1 J
i e0 0 r\
(19)
0-s/2
0
0
es/2
10 010 0 0 1/
+) as a Lie algebra and as a Lie group. Then the mappings
V
0 0 0
(20)
t g (R, +) ^
(-0:)
t G (R, +) ^
(
cos t sin t — sin t cos t
)
(21)
are correspondingly the isomorphism of Lie algebras and respective universal covering epimorpism of Lie groups. Then there exists unique isomorphism 4 of the Lie group (R, +) x Sol(2) onto the Lie group G3, with properties (19), (20), and ^(t) = t for t G (R, +). It follows from previous considerations that we shall realize ( S0, ds2) as the Lie group (R, +) x Sol(2) with left-invariant Lorentz metric if components of this metric in the basis {f0, — fi/2, f2] of its Lie algebra will be as in (10).
— s
The corresponding orthonormal basis is
* = fo, Y' = —/i/2, Z' = V2(/q - /2), (22)
which we can change by
X = fo, Y = fi/2, Z = V2(f2 — fo). (23)
Then
[Y, Z] = [Y', Z'] = V2/2 = Z + V2x. (24)
■
Remark 1. The basis (23) has a form
* =( —> = KJ l) ^ = 4111) . (25)
The group Sol(2) is isomorphic to the Lie group of real lower triangular (2 x 2)—matrices with unit determinant.
5. Left-invariant Lorentz metrics on SO(2) x Sol(2) and SL(2, R)
In [3], the authors consider the Lie group SL(2, R) with left-invariant Lorentz metric and orthonormal basis with the signature (—, +, +) of the form
* = £ ( —111 ) ^ = ( 1 — ) ^ = ( 11 ) - -ftR). (26)
We shall consider this basis as orthonormal with the signature (+, —, —).
Theorem 3. 1) For the Lorentz metric on SL(2, R) from [3] with orthonormal basis (26) and for corresponding by the Iwasawa diffeomorphism basis on SO(2) x Sol(2), the curvatures k(X,Y) = k(X,Z) = k(Y,Z) = —2, while for the orthonormal basis (23) on (R, +) x Sol(2), k(X, Y) = k(X, Z) = k(Y, Z) = —1.
2) One-parameter subgroups in SL(2, R), defined by X, Y, Z, are geodesies.
Proof. For any (pseudo-)Riemannian manifold M with (pseudo-)metric tensor (■, ■), the Levi-Civita connection V, and smooth vector fields X, Y, Z is valid the following equation (3.5.(7)) from [8]:
(Vxy, ^) = 2 [^(Y, Z) + Y (Z, X) — Z(X, Y) + (Z, [X, Y]) + (Y, [Z, X]) — (X, [Y, Z])].
(27)
As a consequence, if (M, (■, ■)) is a Lie group G with left-invariant (pseudo-)metric (■, ■) of the signature (+, —, —) and X, Y, Z are left-invariant, then
(Vxy, ^) = 2[(Z, [X,y]) + (y, X]) — (X> [y, Z])]. (28)
It follows from (26) that
[X, Y] = —V2z, [X, Z] = V2y, [Y, Z] = 2^2X. (29)
Let us apply (28) and (29) in the further computations.
(VXY, Z) = 0, (VXY, X) = (VXY, Y) = 0, VxY = 0,
(VyY, X) = (VyY, Y) = (VyY, Z) = 0, VyY = 0, (VZY,X) = —V2, (VZY,Y) = (VzY,Z) = 0, V ZY = —V2X,
VyZ = VZY + [Y, Z] = V2X, R(X, Y )Y = Vx VyY — Vy VxY — V[x,y]Y = V2VZY = —2X. (30) Then k(X, Y) = (R(X, Y)Y, X) = —2. Analogously, we obtain
VxZ = 0, VxX = 0, VzZ = 0, k(X,Z ) = (R(X,Z )Z,X ) = —2,
R(Y,Z )Z = Vy VzZ—Vz VYZ—V[y,z]Z = —V2VzX = —V2(VXZ +[Z,X ]) = 2Y,
k( Y, Z) = ( R( Y, Z) Z, Y) = —2. 2) [Y, Z] = 2Z + 2^2X inso(2) © soI(2).
We look for Z as af2 + [3X, see (15). It is easy to see that a = 2, ft = — v^2, [Y, Z] = [Y, 2f2 — V2X] = 2[A, /2] = 4 j2 = 2(Z + ^2X) = 2Z + 2^2X.
(VXY, Z) = —V2, (VXY, X) = (VXY, Y) = 0, VXY = V2Z, (VyY, X) = (VyY, Y) = (VyY, Z) = 0, VyY = 0, (VZY,X) = —V2, (VzY, Y) = 0, (VzY, Z) = 2, VzY = —^2X — 2Z,
VyZ = VZY + [Y, Z] = V2X, R(X, Y )Y = Vx VyY — Vy VxY — V[x,y]Y = —V2VyZ = —2X. Then k(X, Y) = (R(X, Y)Y, X) = —2. Analogously, we obtain
VxZ = —V2Y, VZX = VxZ = —V2Y, VzZ = 2Y, k(X,Z) = (R(X,Z)Z,X) = —2, R(Y, Z)Z = VyVzZ — VzVYZ — V{Y)z\Z = Vy (2Y) — Vz (V2X) — V(2Z+2^2X )Z = 2Y — 2VzZ — 2^2VxZ = 2Y — 4Y + 4Y = 2Y, k( Y, Z) = ( R( Y, Z) Z, Y) = —2. Let us compute analogous expressions for ( S0,ds2), using results of Section 4..
V2
V2,
(VXY,Z) = ^, (VXY, X) = (VXY,Y) = 0, VxY = ^—Z
(VyY, X) = (VyY, Y) = (VyY, Z) = 0, VyY = 0, (Vz Y,X ) = — ^, (Vz Y,Y) = 0, (Vz Y, Z) = 1, Vz Y = — ^ X — Z,
Vy Z = VzY + [Y, Z] = ^rX,
tf (X, Y )y = Vx Vy Y — Vy Vx y — V[x,y]Y = — ^ Vy Z = — 1X.
2
Then fc(X, y) = (R(X, Y)Y, X) = — 1. Analogously, we obtain
£2
£2
VxZ = Y, VzX = VxZ = — Y, VzZ = y, k(X,Z) = (R(X,Z)Z,X) = —-
2 ' " ^ 2
R(Y, Z)Z = VyVzZ — VzVyZ — V[yz]Z =
Vy (y) — V^ £ xj — =
1 y — Vz Z — Z =2 y — y + Y = 1 y,
^ ) = (R(Y,Z )Z,Y ) = — 2. Obtained equalities imply all statements of Theorem 3.
■
Remark 2. It is similar, that the left-invariant Lorentz metric on SL(2, R) from [3] is isometric to Godel metric (S0, d^) induced by Godel metric (1) with a = ^.
6. Isometry of non-isomorphic sub-Riemannian Lie groups
Let us change notation x1 o x2,x0 ^ x3. Then the mapping
/ e-X2 0 0 x1 \
0 1 0 X2
0 0 1 X3
V 0 0 0 1 J
0 x1
0 0
1 X3 01
G A+(R) x (R, +) is an isomorphism of matrix Lie groups with the basis of the Lie algebra
(31)
0 0 1 \ /—100 e1 = | 0 0 0 I ,e2 = I 0 0 0 0 0 0 0 0 0
,e3
000 0 0 1 000
such that only [e1, e2] = — [e2, e1] = e1 are unique nonzero Lie brackets. Analogously to (19) and (20), the mapping
,-X2
F
0 0
0 x1
1 X3 01
—2/2
X1
eX2/2
) • (
cos x3 — sin x3
sin x3 cos x3
(32)
2
0
is the universal covering epimorphism of A+(R) x (R, +) onto Sol(2) x SO(2).
The standard left-invariant sub-Riemannian structure on A+(R) x (R, +) is defined in [4] by the orthonormal frame A = span{e2, e1 +e3}. Then there is unique sub-Riemannian structure on Sol(2) x SO(2) such that F is a local isometry; it is defined by the orthonormal frame A = span{e2, el + e3}, where
ei=(00),e 2 =( o/2 i/o,e 3 =( -10. (33)
Now we follow [4]. Let a = e-X2 and b = x1 in the second matrix of (31).
The subgroup A+(R) is diffeomorphic to the half-plane {(a, b) e R2, a > 0}, which is desrcibed in the standard polar coordinates as {(p, 9)\p > 0, — n/2 < 9 < n/2}.
Theorem 4. [4]. The diffeomorphism tf : A+(R) x S1 ^ SL(2, R) defined by
T( n ) = 1 ( COS P sin p \ (34)
tf(p,u,p) = 7^OsHpsin(0 — p) pcos(9 — P)) , (34)
where (p, 9) e A+(R) and p e S1, is a global sub-Riemannian isometry.
Remark 3. Using the above locally isometric covering F, we can and will understand tf as the global isometry between Sol(2) x SO(2) and SL(2, R) supplied with sub-Riemannian metrics defined by the same frame A.
Corollary 4. A+(R) x (R, +) with sub-Riemannian metric, defined by the frame A,
is isometric to the universal covering SL(2, R) of SL(2, R) with sub-Riemannian metric
such that the natural universal covering epimorphism of SL(2, R) onto SL(2, R) with sub-Riemannian metric, defined by the frame A, is a local isometry.
Proposition 3. The global isometry tf in the sense of Remark 3 is the Iwasawa diffeomorphism of Sol(2) x SO(2) onto SL(2, R) of the view ( na, k) e NA x SO(2) ^ nak e NAK = SL(2, R), where
n = (101 ,a = ( a-T 0/^ = ( Cosp SinM,a = pcos9,b = psin9. V b 1 J V 0 a1/2 J V — sin p cos p J
Proof. One needs simply to check that nak is equal to the matrix in (34). ■
Remark 4. Notice that n = exp(te1), a = exp(se2), where e1 = (e 1)T, T is the sign of transposition, b = t, and a1/2 = es/2. Also [e1, e2] = — e1.
Thanks
The work was carried out within the framework of the State Contract to the IM SB RAS, project FWNF-2022-0006.
References
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2. Berestovskii V.N. Timelike and isotropic geodesies of the Godel Universe as a Lie group with left-invariant Lorentz metric // Siber. Math. Journ. 2024. V. 65, N. 5. To be published.
3. Hilgert Jo., Neeb K.-H. Lie semigroups and their Applications. Lect. Notes Math., 1552. Springer-Verlag: Berlin, Heidelberg, 1993.
4. Agrachev A., Barilari D. Sub-Riemnnian structures on 3d Lie groups // Journal of Dynamical and Control Systems. 2012. V. 18, N. 1. P. 21-44.
5. Falbel E., Gorodski C. Sub-Riemannian homogeneous spaces in dimensions 3 and 4 // Geom. Dedicata. 1996. V. 62, N. 3. P. 227-252.
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ВСЕЛЕННАЯ ГЁДЕЛЯ КАК ГРУППА ЛИ С ЛЕВОИНВАРИАНТНОЙ ЛОРЕНЦЕВОЙ МЕТРИКОЙ И РАЗЛОЖЕНИЕ ИВАСАВЫ
В. Н. Берестовский
д-р физ.-мат. наук, профессор, e-mail: [email protected]
Институт математики им. С. Л. Соболева СО РАН, г. Новосибирск, Россия
Аннотация. В работе рассматриваются модели Вселенной Гёделя как группы Ли с ле-воинвариантной метрикой Лоренца для двух односвязных четырехмерных групп Ли, разложение Ивасавы для полупростых групп Ли и левоинвариантная метрика Лоренца на SL(2, R), следуя К.-Х. Нибу. Показано, что изометрия между двумя неизоморфными субримановыми группами Ли, построенная А. Аграчевым и Д. Барилари, индуцируется некоторым разложением Ивасавы для SL(2, R).
Ключевые слова: Вселенная Гёделя, разложение Ивасавы, левоинвариантная лорен-цева метрика, левоинвариантная субриманова метрика, алгебра Ли, группа Ли.
Дата поступления в редакцию: 15.08.2023
