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Piotr Godlewski FIELD of COMPLEX LINEAR FRAMES on REAL DEMONSTRATIO MATHEMATICA Vol. XXXVIII No 4 2005 Piotr Godlewski FIELD OF COMPLEX LINEAR FRAMES ON REAL SPACE-TIME MANIFOLD AS DYNAMICAL VARIABLE FOR GENERALLY-COVARIANT MODELS Abstract. A model of the self-interacting field of complex linear frames E on an n- dimensional real manifold is investigated. The model is generally-covariant and GL{n, C)— invariant. If n = 4, the components of E can be interpreted as dynamical variables for the gravitational field. A Lagrangian of E is constructed, the Euler-Lagrange equations are derived and a wide class of their solutions is found. The solutions are built of left invariant vector fields on a real semisimple Lie group "deformed" by a complex factor of a natural exponential structure. 1. Introduction In [9], [10] and [11], the author presented an alternative model of the gravitational field in which degrees of freedom are described by components of a field of real linear frames e on a " space-time" manifold M of dimension n (a tetrad field if n = 4). The group GL(n, M) can act on e. In contrast to the Einstein theory (formulated in terms of the tetrad) or the general metric- teleparallel theories of gravitation ([4], [5], [6], [7]), which are invariant under the action of pseudo-Euclidean subgroups SO(k,n — k) (50(1,3) in the physical case), the model proposed by Slawianowski is GL(n,R) - invariant. A price one has to pay for this extension of the symmetry group is a very strong nor linearity of the model. More precisely, the Lagrangian density of e has the form (1.1) where t^ are components of a two-fold covariant tensor field on M, built algebraically of e and its first derivatives. The square root structure of (1.1) resembles the Born-Infeld electrodynamics [8]. Hence it is called the modified Born-Infeld-type nonlinearity. Key words and phrases: Euler-Lagrange equations; field of complex linear frames; semisimple Lie group; teleparallelism connection. 820 P. Godlewski In our first paper concerning the field of linear frames [1], we extended the model suggested by Slawianowski. We constructed a new family of La- grangians of the self-interacting field e. We classified them and derived their general properties. We proved the existence of a wide class of solutions of the Euler-Lagrange equations for the Lagrangians belonging to the family we constructed. These solutions exist if the manifold M is (locally) diffeo- morphic to R x G, where G is a semisimple Lie group. The field of real linear frames e satisfying the Euler-Lagrange equations can be then identified with a basis for the Lie algebra of left invariant vector fields onMxG deformed by some factor of a natural exponential structure. In [2], [3] one finds a further extension of Slawianowski's theory. This is the model of the field of real linear frames e interacting with the complex scalar field or the multiplet of complex scalar fields. The latter may be interpreted as a quasiclassical model of matter. After the self-interaction of the real field e and the mutual interaction of e with the complex scalar fields was discussed, the case where the real field e is replaced by the complex one seems to be the next step of our investiga- tions. And so, in this paper we present a model of the self-interacting field of complex linear frames E on the real manifold M. The model is invariant under the action of GL(n, C). In Section 4 we modify definitions of basic geometric objects built of the real field e to adapt them to the complex field E. As in the case of the real field e, the Lagrangian C[E] of E is of the Born-Infeld-type (see (1.1)). In Section 5 we derive the Euler-Lagrange equations from C\E\. Equa- tions written in generally-covariant form are linear with respect to compo- nents of some tensor density H of weight one and its covariant differential VH. For the real field e, V is the covariant differentiation corresponding to the teleparallelism connection induced by e. Hence VH[e] is determined uniquely. In contrast to e, the complex field E induces two teleparallelism connections. Consequently, VH[E] can be defined in various ways. Thus we discuss conditions which determine uniquely the form of VH[E}. In Section 6 we construct a class of solutions of the Euler-Lagrange equations derived from C[E], It is an extension of the family of the solutions for the real field e we mentioned above. We also discuss a spatio-temporal metric being a byproduct of the solutions we constructed. We finish with physical comments and a discussion of expected physical tests (Section 7). In a sense, the model presented here is a complexification of tetrad ap- proaches studied in [9], [10] and [11]. It is a well-known fact that complexfi- cation very often leads to interesting physical results (e.g. poles of the scat- tering amplitude analytically extended onto the complex energy plane give Field of complex linear frames 821 energy levels of bounded states). We expect that complexified tetrad field- theoretic models can shed new light onto the theory of generally-relativistic spinors and the conformal U(2,2)-symmetry ([12], [13]). 2. Notations and preliminaries Let M be an n-dimensional real differentiate manifold of class C°°. Let JF(p, C) be the set of all complex functions of class C1 defined in a neighborhood of a point p € M. The differentiability of / 6 T(p, C) is understood as the differentiability in the real domain. For X, Y € TPM, the complex tangent vector X + iY at p is a C— linear mapping F(p, C) —» C given by (X + iY)f = Xf + iYf. The complex tangent space TP(M, C) of M at p is the C—linear space of all X + iY. The complex conjugate of X + iY is (2.1) X + iY = X — iY. The complex cotangent space TP*(M, C) of M at p is the C—dual space of TP(M, C). Every ui e TP*(M, C) can be uniquely written as UJ = a + i{3, where a, ¡3 € TP*M. The complex conjugate of u> is (2.2) ul = a-i(3. An ordered basis of TP(M, C) (resp. TPM) will be called the complex (resp. real) linear frame at p. The principal bundle of complex (resp. real) linear frame over M will be denoted by F(M, C) (resp. F(M)). Similarly, by F*(M, C) and F*(M) we shall denote the principal bundles of complex and real linear coframes over M, respectively. REMARK 2.1. Obviously, we shall regard F(M, C) and F*(M, C) as real manifolds. Similarly, the structure group of the bundles (i.e., GL(n, C)) will be treated as a real Lie group. 3. Field of real linear frames In this section we shall review basic concepts and results that one finds in [9, 10] and [1]. Let e = (e^) : M —» F(M), A = 0,1,..., n — 1, be a cross section of class C°°. In other words, e is a field of real linear frames on M. Let e = (eA) : M F*(M) be the field of real linear coframes R—dual to e, A A i.e., (e ,eB) = S B• 822 P. Godlewski REMARK 3.1. As a matter of fact, we assume that M is parallelizable, i.e., there exist smooth cross sections of F(M) defined over M. At the end of this section and in Section 6, we shall discuss a case which satisfies this assumption. The structure group of the bundles F(M) and F*(M), i.e., GL(n,R), acts on e and e as follows: e = (e^) eL = {{eL)A) = (LBAeB), e = (eA) ->HL = ((eL)A) = {{L^)ABeB) for L = [LaB] € GL(n, R), where L~l denotes the inverse matrix of L. We define a system of real functions 7j4.ec[e]> A, B,C, = 0,1,... ,n — I, of class C°° on M by (3-1) [eBj ec] = 1A Bc\e\e A- We call yABcle} the structural functions. They are skew-symmetric with respect to the lower indices: (3-2) tV[«] = -lACB[e]. From 7ABc\e\ we build a tensor field 5[e] of type (1,2) and a symmetric covariant tensor field 7[e] of degree 2: (3.3) S[e] = \iAbcWa ®eB® ec, (3.4) 7[c] - 7ABWa ® eB, lAB[e] = 7CAD[e\^DBC[e\. In terms of a local coordinate system (x^), // = 0,1,..., n—1, in a coordinate neighborhood U C M, (3.5) S[e]\U = SVle]^®^®^, = and (3.6) 7[e]|U = T^[¿[dx" ® dx\ 7/ll/[e] = 45Aw[e]5^A[e], Here (•),„ denotes partial differentiation with respect to xv. Let us denote by DiffM the group of all diffeomorphisms of M onto itself of class C°°. The fields 5[e] and 7[e] are DiffM-covariant (generally- covariant) and invariant under the action of GL(n, R), i.e., for any ip € DiffM and L e GL{n, R), S[p.e] = £*(S[e]), S[eL] = S[e] and (3.7) 7 [</>*e] = ^(7[e]), 7 [eL] = j[e], where <p*e = (((p*e)A) = i}p*eA) and denotes the automorphism of the tensor algebra of all tensor fields on M induced by <£>#. Field of complex linear frames 823 Since e is a cross section of the bundle F{M), it induces a flat linear connection Ttei[e] in F(M), called the teleparallelism connection.
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