DILATON-DEPENDENT α CORRECTIONS IN GAUGE THEORY OF GRAVITATION

Transcription

1 FIELD THEORY, GRAVITATION AND PARTICLE PHYSICS DILATON-DEPENDENT α CORRECTIONS IN GAUGE THEORY OF GRAVITATION S. BABETI Department of Physics, Politehnica University, Timiºoara, România, Received February, 005 Some solutions without singularities to an effective action derived from low-energy heterotic string theory with dilaton, α corrections and a potential for the dilaton are presented in a de-sitter gauge theory of gravitation over a spherical symmetric Minkowski space-time. The higher derivative terms of α corrections are constructed so that they vanish when the spacetime is de-sitter in the equivalent dilaton gravity. With a particular choice of the gauge fields and of the action with α expansion, which is typical of strings and disappears in the point-like limit, the field equations are derived. A singularity free solution, presented in this paper, is that which at late times is asymptotically Minkowski and the dilaton is frozen and at t 0 the univers enters a de-sitter phase. The calculations are performed using an analytical program conceived in GRTensorII for Maple 8.. INTRODUCTION In string theory a point is replaced by a one-dimensional object known as a string (open or closed string). Correspondingly, a local field which is a function of a single space-time point is replaced by a nonlocal string field. However, the extension of this one-dimensional object is of the order of Planck length (~ 0 35 m). The spectrum of vibrations of the open and closed free strings are represented by a linearly growth of spin (α) with the mass (m ) the Regge trajectory. The slope of the linear plot is α (the Regge slope) and is given by α dα ~ G ~(09 GeV), d N where G N is the Newtonian constant of m gravitation. The open string contains a massless spin state (to be identified with a gauge boson) plus a tower of supermassive excitations ( m 09 GeV) of higher spins. The closed string has a massless spin state (to be identified with a graviton) followed by supermassive higher spin states. The dynamics of a point particle is determined by the action integral which has the constant coefficient of mass dimension whereas the dynamics of a string Rom. Journ. Phys., Vol. 50, Nos. 3 4, P. 3 39, Bucharest, 005

2 3 S. Babeti is determined by an action integral with the constant coefficient of the dimension of ( mass ) or equivalently of energy per unit length called the string tension T: T =. πα Hence, the particle spectrum of a string theory consists of a finite number of massless states and an infinite tower of massive excitations at a mass scale characterized by a fundamental parameter the string tension or the Regge slope. This parameter must be [] of order the Planck mass (0 9 GeV) so that the graviton interacts with the usual Newtonian strength. If one wishes to give a phenomenological description of the consequences of string theory for low-energy physics, it should not be necessary to describe explicitly what the massive states are doing. It is natural, instead, to formulate an effective action based entirely on fields that correspond to massless, or at least very light, degrees of freedom only. The infinite set of point-particle fields that arise in string theory consists of a finite number of massless fields, which we can collectively represented by φ 0, and an infinite number of heavy fields collectively represented by φ H. In principle, it must be possible to describe string theory by a classical action S( φ,φ 0 H ) (or, at quantum level, a quantum effective action) governing these fields. At present, we do not have really satisfactory ways to formulate the exact classical action S( φ 0,φ H ). The idea of constructing an effective action S( φ 0 ) for the massless fields φ 0 is that although the exact formula would be a complicated and artificial rewriting of the exact theory, useful formulas can be obtained by a systematic expansion in the number of derivatives. In the extreme low-energy limit the leading terms in the effective action can be constructed just from invariance principles from gauge invariance and local supersymmetry. If we choose to work in heterotic string theory the action includes a gauge vector boson A µ (with field strength tensor F µν ) and an antisymmetric tensor H µνρ (tree form) related to a antisymmetric tensor potential B µν and the gauge field A µ by H = db A F, but we can consider only the most relevant massless modes, the dilaton φ and the graviton g µν. In order to obtain solutions without singularities a higher-derivative correction of effective string theory action can be obtained in the context of models implementing the so-called limiting-curvature hypotesis [7, 4, 8] and another higher derivative correction is the α expansion of effective action [9, 0]. The second one is used below in order to obtain solutions without singularities (curvature singularities). First, we present an effective action in low-energy heterotic string theory with dilaton in string and Einstein frame with possible α corrections. Section 3 contains the de-sitter gauge model on a spherical symmetric Minkowski space-time with the

3 3 Dilaton-dependent in gauge theory of gravitation 33 particular ansatz of gauge fields. The general expressions for the components of the strength tensor F A µν are used to compute the particular ones in the case of null torsion. In Section 4 we present the effective action with α correction and with a potential for dilaton and we write the field equations with a solution without singularities of de-sitter gauge theory of gravitation. The analytical program written in the pakckage GRTensorII running on the Maple 8 platform is also presented in Section 4.. THE ACTION In D-dimensions, the string tree-level (i.e., lowest order) effective action for the massless boson sector (in the string frame) is [0]: S = ddx ge φ R+ 4( φ ) +ξ0α 6πG L D () ( D 0) + O( α R 4 ), 3α where φ ( t) is the dilaton (a homogeneous free massless scalar field), tilde indicates that we are using string-frame metric g µν ( µ, ν = 0, D ) and ξ 0 takes on the values 4,, 0 for the bosonic, heterotic and type II superstring 8 theories, respectively. We choose to work in heterotic string theory and to consider only the most relevant massless modes, the dilaton φ and the graviton g µν. Therefore, the axion field or Maxwell field are not contained in the L expression. In order to compare with general relativity, it is convenient to express the action with Einstein metric, but as the string frame is the fundamental frame with respect to string theory, sometimes it is important to perform the calculations directly in the string frame. The Einstein frame metric g µν is related to the string frame metric g µν via a conformal tranformation g = exp( φ) g () µν µν In the Einstein frame, the low energy effective four-dimensional action of heterotic string with dilaton and graviton and some higher derivative terms, like α corrections, is given by: S = x g R φ + α φ + α R +, 6πG 8 d4 ( ) exp( ) 4 ( ) L O (3)

4 34 S. Babeti 4 where L contains Riemann squared term Rµνρσ [0] and terms such as R and Rµν R µν may be added and substracted from it. In some papers [9], L is the Gauss-Bonnet invariant R 4 GB = R Rµνρσ R Rµν µνρσ µν + R which lead to a second order equations of motion. In order to obtain nonsingular solutions it is useful to design something other than the Gauss-Bonnet form for L. With an invariant L which vanish for a nonsingular spacetime and an potential for the dilaton field we can look for solutions that approach this nonsingular manifold in the large curvature regime [0]. A proper choice of L is: R µνρσ L = R Rµνρσ µνρσ R (4) 6 The four-dimensional action in Einstein frame becomes: S = d4x g R ( φ ) + 6πG exp( ) R Rµνρσ R + α φ µνρσ V( φ ), 8 6 where V( φ ) is the potential for the dilaton field. In the Section 4 we rewrite this action for de-sitter gauge theory of gravitation and we obtain the field equations. (5) 3. GAUGE THEORY OF GRAVITATION We consider a gauge theory of gravitation having de-sitter (DS) group as local symmetry. Let X A, A =,,, 0, be a basis of DS Lie algebra with the corresponding equations of structure given by [5]: C AB C A B AB C X, X = if X, (6) where f = f C BA are the constants of structure whose concrete expressions will be given below [see eq. (9)]. We envisionage the space-time as a fourdimensional manifold M 4 ; at each point we have a copy of DS group. Introduce, as usually, 0 gauge fields ha µ ( x), A =,,..., 0, µ = 0,,, 3, were ( x ) denotes the local coordinates on M 4. Then, we construct the tensor of the gauge fields (strength tensor) F A µν = FµνX A which takes its values in the Lie algebra of the DS group (Lie algebra-valued tensor). The components of this tensor are given by: A A A A B C µν µ ν ν µ BC µ ν F = h h + f h h. (7)

5 5 Dilaton-dependent in gauge theory of gravitation 35 In order to write the constants of structure f AB C, we use the following notation for the index A: a = 03,,,, A = (8) [ ab] = [0],[0],[03],[],[3],[3]. This means that A can stand for a single index like as well as for a pair of indices like [0], [], etc. The infinitesimal generators X A are interpreted as: Xa Pa (energy-momentum operators) and X [ ab] M ab (angular momentum operators) with property Mab = Mba, ab, = 03,,, [3]. For the constants of structure f we find the following expressions: C AB a [ ab] a bc = c[ de] = [ bc][ de] = 0, [ ab] [ ] cd 4 d d dc f f f f b a a b ab ( c c ) f ( bc bd c ) = λ δ δ δ δ =, a a a a bcd [ ] f[ cdb ] d f = = η δ η δ, [ ef ] e f e f e f e f f[ ab][ cd] = ηbcδδ a ac ad c bd a c 4 d η δδ b d +η δδ b η δδ e f, where λ is a real parameter, and η ab = diag(,,, ) is the Minkowski metric. In fact here we have a deformation of DS Lie algebra having λ as parameter. When λ 0, we obtain the Poincaré Lie algebra, i.e., the DS group contracts to the Poincaré group. We will denote the gauge fields (or potentials) ha µ ( x) by ea µ ( x) (tetrad fields) if A = a and by [ ab ]( x [ ba] ) ( x) ω = ω (spin connection) if A= [ ab]. Then, µ µ introducing the relations (9) into the definition (7), we find the following expressions of the strength tensor components: ( ) bc ( ) cd 4 ( ) a a a ab c ab c µν µ ν ν µ µ ν ν µ F = e e + ω e ω e η, (0) ab ab ab ac db ac db a b a b µν µ ν ν µ µ ν ν µ µ ν ν µ F = ω ω + ω ω ω ω η λ e e e e. () The action associated to the gravitational gauge fields, quadratic in the components F A µν, is writen in the form [6]: g (9) d4 µνρσ A B ε µν ρσ AB () S = x F F Q, where ε µνρσ is the Levi-Civita symbol of rang four. This action is independent of any specific metric on M 4. The quantities Q AB are constants, symmetric with respect to the indices A, B: QAB = QBA. If we chose [3]:

6 36 S. Babeti 6 Q AB ε = abcd, for A= [ ab], B= [ cd] 0 otherwise (3) then we obtain the action of the General Relativity. For our purpose we develop a gauge theory of the DS group in a 4-dimensional Minkowski space-time M 4, endowed with spherical symmetry: ( ) ds = dt dr r dθ + sin θdϕ (4) We chose a particular form of spherically gauge fields of the DS group ea ( x) and ab ( x) ω given by the following ansatz: µ respectively 0 at () eµ = ( N()000 t,,, ), eµ = 0,, 00,, kr e =,, rat,, e =,,, rat θ ( 00 ()0) ( 000 ()sin ) 3 µ µ 0 at () 0 () rat ω µ = 0 0 0,,, (), ω µ =,,, Nt kr Nt (), 03 ra () t ω 000 sin µ =,,, θ, ω µ = ( 00,, kr, 0 Nt () ), ω =,,, θ, ω =,,, θ, ( 000 kr sin ) ( 000cos ) 3 3 µ µ µ (5) (6) where Nt ( ) and at ( ) are functions only of the time variable t, k is a constant and a is the derivative of at ( ) with respect to the variable t. The choice (6) of gauge fields ω ( ) assures that all components of the strength tensor a µ ab x vanish [8]. If we remember the Riemann-Cartan theory of gravitation, then the torsion tensor Tρ eρ a µν = a Fµν vanish, in accord with GR theory. Here, e ρ a denotes the inverse of e a µ with the properties: a µ a a b b eeν ν µ µ a µ ee F µν =δ, =δ. (7) We use the above expressions of gauge fields ea µ ( x) and µ ab ( x) ω to compute the components of the tensor F ab µν. The calculations are performed using an analytical program conceived by us and presented in the Section 4. ( 4 ) F = r kn + λ a N + a, N kr

7 7 Dilaton-dependent in gauge theory of gravitation 37 ( 4 ) F3 3 = r sinθ kn + λ a N + a, N kr 3 F0 3 0 = an an + 4λ an, F sin 3 = r θ ( kn + 4λ a N + a ), N kr N F0 r 3 03 sin 3 0 = ( an an + 4λ an r ), F θ 03 = ( an an + 4λ an ), N N (8) 4. FIELD EQUATIONS AND SOLUTIONS In order to obtain solutions without singularities of DS gauge theory of gravitation we can use the action () which have α corrections, time-dependent homogeneous dilaton φ ( t) and a potential for dilaton V( φ ) : S = xe F φ + α φ V φ, 6πG 8 d4 ( ) exp( ) ( ) L (9) where F = Fabeµ a eν µν b and e= det( e a µ ); with e ν a the inverse of e a µ defined by eq. (7). A proper choice of invariant L is ab µν L = Fµν F ab F (0) 6 Using the definitions of F, L and the expressions of we obtain for F and L the following forms: 3 3 F = 6 aan aan + kn + a N + 8λ a N, an 3 L ( kn 3 + a N aan + aan ) = 6, an 4 6 ab F µν from eqs. (8) () () where a is the second derivative of at ( ) with respect to the variable t. Introducing these expressions into action (9) and imposing the variational principle δ S = 0 with respect to φ ( t) and Nt ( ) we obtain the corresponding field equations. These field equations for the particular case Nt ( ) = are: ( ) α k V H exp( ) φ 4 H 8 = φ + φ+ φ, a φ (3) 6 α H HH 6HH 4HH k H H H k + φ+ φ+ = 8 a a

8 38 S. Babeti 8 = exp( φ) V( ) 6H 4 k φ + φ + Λ+ a (4) with H = a, H = d H = aa a (5) a dt a where dots denote derivatives with repect to time t, and Λ= λ is interpreted as cosmological constant [6]. If we consider the limit λ 0 or equivalently Λ= 0, we obtain the results in Ref. [0]; but, for Λ 0 we can study in addition the dependence on the cosmological constant of the solutions (without singularities) obtained for eqs. (3) (4). The solution of Eqs. (3) (4) includes a dependence on the cosmological constant Λ. For the case with k = 0 the equations of motion becomes: ( ) α V H φ = exp( φ ) + 4φ+ Hφ, 8 φ (6) 6 α ( H HH 6H H + 4HH φ ) = exp( φ) ( V( φ ) + φ + 6H 4Λ 8 ) (7) For nonsingular solutions of equations of motion we set φ ( t) and V( φ ) to have a proper expression with some properties. At late time we demand φ φ 0 and V( φ) 4 Λ. In this case a de-sitter type solution at ( ) = a0 exp( Ht) which correspond to constant H = H0 satisfy the equations of motion (6) (7) and the constraint equation (imposing the variational principle δ S = 0 with respect to at ()) is H = 0 and, hence at ( ) = const. At t = 0 we will force the solutions to be with at () = a0 exp( Ht). In accord with all the above properties we choose to set the dilaton φ ( t) [0] to be: () 0 tanh t t t φ =φ, t 0 and the potential V( φ ) in the form: (8) V( φ ) = V 0 ( φ+ ) 4 48λ. (9) Setting φ 0 =, V 3 0 = H0, t 0 = and t large enough so that φ ( t = 0) and inserting (8) and (9) into (6), at t 0 we obtain: H = 0, (30) which imply H = const and using (7) we conclude that the parameter λ have the order of H 0 (interpreted in [6] as a Planck scale of the model). Therefore, in our example the Planck scale is related to the cosmological constant Λ and the solution has no singularities and it is valid if the cosmological constant is negative.

9 9 Dilaton-dependent in gauge theory of gravitation 39 Program withalphaprime.mws restart: grtw(): grload(minknou, `c:/grtii(6)/metrics/minknou.mpl`); grdef(`ev{^a miu}`); grcalc(ev(up,dn)); grdef(`omega{[^a ^b] miu}`); \ grcalc(omega(up,up,dn)); grdef(`eta{(a b)}`); \ grcalc(eta(dn,dn)); grdef(`famn{^a miu niu} := ev{^a niu,miu} - ev{^a miu,niu} + omega{^a ^b miu}*ev{^c niu}*eta{b c} - omega{^a^b niu}*ev{^c miu}*eta{b c}`); grcalc(famn(up,dn,dn)); grdisplay(_); grdef(`fabmn{^a ^b miu niu} := omega{^a ^b niu, miu} -omega{^a ^b miu, niu} + (omega{^a ^c miu}*omega{^d ^b niu} -omega{^a ^c niu}*omega{^d ^b miu})*eta{c d} -4*lambda^*(ev{^a miu}*ev{^b niu}-ev{^b miu}*ev{^a niu})`); grcalc(fabmn(up,up,dn,dn)); grdisplay(_); grdef(`einv{a ^miu }`); grcalc(einv(dn,up)); grdef(`fscal:=fabmn{^a ^b miu niu}*einv{a ^miu}*einv{b ^niu }`); grcalc(fscal); grdisplay(_); grdef(`l:=fabmn{^a ^b miu niu }*Fabmninv{a b ^ miu ^niu}- (/6)Fscal^ )`); grcalc(i); grdisplay(_); grdef(`de`); grcalc(de); grdef(`lg:=(fscal-*diff(phi(t),t)^ + c*exp(-*phi(t))*l- V(phi(t))*de`); grcalc(lg)); grdisplay(_); REFERENCES. M. B. Green, J. H. Schwarz, E. Witten, Superstring theory, Cambridge University Press, G. T. Horowitz, The Dark Side of String Theory: Black Holes and Black Strings, 99 Trieste Spring School on String Theory and Quantum Gravity, hep-th/ S. W. MacDowell, F. Mansouri, Unified Geometric Theory of Gravity and Supergravity, Phys. Rev. Lett. 38, No. 4, 739 (977). 4. D. A. Easson, R. H. Brandenberger, Nonsingular Dilaton Cosmology in the String Frame, JHEP 9909, 003 (999), hep-th/ G. Zet, V. Manta, S. Babeti, De-Sitter gauge theory of gravitation, Int. J. Mod. Phys. C, Vol. 4, No., pp (003). 6. G. Zet, C. D. Opriºan, S. Babeti, Solutions without singularities in gauge theory of gravitation, Int. J. Mod. Phys. C, Vol. 5, No. 8, (004). 7. R. Brandenberger, R. Easther, J. Maia, Nonsingular Dilaton Cosmology, JHEP 9808, 007 (998) gr-qc/ S. Babeþi, G. Zet, Solutions in dilaton gauge theory of gravitation, submitted to Anal. Univ. de Vest Timiºoara, (004). 9. M. Gaperini, G. Veneziano, The Pre-Bing Bang Scenario in String Cosmology, hep-th/ D. A. Easson, Towards a Stringy Resolution of the Cosmological Singularity, hep-th/