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Conformal Hamiltonian Dynamics of General Relativity ∗ A.B Physics Letters B 691 (2010) 230–233 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Conformal Hamiltonian dynamics of general relativity ∗ A.B. Arbuzov a,B.M.Barbashova, R.G. Nazmitdinov a,b, V.N. Pervushin a, , A. Borowiec c, K.N. Pichugin d,A.F.Zakharove a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia b Department de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain c Institute of Theoretical Physics, University of Wrocław, Pl. Maxa Borna 9, 50-204 Wrocław, Poland d Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia e Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya str. 25, 117259 Moscow, Russia article info abstract Article history: The General Relativity formulated with the aid of the spin connection coefficients is considered in the Received 17 March 2010 finite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution Accepted 23 June 2010 of the General Relativity describes the vacuum creation of the matter in the empty Universe at the Available online 3 July 2010 electroweak epoch and the dilaton vacuum energy plays a role of the dark energy. Editor: S. Dodelson © 2010 Elsevier B.V. All rights reserved. Keywords: General relativity Cosmology Dilaton gravity The distance-redshift dependence in the data of the type Ia su- Within the Dirac approach the Einstein–Hilbert action takes the pernovae [1] is a topical problem in the standard cosmology (SC). form As it is known, the SNeIa distances are greater than the ones pre- 4 1 dicted by the SC based on the matter dominance idea [2].There W Hilbert =− d x |−e| R(e) are numerous attempts to resolve this problem with a various de- 6 e=e−D e˜ − gree of success (see for review [3]). One of the popular approaches |−e˜|e 2D =− 4 ˜ − −D |−˜|˜ μν −D is the -Cold-Dark-Matter model [4]. It provides, however, the d x R(e) e ∂μ e g ∂νe . − 6 present-day slow inflation density that is less by factor of 10 57 than the fast primordial inflation density proposed to include the (1) Planck epoch. 2 3 = Hereafter, we use the units MPlanck 8π 1. The interval is defined μ Approaches to the General Relativity (GR) with conformal sym- via diffeo-invariant linear forms ω(α) = e(λ)μ dx with the tetrad metry provide a natural relation to the SC [5]. The Dirac version coefficients [6] of the geometry of similarity [7] is an efficient way to include 2 μ ν the conformal symmetry into the GR. In fact, the latter approach ds = gμν dx dx = ω(α)(d) ⊗ ω(β)(d)η(α)(β); allows to explain the SNeIa data without the inflation [8].Inthe η = Diag(1, −1, −1, −1). (2) present Letter, the Dirac formulation of the GR in the geometry of (α)(β) similarity is adapted to the diffeo-invariant Hamiltonian approach The geometry of similarity [6,7] means the identification of mea- by means of the spin connection coefficients in a finite space–time, sured physical quantities F (n), where (n) is the conformal weight, − developed in [9]. In this way we study a possibility to choose with their ratios in dilaton units e nD variables and their initial data that are compatible with the ob- ˜ (n) nD (n) ˜ 2 2D 2 servational data associated with the dark energy content. We find F = e F , ds = e ds . (3) integrals of motion of the metric and matter fields in terms of the We define the measurable space–time coordinates in the GR as variables distinguished by the conformal initial data. the scale-invariant quantities in the framework of the Dirac–ADM 4 = 1 + 3 foliation [10,11] ˜ 2 ˜ ˜ ˜ ˜ ds = ω(0) ⊗ ω(0) − ω(b) ⊗ ω(b), (4) Corresponding author. * μ E-mail address: [email protected] (V.N. Pervushin). where the linear forms ω˜ (α) = e˜(α)μ dx are 0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2010.06.042 A.B. Arbuzov et al. / Physics Letters B 691 (2010) 230–233 231 ˜ = −2D 0 R = [ − ] ω(0) e Ndx , (5) where ε(ab)(k) diag 1, 1, 0 in the orthogonal basis of spatial ± ˜ j 0 vectors [ε (1)(k), ε (2)(k), k]. Here, g are the holomorphic variables ω(b) = ω(b) + e(b) j N dx , (6) √ ofthesingledegreeoffreedom,ω = k2 is the graviton energy = i k ω(b) e(b)i dx . (7) normalized (like a photon in QED) on the units of a volume and j time Here N is the Dirac lapse function, N are the shift vector com- √ ponents, and e are the triads corresponding to the unit spatial (b)i ± 8π ± (3) g = g . (16) metric determinant |g˜ |≡|e e |=1. k 1/2 k ij (b) j (b)i M V (3) Planck 0 The Dirac dilaton D =−(1/6) log |g |=D+D,istaken ij The triad velocities in the Lichnerowicz gauge [12]. The Dirac lapse function N = 0 N −1 = −1 1 N0(x ) (τ , x) is split on the global factor N0 N which + l ⊥ ⊥ v(ab) = ω ∂0 − N ∂l + ∂(a)N + ∂(a)N (17) determines all time intervals used in the observational cosmol- N (a)(b) (b) (b) ogy: the redshift interval dτ = N dx0 [13], the conformal one + 0 depend on the symmetric forms , and the shift vector compo- = −2D = −D = −3D ω(ab) dη dτ e , and the world interval dt e dη dτ e . ⊥ −1 3 = In this case the dilaton zeroth mode D=V d xD (defined nents ∂(b)N(b) 0 are treated as the non-dynamical potentials. This 0 V 0 − in the finite diffeo-invariant volume) coincides with the logarithm means that the anti-symmetric forms ω(ab) are not dynamically in- of the redshift of spectral line energy Em dependent variables but are determined by a matter distribution. Following Dirac [10,16] one can define such a coordinate sys- D=log(1 + z) = log Em(η0 − η)/Em(η0) , (8) tem, where the covariant velocity v D of the local volume element and the momentum where η0 is the present-day conformal time interval, and η0 − η = r/c is the SNeIa distance. In accord with the new Poincaré group 2 l l P = 2v = ∂0 − N ∂l D + ∂l N /3 = 0 (18) classification, the “redshift” (8) is treated as one of the matter D D N components, on the equal footing with the matter. The key point of our approach is to express the GR action di- are zero. As a result, the dilaton deviation D can be treated as a rectly in terms of the redshift factor. The action can be represented static potential. The dilaton contribution to the curvature (11) with as a sum of the dilaton and the graviton terms: matter sources yield the Schwarzschild solution of classical equa- tions [exp{−7D/2}N ]=0 and exp{−D/2}=0. The solutions 2 {− }N = + {− }= − 0 (∂0 D ) −2D are exp 7D/2 1 rg /(4r) and exp D/2 1 rg /(4r) in W Hilbert = dx − + N0e Lg , (9) N0 the isotropic coordinates of the Einstein interval ds, where rg is the gravitation radius of a matter source. These solutions double v2 (3) 2D 3 2 (ab) −4D R the angle of the photon beam deflection by the Sun field, exactly Lg = e d x N −(v ) + − e . (10) D 6 6 as the Einstein’s metric determinant. Note that the GR theory pro- Here, vides also the Newtonian limit in our variables (see details in [9]). Furthemore, in empty space without a matter source (rg = 0),the 4 − N = = l = R(3) = R(3)(e) − e7D/2 e D/2, (11) mean field approximation ( 1, D 0, N 0) becomes exact. 3 If there are no matter sources one can impose the condition − + (3) = is the curvature, where R (e) is expressed via the spin-connection ω(a)(b) 0, since the kinetic term (17) depends only on ω(ab) com- coefficients ponents. In this case the curvature (11) takes the bilinear form + + ± 1 j j i i (3) = ω (∂(c)) = e ∂(c)e ± e ∂(c)e , (12) R (e) ω(ab)(∂(c))ω(ab)(∂(c)). (19) (ab) 2 (a) (b) (b) (a) j The variation of the Hilbert action with respect to the lapse and = ∂ [ei e ∂ ] is the Laplace operator. i (a) (a) j function leads to the energy constraint [17] The dependence of the linear forms 2 − ∂ D = ρ Ω + e 2 D H /V , (20) = i = − i τ cr D g 0 ω(b)(d) e(b)i dx dX(b) X(c)e(c) de(b)i (13) where the dilaton integral of motion ρcrΩD is added, ρcr = i i 2 2 on the tangent space coordinates X(b) ≡ dx e(b)i = x e(b)i by H0 MPl3/(8π) is the critical density, and means of the spin connection coefficients can be obtained by − = − e 4D R(3) virtue of the Leibniz rule AdB d(AB) (AB) d log(A) (in par- = 2D 3 N 2 + [ i ] T = [ i T ]− i [ T ] Hg e d x 3p(ab) (21) ticular d x ebi d x ebi x d ebi ). The difference between this 6 approach to gravitation waves and the accepted one [14,15] is that is the graviton Hamiltonian, p = v /3 is a canonical momen- the symmetry with respect to diffeomorphisms is imposed on spin (ab) (ab) tum (see Eq.
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