Nematic World Crystal Model of Gravity Explaining Absence of Torsion in Spacetime

Physics Letters A 324 (2004) 361–365 www.elsevier.com/locate/pla

Nematic world crystal model of gravity explaining absence of torsion in spacetime

H. Kleinert a,∗, J. Zaanen b

a Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany b Instituut-Lorentz for Theoretical Physics, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands Received 2 January 2004; received in revised form 4 March 2004; accepted 15 March 2004 Communicated by P.R. Holland

Abstract We attribute the gravitational interaction between sources of curvature to the world being a crystal which has undergone a quantum phase transition to a nematic phase by a condensation of dislocations. The model explains why spacetime has no observable torsion and predicts the existence of curvature sources in the form of world sheets, albeit with different high-energy properties than those of string models.  2004 Elsevier B.V. All rights reserved.

1. Introduction two decades. In 1987, he proposed a simple three- dimensional Euclidean world crystal model of gravita- Present-day string models of elementary particles tion in which dislocations and disclinations represent are based on the assumption that relativistic physics curvature and torsion in the geometry of spacetime [5]. will prevail at all energy scales and, moreover, show A full theory of gravity with torsion based on this pic- recurrent particle spectra at arbitrary multiples of the ture is published in the textbook [6] (see also [7,8]). Planck mass. Disappointed by the failure of these The simple 1987 model had the somewhat unaes- models [1] to explain correctly even the low-lying ex- thetic feature that the crystal possessed only second- citations, and the apparent impossibility of ever ob- gradient elasticity to deliver the correct forces between serving the characteristic recurrences, an increasing the sources of curvature, which for an ordinary first- number of theoreticians is beginning to suspect that gradient elasticity grow linearly with the distance R God may have chosen a completely different exten- and are thus confining. In this Letter we would like to sion of present-day Lorentz-invariant physics to ex- point out that the correct 1/R-behavior can also be ob- tremely high energies [2–4]. This philosophy has been tained in an ordinary world crystal with first-gradient advocated by one of the authors (H.K.) for almost elasticity by assuming that the dislocations have pro- liferated. This explains also why the theory of general relativity requires only curvature for a correct descrip- * Corresponding author. tion of gravitational forces, but no torsion. Such a state E-mail address: [email protected] (H. Kleinert). of the world crystal bears a close relationship with the

0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.03.048 362 H. Kleinert, J. Zaanen / Physics Letters A 324 (2004) 361–365 nematic quantum liquid crystals of condensed matter The densities satisfy the conservation laws physics, first suggested by Kivelson et al. [9], and be- ∂ α =− θ ,∂θ = 0. (1.6) lieved to be of relevance both for the quantum Hall i ik kmn mn i il effect [10] and in high-Tc superconductors [11]. Dislocation lines are either closed or they end in Our model will be formulated as before in three disclination lines, and disclination lines are closed. euclidean dimensions, for simplicity. The generaliza- These are Bianchi identities of the defect system. tion to four dimensions is straightforward. The elastic An important geometric quantity characterizing energy is expressed in terms of a material displace- dislocation and disclination lines is the incompatibility ment field ui (x) as or defect density p λ η (x) = ∂ ∂ u (x). (1.7) = 3 2 + 2 ij ikl jmn k m ln E d x µuij (x) uii(x) , (1.1) 2 It can be decomposed into disclination and dislocation where density as follows [6]: 1 1 η (x) = θ (x) + ∂ α (x) + (i ↔ j) uij (x) ≡ ∂i uj (x) + ∂j uj (x) (1.2) ij ij m min jn 2 2 − α (x) . (1.8) is the strain tensor and µ, ν are the elastic shear mod- ij n mn uli. The elastic energy goes to zero for infinite wave This tensor is symmetric and conserved length since in this limit u (x) reduces to a pure trans- i ∂ η (x) = 0, (1.9) lation under which the energy of the system is in- i ij variant. The crystallization process causes a sponta- again a Bianchi identity of the defect system. neous breakdown of the translational symmetry of the It is useful to separate from the dislocation density system. The elastic distortions describe the Nambu– (1.4) the contribution from the disclinations which Goldstone modes resulting from this symmetry break- causes the nonzero right-hand side of (1.6). Thus we down. Note that so far the crystal has an extra longi- define a pure dislocation density tudinal sound wave with a different velocity than the αb (x) ≡ α (x) − αΩ (x), (1.10) shear waves. ij ij ij b = A crystalline material always contains defects. In which satisfies ∂i αij 0. Accordingly, we split their presence, the elastic energy depends only on the = b + Ω difference of the total distortion from the so-called ηij (x) ηij (x) ηij (x), (1.11) p p plastic distortion ui (x).Ifuij (x) denotes the plastic where part of the strain tensor, the energy reads 1 ηb (x) = αb (x) + (i ↔ j)− αb (x) , ij min jn ij n mn 3 p 2 λ p 2 2 E = d x µ uij − u + uii − u . (1.3) (1.12) ij 2 ii and the pure disclination part of the defect tensor looks In general, the crystal may contain a grand-canoni- like (1.8), but with superscripts Ω on ηij and αij . cal ensemble of line-like defects with a dislocation The tensors αij ,θij ,andηij are linearized versions density of important geometric tensors in the Riemann–Cartan space of defects, a non-Euclidean space with curvature = p = ; + αil ij k ∂j ∂kul (x) δi(x L)(bl lqrΩq xr ), (1.4) and torsion. Such a space can be generated from a flat and a disclination density space by a plastic distortion, which is mathematically represented by a nonholonomic mapping [7,8] xi → p p θil = ij k ∂j pmn ∂num(x) − ∂mun(x) xi + ui (x). Such a mapping is nonintegrable. The displacement fields and their first derivatives fail to = δi(x; L)Ωl, (1.5) satisfy the Schwarz integrability criterion: where bl and Ωl are the so-called Burgers and Franck ; ≡ ¯ −¯ (∂i∂j − ∂j ∂i)u(x) = 0, vectors of the defects and δi(x L) L dxi δ(x x) are δ-functions on the lines L. (∂i∂j − ∂j ∂i)∂kul(x) = 0. (1.13) H. Kleinert, J. Zaanen / Physics Letters A 324 (2004) 361–365 363 1 ν The metric and the affine connection of the geometry Estress + Edef ≡ d3x G2 − G2 4µ ij 1 + ν ii in the plastically distorted space are gij = δij + ∂i uj + ∂j ui and Γij l = ∂i∂j ul, respectively. The + ihij ηij , (1.15) noncommutativity of the derivatives in front of ul(x) implies a nonzero torsion, the torsion tensor being where the defect tensor (1.8) has the decomposition Sij k ≡ (Γij k − Γjik)/2. The dislocation density αij is = = Ω + b equal to αij iklSklj . ηij ηij ∂mminαjn. (1.16) The noncommutativity of the derivatives in front The defects have also core energies which has been ig- of ∂ u (x) implies a nonzero curvature. The discli- k l nored so far in this continuum formulation of defects. nation density θij is the Einstein tensor θij = Rji − 1 They can properly been taken into account only in a 2 gjiR of this Einstein–Cartan defect geometry. The lattice formulations. It has been shown in the textbook tensor ηij , finally, is the Belinfante symmetric energy– [6] that these give rise to leading quadratic terms in momentum tensor, which is defined in terms of the dislocation and disclination densities. If we focus at- canonical energy–momentum tensor and the spin den- tention on the dislocation part of the defect density sity by a relation just like (1.8). For more details on (1.16), which is relevant for the phase transition to be the geometric aspects see Part IV in Vol. II of [6], studied, their fluctuations are governed by an energy where the full one-to-one correspondence between defect systems and Riemann–Cartan geometry is devel- disl 3 b c b 2 E = i d x imn∂mhij α + α . (1.17) oped as well as a gravitational theory based on this jn 2 jn analogy. We now assume that the world crystal has undergone Let us now show how linearized gravity emerges a transition to a phase in which dislocations are con- from the energy (1.3). For this we eliminate the densed. Actually, to reach such a state, whose exis- jumping surfaces in the defect gauge fields from the tence was conjectured for two-dimensional crystals partition function by introducing conjugate variables in Ref. [12], without a simultaneous condensation of and associated stress gauge fields. This is done by disclinations in a first-order melting transition, the rewriting the elastic action of defect lines as model requires a modification by an additional higher- gradient rotational energy. This was shown in [13] and 1 ν verified by Monte Carlo simulations in [14]. The three- E = d3x σ 2 − σ 2 4µ ij 1 + ν ii dimensional extension of the extended model is described in [6]. The expressions are to long to be writ- p ten down in the Letter, and their explicit forms are not + iσij uij − u , (1.14) ij relevant for the present discussion. Here we only need the outcome of such a construction that it is possible ≡ + where ν λ/2(λ µ) is Poisson’s ratio, and form- to have an elastic plus plastic energy which allows for ing the partition function by integrating the Boltz- a phase transition in which only dislocations condense −E/kB T mann factor e over σij , ui , and summing while the disclinations remain dilute. over all jumping surfaces S in the plastic fields. The condensed phase is described by a partition The integrals over ui yield the conservation law function in which the discrete sum over the pure ∂i σij = 0. This can be enforced as a Bianchi iden- b dislocation densities in αjn is approximated by an tity by introducing a stress gauge field hij and writ- ordinary functional integral. This has been shown in = ≡ ing σij Gij ikljmn∂k∂mhln. The double curl Ref. [7]. The general integration rule is on the right-hand side is recognized as the Einstein 2 tensor in the geometric description of stresses, ex- −βl2 −a d3lδ(∂ · l) exp + ila = exp T , pressed in terms of a small deviation hij ≡ gij − δij 2 2β of the metric from the flat-space form. Inserting Gij (1.18) into (1.14) and using (1.7), we can replace the en- where aT has the components ergy in the partition function by E = Estress + Edef 2 where aTi ≡−iij k ∂j ak/ −∂ . 364 H. Kleinert, J. Zaanen / Physics Letters A 324 (2004) 361–365

The Boltzmann factor resulting in this way from In the non-relativistic context, a dislocation con- Estress plus (1.17) has now the energy densate is characteristic for a nematic liquid crystal, whose order is translationally invariant, but breaks ro- 1 ν E = d3x G2 − G2 tational symmetry (see [6,12] in the two dimensions 4µ ij 1 + ν ii + and [15] in the (2 1)-dimensional quantum theory). 1 1 The Burgers vector of a dislocation is a vectorial topo- + G G . (1.19) ij 2 ij logical charge, and nematic order may be viewed as an 2c −∂ ordering of the Burgers vectors in the dislocation con- The second term implies a Meissner-like screening densate. Such a manifest nematic order would break of the initially confining gravitational forces between the low energy Lorentz-invariance of spacetime. We the disclination part of the defect tensor to Newton- may, however, imagine that the stiffness of the direc- like forces. For distances longer than the Planck scale, tional field of Burgers vectors is so low that, by the cri- we may ignore the stress term and find the effective terion of Ref. [16], they have undergone a Heisenberg- gravitational action for the disclination part of the type of phase transition into a directionally disordered defect tensor: phase in an environment with only a few disclinations. In three dimensions, dislocations (and disclinations) ≈ 3 1 1 + Ω E d x Gij 2 Gij ihij ηij . (1.20) are line-like. This has the pleasant consequence, that 2c −∂ they can be described by the disorder field theories de- A path integral over hij and a sum over all line veloped in [17] in which the proliferation of disclina- − ¯ ensembles applied to the Boltzmann factor e E/h is a tions follows the typical Ginzburg–Landau pattern of simple Euclidean model of pure quantum gravity. The the field expectation acquiring a nonzero expectation Ω line fluctuations of ηij describe a fluctuating Riemann value. A cubic interaction becomes isotropic in the geometry perforated by a grand-canonical ensemble continuum limit [18] (this is the famous fluctuation- arbitrarily shaped lines of curvature. As long as the induced symmetry restoration of the Heisenberg fixed loops are small they merely renormalize the first term point in a φ4-theory with O(3)-symmetric plus cubic in the energy (1.20). Such effects were calculates in interactions [19]). The isotropic phase is similar to the closely related theories in great detail in Ref. [21]. topological form of nematic order identified by Lam- They also give rise to post-Newtonian terms in the mert et al. [20] as the Coulomb phase in the gener- above linearized description of the Riemann space. alized Z2 gauge theory of nematic order: rotational We may now add matter to this gravitational en- (Lorentz) invariance is restored even though there is no vironment. It is coupled to hij by the usual Einstein condensate of disclinations. A similar isotropic phase interaction is also found in the theory of non-relativistic elasticity in 2 + 1 dimensions [15] where one learns that int 3 ij E ≈ d xhij T , (1.21) any microscopic rotational anisotropy will render the topological order unstable towards a full nematic or- where T ij is the symmetric Belinfante energy–mo- der. The world crystal should certainly lose all infor- mentum tensor of matter. Inserting for Gij the double- mation on its crystal axes. curl of hij we see that the energy (1.20) produces the The generalization to four Euclidean spacetime di- correct Newton law if the core energy is c = 8πG, mensions changes mainly the geometry of the defects. where G is Newton’s constant. In four dimensions, they become world sheets, and a Note that the condensation process of dislocations second-quantized disorder field description of surfaces has led to a pure Riemann space without torsion. has not yet been found. But the approximation of rep- Just as a molten crystal shows residues of the origi- resenting a sum over dislocation surfaces in the prolif- nal crystal structure only at molecular distances, rem- erated phase as an integral as in Eq. (1.18) will remain nants of the initial torsion could be observed only near valid, so that the above line of arguments will survive, the Planck scale. This explains why present-day gen- this being a natural generalization of the Meissner– eral relativity requires only a Riemann space, not a Higgs mechanism. The disclination sources of curva- Riemann–Cartan space. ture will be world sheets, as an attractive feature for H. Kleinert, J. Zaanen / Physics Letters A 324 (2004) 361–365 365 string theorists. However, the high-energy properties [7] H. Kleinert, Theory of fluctuating nonholonomic fields and will be completely different. On the one hand, these applications: statistical mechanics of vortices and defects and surfaces behave nonrelativistically as the energies ap- new physical laws in spaces with curvature and torsion, in: Proceedings of NATO Advanced Study Institute on Formation proach the Planck scale, on the other hand, they will and Interaction of Topological Defects at the University of not have the characteristic multi-Planck recurrences of Cambridge, Plenum, New York, 1995, pp. 201–232, cond- the common strings. Although the latter property may mat/9503030. never be verified in the laboratory, the deviations from [8] H. Kleinert, Gen. Relativ. Gravit. 32 (2000) 769, http://www. relativity at high energies or short distances may come physik.fu-berlin.de/~kleinert/258/258j.pdf. [9] S.A. Kivelson, E. Fradkin, V.J. Emery, Nature 393 (1998) 550. into experimentalists reach in the possibly distant fu- [10] E. Fradkin, S.A. Kivelson, E. Manousakis, K. Nho, Phys. Rev. ture. Lett. 84 (2000) 1982. 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