Brazilian Journal of Physics, vol. 35. no. 2A, June, 2005 359

Emerging from Defects in World Crystal

H. Kleinert Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, D14195 Berlin Received on 25 January, 2005 I show that Einstein Gravity can be thought of as arising from the defects in a world crystal whose lattice −33 spacing is of the order of the Planck length lP ≈ 10 cm, and whose elastic energy is of the second-gradient type (floppy crystal). No physical experiment so far would be able to detect the lattice structure.

I. INTRODUCTION Volterra surfaces via δ-functions on these surfaces [9Ð11]. The energy density is invariant under the single-valued defect One of the most important features of string theories is that gauge transformations they assume the validity of Lorentz invariance for all energies p → p +(∂ λ + ∂ λ )/ , → + λ . in the trans-Planckian regime [1]. In this lecture we would uµν uµν µ ν ν µ 2 uµ uµ µ (4) like to point out that an entirely different scenario is possi- Physically, they express the fact that defects are not affected ble. It could be that we live in a world crystal with a lattice by elastic distortions of the crystal. Only multivalued gauge constant of the order of the Planck length, without being able p functions λ would change the defect content in u ν. to notice this. None of the present-day relativistic physical µ µ We now rewrite the action (5) in a canonical form introduc- laws would have to be observably violated. The gravitational ing an auxiliary symmetric stress tensor field σ ν as forces could arise from variants of ordinary elastic forces in µ   this world crystal, and the observed in gravitational Z 4 1 µν µν p spacetime could be just a signal of the presence of disclina- A = d x σµνσ + iσ (uµν − uµν) . (5) 4µ tions in the world crystal. Matter would be sources of discli- nations. After a partial integration and extremization in u µ, the middle terms yield the equation

µν II. PURE GRAVITY ∂νσ = 0. (6)

For simplicity, we shall present such a construction only for This may be guaranteed identically, as a Bianchi identity, by a system without torsion [2]. The idea goes back to a 1987 lec- an ansatz ture of mine held in the Einstein house in Caputh [3]. Present κλσ κλτ σ = ε ε ∂ ∂  χ . interest in Emerging Gravity [4Ð7] instigated me to revitalize µν µ ν λ λ στ (7) it. The field χστ plays the role of an elastic gauge field. It is If the world crystal is distorted by an infinitesimal displace- ( ) ment field the analog of the vector potential A x in magnetism. Inserting (7) into (5), we obtain µ → µ = µ + µ( ),  x x x u x (1) Z   2 4 1 µκλσ νκλτ A = d x ε ε ∂λ∂λ χστ it has a strain energy 4µ 

Z νκλσ µκλτ p µ 4 2 +iε ε ∂λ∂λ χστuµν . (8) A = d x(∂µuν + ∂νuµ) , (2) 4 where µ is some elastic constant. If part of the distortions are A further partial integration brings this to the form  of the plastic type, the world crystal contains defects defined Z   2 4 1 µκλσ νκλτ by Volterra surfaces, where crystalline sections have been cut A = d x ε ε ∂λ∂λ χστ out. The displacement field is multivalued, and the action (2) 4µ  is the analog of the magnetic action in the presence of a cur- σκλν τκλ p + χ ε ε µ∂ ∂  . rent loop [8]. In order to do field theory with this action, we i στ λ λ uµν (9) have to make the displacement field single-valued with the help of δ-functions describing the jumps across the Volterra This is a double-gauge theory invariant under the defect gauge surfaces, by complete analogy with the gradient representa- transformation (4) and under stress gauge transformations tion of the magnetic energy [8]:

χστ → χστ + ∂σΛτ + ∂τΛσ. (10) Z 4 p 2 = µ d x(u ν − u ν) , (3) A µ µ This can be rewritten as

  Z where uµν ≡ (∂µuν + ∂νuµ)/2 is the elastic strain tensor, and 4 1 µν µν p A = d x σµνσ + iχµνη , (11) uµν the gauge field of plastic deformations describing the 4µ 360 H. Kleinert

where ηµν is the four-dimensional defect density and the interaction (18) of an arbitrary distribution of defects would become κλσ κλτ p ηµν = εµ εν ∂λ∂λ uστ. (12) Z 4 p µν A = µ d xu µν(x)η (x). (21) It is invariant under defect gauge transformations (4), and sat-

isfies the conservation law This coincides with the linearized Einstein action Z µν 1 √ ∂νη = 0. (13) A = − d4x −gRø (22) 2κ p We may now replace uστ by half the metric field gµν, and where κ is the gravitational constant. Indeed, in the linear the tensor ηµν becomes the Einstein tensor associated with the ν ν ν ν approximation gµ = δµ + hµ with |hµ |1, where the metric tensor gµν. Christoffel symbols can be approximated as Let us eliminate the stress gauge field from the action (11).   For this we rewrite the stress field (7) as λ 1 Γø µν ≈ ∂µhνλ + ∂νhµλ − ∂λhµν , (23) κλσ κλτ 2 σ ν = ε εν ∂λ∂λ χστ µ µ the Riemann curvature tensor becomes 2 λ λ λ = −(∂ χ ν + ∂ ∂νχλ − ∂ ∂λχ − ∂ν∂λχ ) µ µ µ µ µ 1 +η (∂2χ λ − ∂ ∂ χλκ). Rø νλκ ≈ ∂ ∂λhνκ − ∂ν∂κh λ − (µ ↔ ν) . (24) µν λ λ κ (14) µ 2 µ µ φ ν ≡ χ ν − 1 δ νχ λ This gives the Ricci tensor Introducing the field µ µ 2 µ λ , and going to the µ ν Hilbert gauge ∂ φµ = 0, the stress tensor reduces to 1 2 Røµκ ≈ (∂µ∂λhλκ + ∂κ∂λhλµ − ∂µ∂κh − ∂ hµκ), (25) 2 2 σµν = −∂ φµν, (15) λ where h is defined to be the trace of the tensor h µν, i.e. h ≡ hλ . and the action of an arbitrary distribution of defects would The ensuing scalar curvature reads become   ø ≈−(∂2 − ∂ ∂ µν) Z R h µ νh (26) 1 ν ν λ = 4 ∂2φµ ∂2φ + φ (ηµ − 1 δµ η ) . A d x µν i µ ν 2 ν λ (16) 4µ so that the Einstein tensor becomes

φµν 1 Eliminating the field yields the interaction of an arbitrary Gø µκ = Røµκ − gµκRø (27) distribution of defects 2

1 λ λ Z 2 ≈−(∂ hµκ + ∂µ∂κh − ∂µ∂λh κ − ∂κ∂λh µ) 4 µ µ λ 1 ν ν λ = µ d x(η ν − 1 δ νη λ) (η − 1 δ η λ). (17) 2 A 2 (∂2)2 µ 2 µ 1 2 νλ + η κ(∂ h − ∂ν∂λh ). 2 µ This is not the Einstein action for a Riemann spacetime. It would be so if the derivatives ∂2 in (16) would be replaced by This can be written as a four-dimensional version of a double ∂. Then the Green function of (∂2)2 would be replaced by the curl Green function of −∂2. An index rearrangement would lead 1 νλ δστ to the interaction Gø κ = ε δ εκ ∂ν∂σhλτ. (28)

µ 2 µ Z λ 1 ν = 4 (ηµ − 1 δµ η ) η . Thus the Einstein-Hilbert action has the linear approxima- A µ d x ν 2 ν λ µ (18) −∂2 tion

η Z The defect tensor µν is composed of the plastic gauge fields 1 4 µν p d xhµνG . (29) uµν in the same way as the stress tensor is in terms of the stress 4κ gauge field in Eq. (14): Recalling the previously established identifications of plastic κλσ κλτ p field and defect density with metric and Einstein tensor, re- ηµν = εµ εν ∂λ∂λ uστ. spectively, the interaction between defects (21) is indeed the = −(∂2 p + ∂ ∂ pλ − ∂ ∂ pλ − ∂ ∂ pλ) uµν µ νuλ µ λuµ ν λuµ linearized version of the Einstein-Hilbert action (22), if we 2 pλ pλκ identify the constant µ with 1/4κ. +ηµν(∂ uλ − ∂λ∂κu ). (19) The world crystal with the elastic energy (5) does not lead pν ≡ pν − 1 δ ν pλ to this action. It must be modified to do so. A first modifi- If we introduce the auxiliary field wµ uµ 2 µ uλ and µ p cation is to introduce two more derivatives and assign to the chose the Hilbert gauge ∂ w ν = 0, the defect density reduces µ crystal the higher-gradient elastic energy

to Z ν ν λ  4 p 2 η = −∂2 p , η − 1 δ η = −∂2 p . A = µ d x[∂(uµν − uµν)] . (30) µν wµν µ 2 µ λ uµν (20) Brazilian Journal of Physics, vol. 35. no. 2A, June, 2005 361

This removes one power of −∂2 from the denominator in the by all particles in the world crystal which are tightly bound to interaction (17). the atoms, and move from site to site only by small tunneling In order to obtain the correct contractions in (18), we must amplitudes. Such particles see precisely the geometry gener- replace the action (16) by ated by the defects.  Z   1 ν ν = 4 − φµ ∂2φ − 1 φ µ∂2φ A d x µν 2 µ µ 4µ  IV. SUMMARY ν λ + φ (ηµ − 1 δµ η ) . i ν ν 2 ν λ (31) We have successfully recovered Einstein gravity from a de- This, in turn, follows from an interaction energy fect model of a crystal in which the leading elastic terms van- ish. For simplicity, this has been done in linearized approxi- Z 4 p 2 µ pν 2 mation only, but the generalization to a full nonlinear theory = µ d x [∂(u ν − u ν)] − 1 [∂(u − u )] . (32) A µ µ 2 µ µ presents no fundamental problem. We do not claim the model to really represent reality but view it only as an illustration of possible short-distance physics in the Planck regime which III. MATTER is completely different from what presently fashionable string models suggest. Both scenarios have in common that there So far, the world crystal contains only the analog of the is no danger of contradicting experiments in that regime for a gravitational field. What is matter in this model? It is supplied long time to come.

[1] For a summary of failures, see: T. Banks, A Critique of Pure Physics, and Financial Markets, third extended edi- : Heterodox Opinions of Diverse Dimensions, tion (World Scientific, Singapore, 2004), pp. 1Ð1450; (hep-th/0306074). (http://www.physik.fu-berlin.de/˜kleinert/b5). [2] The generalization to spaces with torsion is obtained by [9] H. Kleinert, Gauge fields in Condensed Matter, adding one more gradient into the elastic energies of the Vol. I: Superflow and Vortex Lines, Disorder Fields, Phase model in H. Kleinert, Lattice Defect Model with Two Suc- Transitions, Vol. II: Stresses and Defects, Differential Geom- cessive Melting Transitions, Phys. Lett. A 130, 443 (1988); etry, Crystal Defects, (World Scientific, Singapore, 1989). (http://www.physik.fu-berlin.de/˜kleinert/174). [10] H. Kleinert, Theory of Fluctuating Nonholonomic Fields and [3] H. Kleinert, Gravity as Theory of Defects in a Crystal with Only Applications: Statistical Mechanics of Vortices and Defects Second-Gradient Elasticity, Ann. d. Physik, 44, 117(1987); and New Physical Laws in Spaces with Curvature and Torsion, (http://www.physik.fu-berlin.de/˜kleinert/172). in: Proceedings of NATO Advanced Study Institute on Forma- [4] G.E. Volovik, Phys. Rep. 351, 195 (2001), (gr-qc/0005091); tion and Interaction of Topological Defects at the University Phenomenology of effective gravity, in: Patterns of Symmetry of Cambridge (Plenum Press, New York, 1995), pp. 201Ð232, Breaking, H. Arodz et al. (eds.) (Kluwer Academic Publishers, (cond-mat/9503030). NY, 2003), p. 381, (gr-qc/0304061). [11] H. Kleinert, Nonholonomic Mapping Principle for Clas- [5] G. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, sical and Quantum Mechanics in Spaces with Cur- Quantum Phase Transitions and the Breakdown of Classical vature and Torsion,, Gen. Rel. Grav. 32 , 769 (2000); General Relativity, gr-qc/0012094. (http://www.physik.fu-berlin.de/˜kleinert/258/258 [6] S.Z. Zhang, To see a world in a grain of sand, (hep- j.pdf). th/0210162). [12] B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978). [7] H. Kleinert and J. Zaanen, Crystal Model of Gravity explain- [13] W. Janke and H. Kleinert, From First-Order to Two Contin- ing Absence of Torsion and the World Sheet Nature of Matter, uous Melting Transitions - Monte Carlo Study of New 2D Berlin preprint (June 2003). Lattice Defect Model, Phys. Rev. Lett. 61(20), 2344 (1988); [8] See Appendix 10A in the textbook: H. Kleinert, Path (http://www.physik.fu-berlin.de/˜kleinert/179). Integrals in Quantum Mechanics, Statistics, Polymer