arXiv:1107.2610v1 [gr-qc] 13 Jul 2011 hc yields which ihtehl faDrcdlafunction Dirac-delta expressed a be of properly help more the should is with but equation Poisson zero, the strictly of not side right-hand the Hence rem. iseneuto ntevacuum, the in equation Einstein oe iil ypromn pta nerloe ∆ over integral spatial be- a This performing singularity. by the visible size of comes strength whose pole charges, the from electric tracted pointlike form a the has to them of simplest The h otiilsltoso h oso qainfrthe for equation from Poisson originating the potential as of electric considered solutions be nontrivial may the nature in fields a of explanation possible are it. we simplest there of note the part this and In propose universe, composition. to the its want of to as energy speculations the mass roughly many of the constitutes evolution of matter the 83% dark invisible of that of model indicates from amounts Friedmann universe distance large The versus to attributed velocities matter. was galaxy, orbital center a of inside the plot mea- stars of from a of motion came where orbital velocities indications the orbital Later of the surements clusters. in in mass” galaxies “missing the explain l otiilsltosi h omo h Schwarzschild the of by form defined the metric in solutions nontrivial ple eu iseneuto,t n ozr eut namely result, nonzero a find here to Also homoge- equation, the Einstein over neous integral metric. spacetime the Kerr calculate may the we generalization, rotating its with ds o eeta bet,testaini ut iia.The similar. quite is situation the objects, celestial For sawr-p e srmme htalsai electric static all that remember us let warm-up, a As to 1933 in Zwicky F. by postulated was matter Dark 2 = − r S g r ≡ 2 µν ( dθ dx 2 ASnmes 95.35+d.04.60,04.20C,04.90 Einst numbers: of PACS quanta gove the is like physics co quantum interact physical its particles actio and new spin-2 nature, massless a the geometric in with purely here a coincides of appear It which dimensions, fluctuations. spacetime quantum their also of solutions the in GM/c − 2 µ eageta ato dr atr sntmd fmte,bto but matter, of made not is matter” “dark of part that argue We 4 sin + dx ∆ πe ntttfu hoeicePyi,FeeUniversit¨at Ber Freie Physik, f¨ur Theoretische Institut φ ν Z φ fe pligGussitga theo- integral Gauss’s applying after , ( ( 2 (1 = x x 2 d ): en h cwrshl ais or radius, Schwarzschild the being = ) ∆ θdϕ 3 G x φ − ( − 2 x 0 ) h ueyGoercPr f“akMatter”– “Dark of Part Geometric Purely The 0 = ) r 4 0 , S πeδ CAe izaedlaRpblc,1 612 ecr,Ita Pescara, -65122, 10 Repubblica, della Piazzale ICRANeT = /r isensvcu edequation field vacuum Einstein’s rs lyrudfr“tigTheory” “String for Playground Fresh A κcM, ) (3) . c G 2 ( dt µν e/r x 2 ) . − ,psesssim- possesses 0, = n sattributed is and , δ (1 (3) − ( e x r as ) a eex- be can S /r ) ae Kleinert Hagen − φ 1 ( dr (2) (1) (3) (4) x ), 2 isentno,the tensor, a Einstein denotes prime a where G spootoa otecasclato fapitlk par- point-like a of action classical ticle: the to proportional is form ae nsaeie hs a eprmtie by parametrized be may These sur- on spacetime. x singularities in possess also faces will equation also Einstein define particle. to point used spin-0 be path a sin- physics can a a quantum spacetime for in the in action [1] line Einstein’s world particle Thus gular spin-0 orbits. clas- all a over same of integral the also physics describes is quantum orbits, that fluctuating the (8), for of different parti- but modification point-like sically, slight a A of action cle. classical the to proportional etnsconstant Newton’s where rt h soitdslto ftehmgnosEin- homogeneous the of equation solution stein the associated the denotes grate dot a where x rjcoyprmtie by parametrized trajectory oetmtensor momentum rm()w dniytems fteojc sbeing as object the of mass the identify we (4) From ˙ T µ 2 µν nadto opitiesnuaiis h homogeneous the singularities, pointlike to addition In ftems on oe hog pctm ln a along spacetime through moves point mass the If ( µν ,ta htisEnti-ibr action Einstein-Hilbert its that that 1, = ,τ σ, T .TerEnti-ibr cingoverns action Einstein-Hilbert Their 0. = ( i’ theory. ein’s ∗ µν y κ ,adtereeg-oetmtno a the has tensor energy-momentum their and ), ) ( i,115Bri,Grayand Germany Berlin, 14195 lin, ∝ stegaiainlcntn endi em of terms in defined constant gravitational the is ndb h aesrn hoy whose theory, string same the by rned y fcoe ooi srns nfour in “strings” bosonic closed of n = ) tx.Tu,pr fdr atris matter dark of part Thus, ntext. Z −∞ ∞ A M κ A G EH dσdτ EH worldline Z ≡ µν h iglrworld-surfaces singular the f −∞ = ∞ 8 G vrsaeie n n,using find, and spacetime, over 0 = πl δ ˙ ( N − fnto ntesraelast a to leads surface the on -function dτ x P 2 rtePac length Planck the or , µ 2 / 1 x x κ ˙ ˙ − ∝ ~ µ σ ν ly Z 8 = ( drvtv.I h associated the In -derivative. − τ τ drvtv.W a inte- may We -derivative. x ˙ ) d Mc x x ′ 4 πG µ µ ν x x ( ( √ Z τ τ ′ N ν ,i a nenergy- an has it ), ) − /c ) δ ds, δ gR, (4) (4) 3 . ( ( y y − − x x ( l ( P τ ,τ σ, )) as , , )) , M (9) (7) (8) (6) (5) . 2 volume integral [4]: standard model explain an important part of the matter in the Friedmann model of cosmological evolution. 4 µ 2 ′ 2 ′2 ′2 d x√ g Gµ d a dσdτ (xx ˙ ) x˙ x . (10) But the main contribution to the energy comes from Z − ∝Z ≡Z p − the above singularities of Einstein’s equation. Soon after By analogy with the line-like case we obtain for such a the universe was created, the temperature was so high singular field an Einstein-Hilbert action (7) that the configurational entropy of the surfaces over- whelmed completely the impeding Boltzmann factors. ~ worldsurface 1 2 2 Spacetime was filled with these surfaces in the same way EH d a = 2 d a. (11) A ∝−2κ Z −16πlP Z as superfluid helium is filled with the world-surface of vortex lines. In hot helium, these lie so densely packed Apart from a numerical proportionality factor of order that the superfluid behaves like a normal fluid [7, 8]. one, this is precisely the Nambu-Goto action of a bosonic The Einstein-Hilbert action of such a singularity-filled closed string in four spacetime dimensions: turbulent geometry behaves like the action of a grand- ~ canonical ensemble of world surfaces of a bosonic closed- 2 NG = d a, (12) string model. Note that here these are two-dimensional A −2πl2 Z s objects living in four spacetime dimensions, and there is where ls is the so-called string length ls, related to the definite need to understand their spectrum by studying ′ ′ string tension α by ls = ~c√α . Note that in contrast to the associated Polyakov action, without circumventing the world lines, there is no extra mass parameter M. the accompanying Liouville field by escaping into unphys- The original string model was proposed to describe ical dimensions color-electric flux tubes and their Regge trajectories It should be noted that in the immediate neighbor- whose slope dl/dm2 = α′ lies around 1 GeV−2. How- hood of the singularities, the curvature will be so high, ever, since the tubes are really fat objects, as fat as pi- that Einstein’s linear approximation (1/2κ)R to the La- − ons, only very long flux tubes are approximately line-like. grangian must break down and will have to be corrected Short tubes degenerate into spherical “MIT-bags” [5]. by some nonlinear function of R, that starts out like Ein- The flux-tube role of strings was therefore abandoned, stein’s, but continues differently. A possible modification and the action (12) was re-interpreted in a completely has been suggested a decade ago [9], and many other op- different fashion, as describing the fundamental particles tions have been investigated since then [10]. of nature, assuming lS to be of the order of lP. Then After the big bang, the universe expanded and cooled the spin-2 particles of (12) would interact like gravitons down, so that large singular surfaces shrunk by emitting and define Quantum Gravity. But also the ensuing “new gravitational radiation. Their density decreased, and string theory” [2] has been criticized by many authors some phase transition made the cosmos homogeneous and [6]. One of its most embarrassing failures is that it has isotropic on the large scale [11]. But it remained filled not produced any experimentally observable results. The with gravitational radiation and small singular surfaces particle spectra of its solutions have not matched the ex- that had shrunk until their sizes reached the levels sta- isting particle spectra. The proposal of this note cures bilized b quantum physics, i.e., when their fluctuating this problem. If “strings” describe “dark matter”, there action decreased to order ~. The statistical mechanics of would be no need to reproduce other observed particle this cosmos is the analog of a spacetime filled with super- spectra. Instead, their celebrated virtue, that their spin- fluid helium whose specific heat is governed by the zero- 2 quanta interact like gravitons, can be used to fix the mass phonons and by rotons. Recall [12], that in this way proportionality factor between the Einstein action action Landau discovered the fundamental excitations called ro- (11), and the string action (12). tons, whose existence he deduced from the temperature It must be kept in mind that just as Mc ds had behavior of the specific heat. In the universe, the role to be modified for fluctuating paths [1], also− theR Nambu- of rotons is played by the smallest surface-like singular- Goto action (12) needs a modification, if the surfaces ities of the homogeneous Einstein equation, whose ex- fluctuate. That was found by Polyakov when studying istence we deduce from the cosmological requirement of the consequences of the conformal symmetry the theory. dark matter. He replaced the action (12) by a new action that is equal The situation can also be illustrated by a further anal- to (12) at the classical level, but contains in D = 26 ogy with a many-body system. The defects in a crystal 6 dimensions another spin-0 field with a Liouville action. whose “atoms” have a lattice spacing lP simulate pre- Since the singularities of Einstein’s fields possess only cisely the mathematics of a Riemann-Cartan spacetime, gravitational interactions, their identification with “dark in which disclinations and dislocations define curvature matter” seems very natural. All visible matter consists and torsion [8, 13, 14]. Thus we may imagine a model of singular solutions of the Maxwell equations and the of the universe as a “floppy world crystal” [15], a liquid- field equations of the standard model. A grand-canonical crystal-like phase [16] in which a first melting transition ensemble of these and the smooth wave solutions of the has led to correct gravitational 1/r-interactions between 3 disclinations. The initial hot universe was filled with [4] For this one takes the delta function on the surface defects—it was a “world-liquid”. After cooling down to as defined by P.A.M. Dirac, Phys. Rev. 74, 817830 the present liquid-crystal state, there remained plenty of (1948) [see his Eq. (15)], and makes use of the ditribu- residual defects around, which form our “dark matter”. tional form of Gauss’s integral theorem as formulated in H. Kleinert, Int. J. Mod. Phys. A 7, 4693 (1992) We know that the cosmos is filled with a cosmic mi- (http://klnrt.de/203/203.pdf), or on p. 253 of the crowave background (CMB) of photons of roughly 2.725 textbook [8]. Kelvin, the remnants of the big bang. They contribute to [5] L. Vepstas and A.D. Jackson, Physics Reports 187, 109 2 the Friedmann equation of motion a constant Ωradh = (1990). (2.47 0.01) 10−5, where h = 0.72 0.03 is the Hub- [6] See the list of critics in the Wikipedia article on string ble parameter,± × defined in terms of the± Hubble constant theory, also B. Schroer, “String theory deconstructed”, (arXiv:hep-th/0611132). H by h H/(100 km/Mpc sec). The symbol Ω denotes ≡ [7] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I the energy density divided by the so-called critical den- Superflow and Vortex Lines, World Scientific, Singapore, 2 −26 2 3 sity ρc 3H /8πGN = 1.88 10 h kg/m [17]. The 1989, pp. 1–744 (http://klnrt.de/b1) ≡ × 2 baryon density contributes Ωradh =0.0227 0.0006, or [8] H. Kleinert, Multivalued Fields, World Scientific, Singa- 720 times as much, whereas the dark matter± contributes pore, 2008, pp. 1–497 (http://klnrt.de/b11) 2 [9] H. Kleinert and H.-J. Schmidt, Gen. Rel. Grav. 34, 1295 Ωdarkh =0.104 0.006, or 4210 as much. If we assume for a moment that± all massive strings are frozen out, and (2002) (klnrt.de/311/311.pdf). [10] See the review by T.P. Sotiriou and V. Faraoni, Rev. that only the subsequently emitted gravitons form a ther- Mod. Phys. 82, 451 (2010) (arXiv:0805.1726). mal background [18] then, since the energy of massless [11] In this aspect there are parallels with Kerson Huang 4 states is proportional to T , the temperature of this back- and collaborators in K. Huang, H.-B. Low, R.-S. Tung, 1/4 ground would be TDMB 4210 8TCMB 22K. In (arXiv:1106.5282v2), (aXiv:1106.5283v2). However, their general we expect the presence≈ of also≈ the other≈ singular turbulent baby universe is filled with tangles of vortex solutions of Einstein’s equation to change this result. lines of some scalar field theory, whereas mine contains only singularities of Einstein’s equation. A bridge may There is an alternative way of deriving the above- be found by recalling that the textbook [7] explains how described properties of the fluctuating singular surfaces tangles of line-like defects can be described by a com- of Einstein’s theory. One may rewrite Einstein’s theory plex disorder field theory, whose Feynman diagrams are as a gauge theory [8, 14], and put it on a spacetime lattice direct pictures of the worldlines. Thus, if Huang et al. [19]. Then the singular surfaces are built explicitly from were to interprete their scalar field as a disorder field plaquettes, as in lattice gauge theories of asymptotically- of the purely geometric objects of my theory, the pal- free nonabelian gauge theories [20]. In the abelian case, lels would be closer. Note that in two papers with K. Halperin [Phys. Rev. Lett., 74, 3526 (1995); Phys. Rev. the surfaces are composed as shown in Ref. [21], for the 53, 3252 (1996)], Huang manages to make his scalar field nonablian case, see [22]. An equivalent derivation could theory asymptotically free in the ultraviolet (though at also be given in the framework of loop gravity [23]. But the unpleasant cost of a sharp cutoff introducing forces of that would require a separate study beyond this letter. infinite range). This property allows him to deduce an ef- Summarizing we have seen that the Einstein-Hilbert fective dark energy in the baby universe. With our purely action governs not only the classical physics of gravi- geometric tangles, such an effect may be reached using a lattice gauge formulation of Einstein’s theory [8, 14], tational fields but also, via the fluctuations of its line- sketched at the end of the text. and surface-like singularites, the quantum physics of dark [12] S. Balibar, Rotons, Superfluidity, and Helium Crystals, matter. A string-like action, derived from it for the (http://www.lps.ens.fr/∼balibar/LT24.pdf). fluctuating surface-like singularites, contains interacting [13] B.A. Bilby, R. Bullough, and E. Smith, Proc. Roy. Soc. spin-2 quanta that define a finite Quantum Gravity. London, A 231, 263 (1955); K. Kondo, in Proceedings of the II Japan National Congress on Applied Mechan- ics RAAG Memoirs of the Unified Acknowledgment: I am grateful to R. Kerr, N. Hunter- , Tokyo, 1952, publ. in Study of Basic Problems in Engineering and Science by Jones, F. Linder, F. Nogueira, A. Pelster, R. Ruffini, B. Means of Geometry , Vol. 3, 148, ed. K. Kondo, Gaku- Schroer, and She-Sheng Xue for useful comments. jutsu Bunken Fukyu-Kai, 1962. [14] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I Superflow and Vortex Lines, World Scientific, Singapore, 1989, pp. 1–744 (http://klnrt.de/b1) [15] H. Kleinert, Ann. d. Physik, 44, 117 (1987) ∗ Electronic address: [email protected] (klnrt.de/172/172.pdf). [1] H. Kleinert, Path Integrals in Quantum Mechanics, [16] H. Kleinert and J. Zaanen, Phys. Lett. A 324 , 361 (2004) Statistics, Polymer Physics, and Financial Markets, 5th (klnrt.de/346/346.pdf). ed., World Scientific, 2009 (klnrt.de/b5). [17] K. Nakamura et al. (Particle Data Group), J. Phys. G [2] M. Green, J. Schwarz, and E. Witten, Superstring the- 37, 075021 (2010) orym Cambridge University Press, 1987. [18] We do this although the weakness of gravitational in- [3] A.M. Polyakov, Gauge Fields and Strings, Harwood Aca- teractions may be an obstacle to thermal equilibriation. demic Publishers, New York, 1987. See S. Khlebnikov and I. Tkachev, Phys. Rev. D 56, 4

653 (1997); J. Garcia-Bellido, D.G. Figueroa, A. Sastre, [23] A. Ashtekar, Phys. Rev. Lett. 57, 2244 (1986); C. Rovelli, Phys. Rev. D 77, 043517 (2008) (arXiv:0707.0839). Quantum Gravity, Cambridge University Press (2004) [19] She-Sheng Xue, Phys. Lett. B 682 (2009) 300; Phys. Rev. (http://www.cpt.univ-mrs.fr/∼rovelli/book.pdf); D 82, 064039 (2010). L. Smolin, Three Roads to Quantum Gravity Basic [20] K. Wilson, Physical Review D 10, 2445 (1974). Books, London, 2003. [21] H. Kleinert and W. Miller, Phys. Rev. D 38, 1239 (1988). [22] J.M. Drouffe and C. Itzykson Phys. Rep. 38, 133 (1975).