1 The vortex nature of the space-time curvature. 1;2∗ 2;3 2 Sergio Manzetti and Alexander Trounev 1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 2. Fjordforsk A/S, Midtun, 6894 Vangsnes, Norway. Email: [email protected] 3. A & E Trounev IT Consulting, Toronto, Canada. 3 February 27, 2020 4 1 Abstract 1 5 Fundamental physical phenomena, such as gravity, electromagnetism and light 6 have been studied over the last centuries for their properties, mechanics and for 7 their relationships to one-another. Bernoulli proposed in 1736 a vortex-sponge 8 theory to describe light in mechanical terms which was later postulated by using 9 Maxwell's vortex equations. In this study we use a supersymmetric Hamiltonian 10 recently developed for quantum systems to describe the vortex nature of space- 11 time curvature arising at gravity sources. We use the solar system as model and 12 describe the gravitational effect on space-time as a sheer result of vorticity in 13 the gravity field, maintaining planets in their elliptic orbit and bending space- 14 time in a similar fashion to Einstein's gravity sink model. It is shown that from 15 the Einstein equations for the post-Newtonian metric, the nonlinear diffusion 16 equation and the Schr¨odingerequation can be derived. The results are analyzed 1Please cite as: Manzetti Sergio and Alexander Trounev (2020). The vortex nature of the space-time curvature. Technical Reports 22, 20, Fjordforsk A/S. pp. 1-20. www.fjordforsk.no 1 17 numerically and the dynamics of the system studied. 18 2 Introduction 19 Vorticity is ubiquitous in nature and arises in all natural phenomena, from vor- 20 tex eddies in water streams [1], to vortices in electromagnetic fields represented 21 by superposition of wavefunctions [2, 3, 4, 5, 6], vortices in the atmosphere [7], 22 vortices in superheated stellar plasma [8] and in many other phases of nature, 23 including vortices in gravity waves [9, 10, 11]. Interestingly, Bernoulli proposed 24 in 1736 that also light is behaving in a vortex-like manner, a theory which was 25 further discussed by Kelly [12] where light was proposed to be a component of 26 a continuous medium present in the universe. Descartes proposed even earlier, 27 in his work "Principiorum Philosophiae" from 1644, that the universe was solely 28 composed of vortices, also noted as Cartesian vortices. However, after Einstein's 29 general relativity theory, the view of the fundamental forces of the universe was 30 greatly simplified by the theory of space-time curvature by gravity and its effect 31 on cosmological objects. 32 In this study, we study the gravity sink (curvature) proposed originally by Ein- 33 stein [13] as a pure consequence of gravitational space-time vorticity, leading 34 to the circulation of planets and generation of the space-time curvature. We 35 suggest hence a theory where gravity originates from a vortex in a space-time 36 continuum, leading to the formation of the gravity curvature emerging from 37 the center of the source of mass. Albert Einstein described this continuum in 38 similar terms, naming it "the affine field” [14], a model which was founded 39 on the idea of Arthur Eddington [15] who later critically discussed the affine 40 field theory by Einstein [16]. The ”affine field” theory was proposed to merge 41 electromagnetism and gravity under one model, however, the theory was later 42 abandoned and not pursued further. It is however of interest that the ”affine 43 field theory" bears similarities to Descartes model from 1644, and to the subse- 44 quently published vortex sponge model by Bernoulli. As defined by Eddington 45 however [15], the necessity of pure mathematical models is deemed critical 2 46 to describe a continuum in the universe, and the latter two fail therefore to 47 give rigorous mathematical modelling to show a reproducible description of the 48 universal continuum where space-time gravity vortices arise spontaneously in a 49 similar fashion to the atmosphere, fluids and superfluids. 50 Here, we devise a supersymmetric wave-equation which we developed recently 51 over several works [3, 4, 5, 6] when studying electron gas in quantized 2D sys- 52 tems. The Hamiltonian of the supersymmetric wave-equation is of particular 53 appeal to model vorticity in a space-time continuum, as it generates vortic- 54 ity spontaneously without need of auxiliary functions and has a particularly 55 appealing spectrum of eigenvalues, where the range of trivial eigenvalues in 56 its spectrum is positive infinite, and the evolution of the spectrum is slowly 57 crescent [17]. The algebraic properties of the operator of the supersymmetric 58 wave-equation [17] suggest therefore that it is valid for both microscopic and 59 quantized systems as well as for macroscopic continuous systems, yielding real 60 eigenvalues towards infinite levels of energy [17]. The equation, which reads: 1 ( =ir − (e=c)A~) · (− =ir − (e=c)A~)Ψ = EΨ (1) 2m ~ ~ 61 was developed using supersymmetry rules on its factors, generating the in- 62 verse sign on the the two inherent factors as given in (1). The original form of 63 this equation developed by Fang and Stiles [18] differs from it as it does not 64 have supersymmetric form of the factors, and therefore generates different re- 65 sults. We consider a dimensionless nonlinear form of (1) describing the vorticity 66 in a quantum system in the form @ 1 2 @ @ 2 ~ 2 (2) @t = 2 r − iΩ(x @y − y @x ) − βj j + (A) 67 Here A~ is a dimensionless vector potential and β; Ω - parameters of the model 68 describing the quasiparticle interaction and the angular velocity, respectively. 69 Note that nonlinear Schr¨odinger equation (Gross-Pitaevskii model) follows from 70 (2) when replacing t ! it. Eq. (2) describes evolution of the wave function 71 from some initial state (x; y; z; 0) = 0(x; y; z) and up to stationary state with 72 E = 0. 3 73 We show that the nonlinear diffusion equation (2) as well as the nonlinear 74 Schr¨odingerequation can be derived from the Einstein equations for the post- 75 Newtonian metric. Put φ = φ(t; x; y; z) as the gravitational potential, then the 76 metric has the form (see the outline in [19]): 2 2φ 2 2 2φ 2 2 2 ; (3) ds = (1 + c2 )c dt − (1 − c2 )(dx + dy + dz ) 77 here c is the speed of light. We put c = 1 and consider the following generaliza- 78 tion of the metric (3) ds2 = eh(t;x;y;z)dt2 − e−h(t;x;y;z)(dx2 + dy2 + dz2) ; (4) 79 where h(t; x; y; z) is some function that is a solution of the Einstein equation. 80 In metric (4), the Einstein tensor component G00 is reduced to the form 2h 2 1 2h 2 3 2 (5) G00 = e r h − 4 e (rh) + 4 (ht) 81 Using the Einstein equation Gik = 8πγTik, where Tik is the energy-momentum 82 tensor and γ is the gravitational constant, we find 2h 2 1 2h 2 3 2 (6) 8πγT00 = e r h − 4 e (rh) + 4 (ht) 2h 83 Without loss of generality, we set 8πγT00 = ρe , then equation (6) is reduced 84 to 2 1 2 3 −h 2 (7) r h − 4 (rh) + 4 (@te ) = ρ 85 In the static case, in the linear approximation, from equation (7) we have New- 86 ton's theory of gravity r2h = ρ (8) 87 In another case, when h = h(t); h(0) = 0, equation (7) describes the collapse (-) 88 or expansion (+) t 3 (@ e−h)2 = ρ ! h = − ln(1 ± p2 R ρ1=2dt) (9) 4 t 3 0 89 Since equation (7) is not quasilinear, we reduce it to the form of a quasilinear 90 equation using the hypothesis of the dominant influence of ρ. Suppose that 4 2 2 91 jρj >> max(jr hj; (rh) ), then it is possible to bring (7) to the form of a 92 parabolic equation p q p 3 −h 2 1 2 1 2 1 2 (10) 2 @t(e ) = ρ − r h + 4 (rh) = ρ(1 − 2ρ r h + 8ρ (rh) + :::) 93 There are two possible forms of equation (10), depending on the sign of ρ. If 94 ρ > 0, then keeping only the first terms of the expansion on the right-hand side 95 of (10), we find the diffusion equation p −h 3 ep @h 1 2 1 2 (11) 2 ρ @t = −1 + 2ρ r h − 8ρ (rh) 2 96 If ρ < 0, then setting ρ = −m , we reduce equation (10) to the form of the 97 nonlinear Schr¨odingerequation p 3 e−h @h 1 2 1 2 (12) −i 2 m @t = 1 + 2m2 r h − 8m2 (rh) 98 The nonlinear diffusion equation (11) is obviously related to equation (2), and 99 equation (12) is related to the nonlinear Schr¨odingerequation. We pose the 100 question if vorticity has arisen by virtue of equations (1-2), then how will this 101 evolve by virtue of equations (11) or (12)? To simplify the problem, we put 102 ρ = const in eq. (11) and m = const in eq. (12). But here other options are 103 possible, which we will discuss below. 104 2.1 Numerical analysis 105 As the initial state for equations (11) and (12), we take the analytical solution 106 obtained in [20] 2 −iE t+imφ+ irp p m 2 m m 2 = C e r L (−i 2r ); (13) m m κ1 p m 1 107 Here Lκ (x) is the generalized Laguerre polynomial, κ = 8 (2i 2mΩ − 4m + p p 2 108 2i 2E − i 2k − 4), and Cm is constant.