See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/340262362 Simulating galactical vorticity using a supersymmetric operator Technical Report · March 2020 DOI: 10.13140/RG.2.2.34254.00323 CITATIONS READS 0 16 2 authors: Sergio Manzetti Alexander Trunev Fjordforsk A/S Likalo LLC, Toronto, Canada 107 PUBLICATIONS 1,125 CITATIONS 101 PUBLICATIONS 109 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Halogenated silica nanoparticles for solar cell technologies. View project Wavefunction studies of molecules and oligomers. View project All content following this page was uploaded by Sergio Manzetti on 29 March 2020. The user has requested enhancement of the downloaded file. Simulating galactical vorticity using a supersymmetric operator. Sergio Manzetti 1;2∗ and Alexander Trounev 1;3 1. Fjordforsk A/S, Midtun, 6894 Vangsnes, Norway. Email: [email protected] 2.Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 3. A & E Trounev IT Consulting, Toronto, Canada. March 29, 2020 1 Abstract 1 Simulations of galactic vorticity require the development of complex systems of equations to generate models that account for the gravity, mass, kinetic, and electromagnetic effects which govern cluster behaviour. In this study, a rela- tionship is outlined through a series of steps, between the Einstein equations, the Ricci equations and the gravitational model of comoslogical bodies. This series of steps are then followed by the generation of a group of equations, in- cluding a supersymmetric equation which describes vorticity under the presence of an electromagnetic field for quantized systems. The relationship between the Einstein equations and the equation for quantum vorticity is ultimately the described and we outline the model of galactic vorticity through a nonlinear 1Please cite as: Manzetti Sergio and Alexander Trounev (2020). Simulating galactical vorticity using a supersymmetric operator. Technical Reports 23, 21, pp. 1-32. Fjordforsk A/S - www.fjordforsk.no 1 cluster model, which has direct similarities to the Kosterlitz-Thouless model for vorticity in quasi-ordered systems. The resulting supersymmetric model for describing galactic vorticity we here show suggests that galactic vorticity broke symmetry already at T=0, and by the effect of the galactic halo around the periphery, generates nonlinear vorticity through the shear property of the wavefunction. Most interestingly, this property of the wavefunction, which gen- erates autonomous vorticity by the mere presence of a halo-boundary of the galactic cluster is independent of temperature. We describe by this that the nonlinear relationship between vortices in the cluster leads to a cumulative ef- fect which generates an exchange interaction at the macroscopic level between clusters which is the source of irregular moment of inertia of galaxies, that has been attributed to dark matter. Our theory is thus that galactic rotations are both fueled by the mass of the galaxy by classical physics means as well as the nonlinear vortex interaction between clusters. 2 Introduction Einstein's general theory of relativity is widely used in cosmology and astro- physics, including for modeling the motion of stars in galaxies. The problem of many bodies in the general theory of relativity was considered in [1, 2, 3, 4, 3, 5, 6] and others. It was shown in [7] that the field equations for empty space are sufficient to describe the motion of matter represented in the form of point singularities. Consequently, the motion of material particles repre- sented by the singularities of the gravitational field is one of the problems of the general theory of relativity. This approach uses only gravitational equations in empty space. As is known, to describe the motion of inert matter in the framework of the general relativity, Einstein and Infeld formulated the theory [8], where they declared that all attempts to represent matter by the energy- momentum tensor are unsatisfactory, and they want to free theory from the special choice of such a tensor. Therefore, theory [8] deals only with gravita- tional equations in empty space, and matter represented by singularities of the 2 gravitational field. Note that in the existing approaches to the description of black holes, the Schwarzschild metric [9] is widely used, which describes the singularity of the gravitational field, which is physically interpreted as the field of a particle of a given mass. In an alternative approach to the description of the motion of material bodies, the energy-momentum tensor is used [10, 4, 6] and [11, 12, 13, 14, 15, 16]. However, in this case, the theory is surrounded by a large number of hypotheses related to the indeterminacy of the concept of matter, especially in light of recent discoveries in astrophysics, indicating a significant contribution of dark matter and dark energy to the dynamics of the expanding Universe [16]. Einstein's equations have the form [17, 18, 19, 20]: 1 8πG R − g R = g Λ + T (1) ik 2 ik ik c4 ik Here Rik; gik;Tik is the Ricci tensor, the metric tensor and the energy-momentum tensor;Λ; G; c - Einstein's cosmological constant, the gravitational constant and the speed of light, respectively. We put Λ = 0 suppose that due to the small in- fluence of this parameter in the problem of many bodies associated with galaxy dynamics. Solutions of Einstein's equations with axial symmetry were consid- ered in [2, 3, 5], [18, 19, 20, 21, 22] and others (a review of publications is given, for example, in [5], [19, 20, 21]). The metric of such spaces, under certain assumptions, can be reduced to ds2 = eµdt2 − e−µ(eν dρ2 + eν dz2) − e−µρ2dφ2 (2) Here µ(ρ, z); ν(ρ, z) are functions that satisfy Einstein's equations. Calculating the components of the Einstein tensor in the metric (2) and assuming for vacuum 1 Gik = Rik − 2 gikR = 0 we find the field equations: @2µ 1 @µ @2µ ! = + + = 0 (3) 1 @ρ2 ρ @ρ @z2 1 ! = ν − ρ(µ2 − µ2) = 0 (4) 2 ρ 2 ρ z !3 = νz − ρµρµz = 0 (5) 3 @2ν @2ν 1 ! = + + (µ2 + µ2) = 0 (6) 4 @ρ2 @z2 2 ρ z It can be verified that not all equations (3)-(6) are independent and that the following two relations are true [19] @! @µ @! ! = 2 + ρω + 3 (7) 4 @ρ 1 @ρ @z @µ @! @! ρω = 2 − 3 (8) 1 @z @z @ρ Therefore, we can choose, for example, the equations (3) and (6) as a system of equations for determining two functions µ(ρ, z); ν(ρ, z), we have @2µ 1 @µ @2µ + + = 0 (9) @ρ2 ρ @ρ @z2 @2ν @2ν 1 + + (µ2 + µ2) = 0; (10) @ρ2 @z2 2 ρ z indicating that !1 and !2 and !4 are linearly dependent, as also is !1 and !2 and !3. By solving the system of these linearly dependent equations (9) and (10), we can come to the determination of static gravity fields in the case of axial symmetry. Model (9), (10) has a unique expansion property in the theory of Ricci flows, which allows us to specify the motion in the system of N bodies. In this case, the gravitational potentials in the system of equations (9), (10) depend on time. To simulate a change in the metric during the movement of singularities, we use Ricci flows, which are described by the equation [23, 24, 25, 26] @g ik = D R (11) @t 0 ik Here D0 is the diffusion coefficient, which in the standard theory [23, 24, 25, 26] is assumed D0 = −2, however, in the metric (2), taking into account the signature, it should be put D0 = 2, then (9), (10) reduces to a system of parabolic equations: @µ @2µ 1 @µ @2µ eν−µ = + + (12) @t @ρ2 ρ @ρ @z2 4 @ν @2ν @2ν 1 eν−µ = + + (µ2 + µ2) (13) @t @ρ2 @z2 2 ρ z Comparing (12) (13) and (9), (10), we find that for steady flows under conditions µt = νt = 0, system (12)-(13) reduces to (9), (10). The justification for such a transition from the static system (9)- (10) to the parabolic system of equations (12)-(13) is similar to the theory developed in the following studies [23, 24, 25, 26] and others. For many practically important problems, such as the merging of black holes [1], it will be enough to know how the metric of the binary system changes (as spacetime is curved by two merging gravity sources) as the centers of gravity approach a given speed. The Ricci flow model (12)-(13) allows one to answer this and other questions related to a change in the metric. Consider the solutions of the system of equations (9), (10) of the form [5, 21] N X mj µ = − (14) p 2 2 j=1 ρ + (z − Lj) N 2 2 1 X mj ρ 1 X mjmk ν = − 2 2 2 − 2 fjk (15) 4 (ρ + (z − Lj) ) 2 (Lj − Lk) j=1 j6=k 2 Here f = p ρ +(z−Lpj )(z−Lk) − 1, m ;L are some constants. Note jk 2 2 2 2 j j ρ +(z−Lj ) ρ +(z−Lk) that expressions (14)-(15) differ in normalization from expressions given, for example, in [5] by virtue of the definition of metric (2). In the particular case, assuming in (14)-(15) that N = 2, we arrive at the expression of the potentials first obtained in [2]. In the particular case of three bodies, the potentials µ, ν were obtained in [3]. In the nonrelativistic case, the potential µ reduces to the expression µ = 2'=c2, where ' is the gravitational potential in Newton's theory of gravity. Since the point masses in metric (2) with potential (14) do not experience mutual displacement, although they should be attracted according to Newton's theory, the author of [2] draws the erroneous conclusion that Einstein's theory of relativity is not true.