The Vortex Nature of the Space-Time Curvature

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 ( /i∇ − (e/c)A~) · (− /i∇ − (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

∂ψ 1 2 ∂ψ ∂ψ 2 ~ 2 (2) ∂t = 2 ∇ ψ − iΩ(x ∂y − y ∂x ) − β|ψ| ψ + (A) ψ

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 ψ(x, y, z, 0) = ψ0(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 ∇ h − 4 e (∇h) + 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 ∇ h − 4 e (∇h) + 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) ∇ h − 4 (∇h) + 4 (∂te ) = ρ

85 In the static case, in the linear approximation, from equation (7) we have New-

86 ton’s theory of gravity ∇2h = 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 ± √2 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 |ρ| >> max(|∇ h|, (∇h) ), then it is possible to bring (7) to the form of a

92 parabolic equation

√ q √ 3 −h 2 1 2 1 2 1 2 (10) 2 ∂t(e ) = ρ − ∇ h + 4 (∇h) = ρ(1 − 2ρ ∇ h + 8ρ (∇h) + ...)

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

√ −h 3 e√ ∂h 1 2 1 2 (11) 2 ρ ∂t = −1 + 2ρ ∇ h − 8ρ (∇h)

2 96 If ρ < 0, then setting ρ = −m , we reduce equation (10) to the form of the

97 nonlinear Schr¨odingerequation

√ 3 e−h ∂h 1 2 1 2 (12) −i 2 m ∂t = 1 + 2m2 ∇ h − 8m2 (∇h)

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]

ir2 √ −iEmt+imφ+ √ m m 2 ψ = C e 2 r L (−i 2r ), (13) m m κ1 √ m 1 107 Here Lκ (x) is the generalized Laguerre polynomial, κ = 8 (2i 2mΩ − 4m + √ √ 2 108 2i 2E − i 2k − 4), and Cm is constant. Here we set the constant from the

109 condition max|ψm| = 1. Consider a superposition of states with 3 diﬀerent m

110 in the form (14) ψ = Cm1 ψm1 + Cm2 ψm2 + Cm3 ψm3

5 111 Variants of the initial states for equations (11)-(12) are shown in Figure 1. For

112 example we use the combination, including the ﬁrst, seventh and thirteenth

113 harmonics (bottom line in ﬁg1). Figure 2 (top row left) shows the modulus of

114 the function ψ(0, x, y) for triplet with m1 = 1, m2 = 7, m3 = 13. This triplet

115 describes a system consisting of 6 vortices around a giant vortex. Figure 2, 3

116 shows the evolution of the initial state in a rectangular region with periodic

117 boundary conditions and with initial data

h(0, x, y) = ±|ψ(0, x, y)| (15)

118

119 Figure 1.The modulus of the wave function at diﬀerent instants of time used

120 as initial state for eqs. (11)-(12)

121

6 122

123 Figure 2.The evolution of the metric with an initial state in the form of a

124 hexagonal ﬂow with a negative sign in (15).

125

126

127 Figure 3.The evolution of the metric with an initial state in the form of a

128 hexagonal ﬂow with a positive sign in (15).

129

7 130 If we take the negative sign in equation (15), then the evolution of the metric

131 proceeds while maintaining the hexagonal structure - see Figure 2. If we change

132 the sign in the initial data (15) then the evolution of the metric is diﬀerent -

133 Fig. 3. This is obviously related to the property of the gravitational potential

134 φ, which should be negative in Newton’s theory. Comparing metrics (3) and 2φ 135 (4) we ﬁnd that in linear theory h = c2 . Therefore, to be consistent with

136 Newton’s theory, we must choose the minus sign in equation (15). In this case,

137 the metric preserves some properties of the quantum micro-structure - Fig. 2.

138 This property is even more evident if, in the initial data, instead of magnitude,

139 we use the square of the modulus of the wave function with a minus sign,

h(0, x, y) = −|ψ(0, x, y)|2 (16)

140 Figure 4 shows the evolution of the metric with initial data in the form (16).

141 In this case, the hexagonal structure of the initial data is also preserved.

142

143 Figure 4.The evolution of the metric with an initial state in the form of a

144 hexagonal ﬂow using eq. (11) and (16).

145

146 We now turn to equation (12), which we could call the equation of quantum

147 gravity. It could be assumed that for equation (12) it is necessary to use the

8 148 wave function as the initial data. However, equation (12) is very sensitive to

149 boundary conditions, while function (13) - (14) does not completely satisfy the

150 periodicity condition. This does not noticeably aﬀect the solution of equation

151 (11); therefore, the wave function was used as initial data in the form of (15)

152 and (16). However, for equation (12) we use the following initial data:

2 2 h(0, x, y) = ψ(0, x, y)e−x −y (17)

153 Figure 5 shows the evolution of the metric calculated using equation (12)

154 with the initial data (17). We see here a completely diﬀerent picture than in

155 Fig. 2 and 4. Firstly, there is a loss of symmetry. Secondly, a peculiar structure

156 is formed that is not similar to the original hexagonal.

157

158 Figure 5. The evolution of the metric with an initial state in the form of a

159 hexagonal ﬂow using eq. (12) with initial conditions in (17).

160

161 A remarkable property of equation (12) with initial data (17) is that the

162 number of vortices increases from the initial 6 + 1 = 7 to 15 at t = 1 - see Fig.

163 5. This vortex growth becomes even more noticeable if we change the initial

9 164 data to

2 2 − x +y h(0, x, y) = ψ(0, x, y)e 2 (18)

165 Fig. 6 shows the evolution of the metric calculated using equation (11) with

166 vital data (18). In this case, we see that the initial number of vortices (6 + 1)

167 turns into 30 vortices arranged in pairs in a circle.

168

169 Figure 6.The evolution of the metric with an initial state in the form of a

170 hexagonal ﬂow using eq. (12) and (17).

171

2h 2h 172 We then put in the equations (11) and (12) ρ = e and m = e conse-

173 quently, we then get the two equations:

√ 3 ∂h 2h 1 2 1 2 (19) 2 ∂t = −e + 2 ∇ h − 8 (∇h)

√ 3 ∂h 2h 1 2 1 2 (20) −i 2 ∂t = e + 2 ∇ h − 8 (∇h)

174 Figure 7 shows the evolution of the metric by virtue of equation (20) with

175 the initial data (17) for a hexagonal ﬂow with periodic boundary conditions.

176 We see that in this case too, the hexagonal structure is preserved.

10 177

178 Figure 7.The evolution of the metric with an initial state in the form of a

179 hexagonal ﬂow using eq. (19) and (16).

180

181 A similar calculation of the metric according to equation (20) with initial

182 data (17) with periodic boundary conditions is shown in Fig. 8. In this case,

183 the result is similar to that shown in Figure 5.

184

11 185 Figure 8.The evolution of the metric with an initial state in the form of a

186 hexagonal ﬂow using eq. (20) with initial conditions in (17) (magnitude of the

187 wave function shown).

188

189 Thus, we have shown that both in the case of a homogeneous and in the

190 case of a nonuniform density distribution, the evolution of the metric leads ei-

191 ther to an increase in the initial quantum vorticity (equations (12) and (20)),

192 or to its conservation (equations (11) and (19)). Finally, note that we have

193 studied the evolution of various initial data indicated in Fig. 1. Some of

194 these data are presented in Fig. 9-11 in 3D format. We see that the initial

195 states with 6 (Fig. 9), 4 (Fig. 10) and 10 vortices (Fig. 11) around the

196 central giant vortex evolve the same way while preserving the original struc-

197 ture. In this case, we used equation (11) with initial data in the form (16).

198

199 Figure 9.The evolution of the metric with an initial state in the form of a

200 hexagonal ﬂow using eq. (11) and (16).

201

202

203 Figure 10.The evolution of the metric with an initial state with m1 = 4, m2 =

12 204 8, m3 = 12 using eq. (11) and (16).

205

206

207 Figure 11.The evolution of the metric with an initial state with m1 = 3, m2 =

208 8, m3 = 13 using eq. (11) and (16).

209

210 2.2 Discussion

211 The evolution of the metric , shown in Figure 2 and 3 bears a series of important

212 implications for the phenomenon of gravity. Taken from a spacetime curvature

213 perspective, the positive sign in the metric attributes to gravity a geometry

214 in spacetime which favors the formation of hexagonal shape, which we do not

215 observe in gravitational orbits, but rather in electromagnetic systems, such as

216 vortices in Bose-Einstein condensates [4, 5, 6, 21, 22, 23, 24] or for macroscopic

217 ﬁne medium (gases) governed by electromagnetic ﬁelds such as on the North

218 Pole of Saturn [25]). We show therefore a preserved vortex model in eqn. (10)

219 by assigning a positive sign to ρ which favors hexagonality in micro-gravity

220 environments and electromagnetic ﬁelds of weak energy, which we describe as

221 an equation for quantum gravity. The quantized state of gravity yields hence

222 a hexagonal shape, composed of six angles that represent six discrete preferred

223 positions for a quantum gravitational vortex model. This is indeed in agreement

224 with that the hexagonal symmetry is the single symmetry that allows the highest

225 number of vortices to arrange about a center and that overlap to form multi-

226 hexagonal lattices, as also other have reported for electromagnetic systems and

227 quantum gases [26, 27, 28]. The model of eqn. (10) with a negative sign explains

13 228 also the inevitable emergence of quantum geometry, where all orientations of

229 orbitals follow quantum rotation, as also proposed by Haldane [29, 30].

230 For macroscale gravity however, we observe directly an agreement with Newtons

231 theory on gravity, which is only satisﬁed by having a minus sign in for ρ in

232 eqn. (10), where the continuum at macroscale yields a spherical shape with

233 a growing elliptical pattern from the center of gravity in the space-dimensions

234 (x,y). The model in (11) and (12) shows also a potential description on how

235 gravity arises and how it evolves during a period of time. The evolution of the

236 gravity potential in spacetime (ﬁg 3) shows that the vortex nature of gravity

237 assembles a hexagonal structure in its very infant states, and then develops an

238 intense gravity-void around the gravity center (Fig 3, t=0.4) which then rapidly

239 disappears as the gravity potential assumes an elliptical shape with a circular

240 form the closer one reaches the center of the gravity. Attributing this model

241 to our solar-system for instance, shows an excellent agreement with the orbit

242 of the planets. Furthermore, the gravity-vortex model suggests also that the

243 gravitational center increases in extension in the space plane (x,y), with a larger

244 area of intensity arising with time (compare t=1.6, 2.0), which can explain the

245 evolution of red giants and supergiants which develop into black holes. Figure

246 3, at time t=0.4 shows also a similarity to the behaviour of black holes at

247 even horizon, when a superﬁne boundary of trapped light is contained outside

248 the center of the black hole, hence neither escaping its gravity pull nor being

249 compressed in the black hole. Figure 3 shows such a distinctive ring, which is in

250 the periphery of the gravity center. The principal diﬀerences between macroscale

251 gravity and quantum gravity, as described by the model showed above, is that

252 macroscale gravity appears to be degenerate (elliptic) while quantum gravity

253 appears to be orthogonal. To our knowledge, only one author has reported

254 degeneracy of universal forces, where Kunz reported recently a model of dark

255 energy with degenerate properties [31], which yields better agreement with the

256 behaviour of supernovas and the angular velocities of rotating galaxies.

257 When we consider Figures 5-11, we focus on the evolution of the metric which

258 explains preferred states of the initial symmetry for gravitational ﬁelds, by their

14 259 quantized parameters m. It is expected that the higher levels of m will yield

260 higher order symmetries which eventually will reach the circle symmetry, as its

261 symmetry states jump from 4 (Fig 10) to 6 (Fig 5, 6 and 9), and further to

262 10 angles (Fig 11) in the symmetrical organization of the vortex shape. To our

263 knowledge, only gravity waves display vortex phenomena at the mascroscopic

264 scale [32, 33, 34], and we can therefore extrapolate that gravity vortices in the

265 atmosphere are the closest empirical example of the vortex signature of gravity

266 at the universal scale. This study relates the supersymmetric wave-equation

267 [3, 4, 5, 6] to the nonlinear diﬀusion equation in (2) and the Einstein equations.

268 Equations (11) and (12) have been published recently by the corresponding

269 author in [35], however the transition from equation (1) and (2) to (11) and

270 (12) is here described for the ﬁrst time. Studies to measure and model the

271 vorticity of gravity ﬁelds should be emphasized in future astrophysics research,

272 as it can explain several nonlinear astrophysical phenomena, such as the higher

273 rotational velocities of galaxies compared to their total mass.

274 2.3 Conclusions

275 We have in this paper, presented a theory which suggests that gravity ﬁelds are

276 governed by vorticity which generate the elliptic orbit of planets, and bear sim-

277 ilarities to gravitational waves in the atmosphere, and yield quantum geometry

278 by quantum gravity. We have also shown that the proposed model separate

279 the quantum scale from gravity at the macroscopic scale by their geometry;

280 the former is orthogonal, while the latter is degenerate and generates elliptic

281 structures. The cause for this principal diﬀerence between quantum gravity

282 and macroscopic gravity lies in the evolution of the respective wave-functions,

283 which suggests that both gravity ﬁelds have the same original start geometry

284 (hexagonal) at time zero, however due to the low-energy of particles governed

285 by quantum gravity, they remain in their discrete levels and retain a quantum

286 geometry. Inversely, macro-scale bodies, having a higher energy of macroscopic

287 elements exert a dynamical component on their gravity pull (i.e. planets) that is

15 288 able to oscillate beyond the orthogonal framework of gravity (complete circular),

289 and without losing orbit, tend to oscillate in elliptic shapes.

290 3 Acknowledgements and correspondence

291 The authors declare that they have no competing ﬁnancial interests.

292 Correspondence and requests should be addressed to Sergio Manzetti,

293 [email protected]).

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