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Modelling the formation of atmospheric vortices on planet earth using a supersymmetric operator.

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Macrophysical phenomena, such as turbulence, vorticity are normally modelled using fluid-mechanical differential equations and geodesics systems. Other methods may also be of importance to the meteorology community, such as quantum physical operators. In this study we use a novel Hamiltonian and study the vortex formation in the atmosphere of planet Earth under the effects of the gravity density stratification and the electromagnetic field of the planet. The results propose that vorticity in the atmosphere (high and low pressure systems) is driven in major part by the interplay between the earths magnetic field and gravity density. The results show that the quantized behaviour of atmospheric vortices lies in their dominant occurrence on the northern and southern hemispheres. The use of quantum mechanical operators in modelling planetary vorticity reveals also that these vortices arise from the core of the planet and manifest in a most pronounced manner on the surface of the earth where gravity density is experiencing an abrupt phase change. Further research is made on combining this model with earths atmospheric parameters, such as ocean temperatures and circulation, terrestrial oscillation and the sun's magnetic field. The results are important for future developments of climate and weather prediction models.
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Modelling the formation of atmospheric vortices
on planet earth using a supersymmetric operator.
Sergio Manzetti 1,2and Alexander Trounev 1,3
1. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway.
2. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596,
SE-75124 Uppsala, Sweden.
3. A & E Trounev IT Consulting, Toronto, Canada.
March 14, 2019
Key points
Main point 1: A supersymmetric Hamiltonian for electromagnetic vortices is
used to describe vorticity on planet earth.
Main point 2: A Master equation is generated which describes the phenomenon
of vortex formation in the atmosphere in a 3D system.
Main point 3 : Interplay between electromagnetic field of the earth and the
stratified gravity yields vorticity.
Keywords
Hamiltonian; magnetism; planetary; gravity; supersymmetry; numerical; anal-
ysis
1
Citing information1
1 Abstract
Macrophysical phenomena, such as turbulence, vorticity are normally mod-
elled using fluid-mechanical differential equations and geodesics systems. Other
methods may also be of importance to the meteorology community, such as
quantum physical operators. In this study we use a novel Hamiltonian and
study the vortex formation in the atmosphere of planet Earth under the effects
of the gravity density stratification and the electromagnetic field of the planet.
The results propose that vorticity in the atmosphere (high and low pressure sys-
tems) is driven in major part by the interplay between the earths magnetic field
and gravity density. The results show that the quantized behaviour of atmo-
spheric vortices lies in their dominant occurrence on the northern and southern
hemispheres. The use of quantum mechanical operators in modelling planetary
vorticity reveals also that these vortices arise from the core of the planet and
manifest in a most pronounced manner on the surface of the earth where gravity
density is experiencing an abrupt phase change. Further research is made on
combining this model with earths atmospheric parameters, such as ocean tem-
peratures and circulation, terrestrial oscillation and the sun’s magnetic field.
The results are important for future developments of climate and weather pre-
diction models.
1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Modelling the formation
of atmospheric vortices on planet earth using a supersymmetric operator.” In: Modeling of
quantum vorticity and turbulence in two- and three-dimensional systems. Report no. 142021.
Copyright Fjordforsk A/S Publications. Vangsnes, Norway. www.fjordforsk.no
2
2 Introduction
Hamiltonians are mathematical operators used for modelling various phenomena
in physics, ranging from quantum physics and quantum field theory to relativis-
tic physics [1, 2, 3, 4]. The advantage of Hamiltonian models over classical
physics models is the derivation of eigenvalues and eigenfunctions that describe
the wave-behaviour of energy, such as in wind waves, ocean waves, acoustic
waves as well as electromagnetic waves. Gravity is also a phenomenon that is
known to occur as waves [5, 6, 7, 8, 9, 10], and describing new models for its
effects on its domain of objects (planets, stars, galaxies etc) can increase the
knowledge of how it shapes and affects planets and its components, solar systems
and larger ensembles of objects. In this study, we are interested in modelling
the effects that gravity has on the atmosphere, by using a model Hamiltonian
derived by applying supersymmetry rules on a Hamiltonian for free particles
rotating around a center [11, 12, 13].
2.1 Gravity and the atmosphere
Recent studies suggest that gravity affects the atmosphere and its dynamics
[14, 15, 16, 17]. The fluctuating character of gravity [18] and its continuously
oscillating nature [19] suggest that the intricate behaviour of atmosphere, be-
having as a fluid, can be modelled using model Hamiltonians which consider also
the gravity fluctuations. The conventional models to model the atmosphere ori-
gin from the first equations published by Edward Lorenz [20] in the early 1960’s.
Since then and with the development of supercomputers, several state of the art
approaches have emerged; such as application of grid-defined symmetry and ge-
ometry for simulating an atmospheric environment in silico [21, 22] simulation
of atmospheric components by Cartesian meshes or other geometric forms [23],
application of Double Fourier Series and other models for simulating non-linear
dynamics [24, 25] use of semi-implicit semi Lagrangian discrete models for simu-
lating the fluid-dynamic behaviour of weather systems [26], and implementation
of other mathematical models to simulate atmospheric fluctuations for weather
3
prediction and atmosphere study [27, 28, 22]. These models and the derived al-
gorithms apply physical qualities to the simulated atmosphere and its dynamical
behaviour is represented by classical vector calculus based on scalar fields [21].
Fluctuations by angular momenta are simulated using continuous equations,
also known as mimetics [29]. Also, large scale atmospheric flows are mod-
elled by fluid mechanics, where large-scale bodies (i.e. a low pressure system)
are adjusted with a dynamical core to represent the non-linear vortical motions
through a mass-conservation scheme [21, 30]. State of the art methods in atmo-
spheric modelling do however not consider the spatio-temporal variations in the
gravitational field and its influence on atmospheric processes. These missing re-
lationships, although quite small at the regional scale, may nonetheless influence
atmospheric processes via highly non-linear phenomena such as instability and
turbulence and promote large scale dynamics on the atmosphere which have not
been mapped before. Both turbulence and instability have been modelled ac-
curately [31], however their linear or non-linear dependency on the gravity flux
has not been proposed before, and thus introduces a novel idea. Furthermore,
gravity’s dynamical fluctuations on the uppermost parts of the atmosphere, also
known as gravity waves [10, 9] and gravity-induced wave propagation across
troposphere and ionosphere [6, 5] have been modelled and studied previously.
Still however, a derivation of linear/non-linear models for gravity-atmosphere
interaction has not been derived, also considering the quantized behaviour of
gravity [19]. Several approaches to describe mesospheric systems including nu-
merical studies [32, 6, 5, 7, 8, 33] do therefore support the effect from gravity
on the atmosphere, including the temperature-inversion patterns from gravity
[34], momentum-alteration of atmospheric bodies and diffusion [35], and also
the triggering of mechanical impulses across the atmosphere [36]. By this, we
are interested in modeling the vorticity that is induced in the atmosphere by
the gravity field, by solving our model Hamiltonian numerically. In this en-
deavour we first focus on describing the Hamiltonian for the physical problem
and then apply mathematical software to solve numerically the corresponding
master equation. The results of the analysis are presented throughout a series
4
of figures and illustrations, and discussion is presented afterwards.
2.2 Supersymmetric Hamiltonian for vortex formation
We present here the derivation of the supersymmetric operator for vortex sys-
tems in a sphere or in a 2D system. We develop the operator from the model
published by Laughlin [37], who defined the Hamiltonian operator for anyons
as
H=|~/i∇ − (e/c)~
A|2,(1)
where ~
A=1
2H0[xˆyyˆx], is the symmetric gauge vector potential, where
H0is the magnetic field intensity. However, presenting this treatise in a first
quantized formalism, we use the regular form:
H= [~/i∇ − (e/c)~
A]2,(2)
where the Hamiltonian is simply the normal square of the covariant deriva-
tive. Supersymmetry rules [38]are applied on eqn. (2), which is composed of
the two first-order differential operators:
H= [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
A],(3)
and we get the superpair of the Laughlin Hamiltonian:
HSU SY = [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
A].(4)
Hence, the SUSY counter-part of the Hamiltonian from (2) becomes:
HSU SY =T T ,(5)
with
T= [~/i∇ − (e/c)~
A],(6)
5
and
T= [~/i∇ − (e/c)~
A],(7)
where Tand Tare the superpair in the SUSY Hamiltonian in (4) and one
anothers Hilbert-adjoint operator.
We consider then Tand T, with γ= (e/c)~
A, in one dimension:
T=h
i
d
dx γ·˜
I
T=h
i
d
dx γ·˜
I.
which commute by the relation:
[T T ][TT] = 0 (8)
making Tand Ttwo commutating complex operators in Hilbert space H.
Both Tand Tare unbounded operators given the condition:
||T x|| c||x||,(9)
Tand Tare also non-linear given the γconstant, being a constant trans-
lation from the origin. Tand Tare non-self-adjoint in Has the following
condition is not satisfied:
hT φ, ψi=hφ, Tψi,(10)
From this, it follows that:
HSU SY :D(HS U SY )H,
and
D(HSU SY )H,
6
where HSU SY is an unbounded linear and non-self-adjoint complex operator
on Hilbert space H=L2[−∞,+]. However, HSUS Y is self-adjoint in its
domain, D(HSU SY ) on L2[a, b].
From supersymmetry theory in quantum mechanics [38] it follows that
HΨ = HΨ=EΨ = EΨ, therefore we can assume that:
HSU SY Ψ = EΨ,(11)
where E is the energy of the system, which generates the boundaries of
D(HSU SY ) on L2[0, L], where the interval of quantization (L) is contained in
the zero-point energy term E= n2~2π2
2mL2.
3 Materials and methods
In Ref. [11, 12], we investigated the formation of vortices in a homogeneous and
alternating magnetic field within the framework of the modified Gross-Pitaevskii
model. To derive the basic equation, we used the supersymmetric Hamiltonian
in (4), which is a supersymmetric generalization of the well-known Hamiltonian
proposed by Laughlin [37] to describe the quantum fractional Hall effect. In
the present paper, we investigate the case of the 3D nonlinear quantum system
in an electromagnetic field and in the gravitational field, taking into account
the stratification of the density of matter. The corresponding master equation
in dimensionless variables has the form:
∂ψ
∂t =1
22ψ+iΩ(x∂ψ
∂y y ψ
∂x )β|ψ|2ψV ψ + ( ~
A)2ψ(12)
Here ~
Ais a dimensionless vector potential, V=V(x, y, z), β =β(x, y, z),Ω -
gravity potential and parameters of the model describing the number of particles
and the angular momentum, respectively. Eq (12) describes evolution of the
wave function from some initial state ψ(x, y, z , 0) = ψ0(x, y, z) and up to the
state describing a certain number of vortices, which depends mainly on the
angular velocity Ω. In the case of a magnetic dipole ~m =m0(sin θ, 0, cosθ) and
7
the electric charge qcombined with it, we assume scalar potential ϕ= 0, and
the vector potential
~
A(x, y, z, t) = [~m~r]
r3qt~r
r3(13)
For the equation (12) we consider the problem of the decay of the initial
state, which we set in the form
ψ(x, y, z, 0) = exp[(x2+y2+z2)/2] (14)
As boundary conditions, we will use the function of the initial state (14) given on
the boundaries of the computational domain. We note that for sufficiently large
dimensions of the domain this is equivalent to zero boundary conditions. We
used the finite element method with a division of the sphere into tetrahedrons
and an explicit Euler in time. Typical calculations used 14286 elementary cells,
in test problems this number increased to 142860 and 1428600, respectively.
The time step varied from 0.002 to 0.02. As the main solver and for visualizing
the data, we used Wolfram Mathematica 11.3.
4 Results
Our model describes vorticity on a sphere and we apply this to study the in-
fluence of gravity on the process of formation of quantum vortices. Gravity is
modelled by the parameter βin the master equation (see materials and meth-
ods), which yields a stratification of the gravitational potential with higher
gravity density at the core of the sphere. The stratification of the parameters β
varies inversely with the density of matter, and hence the stratification leads to
an increase in the quantum vorticity in the spherical model which arises from
the nucleus. The results show that first phenomenon of importance is that the
gravity of the planet suppresses vorticity in the spherical system, which can be
seen over several figures of the solutions to the master equation (Figure 1-4, 9-
11). The results show furthermore that vorticity is fueled by the electromagnetic
field of the planet, and hence counterbalanced by the gravitational field. We
8
modelled the electromagnetic field by (13), which leads to that vorticity arises
entirely from the nucleus of the planet and is projected towards the atmosphere
(Figure 5-8). In order to conceive in detail these results we look deeper at the
analysis behind this model. Let us first consider the effect of stratification on
quantum vorticity. Suppose that the beta parameter changes as the earth’s
density, the gravitational potential is zero, and the remaining parameters of the
model are Ω = 2, m0= 6, θ = 0, q =6 (Fig. 1-4). By following this we see
in Fig. 1 the distribution of the βparameter (gravity - on the left) and the
quantum vorticity near the core of the planet (right). Following, fig. 2 shows
the quantum vorticity near the core of the planet.
.
Figure 1 The beta parameter (on the left) and the vorticity distribution around
a dense nucleus (right) computed for t= 1 with Ω = 2, m0= 6, θ = 0, q =6.
.
9
Figure 2 The quantum vorticity near the core of the planet (on the left) and
the vortex output to the planet’s surface (right).
Here we see the emergence of the vortices that we propose are the driv-
ing force behind the cyclonic and anti-cyclonic vortices behind low and high
pressure systems manifested in the troposphere. The arise of the vortexes is
interestingly projected towards the poles of the planet, as shown in figure 3,
which respects the patterns of the distribution of low and high pressure systems
frequently circulating on the northern and southern hemispheres [39]. Fig 3
shows furthermore in the cross-sections that vorticity is strongest closer to the
nucleus, and thus the troposphere and mesosphere manifest the most marked
atmospheric phenomena of vorticity (hurricanes, cyclones, low pressures) com-
pared to the thermosphere, ionosphere and stratosphere. The stratification of
the gravity gives therefore a good model for modelling vorticity on planet earth
using quantum mechanical operators. The regions on figure 3 correspond to
planetary latitudes of 50 to 60 degrees. We wish further to study how the wave-
function solution behaves over time, and by modelling the amplitude from the
initial Gaussian distribution one can follow how the wavefunction is at the ini-
tial stage a dense core, which then decays to form a quantum vorticity. This is
of particular interest, because the density of the core is critical for generating a
dynamo effect on the planet, which is a driving force of the vortex formation.
Planetary dynamos require a particular density of the core [40], which we model
by studying the evolution of the wavefunction (Fig. 4).
10
.
Figure 3 The amplitude of the wave function at different cross sections calcu-
lated for t= 1 with Ω = 2, m0= 6, θ = 0, q =6. z= 0,0.5,1,1.5,2,2.5,3,3.5
.
Figure 4 The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, m0= 6, θ = 0, q =6.
We furthermore move to the second problem, where we consider a system
with an inverse stratification of the beta parameter, when beta increases from
the center to the periphery with a jump from 10 to 1000 at r= 2. This is
done in order to model the surface of the planet, where the density of matter is
11
drastically reduced. At this level of the planet, the quantum vortices distribute
in an entirely arbitrary fashion, without any particular symmetry (Fig 5). The
model shows a nucleus of the origin of the vortices at the rupture region to
which the vortices adjoin. The vortices gain here a more pronounced tubular
structure (Fig 5). The amplitude of the wavefunction for this problem shows
further a more delocalized pattern of vortices forming across the globe, without
a predominance of higher latitudes (Fig 6). When a comparison is made of this
figure with the first four figures, one can see the structure of the nucleus, whose
density reaches 100, is well seen at the top, whereas in the region of the existence
of vortices the typical density (the square of the amplitude) is of the order of
0.01. This suggests that vortices manifest in less dense areas of the planet, as
in the atmosphere and vanish in the higher strata of the planet.
.
Figure 5 The distribution of quantum vortices in the half sphere (left) and the
vorticity distribution around a dense nucleus (right) in a system with an inverse
stratification of the beta parameter computed for t= 1 with Ω = 2, m0= 6, θ =
0, q =6.
12
.
Figure 6 The amplitude of the wave function at different cross sections calcu-
lated for t= 1 with Ω = 2, m0= 6, θ = 0, q =6.
The wavefunction which describes these vortices evolves over time from a
dense gravity nucleus to a more pronounced state of chaos and turbulence to
higher levels of the planet, with a higher density of vortices at time 0.5-0.8 (Fig.
7).This suggests that the interplay between the compressive force of gravity in
a sphere and the effect of magnetism forms a basis for vorticity on the planet,
which is finely tuned by the density of the strata. This can be seen more
clearly in figure 8, where the vorticity distributes spherically from the nucleus.
When stratification is applied on gravity (β) it is proportional to the density of
the planet and hence in the presence of a gravitational potential, the quantum
vorticity is less pronounced.
13
.
Figure 7 The amplitude of the wave function at different tin cross sections
z= 0 calculated with Ω = 2, m0= 6, θ = 0, q =6.
.
Figure 8 The quantum vorticity near the core of the planet (on the right) and
the vortex output to the planet’s surface (left).
The distribution of the density across the planetary globe and the gravi-
14
tational potential play therefore a major role on the vortex formation in the
vicinity of the nucleus (Fig 9). Comparing these data with the data in Fig. 1,
2, 5, and 8, we find the difference in the form of vortices, due to gravitational
potential. We can therefore assume that by this model other phenomena such
as the Jupiter’s spot and other planetary atmospheres can be modelled by using
the combination of quantum mechanical operators and stratification of gravity.
We furthermore calculated the the distribution of the vortices in different cross
sections, z= 0,0.5,1,1.5,2,2.5,3,3.5 for this case, where the vortices also reach
the surface at a latitude between 50 and 60 degrees (Fig 10).
.
Figure 9 The density of the planet ρand β=ρ(top left), gravitational poten-
tial (top right), and the vorticity distribution around a dense nucleus (at the
15
bottom) computed for t= 1 with Ω = 2, m0= 6, θ = 0, q =1.
.
Figure 10 The amplitude of the wave function at different cross sections cal-
culated for t= 1 with Ω = 2, m0= 6, θ = 0, q =1.
By increasing the beta parameter by two orders of magnitude with the grav-
itational potential unchanged, we find a new state in which the vortex tubes are
thinned out (Fig. 11) and finally, as the gravitational potential is increased by
two orders of magnitude, vorticity completely disappears (data not shown). In
conclusion, we note that a numerical solution to the supersymmetric Hamilto-
nian by using the finite element method makes it possible to visualize vorticity
with a good resolution (Fig. 12).
.
16
Figure 11 The quantum vorticity near the core of the planet in the state with
β= 100ρin the gravitational potential as in Fig. 9 computed for t= 1 with
Ω=2, m0= 6, θ = 0, q =6.
.
Figure 12 Spherical mesh and visualization of quantum vorticity in a homoge-
neous magnetic field parallel to the axis of rotation.
However, in the three-dimensional problem the solution has no symmetry,
although the initial data are symmetric. This is explained by the fact that
the vortex system is unstable in 3D, since here we have the problem of many
bodies.The results obtained earlier for planar systems show the presence of
symmetry in the distribution of vortices. This symmetry, obviously, disappears
as we go over to 3D, since the vortices interact in a volume in a different way
than in the plane.This problem is closely related to the problems of quantum
vortex turbulence and hydrodynamic turbulence. In this sense, model (12) is a
convenient tool for investigating these problems.
5 Discussion
Our model shows a representation of the vorticity in the atmosphere on the
earths surface, with a pronounced vorticity arising from the nucleus of the
17
planet, localizing over the northern and southern hemispheres according to a
quantized form without following a particular symmetry. The model accounts
for a stratification of the gravity field with the variable β. Compared to other
models, such as the models for simulating non-linear dynamics [24, 25] and
the Lagrangian discrete models for simulating the fluid-dynamic behaviour of
weather systems [26] and other models [27, 28, 22] we differentiate by a simpli-
fication of the ensemble of vortices, which from our model solely arise from one
source (the interplay of magnetism and gravity). This simplification can give
advantages when modelling atmospheric behaviour on other planets ab initio,
by accounting for only basic parameters as the strength of the gravity field, the
properties of the core and the intensity of the magnetic field, which can also give
an efficient computational time for atmospheric predictions. When modelling
weather on planet earth however, the implementation of parameters such as
ocean temperatures, sun light reflection, circulation of the great conveyor belt
as well as the circulation of the trade winds are required to provide exact fore-
cast data and differentiate between the vortices accurately. The Hamiltonian
used in this study origins from particles oscillating in an electromagnetic field
in a quantized system [12, 11, 13] and by describing quantized phenomena can
suggest that the results presented by [19] on the quantized behaviour of gravity
can have impact on the vorticity in the atmosphere on planet earth and other
planets as well. The quantization manifests at macroscale by a dominant popu-
lation of vortices forming closer to the poles, which is observed on planet earth
indeed [40]. It grants however further scrutiny to study atmospheric phenomena
using quantum models, as many of the macroscopic phenomena are triggered
by ensembles of classical mechanic effects (i.e. thermal diffusion), which play
a marked role on the dynamics of atmospheres. A combination of quantum
representation of major effects on planetary systems with classical models may
therefore open for a new model to study atmospheric dynamics.
18
6 Conclusions
This study shows that a supersymmetric quantum mechanic operator (4) solved
by using the master equation in (12) can be used as a foundation to model
vorticity on planet earth, considering only gravity as a stratified potential and
the electromagnetic field. The results require further development to include
parameters such as ocean temperatures and circulation, reflection, planetary
oscillation around the sun, as well as magnetic effects from the moon and the
sun’s magnetic fields. The evolution of the wavefunction requires furthermore
localizing the exit of the vortices on the surface in detail, so a coordinate sys-
tem can be attributed to the supersymmetric Hamiltonian model for planetary
vorticity. The authors are working further to include these properties in future
models.
Acknowledgements
All data related to this project can be accessed at http://www.fjordforsk.no
19
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... Vorticity is ubiquitous in nature and arises in all natural phenomena, from vortex eddies in water streams [1], to vortices in electromagnetic fields represented by superposition of wavefunctions [2,3,4,5,6], vortices in the atmosphere [7], vortices in superheated stellar plasma [8] and in many other phases of nature, including vortices in gravity waves [9,10,11]. Interestingly, Bernoulli proposed in 1736 that also light is behaving in a vortex-like manner, a theory which was further discussed by Kelly [12] where light was proposed to be a component of a continuous medium present in the universe. ...
... Here, we devise a supersymmetric wave-equation which we developed recently over several works [3,4,5,6] when studying electron gas in quantized 2D systems. The Hamiltonian of the supersymmetric wave-equation is of particular appeal to model vorticity in a space-time continuum, as it generates vorticity spontaneously without need of auxiliary functions and has a particularly appealing spectrum of eigenvalues, where the range of trivial eigenvalues in its spectrum is positive infinite, and the evolution of the spectrum is slowly crescent [17]. ...
... The evolution of the metric , shown in Figure 2 and 3 bears a series of important implications for the phenomenon of gravity. Taken from a spacetime curvature perspective, the positive sign in the metric attributes to gravity a geometry in spacetime which favors the formation of hexagonal shape, which we do not observe in gravitational orbits, but rather in electromagnetic systems, such as vortices in Bose-Einstein condensates [4,5,6,21,22,23,24] or for macroscopic fine medium (gases) governed by electromagnetic fields such as on the North Pole of Saturn [25]). We show therefore a preserved vortex model in eqn. ...
Technical Report
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Fundamental physical phenomena, such as gravity, electromagnetism and light have been studied over the last centuries for their properties, mechanics and for their relationships to one-another. Bernoulli proposed in 1736 a vortex-sponge theory to describe light in mechanical terms which was later postulated by using Maxwell's vortex equations. In this study we use a supersymmetric Hamiltonian recently developed for quantum systems to describe the vortex nature of space-time curvature arising at gravity sources. We use the solar system as model and describe the gravitational effect on space-time as a sheer result of vorticity in the gravity field, maintaining planets in their elliptic orbit and bending space-time in a similar fashion to Einstein's gravity sink model. It is shown that from the Einstein equations for the post-Newtonian metric, the nonlinear diffusion equation and the Schrödinger equation can be derived. The results are analyzed numerically and the dynamics of the system studied.
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