Significance of gyropotentials in metric theory of inertia

Igor É. Bulyzhenkov

Levich Institute for Time Nature Explorations, Moscow, Russia

Abstract. The constancy of orbital velocities of peripheral stars in a spiral
galaxy points to a potential regime of co-rotation together with the
interstellar densities of the galactic disk. The Einstein gyropotential rises
to the evolutionary constant of a self-rotating metric profile, while the
Newtonian potential falls radially. Gyropotential in vortex self-organizations
creates centripetal gyroforces that prevail over Newtonian attraction of probe
bodies at large distances from the axis of rotation. For practical
verifications, the 1914 geodesic relations of Einstein predict the inverse
distance velocities for retrograde stars. For the solar system, gyrometric
corrections of Newtonian forces under transitions to the potential circulation
in material vortexes can be sought in the Kuiper Belt, the peripheral rings of
Saturn and high equatorial orbits around other planets. In conclusion, the
mechanical analogy with the Aharonov–Bohm effect and a possible transition to
the monistic theory of metric inertia are discussed.

PACS: 95.35.+d; 98.62.Dm; 98.62.Gq

##  1 Introduction

Non-Keplerian orbits at the galactic periphery can be demystified by
observational study of retrograde stars in those vortex organizations where
the gyropotentials of Einstein’s metric theory begin to make a significant
contribution to the low-velocity limit of motion. The geodesic relations for
probe bodies rely on four metric potentials [1], while incomplete Newtonian
gravity allows only one-component potential even for rotating sources. To
explain non-Keplerian orbits, dark matter profiles are persistently added to
Newtonian physics to model the observed constancy of the azimuth velocity of
stars at large distances from the axis of a rotating galaxy.

My domestic opponents firmly support the existence of dark matter as “a well-
argued fact” [2]. Their instrumental infrastructure to search for dark matter
in space and underground assumes infallibility of Newtonian mechanics for the
slow motion in weak fields. It came down to a priori statements that
relativistic physics is not able to get around the requirement of dark matter
due to the far-fetched assumption that Einsteinian theory should allegedly
repeat Newtonian one in “the area of joint applicability” i.e. when mechanical
bodies can move freely in weak accelerating fields.

The official plan of the Russian Academy of Sciences for 2006-2025 prescribed
fundamental research not only “to develop cosmological models that should take
into account dark matter and seek to explain its nature” analytically, but
also to spend on the costly “solution of the problem of finding the hidden
baryonic matter”. The purpose of this work is to undeline a measurable
difference between Newton’s dynamics and Einstein’s geodesics in weak fields
of rotating masses. And to reveal the principle Newtonian errors in agreement
with Kuhn’s conceptual conclusion [3]: “Einstein’s theory can be accepted only
with the recognition that Newton’s was wrong”. This theoretical part of the
domestic discussion with dark matter proponents may be of interest to a broad
readership in those countries that were also involved in expansive searches
for dark wonders.

After discussing the incompleteness of the Newtonian model in the case of
vortex self-organization, an attempt will be made to reorient collective
research from the dark matter highway to an informative analysis of retrograde
stars and potential forms of cyclic motion in vortex distributions according
to Einstein’s physics with a monistic field-matter, as in quantum mechanics.
Briefly, the rotating self-organization of the galactic medium can generate
three metric gyropotentials at the disk periphery in addition to the Newtonian
scalar potential. With large radii of the equatorial orbits, Einstein’s
gyroforces begin to supplement the centripetal forces from the inverse square
law. It can be expected that possible observations of retrograde bodies in
distant orbits around elliptical galaxies or even individual planets will
translate Einstein’s metric formalism for the macro- and megacosm into a non-
local language of the material continuum with instantaneous correlations.

##  2 Incompleteness of Newtonian gravity for fields in rotating distributions

To make it easier to understand Kuhn and the conceptual difference between
orbits in one-component and four-component potentials, we will first try to
think only according to Newton, as taught in high school. One can model a
spiral galaxy with a summary mass $M_{\Sigma}=M_{b}+M_{d}$ of a spherical
bulge and a thin disk with two-dimensional mass density $\sigma=M_{d}(L)/\pi
L^{2}=const$. Let the disk significantly go beyond the radius $l$ of the bulge
containing a constant (for simplicity) mass density $\mu=3M_{b}(l)/4\pi
l^{3}=const$. Spherically symmetric potential
$\varphi_{b}(|{\bm{x}}|)=-GM_{b}[(3l^{2}-|\bm{x}|^{2})/2l^{3}]\Theta(l-|\bm{x}|)-GM_{b}\Theta(|\bm{x}|-l)/|\bm{x}|$
from the homogeneous distribution of the bulge mass $M_{b}$ is additively
superimposed in Newton’s theory on the gravitational potential
$\varphi_{d}({\rho},{z})$ of the thin disk distribution of mass $M_{d}$ within
the entire circle with radius $L\gg l$.

To understand the force gradients along the plane of the disk, let us first
compare the scalar potentials $\varphi_{d}({\rho}=0,{z}\Rightarrow 0)=-2\pi
G\sigma({\sqrt{L^{2}+z^{2}}-|z|})\Rightarrow-2\pi G\sigma L$ in the center of
the disk and on its edge,

$\varphi_{d}(L,0)=-G\\!\\!\int\\!\sigma dS/r=-G\sigma\\!\\!\int 2\theta
dr=4\pi G\sigma L\\!\\!\int^{0}_{\pi/2}\\!\theta\sin\theta d\theta=-4G\sigma
L.$ (1)

Here we used a geometric constraint for the integration segment from the edge
of the disk ($r=2Lcos\theta$, $dr=-2Lsin\theta d\theta$) and integrated by
parts with respect to the angular variable $\theta$.

The Newtonian potential at the center of a thin disk is smaller than at the
periphery, since $-2\pi G\sigma L<-4G\sigma L$. The monotonic variation of
this potential along the disk radius can be approximated by the linear
function $\varphi_{d}(0\leq\rho\leq L,z=0)\approx-2\pi G\sigma
L+2G(\pi-2)\sigma\rho$. Therefore, in addition to the radial gradients
$\varphi_{b}(|{\bm{x}}|)$, the static model of the galaxy also creates
centripetal forces from the gradients of the disk potential $\varphi_{d}$. In
the simplest case of circular orbits around the axis of such a model galaxy,
Newtonian physics leads to a radial balance of accelerations and specific
forces in the total potential of a spherical bulge and a thin disk,

$\displaystyle\frac{v^{2}}{\rho}=\partial_{\rho}(\varphi_{b}\\!+\\!\varphi_{d})\Rightarrow\frac{GM_{b}\rho}{l^{3}}\Theta(l-|\bm{x}|)+\frac{GM_{b}}{\rho^{2}}\Theta(|\bm{x}|-l)+2G(\pi-2)\sigma.$
(2)

This classical balance corresponds to Kepler orbits at moderate distances from
the axis, when $\rho\ll L\sqrt{M_{b}/M_{d}}$, but on the homogeneous disk
periphery it unexpectedly predicts even a radial increase in the orbital
velocity $v=$ ${\sqrt{2G(\pi-2)\sigma\rho}}$. It is clear that a uniform disk
profile of interacting masses is unstable without adjusting pressure or some
artificial kind of centrifugal forces. In the reality of differential auto-
rotations, such a uniform profile cannot be formed either in in statics or in
dynamics. Nonetheless, the simplicity of calculations for static profiles,
such as the Mestel disk with a radially decreasing density [4], makes it
possible to quickly assess the key points in modeling of inhomogeneous
distributions.

Far from the bulge boundaries, Newtonian balance (2) could also predict the
radial constancy of the orbital velocities of probe bodies in a static disk
with a monotonically decreasing density
$\sigma(\rho)\approx\sigma(\rho_{o})\rho_{o}ln(e\rho/\rho_{o})/\rho$ on the
far periphery of radii $\rho$, when $l\leq\rho_{o}\leq\rho\leq L$. But how to
keep such a bulge-disk profile of a stellar medium in statics? Numerical
simulations of spiral galaxies through inhomogeneous density distributions for
scalar potentials cannot abandon the classical balance for centripetal
accelerations and Newtonian forces
${v^{2}}/{\rho}=\partial_{\rho}(\varphi_{b}\\!+\\!\varphi_{d})$. Deviation
from the spherical symmetry of the static profiles of galaxies without
rotations would raise questions about the principles of their self-
organization, albeit with dark matter.

Not only a disk, but also an ellipsoidal profile of mass-energy implies
stationary motions of Newtonian sources along stable orbits. However, there
are no Lorentz-type forces in Newtonian gravity. And it is not surprising that
Newton’s scalar potential failed to quantitatively match the observed
luminosities in rotating galaxies with their calculated profiles. To get rid
of the mystical forces of dark matter in modern physics, it is necessary to
move from Newtonian gravi-mechanics with one scalar potential to the Einstein
transport of probe bodies in metric space-time with 4 geodesic potentials as
in Maxwell’s theory with the electromagnetic four-potential.

##  3 Potential circulations of geodesic bodies in metric space-time with
gyro-potentials

The budgeting of mega-science infrastructures for the search and scientific
verification of dark matter in an era of non-local quantum distributions and
well-defined affine connections in the geometric organization of space-time
raises many questions. It would seem that in modern mechanics it is necessary
to comprehend not medieval ideas about dark forces in the surrounding cosmos,
but the conceptual flaws of the dual world palliative when transferring the
monistic distribution of quantum mass-energy to macroscopic scales.

Before introducing dark matter into the galactic halo of the Newtonian void,
let’s recall the well-known differences between the Newton model (massless
fields between massibe particles) and the mass-energy unity in quantum fields.
And, let us confirm quantitatively that the nonrelativistic accelerations in
the four metric potentials of Einstein, $c^{2}g_{\mu}\equiv
c^{2}g_{o\mu}/{\sqrt{g_{oo}}}$, are by no means reducible to slow motion
referents in the scalar potential of Newton,
$\varphi_{\Sigma}\equiv-G\sum(m_{k}/r_{k})=c^{2}{\sqrt{g_{oo}}}-c^{2}\leq 0$.
Everyone knows that back in 1914, Einstein introduced the geodesic
4-acceleration $a_{\mu}\equiv
c^{2}(du_{\mu}/ds)-c^{2}{\Gamma^{\lambda}_{\mu\nu}}u_{\lambda}u^{\nu}\equiv
c^{2}u^{\nu}\nabla_{\nu}u_{\mu}=0$ for the free motion of the probe mass (or
its scalar density $\mu_{p}=const$) in arbitrary metric fields [1].

In the absence of dissipation or non-metric external forces, the probe mass
density (or an idealized point mass) is forced to move along the spatial
coordinates, $dx_{p}^{i}\neq 0$, according to the geodesic 4-acceleration with
a relativistic 4-velocity
$cu_{\mu}\equiv\\{c\gamma{\sqrt{g_{oo}}};-\gamma[v_{i}-c(g_{oi}/{\sqrt{g_{oo}}})]\\}\equiv
cg_{\mu\nu}dx_{p}^{\nu}/ds$, with $u^{\nu}\nabla_{\mu}u_{\nu}\equiv 0$. Here,
the well-known factor $c^{2}\gamma\equiv c^{2}/{\sqrt{1-v_{i}v^{i}/c^{2}}}$ is
the variable kinetic potential acquired by the probe mass density $\mu_{p}$
during motion. Moreover, the physical 3-velocity $v^{i}\equiv
cdx_{p}^{i}/\gamma ds\equiv cdx_{p}^{i}/g_{\nu}dx^{\nu}$ for geodesic transfer
along three spatial coordinates $x^{i}$ is determined in General Relativity
not by the Minkowski world coordinate $x^{o}/c$, but by a specific rate of
physical time
$g_{\mu}dx^{\mu}/c\equiv[{\sqrt{g_{oo}}}dx^{o}+(g_{oi}/{\sqrt{g_{oo}}})dx_{p}^{i}]/c$
for the moving density $\mu_{p}$. A motionless observer operates with the
universal local time ${\sqrt{g_{oo}}}dx^{o}/c$ to describe moving particles.
Therefore, the locally measured velocity of any probe particle becomes
$dx_{p}^{i}/({\sqrt{g_{oo}}}dx^{o}/c)$. The physical and measured velocities
coincide only under the local condition $g_{oi}dx_{p}^{i}/{\sqrt{g_{oo}}}=0$,
which emphasizes the importance of the Einstein gyropotential
$c^{2}g_{i}\equiv c^{2}g_{oi}/{\sqrt{g_{oo}}}$ for a correct description of
the moving probe masses even in the negative potentials of Newton (for always
positive values of the corresponding Einstein potential, $0<c^{2}g_{o}\leq
c^{2}$).

In Einstein’s physics, the free motion of the probe scalar mass (or its local
density $\mu_{p}c^{2}$) obeys the strict geodesic balances
$\mu_{p}c^{2}u^{\nu}\nabla_{\nu}u_{\mu}=0$ and the metric path identities
$(-\mu_{p}c^{2})u^{\nu}\nabla_{\mu}u_{\nu}\equiv 0$ for local 4-velocities
$u_{\mu}\equiv dx_{\mu}/ds$ and $u^{\mu}\equiv dx^{\mu}/ds$ with the adaptive
coupling $u_{\mu}u^{\mu}\equiv 1$:

$\displaystyle\cases{\mu_{p}c^{2}u^{\nu}(\nabla_{\nu}u_{\mu}-\nabla_{\mu}u_{\nu})\equiv\mu_{p}c^{2}u^{\nu}(\partial_{\nu}u_{\mu}-\partial_{\mu}u_{\nu})=0\
$or by components:$\cr-\mu_{p}\gamma
v^{j}[c^{2}\partial_{j}(-\gamma{\sqrt{g_{oo}}})-c\partial_{o}\gamma(v_{j}-cg_{j})]\equiv-\mu_{p}\gamma
g_{o}{\bm{v}}\cdot{\bm{E}}=0,\cr\mu_{p}u^{o}\\![c^{2}\partial_{i}(-{\gamma\sqrt{g_{oo}}})\\!-\\!c\partial_{o}\gamma\\!(\\!v_{i}\\!-\\!cg_{i})]\\!+\\!\mu_{p}\gamma
v^{j}[\partial_{i}\gamma\\!(\\!v_{j}\\!-\\!cg_{j})\\!-\\!\partial_{j}\gamma\\!(v_{i}\\!-\\!cg_{i})]=0\cr$or
$\
\mu_{p}u^{o}[c^{2}{\bm{\partial}}(-\gamma{\sqrt{g_{oo}}})-c\partial_{o}(\gamma{\bm{v}}\\!-\\!c\gamma{\bm{g}})]+\mu_{p}\gamma{\bm{v}}\times
curl[({\bm{v}}\\!-\\!c{\bm{g}})\gamma]\cr\equiv\mu_{p}(u^{o}g_{o}{\bm{E}}+\gamma{\bm{\beta}}\times{\bm{B}})\equiv\mu_{p}u^{o}g_{o}[{\bm{E}}+(d{\bm{x}}/g_{o}dx^{o})\times{\bm{B}}]=0.}$
(3)

Here ${\bm{E}}\equiv
c^{2}g^{-1}_{o}{\bm{\partial}}(-\gamma{{g_{o}}})-cg^{-1}_{o}\partial_{o}\gamma({\bm{v}}-c{\bm{g}})$,
${\bm{B}(x)}\equiv curl(c{\bm{v}}\gamma)-curl(c^{2}{\bm{g}}\gamma)$, and the
3-vector $(curl\bm{a})^{i}\equiv
e^{ijk}(\partial_{j}a_{k}-\partial_{k}a_{j})/2{\sqrt{|g_{ps}|}}$ is dual to
the tensor $(\nabla_{j}a_{k}-\nabla_{k}a_{j})$. Four geodesic relations (3)
for moving probe densities $\mu_{p}$ generalize non-dissipative Bernoulli
flows with conservation of energy along fluid streamlines
${\bm{v}}\cdot{\bm{E}}=0$. This geodesic conservation for the adaptive motion
in metric fields with four relativistic potentials $c^{2}g_{\nu}\equiv
c^{2}g_{o\nu}/g_{o}$ is a direct mathematical consequence of three last
equations for the kinetic-metric tensions ${\bm{E}}$ and ${\bm{B}}$.

The tension ${\bm{B}}$ is responsible in (3) for the relativistic imbalance
between the Euler-Lagrange centrifugal force $\mu_{p}\gamma{\bm{v}}\times
curl(-{\bm{v}}\gamma)$ and the Einstein centripetal gyro-force
$\mu_{p}c\gamma{\bm{v}}\times curl({\bm{g}}\gamma)$ in metric organizations of
vortex fields. For the metric gyro-force, the analogy with the current
components of the Lorentz force is obvious when electric charges move in an
external magnetic field. Metric gyro-potentials
$c^{2}{\bm{g}}\equiv\\{c^{2}g_{1},c^{2}g_{2},c^{2}g_{3}\\}$ for probe masses
were not taken into account by Euler in the classical model of an ideal fluid.
And Newton’s scalar potential stayed in hydrodynamic applications not
supplemented by Einstein’s gyro-potential for swirling mass-energy flows.
Newtonian mechanics and Euler forces do not correspond to Einsteinian
accelerations of probe bodies even in slowly rotating distribution of inertial
mass-energy. Einstein’s four-potential geodesics had not been not falsified by
experiment, while the Euler/Navier-Stokes equation does not work in many
applications to rotating fluids. In no case can the Newtonian dynamics with
one-component potential claim to be complete in modeling the material profiles
of a vortex galaxy.

The key misinterpretation of metric spacetime after the dual worldview with
massive particles and massless fields is that the metric field inside the
galaxy disk belongs to the void with zero local inertia. Such spatial voids
cannot contribute to the local creation of field gyropotentials. In 1938,
Einstein and Infeld [7] proposed a non-dual reading of mechanical events in
purely field terms for extremely high and low densities in the mass-energy
continuum. Such monistic worldview maintains the nonlocal organization of
continuously distributed inertia [9] through correlated metric connections and
volumetric mass-energy conservations on all time intervals.

If the metric self-organization of a distributed mass-energy (called a galaxy)
rotates around a fixed axis, then a motionless probe density $\mu_{p}$ with
$dx_{p}^{i}/g_{\mu}dx_{p}^{\mu}=dx_{p}^{i}/g_{o}dx^{o}=0$ and
$u^{o}\equiv\gamma dx^{o}/g_{\mu}dx_{p}^{\mu}=1/g_{o}$ will undergo according
to (3) only universal metric forces
$\mu_{p}(-c^{2}{\bm{\partial}}{lng_{o}}+g^{-1}_{o}c^{2}\partial_{o}{\bm{g}})$.
This does not mean, as already noted, that any moving density
$\mu_{p}\gamma_{p}$ will also experience only universal force actions. Moving
probe bodies are subjected to very complex accelerations from the geodesic
tensions (3) because of the specific potentials $\gamma_{p}c^{2}$. Thus,
geodesic accelerations depend both on the universal metric potentials
$c^{2}g_{o\mu}/{\sqrt{g_{oo}}}$ and on specific 3-velocities
$v_{p}^{i}=cdx_{p}^{i}/g_{\mu}dx_{p}^{\mu}$ for each probe particle.

One can use the vector identity $curl\ grad\ \equiv 0$ for the metric-kinetic
balance (3). This identity leads to the temporal dependence of the geodesic
tension ${\bm{B}}$ for moving probe densities in a metric vortex,

$\displaystyle\cases{\partial_{o}{\bm{B}}\equiv\partial_{o}curl[({\bm{v}}\\!-\\!c{\bm{g}})\gamma
c]=curl[(\gamma{\bm{v}}/cu^{o})\times
curl({\bm{v}}\gamma-c{\bm{g}}\gamma)c]\cr\cr\int_{x^{o}_{1}}^{x^{o}_{2}}\\!dx^{o}\partial_{o}{\bm{B}}\\!\equiv
c[curl\\!({\bm{v}}\\!-\\!c{\bm{g}})\gamma]_{x^{o}_{1}}^{x^{o}_{2}}\\!=\\!c\\!\int_{x^{o}_{1}}^{x^{o}_{2}}curl\\!\left[\frac{d{\bm{x}}}{dx^{o}}\\!\times\\!curl({\bm{v}}\gamma\\!-\\!c{\bm{g}}\gamma)\right]\\!dx^{o}.}$
(4)

Einsteinian geodesics (3) leads to mathematical consequences (4), which allow
us to understand the physical meaning of the coordinate components
$cu_{i}=-({v_{i}}-c{g_{i}})\gamma$ in the covariant 4-velocity $cu_{\mu}$. It
can be seen from the first equation in (4) that if $curl[\gamma
c({\bm{v}}-c{\bm{g}})]_{x^{o}_{1}}$ = 0 everywhere at the initial time
$x^{o}_{1}$, then the time derivative of this value also vanishes. This means
that, according to the second Lagrange theorem, ${\bm{B}}(x^{o}>x^{o}_{1})=0$
at all subsequent times. The stable vanishing
$curl[\gamma({\bm{v}}-c{\bm{g}})]=0$ of the kinetic-metric tension in self-
organized vortexes of matter was noticed, in particular, for the superfluid
(and dissipation free) motion of helium. These steady circulations of liquid
densities were called the potential motion of vortex matter.

The geometric (holonomic) organization of space-time with geodesics (3) must
support potential vortexes for both weak and strong metric fields, since the
balanced kinetic-metric tension ${\bm{B}}\equiv curl[\gamma
c({\bm{v}}-c{\bm{g}})]=0$ can exist, according to (4), for a long period of
the coordinate parameter $x^{o}$. According to Stokes’s theorem, the
corresponding Kelvin circulation along any closed line can also exists during
the same period of time. In general geodesic motions (3), the potential
transfer of probe densities is realized with simultaneous zeroing of both
kinetic-metric tensions: ${\bm{E}}={\bm{B}}=0$. In this case, the steady
Kelvin circulation allows a non-stationary change in the observable kinetic
fraction (or tornado), $\mu_{p}\oint\gamma{\bm{v}}({\bf
x},x^{o})d{\bm{l}}=\mu_{p}c\oint\gamma{\bm{g}}({\bf x},x^{o})d{\bm{l}}\neq 0$,
due to compensatory changes in the Einstein gyropotential $c^{2}{\bm{g}}({\bf
x},x^{o})$ along the closed contour. In many isolated galaxies, the channels
of dissipation of the integral mass-energy can be neglected. Then a stationary
tornado of the observed matter at the periphery of the vortex galaxy can be
accompanied by a finite gyropotential ${\bm{g}}$, with
$\oint\gamma{\bm{g}}d{\bm{l}}\neq 0$, related to the metric self-organization
of galactic fields and mass-energy currents.

##  4 Einstein’s gyroforces instead of dark options

For Keplerian circulations of probe bodies in the central part of the galaxy
and in its near periphery, a non-potential kind of non-relativistic motion is
realized, when one can set in the geodesics (3) that
${\bm{v}}({\bm{x}},x^{o})\approx cd{\bm{x}}/dx^{o},$ $v^{2}\ll c^{2}$,
${\bm{B}}\approx curl(c{\bm{v}}-c^{2}{\bm{g}})\neq 0$,
$\partial_{i}\gamma\approx\partial_{i}v^{2}/2c^{2}$, ${\bm{g}}^{2}\ll 1$,
$c^{2}(1-{\sqrt{g_{oo}}})=-(\varphi_{b}+\varphi_{d})\equiv-\varphi_{{}_{\Sigma}}\ll
c^{2}$,
${\bm{E}}\approx-{\bm{\partial}}(\varphi_{{}_{\Sigma}}+v^{2}/2)-c\partial_{o}(\\!{\bm{v}}-c{\bm{g}})\neq
0$. Let us consider the simplest case of a common axis of symmetry $z$ for
metric fields and circular orbits of probe masses, when
${\bm{g}}(\rho,z)={\hat{\alpha}}g(\rho,z)$, and
${\bm{v}}(\rho,z)={\hat{\alpha}}v(\rho,z)$. Then, specifying the unknown
metric potentials in general form as
$c^{2}g_{o}(\rho,z)=c^{2}+\varphi_{{}_{\Sigma}}$ and
$c^{2}g_{i}(\rho,z)=\\{0,c^{2}g(\rho,z),0\\}$, one can start searching for
orbital velocities for the nonrelativistic limit in the Einstein geodesic (3).

Kepler’s equatorial orbits in the $z=0$ plane are supported by two centripetal
forces
$-m_{p}{\hat{\bm{\rho}}}{\partial}_{\rho}v^{2}/2=-m_{p}\omega^{2}{\bm{\rho}}$
and $-{\hat{\bm{\rho}}}m_{p}{\partial}_{\rho}\varphi_{{}_{\Sigma}}$ (from the
kinetiv-metric tension ${\hat{\bm{\rho}}}E_{\rho}\neq 0$) counteracting the
Euler-Lagrange centrifugal force $m_{p}{\bm{v}}\times
curl{\bm{v}}=+2m_{p}\omega^{2}{\bm{\rho}}$. This dynamic confrontation of
radial forces from non-potential geodesic tensions leads to the vector sum
$-m_{p}\omega^{2}{\bm{\rho}}$ and the resulting acceleration
$-\omega^{2}{\bm{\rho}}$ towards the center of the Kepler orbit. Einstein’s
centripetal gyroforce $m_{p}{\bm{v}}\times curl(-c{\bm{g}})$ cannot compete in
(3) with Newton’s gravitational force
${\hat{\bm{\rho}}}m_{p}(-{\partial}_{\rho}\varphi_{{}_{\Sigma}})$ at small and
medium radial distances.

In the rotating Solar system, for example, mass-energy densities are grouped
almost spherically near the origin, and the radial growth of the relatively
weak gyropotential $c^{2}g(\rho,z)\ll c^{2}$ cannot strongly perturb the
Keplerian rotations of the first nine planets obeying the finite tensions
${\bm{E}}\neq 0$ and ${\bm{B}}\neq 0$ in the geodesics (3). The trans-
Neptunian area and the Kuiper Belt at 30–55 AU from the Sun is practically not
studied because of the low luminosity. In disk galaxies, well-observed
peripheral stars co-rotate together with the interstellar medium at kiloparsec
distances, where the differential rotation of the mass-energy profile brings
the evolutionary gyropotential to the saturation level. This equilibrium
gyropotential at the far periphery of a spinning galaxy can correct and even
completely compensate for the Euler-Lagrange centrifugal force,
$m_{p}{\bm{v}}\times curl({\bm{v}}-c{\bm{g}})\rightarrow 0$:

$\displaystyle\cases{c\partial_{o}({\bm{v}}-c{\bm{g}})+{\bm{\partial}}[(\frac{v^{2}}{2})+c^{2}({\sqrt{g_{oo}}}-1)]={\bm{v}}\times
curl({\bm{v}}-c{\bm{g}}),\cr\cr\partial_{\rho}[\frac{v^{2}(\rho)}{2}+\varphi_{{}_{\Sigma}}]=\frac{v(\rho)}{\rho}\partial_{\rho}[\rho
v(\rho)-c\rho g(\rho)],\cr\cr\
v^{2}(\rho)=\rho\partial_{\rho}\varphi_{{}_{\Sigma}}+v(\rho)c\partial_{\rho}[\rho
g(\rho)],\cr\cr v_{1,2}=\frac{1}{2}\partial_{\rho}(c\rho
g)\pm\frac{1}{2}{\sqrt{[\partial_{\rho}(c\rho
g)]^{2}+4\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}}=\cr\cr\cases{{\sqrt{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}}+\frac{c}{2}\partial_{\rho}(\rho
g)\ $and$\ \frac{c}{2}\partial_{\rho}(\rho
g)-{\sqrt{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}},\ $if$\
[c\partial_{\rho}(\rho
g)]^{2}\ll{{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}}\cr\cr
c\partial_{\rho}(\rho
g)+\frac{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}{c\partial_{\rho}(\rho g)}\
$and$\ -\frac{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}{c\partial_{\rho}(\rho
g)},\ $if$\
{{\rho\partial_{\rho}\varphi_{{}_{\Sigma}}}}\ll[c\partial_{\rho}(\rho
g)]^{2}}.}$ (5)

Strong radiative losses or intensive gas exchange with the intergalactic
medium can lead to non-stationary changes in the Newtonian potential for
peripheral orbits due to temporal changes in the inter-orbit mass integral. In
such a non-stationary galaxy, the temporal derivatives of
$\partial_{o}\\{...\\}$ can give additional centripetal accelerations to
perturb Keplerian orbits due to the intervention of dissipation derivatives
into Newton’s gravimechanics. Now, we neglected evolutionary dynamics in (5)
for the orbital velocity solutions $v_{1,2}$, by considering the stationary
case of motion with $\partial_{o}({\bm{v}}-c{\bm{g}})=0$.

In contrast to the radially decreasing gyropotential in the Kerr metric
$g_{Kerr}$ $\propto const/\rho^{2}$ for the massive singularity with spin [5],
we admit the massive medium in the rotating disk periphery where
$\rho\partial_{\rho}\varphi_{{}_{\Sigma}}/c^{2}$ can be compared with
$[\partial_{\rho}(\rho g)]^{2}\approx const\ll 1$. Then Keplerian rotations
can be perturbed in (5) due to local damping of the Euler-Lagrange centrifugal
force by the Einstein centripetal gyro-force. The post-Keplerian regime of
orbital circulations in the Einstein geodesic starts at
$(c\partial_{\rho}[\rho g(\rho)])^{2}\geq
4\rho\partial_{\rho}\varphi_{{}_{\Sigma}}$, which allows us to estimate from
observations the peripheral gyropotential for the most of disk galaxies as
$g\approx 10^{-3}-10^{-2}$ .

In slowly circulating distributions of galactic mass-energy, the gyropotential
increases monotonically with distance from the rotation axis, but should reach
its limiting saturation at non-relativistic values,
$g(\rho\geq\rho_{*},z=0)\Rightarrow g_{*}=const\ll 1$. After this saturation,
the orbital speed of disk densities cannot grow with radius and remains small
compared to the speed of light. In principle, the two-variable function
$g(\rho,z)$ must be consistent with the metric solutions
${\sqrt{g_{oo}(\rho,z)}}$ for the field - matter organization in the inertial
profile of a rotating disk. This metric self-organization has not yet been
satisfactorily described a rotating medium with constant mass-energy and spin
integrals due to non-locally organized affine connections. However, non-
relativistic geodesic (5) and astronomical observations of retrograde stars in
vortex galaxies together can shed new light on the spatial gradients of the
metric 4-potential $c^{2}g_{\mu}\equiv c^{2}g_{o\mu}/{\sqrt{g_{oo}}}$ in
rotating self-assembly of inertial matter.

##  5 Potential motion on high orbits around elliptical galaxies and planets

The Newtonian potential beyond an elliptical galaxy can be approximated in the
equatorial plane by the textbook function
$\varphi_{{}_{\Sigma}}(\rho,z=0)=-GM/\rho$. Can one expect the geodesic curve
to be constant beyond the visible frames of compactly rotating systems, as was
claimed from (5) for the massive disk medium of a spiral galaxy?

Geodesic relations (3) at very large radial distances can always have a
stationary solution ${\bm{B}}=curl({\bm{v}}-c{\bm{g}})c\Rightarrow 0$ and
${\bm{E}}=-c^{2}\gamma{\bm{\partial}}ln(\gamma g_{o})\Rightarrow 0$ for
potential circulations around a slowly rotating galaxy, since it always
generates a finite, albeit very small, gyropotential
${g(\rho\gg\rho_{*})\approx const\equiv g_{*}}$. We try the orbital velocity
for such potential circulations in the $z=0$ plane as follows:
${\bm{v}}(\rho,z=0)/c={\hat{\bm{\alpha}}}{g}(\rho,z=0)+a[{\hat{z}}\times{\hat{\rho}}]/\rho$
or $v(\rho)=cg_{*}+c(a/\rho)$ for $\rho\geq\rho_{*}$. The so-far undefined
scale $a=const$ will be specified by nonrelativistic geodesic relations:

$\displaystyle\cases{curl[{\bm{v}}(\rho)-c{\bm{g}}(\rho)]=0,\
v(\rho)=cg_{*}+c\frac{a}{\rho}\ll
c,\cr{\bm{\partial}}[\varphi_{{}_{\Sigma}}(\rho)+\frac{v^{2}(\rho)}{2}]=0,\
2\varphi_{{}_{\Sigma}}(\rho)+c^{2}[g_{*}+\frac{a}{\rho}]^{2}=v^{2}(\infty)=c^{2}g^{2}_{*},\cr
g_{*}=\sqrt{g_{*}^{2}+\frac{(-2\varphi_{{}_{\Sigma}})}{c^{2}}}-\frac{a}{\rho}\approx
g_{*}+\frac{(-\varphi_{{}_{\Sigma}})}{c^{2}g_{*}}-\frac{a}{\rho},\cr
a=\frac{(-\varphi_{{}_{\Sigma}})\rho}{g_{*}c^{2}}\Rightarrow\frac{GM}{g_{*}c^{2}}=const,\
\ \frac{GM}{c^{2}\rho}\ll g^{2}_{*}\ll 1}$ (6)

This potential solution $v(\rho)=cg_{*}+({GM}/{g_{*}c\rho})\approx
cg_{*}=const$ acquires the constancy of radial curves at
$\rho\gg{GM}/{g^{2}_{*}c^{2}}\equiv\rho_{*}$. The high orbit velocity
$+{\hat{\bm{\alpha}}}cg_{*}$ in the $z=0$ plane corresponds to the angular co-
rotation of the visible galaxy and probe bodies. A retrograde body in such
high orbits must have the backward velocity
${\bm{v}}_{b}(\rho)\equiv{\hat{\bm{\alpha}}}\rho\omega(\rho)=-{\hat{\bm{\alpha}}}GM/cg_{*}\rho$
from (5) with $\partial_{\rho}(g\rho)\Rightarrow g_{*}$ and
$\rho\partial_{\rho}\varphi_{{}_{\Sigma}}\Rightarrow GM/\rho$ for the textbook
potential of an elliptical galaxy. The backward motion is not potential, since
${\bm{B}}=-{\hat{\bm{z}}}g_{*}/\rho\neq 0$ and
${\bm{E}}=-{\hat{\bm{\rho}}}GM/\rho^{2}\neq 0$ in the geodesic relations
${\bm{E}}+{\bm{v}}_{b}\times{\bm{B}}=0$ for the high equatorial orbits at
$\rho\gg\rho_{*}$. Needless to say, that these geodesic relations also
maintain the Kepler law $v^{2}(\rho)\equiv\rho^{2}\omega^{2}(\rho)=GM/\rho$
for the low orbits at $\rho\ll\rho_{*}$.

Potential circulation in the $z=0$ plane can be realized not only for
perepherial orbits around elliptical galaxies, but also for high equatorial
orbits around rotating planets. Here the equatorial gyropotentials also
acquire finite, albeit small, values $c^{2}g(\rho\rightarrow\infty)=const\neq
0$. The metric fields of the spinning mass-energy are inhomogeneous along the
$z$ axis, which significantly complicates their relativistic calculation.
Nevertheless, upon transition to the potential regime of stationary
circulations in high equatorial orbits, one should expect
${\partial}_{z}[\varphi_{(}\rho,z)+{v^{2}(\rho,z)}/{2}]=0$ from the
potentiality condition ${\bm{E}}=0$. Such a potential motion requires gradient
forces along the $z$ axis, which push the circulating bodies toward the $z=0$
plane. These gradient forces arise even for the middle orbits, where
$\rho\approx\rho_{*}$, by making, for instance, the peripheral rings of
Saturn’s disk extremely thin.

Destroying the nonlocal organization of gyrometic fields for potential
circulations in the disk of Saturn by local mechanical influences is
apparently as difficult as destroying the nonlocal self-organization of a
tornado in the atmosphere or Kelvin circulation along closed macroscopic
contours in an ideal fluid. One could expect non-dissipative reflection of
radio waves by inert bodies on the potential orbits, anomalously slow
relaxation of disk dislocations after collisions with external asteroids, and
suppression of paired interactions between potentially circulating bodies on
neighboring rings of Saturn’s disk. For example, the co-orbital moons Janus
and Epimetheus have roughly equal masses of about $10^{18}kg$, but their
orbits (only a few kilometers apart along the semi-major axis) do not result
in the mutual collision expected from Newtonian gravity. Potential motion in
Einstein’s vortex fields allows only slight fluctuations of orbits, which are
observed every four years [6].

The geodesic relations
${\bm{E}}+({\bm{d}{\bm{x}}}/g_{o}dx^{o})\times{\bm{B}}=0$ are applicable to
any orbits of the Solar System, but the potential case ${\bm{E}}={\bm{B}}=0$
should only be expected at the far periphery of metric vortexes, including the
Kuiper Belt. It can also be recommended from (5) to search for the radial
constancy of the azimuthal velocities of co-rotating satellites and the
inverse radial curves for the backward velocities of retrograde satellites in
different high orbits around Jupiter, Uranus, Neptune and other spinning
concentrations of inertia. Such a directed search will help to abandon the
dark matter hypothesis and to justify Einstein’s geodesics for the ‘anomalous’
rotation in cosmology.

##  6 Conclusion

The main conclusion from the strict mathematical revisit of the relativistic
geodesics is that the non-relativistic limit for metric forces and paired
Newtonian interactions give different predictions for observations. Fields
with a 4-component potential and a one-component potential should lead to
measurable differences in the practice of vortex motions. Moreover, Einstein’s
theory of General Relativity quantitatively and qualitatively differs from
Newton’s physics even at zero speeds of spatial translations because of the
internal degrees of kinetic freedom (with variable rest-energy deposits under
the spatial transport). After all, Einstein’s heated stone should weight more,
while Newton’s has the same weight.

Kuhn has been right indeed in saying that “Einstein’s theory can be accepted
only with the recognition that Newton’s was wrong”. In Einstein’s metric
description of the united energy-momentum there is no independent concept of
negative (gravitational) energy, but there are coordinate shifts of the
unexposed kinetic fraction $mc^{2}$ (which can be extracted, for instance, by
nuclear explosions). The total energy content of any probe mass $m_{p}$ is not
a dual sum of kinetic, $m_{p}c^{2}\gamma>0$, and gravitational,
$m_{p}\varphi_{\Sigma}\leq 0$, contributions of different nature, but the
monistic whole of unexposed and exposed kinetic contributions to the always
positive product, $\mu_{p}c^{2}{\sqrt{{g_{oo}}/(1-\beta^{2})}}>0$. There is
only one (kinetic) four-vector composition for the energy-momentum of
mechanical bodies in General Relativity. And there are no independent four-
vector compositions in co-variant transformations to support the dual concept
of kinetic and potential entities.

Einstein’s metric counterpart of Newton’s scalar potential is always and
everywhere positive $c^{2}{\sqrt{{g_{oo}(x)}}}\equiv
c^{2}+\varphi_{\Sigma}(x)>0$, even for strong external fields. This geometric
potential changes only positive values of the monistic (kinetic) energy along
the body trajectory. At the same time, Einstein’s metric tension are
everywhere local in confirmation of their kinetic (pushing) nature in the
geometric organization of holonomic space-time.

Not only Plato, Aristotle, Descartes, Hegel, Mi, all Russian and Eastern
cosmists proposed to interpret mechanical phenomena in monistic terms of
material fields, but also Einstein and Infeld called for a non-dual theory of
the material field [7]. Everyone understands that the motion of mass-energy
densities determines metric fields. And that the local responses of metric
fields or their tensor stresses can also determine the motion of densities
themselves. Further development of such relativistic ideas about instantaneous
feedback for the adaptive correlation of mechanical motion and the
corresponding metric fields will fail in the dual concept of localized masses
and distant fields in void. An advanced metric theory of inertial densities
does not have to follow the Newtonian declarations, but must self-consistently
calculate an adaptive geometry of space-time for a rotating mass-energy
distribution, i.e. with correlated affine connections and metric components at
all values of the world parameter $x^{o}$. Indeed, the measured conservation
of the system integrals at different times requires strict nonlocal
correlations in the corresponding distributions of mass-energy, momentum, and
angular momentum. In problems of metric self-organization of a rotating
galaxy, what is of interest is not the Kerr field for emptiness (presumably
far somewhere from a rotating mass), but a self-consistent geometry for an
adaptive mass-energy medium without voids for its inertia.

If the peripheral coexistence of gyrofields and intense mass currents in the
visible galactic disk could somehow save the dual model (as in the empty-space
electricity of point charges), then the constancy of orbital velocities around
an elliptical galaxy or planet suggests to assume invisible mass-energy
currents even on the very high orbits. Here centripetal giro-forces should be
supported by local ‘displacement currents’, but not Newtonian void, which
cannot circulate at all. The existence of potential motion around spinning
galaxies and planets, as well as the corresponding retrograde anomalies in
mechanical systems, duplicates the Aharonov-Bohm phenomenon [8] of material
circulations along the electromagnetic 3-potential instead of presumed
Newtonian void beyond the solenoid.

In general, the observed constancy of cosmological curves and the predicted
inverse curves of retrograde circulations can substantiate the local inertia
of material space in monistic physics on mega and macro scales in the same
nonlocal way as in the quantum microcosm. The metric theory of continuous
inertia in such matter-space with the adaptive local time for macroscopic
distributions of non-local mass-energy can also be checked or falsified in the
laboratory due to the appropriate modification [9] of the Euler/Navier-Stokes
dynamics. The monistic theory of metric inertia (but not gravity) justifies
the non-local nature of turbulence and offers high-order temporal derivatives
for non-relativistic fluids in simple laboratory probes. The non-Newtonian
motion of inertial space along its gyro-potentils implies different energy
consumption for secondary restarts of gyroscopes compared to the first launch.
Less energy may be required to restart in the same direction of rotation, and
more - in the opposite direction.

The dual theory of many bodies and their gravitational field encountered
unresolved difficulties in calculating the self-consistent profile of a galaxy
with differentially rotating substance. Overcoming these difficulties is more
promising in the monistic theory of an adaptive metric field, which, due to
its non-local affine connections-correlations, regulates the equilibrium
distribution of rotating mass-energy fluxes according to the hydrodynamic
scenario for non-homogeneous continuum of purely kinetic energy (with thermal
and ethereal fractions of inertia). Here, retrograde stars can provide an
important tool for further understanding of monistic physics for material
space with nonlocal metric vortexes in cosmology, the solar system, and the
earth laboratory. By closing, the methodical transition from the dual model of
separated particles and fields to material fields and waves with quantization
in a continuous distribution of non-local mass-energy deserves more attention
than the costly search for dark matter in the emptiness of Newtonian space.

## References

  * [1] A. Einstein, Die formale Grundlage der allgemeinen Relativitatstheorie. Sitzungsber. Preuss. Acad. Wiss. Berlin (Math. Phys.) 2, 1030-1085 (1914)
  * [2] A.V. Zasov, A.S. Saburova, A.V. Khoperskov, S.A. Khoperskov, Dark matter in galaxies. IOP Publishing Physics-Uspekhi 60, N 1, 3-39 (2017)
  * [3] T. Kuhn, The Structure of Scientific Revolutions, (The University of Chicago Press, Chicago, IL, 1962)
  * [4] L. Mestel, On the galactic law of rotation. Month. Not.Royal Astr. Soc. 126, 553 (1963)
  * [5] R.P. Kerr, Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters. 11 (5), 237–238 (1963)
  * [6] J.N. Spitale, R.A. Jacobson, C.C. Porco, W.M. Owen, Jr. (2006). ”The orbits of Saturn’s small satellites derived from combined historic and Cassini imaging observations”. The Astronomical Journal 132 (2), 692–710 (2006).
  * [7] A. Einstein, L. Infeld, The Evolution of Physics, Edited by C.P. Snow, (Cambridge University Press, 1938)
  * [8] Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in quantum theory. Physical Review. 115 (3), 485–491 (1959)
  * [9] I.É. Bulyzhenkov, Journal of Turbulence 22, 623-639 (2021)


# CDCGen: Cross-Domain Conditional Generation via Normalizing Flows and
Adversarial Training

Hari Prasanna Das  Ryan Tran  Japjot Singh  Yu-Wen Lin  Costas J. Spanos

###### Abstract

How to generate conditional synthetic data for a domain without utilizing
information about its labels/attributes? Our work presents a solution to the
above question. We propose a transfer learning-based framework utilizing
normalizing flows, coupled with both maximum-likelihood and adversarial
training. We model a source domain (labels available) and a target domain
(labels unavailable) with individual normalizing flows, and perform domain
alignment to a common latent space using adversarial discriminators. Due to
the invertible property of flow models, the mapping has exact cycle
consistency. We also learn the joint distribution of the data samples and
attributes in the source domain by employing an encoder to map attributes to
the latent space via adversarial training. During the synthesis phase, given
any combination of attributes, our method can generate synthetic samples
conditioned on them in the target domain. Empirical studies confirm the
effectiveness of our method on benchmarked datasets. We envision our method to
be particularly useful for synthetic data generation in label-scarce systems
by generating non-trivial augmentations via attribute transformations. These
synthetic samples will introduce more entropy into the label-scarce domain
than their geometric and photometric transformation counterparts, helpful for
robust downstream tasks.

Machine Learning, ICML

##  1 Introduction

A large majority of the real-world signals obtained are unlabeled, and require
significant human effort or machine intelligence for labeling. This has led to
a surge in popularity of unsupervised learning algorithms. A prominent branch
of such algorithms, generative modeling, has proven to be efficient in
transferring knowledge gained from one (or multiple) domain(s) to other
domain(s). Variants of such approaches include cross-domain translation (Zhu
et al., 2017; Isola et al., 2017), domain adaptation for classification
(Hoffman et al., 2017; Zou et al., 2019c) etc. By jointly modeling the data
samples and their labels/attributes, variations of conditional synthesis
methods have been proposed (Mirza & Osindero, 2014; Odena, 2016; Liu et al.,
2019a), which during inference phase, can generate synthetic conditional
samples. We combine both the above avenues of cross-domain translation along
with conditional synthesis and propose a framework capable of generating
conditional samples for a domain without utilizing its labels/attributes.

Prior works on cross-domain translation involve construction of a mapping
between two (or more) unpaired domains. The translation consistency is
maintained by introducing some form of inductive bias terms such as cycle
consistency (Zhu et al., 2017), semantic consistency (Royer et al., 2020),
entropic regulation (Courty et al., 2017) etc. Most of the proposed models for
domain translation are generative adversarial network (GAN) (Goodfellow et
al., 2014) based and involve many-to-one/one-to-many mappings, making the
cycle consistency only approximate. A recent work, Alignflow (Grover et al.,
2019) achieves exact cycle consistency by modeling the domains with
normalizing flows via a common latent space. Normalizing flows (Dinh et al.,
2016; Kingma & Dhariwal, 2018) are a class of generative models which map an
unknown and complex data distribution to a latent space with a simple (e.g.
standard gaussian) prior distribution via invertible mappings. Another benefit
with having flow model mappings is that they offer a rich latent space, which
is suitable for a number of downstream tasks, such as semi-supervised learning
(Odena, 2016), synthetic data augmentation and adversarial training (Cisse et
al., 2017), text analysis and model based control etc.

Table 1: Comparison of CDCGen with state-of-the-art cross domain translation
and conditional syntheis models. Across the board, CDCGen features all the
advantages over other models.

Model |  Cross-Domain Translation |  Cycle Consistency |  Independent Conditional Synthesis |  Availability of Latent Space Embeddings
---|---|---|---|---
XGAN (Royer et al., 2020) | ✓ | Approximate | ✗ | ✗
CycleGan (Zhu et al., 2017) | ✓ | Approximate | ✗ | ✗
Taigman et al. (2016) | ✓ | Approximate | ✗ | ✗
Alignflow (Grover et al., 2019) | ✓ | Exact | ✗ | ✓
CGAN (Mirza & Osindero, 2014) | ✗ | – | ✓ | ✗
ACGAN (Odena et al., 2017) | ✗ | – | ✓ | ✗
CAGlow (Liu et al., 2019a) | ✗ | – | ✓ | ✓
CDCGen (ours) | ✓ | Exact | ✓ | ✓

Conditional synthesis has been explored by CGAN (Mirza & Osindero, 2014) by
augmenting the conditions with the data and processing it via GAN and by ACGAN
(Odena, 2016) by introducing an auxilliary classifier for the conditions. This
becomes challenging for flow models which are bijective in nature, and hence
indirect methods must be adopted to jointly model data and the conditions. Liu
et al. (2019a) propose an encoder-discriminator-classifier-decoder based
approach on flow latent space which can generate synthetic samples for a
domain by passing its conditions via encoders to the data via a flow network.
They show improvements in varying the quality of generated images for handles
relating to various features.

We present CDCGen, a generative framework that is capable of transferring
knowledge across multiple domains and using it to generate synthetic samples
for domains lacking information about labels/attributes. We model the
label/attribute scarce domain as the target, and a related domain with
available information about its labels/attributes as the source. We model the
source and target domain via normalizing flows with a common latent space. For
conditional synthesis, we introduce a variant of ACGAN by using it on the
learned latent space rather than the data space, and train it with only the
data and available labels from the source domain. The features can be
manipulated easily in the latent space, which is learnt by the conditional
synthesis network. During inference phase, CDCGen offers independently
specifying conditions, encoding them to a common latent space and moving
through the inverse flow to generate conditional synthetic samples in the
target domain. Table 1 summarizes the comparison between CDCGen and other
related models for different feature availability. CDCGen comes out to be an
amalgamation of all features available among the model selections. We
establish the CDCGen framework and conduct empirical evaluations with
benchmarked image datasets. CDCGen shows encouraging performance in domain
alignment, as well as conditional generation for all source and target
combinations.

##  2 Preliminaries

Flow-based generative models and generative adversarial networks constitute
the major building blocks for proposed CDCGen. The functioning of both the
classes of generative models are included in the Appendix.

##  3 Related Work

We discuss the related work from two perspectives relevant to the CDCGen
framework, namely cross-domain translation and conditional synthesis.

###  3.1 Cross-Domain Translation

Cross-domain translation involves construction of mappings between two or more
domains, by training on unpaired data samples in both the domains. Such a
problem is under-constrained and involves aligning the domains in feature
space via mappings. A number of research in this space (Zhu et al., 2017;
Royer et al., 2020; Liu & Tuzel, 2016; Yi et al., 2017; Tzeng et al., 2017)
introduce a form of cycle consistency loss which ensures that by translating
an image from one domain to another domain via mappings and then applying
reverse mappings to translate back yields the same image. XGAN (Royer et al.,
2020) uses additional loss terms to incorporate semantic consistency across
domains, to match the subspace for embedding from multiple domains and prior
knowledge via pre-trained models. However, since all the above models involve
GAN based architectures, they lack a latent space embedding useful for
downstream manipulation tasks (Kingma & Dhariwal, 2018). Moreover, since the
mappings are not guaranteed to be invertible, the cycle consistency is only
approximate.

Alignflow (Grover et al., 2019) involves modeling each of the domains via
normalizing flow mappings to a common latent space (Dinh et al., 2016; Das et
al., 2019a, b). It has a hybrid training objective constituting both maximum
likelihood estimation and adversarial training. Moreover, since flows are
invertible mappings, Alignflow achieves exact cycle consistency. However, flow
models, by virtue of the training procedure, face a challenge to align domains
which are apart in terms of semantics and/or style, apparent from the
generated samples quality in comparison with GANs. For CDCGen, we use the best
of both worlds: flow model mappings for the domains to a common latent space,
along with loss terms useful to align the domains in the embedding space.
CDCGen offers a rich latent space, which is utilized for conditional synthesis
in attribute scarce domains.

###  3.2 Conditional Synthesis

Conditional generative models have been introduced to generate desired
synthetic data by incorporating conditions information in model design. From
CGAN (Mirza & Osindero, 2014) which is a modification of conventional GANs and
works by feeding the label/attribute information to the generating block,
conditional synthesis has seen different algorithmic variations (Hong et al.,
2018; Wang et al., 2018; Odena, 2016). A notable work, ACGAN (Odena, 2016)
employs an auxilliary classifier for the discriminator to classify the class
labels. A recent work, CAGlow (Liu et al., 2019a) proposes a variant of ACGAN
with an encoder-decoder network, adding ability to model unsupervised
conditions. Additionally, above works deal with conditional generation in a
single domain. We use a variant of ACGAN over a shared latent space for
multiple domains, thereby transferring knowledge from label-rich domains to
perform conditional synthesis in label-scarce domains.

##  4 The CDCGen Framework

(a) CDCGen Training Schematic (b) CDCGen Inference Schematic

Figure 1: Illustration of training and inference methods in CDCGen. The
networks inside the dashed box are for domain alignment (Sec 4.1) and those
outside are for conditional synthesis (Sec 4.2).

In this section, we will present the CDCGen framework capable of generating
conditional synthetic samples for a domain in an unsupervised setting. We
select a domain with availability of information about the labels/attributes
(namely, the source domain) and has shared attributes with the domain for
which we don’t have information about labels/attributes (namely, the target
domain). Under this setting, the framework consists of two major networks: one
for domain alignment and another for conditional synthesis. We consider the
case of two domains, but under the assumption of having shared attributes
between the source and target domains, the proposed method generalizes to
multiple domains seamlessly.

###  4.1 Domain Alignment

The first step in CDCGen is to align the source and target domains. Let the
source and target domain be denoted by $\mathcal{D}_{s}$ and $\mathcal{D}_{t}$
with unknown marginal density $p_{s}$ and $p_{t}$ respectively. Both the
domains are mapped via invertible transformations (normalizing flows)
$\mathcal{F}_{s}$ and $\mathcal{F}_{t}$ to a common latent space $Z$, which
serves as a shared feature space for alignment. We assume the shared latent
space follows a normal gaussian distribution $p(z)$, common for training of
most of the state-of-the-art flow models. The relationship between the sample
space and latent space can be represented as,

$\displaystyle\mathcal{D}_{s}\xrightarrow[]{\mathcal{F}_{s}}Z\xleftarrow[]{\mathcal{F}_{t}}\mathcal{D}_{t}$

Note that the invertible nature of the flow model is helpful in two different
ways,

  * •

It provides a mechanism to translate between source and target domains, with
invertible mappings $\mathcal{F}_{s\rightarrow
t}=\mathcal{F}_{t}^{-1}\circ\mathcal{F}_{s}$ and $\mathcal{F}_{t\rightarrow
s}=\mathcal{F}_{s}^{-1}\circ\mathcal{F}_{t}$.

  * •

It helps achieve exact cycle consistency (as introduced in CycleGAN (Zhu et
al., 2017) to ensure accurate representation of the mappings) between the
domains, since $\mathcal{F}_{s\rightarrow t}\circ\mathcal{F}_{t\rightarrow
s}=\mathcal{F}_{t}^{-1}\circ\mathcal{F}_{s}\circ\mathcal{F}_{s}^{-1}\circ\mathcal{F}_{t}=I$,
where $I$ is the identity matrix.

We use a hybrid training objective involving both maximum likelihood
estimation and adversarial training. Flow models are trained with an
unsupervised maximum likelihood loss, with a normal gaussian prior on the
latent space $Z$. Since there are two flow models involved for the two
domains, the maximum likelihood loss is expressed as,

$\displaystyle\mathcal{L}_{MLE}(\mathcal{F}_{s})+\mathcal{L}_{MLE}(\mathcal{F}_{t})$

For cross-domain mappings, adversarial loss terms are introduced. These terms
introduce inductive bias required for cross domain translation (Zhu et al.,
2017). We employ critics $\mathcal{C}_{s}$ and $\mathcal{C}_{t}$ for source
and target domains respectively, which distinguish between real samples
(sampled from the same domain) vs. generated samples (obtained via cross-
domain mappings). For example, the adversarial loss for source domain can be
expressed as,

$\displaystyle\mathcal{L}_{ADV}(\mathcal{C}_{s},\mathcal{F}_{t\rightarrow
s})=$ $\displaystyle\mathbb{E}_{x_{s}\sim p_{s}}[\log\mathcal{C}_{s}(x_{s})]$
$\displaystyle+$ $\displaystyle\mathbb{E}_{x_{t}\sim
p_{t}}[\log(1-\mathcal{C}_{s}(\mathcal{F}_{t\rightarrow s}(x_{t})))]$

We also use a domain-adversarial loss (Ganin et al., 2016) which forces the
embeddings learnt by the flow models $\mathcal{F}_{s}$ and $\mathcal{F}_{t}$
to lie in the same subspace. This is achieved by training a classifier
$\mathcal{C}_{DAL}$ which takes the latent space embeddings for each domain
and classifies the sample to be coming from $\mathcal{D}_{s}$ or
$\mathcal{D}_{t}$. It is trained in an adversarial manner, with a
classification loss function $\ell(\cdot,\cdot)$, such as cross-entropy.
$\mathcal{L}_{DAL}$ can be expressed as,

$\displaystyle\mathcal{L}_{DAL}(\mathcal{F}_{s},\mathcal{C}_{DAL})=\mathbb{E}_{x_{s}\sim
p_{s}}\ell(\mathcal{D}_{s},\mathcal{C}_{DAL}(\mathcal{F}_{s}(x_{s})))$

Finally, for domain alignment, the overall loss term is,

$\displaystyle\mathcal{L}_{Domain\;Alignment}(\mathcal{F}_{s},\mathcal{F}_{t},\mathcal{C}_{s},\mathcal{C}_{t},\mathcal{C}_{DAL};\lambda_{s},\lambda_{t},\gamma_{s},\gamma_{t})=$
$\displaystyle\mathcal{L}_{ADV}(\mathcal{C}_{s},\mathcal{F}_{t\rightarrow
s})+\mathcal{L}_{ADV}(\mathcal{C}_{t},\mathcal{F}_{s\rightarrow
t})+\gamma_{s}\mathcal{L}_{DAL}(\mathcal{F}_{s},\mathcal{C}_{DAL})$
$\displaystyle+\gamma_{t}\mathcal{L}_{DAL}(\mathcal{F}_{t},\mathcal{C}_{DAL})-\lambda_{s}\mathcal{L}_{MLE}(\mathcal{F}_{s})-\lambda_{t}\mathcal{L}_{MLE}(\mathcal{F}_{t})$

where, hyperparameters $\lambda_{s}$ and $\lambda_{t}$ dictate the relative
contribution of maximum likelihood loss, and $\gamma_{s}$ and $\gamma_{t}$
correspond to contribution of domain adversarial loss, both as compared to the
adversarial loss. The objective is minimized w.r.t. the parameters of the flow
models $\mathcal{F}_{s}$ and $\mathcal{F}_{t}$ and maximized w.r.t. parameters
of $\mathcal{C}_{s}$, $\mathcal{C}_{t}$ and $\mathcal{C}_{DAL}$. This
procedure is illustrated in the dashed box in Fig. 1(a).

(a) Result with MNIST as source and USPS as target

(b) Result with USPS as source and MNIST as target

Figure 2: Results for domain alignment between source and target domains. The
top row has original samples from the source domain. The middle row is the
corresponding latent space mapping and the bottom row is the sample obtained
by translating it to the target domain. The USPS images are slightly blurred
due to the upscaling applied as standard pre-processing.

Figure 3: t-SNE representation of shared latent space for MNIST
$\leftrightarrow$ USPS. For each digit, points for USPS are visualized with
the darker colors, and points with lighter colors correspond to MNIST.

###  4.2 Conditional Synthesis

For conditional synthesis, we propose a variant of ACGAN (Odena et al., 2017).
Instead of using class/attribute conditioning on the sample space as done in
ACGAN, we use it in the shared latent space. Under the setting of our problem,
we don’t have any information about the labels/attributes in the target
domain. So, for the conditional synthesis part, only the attributes available
from the source domain are used for training.

We denote the available source attributes/conditions as $c_{s}\sim p(c_{s})$,
represented as one-hot encodings. Our network consists of an encoder to model
the conditions, a critic to differentiate between the real and generated
latent vectors, and an auxiliary classifier to classify the encoded
conditions. We will introduce each of the above components and their
associated loss functions separately.

Encoder: An encoder network $E$ encodes the conditions $(c_{s},\epsilon)$ into
a latent space $Z^{*}$ (separate from the shared latent space $Z$ for aligned
domains), where $\epsilon$ is sampled from standard gaussian distribution
$(p(\epsilon))$ and is helpful for incorporating stochastic behavior among
condition vectors. Let the distribution for the above mentioned latent space
be denoted as $p^{*}(z)$. Our objective is to minimize the Jensen-Shannon (JS)
divergence between the encoded distribution $p^{*}(z)$ and the shared latent
distribution $p(z)$ for aligned domains $\mathcal{D}_{s}$ and
$\mathcal{D}_{t}$. So, the encoder loss is represented as,

$\displaystyle\mathcal{L}_{E}=\mathbb{E}_{\epsilon\sim p(\epsilon),c_{s}\sim
p(c_{s})}[\log\mathcal{C}(E(c_{s},\epsilon))]$

where, $\mathcal{C}$ is a critic, more about which we describe now.

Critic: A critic $\mathcal{C}$ discriminates between the latent vectors coming
from generated conditional distribution $p^{*}(z)$ and real shared latent
distribution $p(z)$ for aligned domains. This is an adversarial loss which is
trained so as it is unable to distinguish the latent vectors at equilibrium,
thus enabling the encoder $E$ to generate latent vectors close to the real
shared latent distribution $p(z)$. The loss function for $\mathcal{C}$ is,

$\displaystyle\mathcal{L}_{CRITIC}=\mathbb{E}_{z\sim
p^{*}(z)}[\log\mathcal{C}(z)]+\mathbb{E}_{z\sim p(z)}[1-\log\mathcal{C}(z)]$

Classifier: A classifier takes the latent vectors ($z\sim p^{*}(z)$ and $z\sim
p(z)$) as input and classifies the conditions ($c_{s}$). The classifier loss
is a cross entropy loss between the predicted and true conditions. If the
class posterior probabilities are $q(c_{s}|z)$, the classifier loss function
can be expressed as,

$\displaystyle\mathcal{L}_{CLASSIFIER}=$ $\displaystyle\mathbb{E}_{z\sim
p^{*}(z),c_{s}\sim p(c_{s})}q(c_{s}|z)$ $\displaystyle+$
$\displaystyle\mathbb{E}_{z\sim p^{(}z),c_{s}\sim p(c_{s})}q(c_{s}|z)$

The overall loss function for the conditional synthesis part is,

$\displaystyle\mathcal{L}_{Conditional\;Synthesis}=$
$\displaystyle\beta_{E}\mathcal{L}_{E}+\beta_{Cr}\mathcal{L}_{CRITIC}$
$\displaystyle+$ $\displaystyle\beta_{Cl}\mathcal{L}_{CLASSIFIER}$

where $\beta_{E},\beta_{Cr},\beta_{Cl}$ are hyperparameters. The critic and
the classifier networks share their parameters except for their output blocks.
Conditional synthesis procedure is illustrated in Fig. 1(a).

###  4.3 Inference

CDCGen can generate conditional samples in the target domain, even when the
training process does not utilize its class/attribute information. To generate
samples with conditions $\tilde{c}_{s}$, a latent vector $\tilde{z}$ is
generated by encoding the one-hot conditions $\tilde{c}_{s}$ and
$\tilde{\epsilon}\sim p(\epsilon)$ via the encoder network, i.e.
$\tilde{z}=E(\tilde{c}_{s},\tilde{\epsilon})$. Then the latent vector
$\tilde{z}$ is passed via the inverse flow $\mathcal{F}_{t}^{-1}$ to generate
the desired sample in the target domain, i.e.
$\mathcal{F}_{t}^{-1}(\tilde{z})$. The inference schematic is illustrated in
Fig. 1(b).

##  5 Experiments

(a) Generated samples for USPS as target

(b) Generated samples for MNIST as target

Figure 4: Conditional synthetic samples generated by CDCGen. The rows
represent conditioned digit classes (0-9) and the columns include more samples
for each class.

In this section, we empirically evaluate CDCGen for synthetic generation in
label scarce domains.

Datasets: We perform experiments on 2 standard image datasets for digits,
namely MNIST (LeCun et al., 1998) and USPS. MNIST contains $60,000$ training
and $10,000$ test images with ten classes corresponding to digits from $0$ to
$9$. USPS has $7291$ training and $2007$ test data with the same classes as
MNIST. To address this data imbalance, for each domain, we sample $542$ images
from the original training set for each class to form the new training set. To
form the test set, we sample $147$ images from the original test sets for each
class. We resize all the images to $32\times 32$ for training and synthesis.

Source and Target Domain Combinations: We consider two cases, first with MNIST
as the source and USPS as the target domain, and second, with the roles
interchanged, i.e. USPS as the source and MNIST as the target. We report
results for domain alignment and as well as subsequent conditional synthesis
in the target domain, all while not using any labels from that domain.

Networks: We use architecture from RealNVP (Dinh et al., 2016) for each of the
domain flows ($\mathcal{F}_{s}$ and $\mathcal{F}_{t}$). Typical configurations
for RealNVP can be specified as a tuple comprising $N_{scales}$ (number of
scales), $N_{channels}$ (number of channels) in the intermediate layers, and
$N_{blocks}$ (number of residual blocks in the scaling and translation
networks of the coupling layers). For MNIST $\leftrightarrow$ USPS case, both
$\mathcal{F}_{s}$ and $\mathcal{F}_{t}$ are set to RealNVP($2$, $64$, $8$).
The critics ($\mathcal{C}_{s}$ and $\mathcal{C}_{t}$) used convolutional
discriminators from PatchGAN (Isola et al., 2017) , each with $16$ filters in
the critic’s first convolutional layer. For conditional synthesis, we
concatenate the one-hot vector of labels with components of random noise as
input to the encoder. The vector then passes through one fully-connected layer
and eight transposed convolutional layers with upsampling scale $2$, $2$, $2$,
$2$, $2$, $1$, $1$, $1$ and channel sizes $256$, $1024$, $512$, $256$, $128$,
$64$, $32$, $16$ respectively. The supervision block contains four
convolutional layers with stride $2$ and channel sizes $64$, $128$, $256$,
$512$. This is followed by two separate fully-connected layers for each
network head, one for outputting probabilities of real or fake and the other
for classifying the one-hot encoded conditions.

Optimizer: For training the domain alignment network, we use the Adam
optimizer with $\beta_{1}=0.5$, $\beta_{2}=0.999$, and learning rate $1\cdot
10^{-6}$. For training the conditional synthesis network, we use the Adam
optimizer with $\beta_{1}=0.5$, $\beta_{2}=0.999$, and learning rate $2\cdot
10^{-5}$.

###  5.1 Domain Alignment

In this section we present the results for domain alignment between source and
target combinations. Fig. 2(a) shows the source MNIST samples and
corresponding USPS samples by translating it via the forward source and
inverse target flows. The middle sample is visualization of corresponding
latent space sample. Fig. 2(b) depicts the same with USPS as the source and
MNIST as the target. It can be observed that the class identity is preserved
with the translation with the style adapted for the target domains. The
sharpness of the translated samples are compromised, which is a result of the
flow model assigning some probability mass to all the samples it is fed. This
is unlike pure GAN models which selectively assign probability mass to
meaningful samples.

Another interesting observation is the appearance of digit class identity in
latent space visualizations. This is particularly useful from the perspective
of CDCGen, since the conditional synthesis network works based on the latent
space mappings from both the domains.

We present the t-SNE embeddings for the shared latent space in our proposed
domain alignment network for MNIST and USPS in Fig. 3. It can be observed that
the visualization has distinct clusters for each digit class, but the
embeddings from both the source and target domain are close and belong to the
same cluster for the overall digit class clustering. The visualization allows
us to infer that the latent space has learned a subspace corresponding to each
digit, and interpolating across this subspace is effectively a conditional
feature-preserving domain transfer.

###  5.2 Conditional Synthesis

We trained the conditional synsthesis part of CDCGen (Section 4.2) with source
labels to generate conditional synthetic samples in the target domain. Fig.
4(a) shows the samples generated with USPS as the target domain and Fig. 4(b)
shows the samples generated with MNIST as the target. Each row corresponds to
the digit classes which are assigned as conditions. It can be observed that
CDCGen is able to generate synthetic samples belonging to the digit class as
conditioned. There are also variations among the samples across different
columns which shows the stochastic nature of generation by CDCGen. The
compromise in sharpness of the samples generated is owed from the domain
alignment mappings by flow models.

##  6 Conclusions

In this work, we proposed CDCGen, a generative framework capable of generating
conditional synthetic samples for domains without the requirement of obtaining
its labels/attributes. We also conducted empirical studies with standard image
datasets to observe feature transfer and independent conditional generation.
We are working on CDCGen implementations for datasets with complex
interactions between features, e.g. facial data. In the future, making the
conditional generation models across multiple domains can be studied with
varying levels of label availability (few-shot learning) for target domain.
CDCGen can also be adapted for other modalities including audio and tabular
data. It can also be used alongside real-world applications (Zou et al.,
2019b, a; Konstantakopoulos et al., 2019; Chen et al., 2021; Periyakoil et
al., 2021; Das et al., 2019c, 2020; Liu, 2018; Liu et al., 2019b; Donti &
Kolter, 2021; Jin et al., 2018) where having access to diverse conditional
data is important, but is hard to obtain, hence the need for synthetic data.

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##  Appendix A Functioning of Flow and GAN based generative models

###  A.1 Flow-based Generative Models

Let X be a high-dimensional random vector with unknown true distribution
$p(x)$. The following formulation is directly applicable to continuous data,
and with some pre-processing steps such as dequantization (Uria et al., 2013;
Salimans et al., 2017; Ho et al., 2019) to discrete data. Let Z be the latent
variable with a known standard distribution $p(z)$, such as a standard
multivariate gaussian. Using an i.i.d. dataset $\mathcal{D}$, the target is to
model $p_{\bm{\theta}}(x)$ with parameters $\bm{\theta}$. A flow,
$\mathcal{F}$ is defined to be an invertible transformation that maps observed
data X to the latent variable Z. A flow is invertible, so the inverse function
${\mathcal{T}}$ maps Z to X, i.e.

$\displaystyle\textit{Z}=\mathcal{F}(\textit{X})={\mathcal{T}}^{-1}(\textit{X})\;\;\text{and}\;\;\textit{X}={\mathcal{T}}(\textit{Z})=\mathcal{F}^{-1}(\textit{Z})$
(1)

The log-likelihood can be expressed as,

$\displaystyle\log p_{\bm{\theta}}(x)$ $\displaystyle=\log
p(z)+\log\left|\det\left(\frac{\partial\mathcal{F}(x)}{\partial
x^{T}}\right)\right|$ (2)

where $\cfrac{\partial\mathcal{F}(x)}{\partial x^{T}}$ is the Jacobian of
$\mathcal{F}$ at $x$. The training of flow models is accomplished via maximum-
likelihood estimation. Confirming with the qualifying properties for a flow as
above, different types of flow models have been introduced to efficiently
estimate the distribution density and generate synthetic samples (Kingma &
Dhariwal, 2018; Dinh et al., 2016, 2014; Chen et al., 2019).

###  A.2 Generative Adversarial Networks (GANs)

GANs (Goodfellow et al., 2014) are a class of implicit generative models which
work based on the principles of a mini-max game. It involves a generator
$\mathcal{G}$ which is tasked to generate synthetic samples from standard
noise distribution and a critic $\mathcal{C}$ which learns to discriminate the
samples generated by $\mathcal{G}$ and samples from original data distribution
$p_{data}$. The training objective for a GAN is given by,

$\displaystyle\min_{\mathcal{G}}\max_{\mathcal{C}}\mathcal{L}(\mathcal{C},\mathcal{G})=$
$\displaystyle\mathbb{E}_{x\sim p_{data}}[\log\mathcal{C}(x)]$
$\displaystyle+$ $\displaystyle\mathbb{E}_{z\sim\text{standard noise
distribution}}[\log(1-\mathcal{C}(\mathcal{G}(z)))]$

At nash equillibrium, the generator and critic are optimal, and the generator
is capable of generating samples resembling original data. GAN based models
have been particularly successful in generating high-fidelity images (Karras
et al., 2018; Brock et al., 2019), manipulating features of images to generate
custom samples (Radford et al., 2015), audio generation (Engel et al., 2019;
Donahue et al., 2018), video generation (Tulyakov et al., 2018) etc. Despite
the potential of GANs in generating qualitative samples, they are hard to
train due to the mini-max optimization. Unlike flow models, they lack a latent
space suitable for a number of downstream applications. Another major problem
with GANs is mode collapse, where the generator starts producing the same
output (or a small set of outputs) over and over again. A number of remedies
have been proposed to tackle this over the years (Arjovsky et al., 2017;
Salimans et al., 2016; Metz et al., 2016).