# A correspondence between Maxwell–Einstein theory and superfluidity

Emil Génetay Johansen  Optical Sciences Centre, Swinburne University of
Technology, Melbourne 3122, Australia<EMAIL_ADDRESS>
###### Abstract

A planar superfluid is considered and interpreted in terms of electromagnetism
and gravity. It has previously been suggested that the superfluid flow can be
regarded as analogous to an electromagnetic field and that a non-vanishing
density gradient give rise to a gravity-like force. The present work seeks to
reconcile these hitherto distinct pictures into a unified exposition of vortex
electrodynamics in a curved analogue space-time. By constructing a theory in
which the planar Maxwell’s equations are coupled to a $(d+1)$-dimensional
conformally curved space-time, we expose a correspondence between the
resulting equations of motion of the embedded fields and the dynamics of a
quantum fluid. Finally, an effective vortex metric whose connection components
are torsional is studied and its effect on the superfluid Maxwell’s equations
is elucidated upon.

##  I Introduction

Einstein’s theory of gravity [1] and Maxwell’s theory of electromagnetic
fields [2] may be regarded as the cornerstones of classical physics. While
general relativity establishes a relationship between energy and space-time
geometry, electromagnetic field theory furnishes a description of how charged
particles interact, and gives rise to the notion of light. In pioneering works
put forth by Schwinger [3, 4], Tomonaga [5] and Feynman [6, 7, 8], Maxwell’s
theory was successfully quantized and unified with special relativity, which
culminated in a fully relativistic theory of quantum electrodynamics (QED).
This crowning achievement marked the beginning of the twentieth century
revolution in elementary particle physics. The ideas upon which QED was
founded were further developed by considering more general unitary gauge
groups $\rm{SU}(N)$. Such theories, collectively known as Yang–Mills theories
[9, 10, 11], are at the heart of the Standard Model of elementary particles
and our contemporary understanding of fundamental physics.

Gravity is significantly weaker than the forces emerging from these theories
though. While the ratio between the strong nuclear force and electromagnetism
is about $10^{-2}$, and electromagnetism and the weak nuclear force about
$10^{-4}$, the gap between the weak nuclear force and gravity is approximately
$10^{-33}$. Explaining this vast discrepancy constitutes one of the central
problems in foundational physics and is known as the hierarchy problem. In the
vast majority of instances, the impact of gravity on electromagnetism, and
conversely, the effects of electromagnetism on gravity, is negligible; gravity
is generally too weak to significantly perturb the electromagnetic fields, and
the energy-momentum carried by the electromagnetic fields is most often too
low to cause a measurable curvature of space-time. The main exception to this
approximation is of course in the context of black holes. It is well
established that the strong gravitational pull by a black hole is causing
light passing by in its vicinity to deflect [12, 13, 14]. In such a scenario,
and in situations alike, electromagnetic field theory and gravity cannot be
treated independently since the coupling between the two is non-negligible.
This is generally also the case when gravity and electromagnetism is simulated
in a superfluid. It was demonstrated by Popov [15] that a superfluid can be
described in terms of relativistic electrodynamics where the superfluid flow
corresponds to the the electric field and the condensate density to the
magnetic field. In addition, Unruh showed [16] that excitations in a
superfluid experience the background system as an effective space-time where a
non-vanishing density gradient accounts for the curvature [16, 17, 18, 19, 20,
21, 22, 23, 24, 25]. Any density landscape can in principle be imprinted with
the technologies available today [26], so the effective gravity in a
superfluid may well be strong enough to affect the properties of the
superfluid electromagnetic fields, just as when light is propagating in the
presence of a black hole. This calls for an extension of the vortex
electrodynamics picture to a complete formulation accounting for this
interaction.
Over the years, several strategies for unifying electromagnetism and gravity
have been proposed including the so-called teleparallel formulation [27, 28]
developed by Einstein, Kaluza–Klein theory [29, 30, 31, 32] and its precursor,
Nordström’s theory [33, 34, 35], and in addition Maxwell–Einstein theory [36,
37]. Teleparallelism relies on the notion of a superpotential and is
constructed in a non-coordinate basis via the introduction of tetrad fields
[38], which makes possible a description of gravity that is quadratic in field
strength, akin to Yang–Mills. Kaluza–Klein theory and Nordström’s theory share
the common trait that gravity and electromagnetism are treated as two parts of
the same object and in Maxwell–Einstein theory the space-time curvature is
viewed as a background gauge-like field to which the field strength tensor is
coupled. It may thus be natural to pose the question of whether the vortex
electrodynamics description can be naturally reconciled with analogue gravity
into a single theory. The present work is devoted to addressing this
particular question. Finally, for the sake of completeness, we also consider a
vortex metric whose connection components have non-vanishing torsion.

##  II Vortex electrodynamics

The wavefunction $\Psi(\text{r},t)$ describing the collective behaviour of a
weakly interacting superfluid is a solution to the Gross–Pitaevskii equation
(GPE) [39]

$i\hbar\frac{\partial\Psi(r,t)}{\partial
t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+g|\Psi(r,t)|^{2}-\mu\right)\Psi(r,t),$
(1)

where $m$ is the mass of the condensate particle, $g$ is the coupling constant
and $\mu$ is the chemical potential. By performing a Madelung transformation
$\Psi(\text{r},t)=|\Psi(\textbf{r},t)|e^{iS(\textbf{r},t)}$, the GPE can be
split into an Euler-like equation

$-\hbar\frac{\partial S}{\partial
t}=\frac{m}{2}v_{\rm{sf}}^{2}+g|\Psi(\textbf{r},t)|^{2}-\mu$ (2)

and a continuity equation

$\frac{\partial n(\textbf{r},t)}{\partial
t}=\nabla\cdot\left(n(\textbf{r},t)\bm{v}_{\rm{sf}}\right),$ (3)

where $n(\textbf{r},t)=|\Psi(\textbf{r},t)|^{2}$ and
$\bm{v}_{\rm{sf}}(\textbf{r},t)=\frac{\hbar}{m}\nabla S(\textbf{r},t)$. In
Ref. [40], it was shown that the following definitions for the superfluid
electric and magnetic field
$\textbf{E}_{\rm{sf}}=mn_{0}\bm{v}_{\rm{sf}}\times\hat{\textbf{e}}_{z}$ (4)
$\textbf{B}_{\textrm{sf}}=\frac{\hbar m}{g}\partial_{t}S\hat{\textbf{e}}_{z}$
(5) with vacuum constants

$\epsilon_{\rm{sf}}=\frac{1}{mn_{0}}$ (6)

$\mu_{\rm{sf}}=\frac{m^{2}}{g}$ (7)

such that the speed of sound satisfies the square root law

$c_{\rm{sf}}=\frac{1}{\sqrt{\epsilon_{\rm{sf}}\mu_{\rm{sf}}}},$ (8)

solve Maxwell’s equations under the assumption that the density
$n(\textbf{r})=n_{0}$ is constant. There are, however, phonons in the system
which in this formulation play the role of photons, which are linear
perturbations of this otherwise flat background. This particular configuration
is just one of an infinite set of possible solutions though. In general, it
holds true that any configuration
$\textbf{E}_{\rm{sf}}=mn_{0}^{x+1}\bm{v}_{\rm{sf}}\times\hat{\textbf{e}}_{z}$
(9) $\textbf{B}_{\textrm{sf}}=\frac{\hbar
mn_{0}^{x}}{g}\partial_{t}S\hat{\textbf{e}}_{z}$ (10) with vacuum constants

$\epsilon_{\rm{sf}}=\frac{1}{mn_{0}^{x+1}}$ (11)

$\mu_{\rm{sf}}=\frac{m^{2}n_{0}^{x}}{g}$ (12)

forms a solution for any arbitrary constant $x\in\mathds{R}$. We shall later
see that this constant has to be an integer in a Maxwell–Einstein formulation
as it relates to the dimension of the pertinent space-time. Now, let us
generalize the above results by allowing for a variable density
$n(\textbf{r})$ while keeping it constant in time.

###  II.1 Generalized Gauss-like law

Gauss’s law in the non-covariant formulation is denoted as

$\nabla\cdot\textbf{E}_{\rm{sf}}=\frac{\rho_{v}}{\epsilon_{\rm{sf}}},$ (13)

where $\rho_{v}$ is the vortex density. Inserting the electric field in Eq.
(9) and expanding the divergence on the left hand side we obtain after some
rearrangements

$n^{x+1}\left(\partial_{r}+(x+1)\frac{\partial_{r}n}{n}\right)v_{\rm{sf}}=\frac{\rho_{v}}{\epsilon_{\rm{sf}}}+\lambda(\textbf{r}),$
(14)

where $\lambda(\textbf{r})$ is an additional source term accounting for the
interaction between electromagnetism and gravity. The physical interpretation
of this term will be clarified in the next section. Now, if we divide both
sides by $n^{x+1}$ and write the superfluid flow as
$v_{\rm{sf}}=n^{-(x+1)}(n^{(x+1)}v_{\rm{sf}})={n^{-(x+1)}}\textbf{E}_{\rm{sf}}$,
we arrive at

$\left(\partial_{r}+(x+1)\frac{\partial_{r}n}{n}\right)n^{-(x+1)}\textbf{E}_{\rm{sf}}=n^{-(x+1)}\frac{\rho_{v}}{\epsilon_{\rm{sf}}}+\tilde{\lambda}(\textbf{r}),$
(15)

where the factor $n^{-(x+1)}$ was absorbed into $\tilde{\lambda}(\textbf{r})$.
The resulting equation has the same form as an electric field (scaled by
${n^{-(x+1)}}$) minimally coupled to a gauge field with coupling $x+1$. This
field is nothing but the velocity $\bm{v}_{\nabla|\Psi|}$ due to the density
gradient

$\bm{v}_{\nabla|\Psi|}(\textbf{r})=\frac{\hbar}{m}\frac{\nabla
n(\textbf{r})}{n(\textbf{r})}.$ (16)

Focusing on the first term $\partial_{r}{n^{-(x+1)}}\textbf{E}_{\rm{sf}}$ a
straight forward calculation gives us the vortex density

$\partial_{r}v_{\rm{sf}}=\nabla\times\bm{v}_{\rm{sf}}\cdot\hat{\textbf{e}}_{z}=\rho_{v},$
(17)

since $n^{-(x+1)}\epsilon_{\rm{sf}}^{-1}=m$. The second term proportional to
the density gradient arises due to the fact that the vortex is moving with a
speed $v_{\nabla|\Psi|}$. Thus, since the conventional Gauss’s law only holds
in the rest frame of the vortex, we must perform a frame transformation into
this frame, which results in a coupling to the gauge field $v_{\nabla|\Psi|}$.
The source of this interaction can therefore be identified as

$\tilde{\lambda}(\textbf{r})=m(x+1)v_{\nabla|\Psi|}(\textbf{r})v_{\rm{sf}}(\textbf{r})$
(18)

###  II.2 Generalized Ampère-like law

Ampère’s law in its non-covariant form can be written

$\nabla\times\textbf{B}_{\rm{sf}}=\frac{\partial\textbf{E}_{\rm{sf}}}{\partial
t}+\mu_{\rm{sf}}\textbf{j}_{v},$ (19)

where $\textbf{j}_{v}=\bm{v}_{v}\rho_{v}$ denotes the vortex current, and
$\bm{v}_{v}$ is the velocity of the source vortex. Going through the same
steps as in the preceding subsection, we find

$\left(\partial_{\theta}+x\frac{\partial_{r}n}{n}\right)n^{-x}\textbf{B}_{\rm{sf}}=n^{-x}\left(\epsilon_{\rm{sf}}\mu_{\rm{sf}}\frac{\partial\textbf{E}_{\rm{sf}}}{\partial
t}+\mu_{\rm{sf}}\textbf{j}+\hat{\lambda}(\textbf{r})\right).$ (20)

Again, we start off by focusing on the first term on the left hand side, which
gives us

$\partial_{\theta}\left(\frac{\hbar m}{g}\partial_{t}\theta\right)=\frac{\hbar
m}{g}\partial_{t}\left(\partial_{\theta}S\right)=\epsilon_{\rm{sf}}\mu_{\rm{sf}}\frac{\partial\textbf{E}_{\rm{sf}}}{\partial
t}.$ (21)

The current $\textbf{j}_{v}$ can be obtained by simply performing a frame
transformation with respect to $v_{v}$ [40], not taking into account
$v_{|\nabla\Psi|}$. In the rest frame of the vortex with speed $v_{v}$, we
thus get

$i\hbar\partial_{t}\Psi=\left(i\hbar\partial_{t}-\textbf{J}_{v}\cdot\textbf{A}\right)\Psi_{v},$
(22)

where
$\textbf{J}_{v}\cdot\textbf{A}=-m\textbf{j}_{v}\cdot\textbf{r}\times\hat{\textbf{e}}_{z}$.
Now, taking the curl of this expression we get the definition of the vortex
current

$\partial_{\theta}\left(m\textbf{j}_{v}\cdot\textbf{r}\right)\hat{\textbf{e}}_{\theta}=\nabla\left(m\textbf{j}_{v}\cdot\textbf{r}\right)\times\hat{\textbf{e}}_{z}=\mu_{\rm{sf}}\textbf{j}_{v}.$
(23)

We also need to perform an additional frame transformation with respect to
$v_{\nabla|\Psi|}$. Doing this, the source accounting for the minimal coupling
is found via identification as

$\hat{\lambda}(\textbf{r})=\frac{\hbar
m}{g}v_{\nabla|\Psi|}\left(\partial_{t}S+\frac{m}{\hbar}\textbf{j}_{v}\cdot\textbf{r}\times\hat{\textbf{e}}_{z}\right)\hat{\textbf{e}}_{z}.$
(24)

###  II.3 Generalized Faraday-like law and Gauss-like B law

The superfluid Faraday-like law is given by

$\nabla\times\textbf{E}_{\rm{sf}}=-\frac{\partial\textbf{B}_{\rm{sf}}}{\partial
t}.$ (25)

Taking the two-dimensional curl of the $\textbf{E}_{\rm{sf}}$ field gives us

$\nabla\times\textbf{E}_{\rm{sf}}=-mn^{x+1}\partial_{\theta}v_{\rm{sf}}\hat{\textbf{e}}_{z},$
(26)

and as for the right hand side, using the Euler-like Eq. (2) and the
continuity equation Eq. (3), we get after applying the time derivative

$\frac{\partial\textbf{B}_{\rm{sf}}}{\partial
t}=-m\left(xn^{x}\partial_{\theta}v_{\rm{sf}}+n^{x+1}\partial_{\theta}v_{\rm{sf}}\right)\hat{\textbf{e}}_{z}.$
(27)

Since we are only considering static density landscapes, this equation must be
satisfied due to the continuity equation
$\partial_{t}n=-mn\partial_{\theta}v_{\rm{sf}}$. Finally, let us look at the
Gauss-like B law. This equation is, just as in the flat case, trivially
satisfied since the planar divergence of a vector pointing in a direction
orthogonal to the plane must vanish, i.e.

$\nabla\cdot\textbf{B}_{\rm{sf}}=\nabla\cdot\frac{\hbar
mn^{x}}{g}\partial_{t}S\hat{\textbf{e}}_{z}=0,$ (28)

A noteworthy remark regarding these two equations is that they seem to be
insensitive to the density distribution. In fact, they are pure geometrical
statements, which together, are commonly referred to as the Bianchi identity.
In the language of differential forms [41], the source-full and source-less
Maxwell’s equations can, respectively, be formulated as

$\displaystyle d\textbf{F}=0$ (29) $\displaystyle\star
d\star\textbf{F}=\textbf{J},$ (30)

where

$\displaystyle\textbf{F}=E_{1}dt\wedge dx+E_{2}dt\wedge dy+E_{3}dt\wedge dz+$
$\displaystyle B_{1}dy\wedge dz+B_{2}dz\wedge dx+B_{3}dx\wedge dy$ (31)

is the field strength 2-form, $\star$ denotes the Hodge operator and J is the
source. Since the Hodge operation is carried out with respect to the
underlying metric, the sourceful equation cannot be invariant under the change
of metric. This is not true for the sourceless equation though, since the
exterior derivative is coordinate free, thus implying a coordinate
independence. In a superfluid, the spatial dependence of the density can be
interpreted as an effective space-time curvature, meaning that the effective
metric is determined by the density. This is the picture we shall adopt in the
next section.

##  III A Maxwell–Einstein formulation

The equations obtained in the preceding section appear reminiscent of a
Maxwell theory minimally coupled to a gauge field $v_{\nabla|\Psi|}$. It is
therefore conceivable that the same set of equations may be obtained via a
Maxwell–Einstein formulation in which the electromagnetic field strength
tensor is minimally coupled to analogue gravity. In this section we endeavour
to demonstrate that this is the case. The basic building blocks we use in our
construction are given by
$\\{g_{\mu\nu}(n),\Gamma^{\lambda}_{\mu\nu}(n),F_{\mu\nu}(n,\nabla,S),J^{\rm{ME}}_{\mu}(n,\nabla,S),g_{i}\\}$,
where $g_{\mu\nu}$ is the effective metric, $\Gamma^{\lambda}_{\mu\nu}(n)$ is
the space of Christoffel symbols corresponding to $g_{\mu\nu}$, $F_{\mu\nu}$
is the superfluid field strength tensor, $J^{\rm{ME}}_{\mu}$ is the
Maxwell–Einstein current, $g_{i}$ are the couplings and $n=n(\textbf{r})$ is
the density. The metric for a $(d+1)$-dimensional conformal analogue space-
time is given by [42]

$g_{\mu\nu}(r)=\Omega(r)^{\frac{2}{d-1}}\left(\begin{array}[]{c|c}-(c_{\rm{sf}}^{2}-v_{\rm{sf}}^{2})&-v_{i}\\\
\hline\cr-v_{j}&\delta_{ij}\end{array}\right),$ (32)

where $\Omega(r)$ is the conformal factor, which is proportional to the
density $n(r)$. In what follows, we shall consider two special cases of this
metric: i) the metric describing the situation in which the vortices are kept
far away from one another so that off-diagonal elements vanish, but with
arbitrary density profile, and ii) the metric corresponding to the analogue
space-time in the neighborhood of a vortex residing in a system with an
otherwise constant density, such that the torsion is non-vanishing. Moreover,
the $(2+1)$-dimensional field strength tensor is given by

$F_{\mu\nu}=\left(\begin{array}[]{ccc}0&\frac{E_{r}}{c_{\rm{sf}}}&\frac{E_{\theta}}{c_{\rm{sf}}}\\\
-\frac{E_{r}}{c_{\rm{sf}}}&0&B_{z}\\\
-\frac{E_{\theta}}{c_{\rm{sf}}}&-B_{z}&0\end{array}\right).$ (33)

In order to couple Maxwell’s equations to gravity, we adopt the covariant
formulation and compute the covariant derivatives $\nabla_{\mu}$ taking into
account the change of coordinates due to the non-trivial structure of the
effective space-time. Maxwell’s equations, in this formulation, may then be
written as

$\nabla_{\mu}F^{\mu\nu}=\mu_{v}J^{\nu}_{\rm{ME}},$ (34)

where

$\displaystyle F^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}$ (35)
$\displaystyle J^{\nu}_{\rm{ME}}=g^{\nu\alpha}J^{\rm{ME}}_{\alpha}$ (36)

and
$J^{\rm{ME}}_{\alpha}=(c_{\rm{sf}}\rho_{\rm{ME}},\textbf{j}_{\rm{ME}})+\delta_{\alpha}(\textbf{r})$
are the covariant components of the Maxwell–Einstein current and
$\delta_{\alpha}$ are the components of the correction term which are to be
identified. The covariant derivative applied to the rank-2 tensor $F^{\mu\nu}$
takes on the form

$\nabla_{\mu}=\partial_{\mu}+g\left(\Gamma^{\alpha}_{\lambda\mu}+\Gamma^{\nu}_{\lambda\mu}\right),$
(37)

where $g$ is the coupling between gravity and the field strength tensor. The
connection coefficients, in absence of torsion, are given by the Christoffel
symbols

$\Gamma^{\lambda}_{\mu\nu}=g^{\lambda\alpha}\left(\partial_{\nu}g_{\alpha\mu}+\partial_{\mu}g_{\alpha\nu}-\partial_{\alpha}g_{\mu\nu}\right).$
(38)

Note that due to the interchange symmetry of the lower two indices of
$\Gamma^{\lambda}_{\mu\nu}$, the field strength tensor is left invariant so
that

$\displaystyle F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\left(\Gamma^{\alpha}_{\mu\nu}-\Gamma^{\alpha}_{\nu\mu}\right)A_{\alpha}$
(39) $\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$

where $A_{\mu}$ are the components of the 3-vector potential. However, as we
shall see in the next section, this is not the case when torsion is present,
which thus necessitates the introduction of a more general connection, which
is defined by Eq. (61).

###  III.1 Non-zero curvature and vanishing torsion

We shall now consider an arbitrarily conformally curved space-time in which
the parallel transport is determined by the symmetric connection in Eq. (38).

####  III.1.1 Gauss–Einstein law

Generalized Gauss’s law can be obtained by setting $\nu=t$ in Eq. (34) which
leads to the equation

$\displaystyle\nabla_{\mu}F^{\mu
t}=\left(\partial_{r}+\Gamma^{r}_{rr}+\Gamma^{t}_{tr}+\Gamma^{\theta}_{r\theta}\right)F^{rt}$
$\displaystyle+\left(\partial_{\theta}+\Gamma^{\theta}_{\theta\theta}+\Gamma^{t}_{t\theta}+\Gamma^{r}_{\theta
r}\right)F^{\theta t}=\mu_{0}J^{t}_{\rm{ME}}.$ (40)

Computing the relevant Christoffel symbol components and using Eq. (35) to
obtain the contravariant field strength components results in the equation

$\nabla_{\mu}F^{\mu
t}=-\left(\partial_{r}+g_{E}\frac{\partial_{r}n}{n}\right)n^{-\frac{d+1}{d-1}}F^{tr},$
(41)

where $g^{tt}g^{rr}=-\frac{d+1}{d-1}$. Moreover, we may compute the
contravariant components of the current by means of Eq. (36), which yields

$J^{t}_{\rm{ME}}=g^{tt}\left(J^{\rm{ME}}_{t}+\delta_{t}\right)=n^{-\frac{d}{d-1}}(c_{\rm{sf}}\rho^{\rm{ME}}+\delta_{t}).$
(42)

We find that the resulting equation is in exact agreement with the generalized
Gauss’s law in Eq. (15), given that we define the Maxwell–Einstein source and
identify the correction as

$\displaystyle\rho_{\rm{ME}}=n^{-\frac{2-d}{d-1}}\rho_{v}$ (43)
$\displaystyle\delta_{t}=n^{-\frac{2-d}{d-1}}\tilde{\lambda},$ (44)

and enforce the following relationship between the spatial dimension $d$, the
parameter $x$ and the coupling $g_{E}$

$\displaystyle x=\frac{2}{d-1}$ (45) $\displaystyle
g_{E}=x+1=\frac{d+1}{d-1}.$ (46)

####  III.1.2 Ampère–Einstein law

Next we consider the Ampère-Einstein law. Here we let $\nu=a$, where $a$ is a
spatial index. Going through the same steps as for $\nu=t$, the left hand side
of Eq. (34) becomes

$\nabla_{\mu}F^{\mu
r}=-n^{-\frac{2}{d-1}}\partial_{t}F^{tr}+\left(\partial_{\theta}+g_{M}\frac{\partial_{r}n}{n}\right)n^{-\frac{2}{d-1}}F^{\theta
r},$ (47)

since the $a=\theta$ equation vanishes. Again we recover the left hand side of
the generalized Ampère’s law in Eq. (20) given the same relationship between
$x$ and $d$ as in Eq. (45), but now with the coupling $g_{M}=x$. Moreover, the
Maxwell–Einstein current and the correction can be identified as

$\displaystyle J^{r}_{\rm{ME}}=n^{-\frac{1}{d-1}}j_{v}$ (48)
$\displaystyle\delta_{r}=n^{\frac{1}{d-1}}\hat{\lambda}_{r}$ (49)

for which the generalized Ampère’s law derived from hydrodynamics is obtained.

####  III.1.3 Einstein–Faraday law and Einstein–Gauss B law

The Faraday and Gauss-like B laws may, as already discussed, be regarded as
geometric identities that cannot be obtained from Noether’s theorem since
there is no current to couple the fields to in the action. In the hydrodynamic
picture, these were trivially invariant under the replacement
$n\longrightarrow n(\textbf{r})$, which in a gravity picture entails that
there is no response to the curvature of space. In the covariant formulation,
these two equations may be summarized as

$\frac{1}{2}\partial_{\mu}\epsilon^{\mu\nu\gamma\delta}F_{\gamma\delta}=\partial_{\gamma}F^{\mu\nu}+\partial_{\mu}F^{\nu\gamma}+\partial_{\nu}F^{\gamma\mu}=0.$
(50)

In order to show that this equation is insensitive to the underlying metric,
we may first lower the indices of the field strength tensor by applying the
metric. Doing this leads to an identical equation but for the covariant
components instead of the contravariant ones, since the right hand side is
zero. Replacing the derivatives with their covariant counterparts, and
collecting the terms, thus gives us

$\displaystyle\left(\partial_{\gamma}F_{\mu\nu}+\partial_{\mu}F_{\nu\gamma}+\partial_{\nu}F_{\gamma\mu}\right)+\left(\Gamma^{\alpha}_{\mu\gamma}-\Gamma^{\alpha}_{\gamma\mu}\right)F_{\alpha\nu}$
(51)
$\displaystyle+\left(\Gamma^{\beta}_{\nu\gamma}-\Gamma^{\beta}_{\gamma\nu}\right)F_{\beta\mu}+\left(\Gamma^{\delta}_{\nu\mu}-\Gamma^{\delta}_{\mu\nu}\right)F_{\delta\gamma}=0.$

Now, since the Christoffel symbols are symmetric under the interchange of the
lower indices, all terms due to the curvature cancel out and we are left with
the equation defined on a flat background, which defines the Bianchi identity
and thereby must vanish too. Note, however, that this is only due to the fact
that the Christoffel symbol is a symmetric connection, which means that it
does not take into account the effects due to torsion. As a consequence of
this, it is generally the case that the homogenous equations do not remain
invariant under the change of metric when torsion is present. We shall return
to this in the following section. We have now worked out the Maxwell–Einstein
equations for the metric given by Eq. (32) and found that they are in exact
agreement with the generalized Maxwell’s equations derived from hydrodynamics,
where $\textbf{E}_{\rm{sf}}$ and $\textbf{B}_{\rm{sf}}$ are defined as in Eq.
(9) and the vacuum constants are given by Eq. (11), provided the relationship
between the parameter $x$ and the spatial dimension $d$ established in Eq.
(45), and couplings $g_{E}=x+1$ and $g_{M}=x$.

###  III.2 Non-zero torsion and vanishing curvature

Let us now consider instead a system in which the density is kept constant but
with a vortex residing in it. Thus, in the vicinity of the vortex, the metric
defined in Eq. (32) is no longer diagonal, which consequently gives rise to a
different Maxwell–Einstein theory that is coupled to the torsion of the
underlying connection, whose equations of motion are derived in this section.
However, before we go on and work these equations out, we shall outline some
pertinent remarks regarding torsion and its relationship to vorticity.

####  III.2.1 On torsion — general remarks

The vanishing of torsion and Lorentz invariance of $\eta_{\mu\nu}$ uniquely
determine the connection to be of a Levi–Civita kind, since the manifold must
be Riemannian under these conditions. However, in the context of non-
Riemannian manifolds, as those with torsion present, a more general asymmetric
connection $\tilde{\Gamma}^{\kappa}_{\mu\nu}$ must be used. The torsion may
then be defined as the asymmetry of the connection as the lower two indices
are interchanged [43]

$T^{\kappa}_{\mu\nu}=\tilde{\Gamma}^{\kappa}_{\mu\nu}-\tilde{\Gamma}^{\kappa}_{\nu\mu}.$
(52)

This property has far reaching consequences for Maxwell’s equations. To begin
with, the field strength tensor $\tilde{F}_{\mu\nu}$ has a more complex
structure since it is no longer invariant under the change of metric as in Eq.
(III). Instead the field strength in its covariant and contravariant basis,
respectively, is given by

$\displaystyle\tilde{F}_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\left(\tilde{\Gamma}^{\alpha}_{\mu\nu}-\tilde{\Gamma}^{\alpha}_{\nu\mu}\right)A_{\alpha}$
(53)

and

$\tilde{F}^{\mu\nu}=g^{\alpha\mu}g^{\beta\nu}\tilde{F}_{\alpha\beta},$ (54)

and secondly, as already touched upon, the homogeneous equations are now
sensitive to the background metric. In addition, due to the non-vanishing of
the off-diagonal components of the metric, mixing of the field components in
the contravariant basis is to be expected. Next we shall discuss the
relationship between vorticity and torsion.

####  III.2.2 Vorticity and torsion

Let us consider the effective metric in the neighbourhood of a vortex [44]. In
this regime the condensate density is approximately constant but the
superfluid flow $v_{\rm{sf}}$ is non-zero. Interestingly, this metric takes on
an identical form to that of a spinning cosmic string [45, 46, 47, 48]. The
vortex can thus be regarded as a torsion defect of the analogue space-time,
which if encircled, leads to the accumulation of a geometric phase. This phase
factor can be computed by interpreting the gravity as gauge theory in which
the torsion plays the role of the field strength of the abelian tetrad fields
$e^{a}_{\mu}$ which decomposes the metric as
$g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\eta_{ab}$, where $\eta_{ab}$ is the flat
Minkowski metric and $e^{a}=e^{a}_{\mu}dx^{\mu}$. Setting the conformal factor
$\Omega=1$, we may factorize the invariant line element as

$ds^{2}=\left(c_{\rm{sf}}\gamma_{\rm{sf}}^{-1}dt+\frac{w\kappa}{c_{\rm{sf}}}\gamma_{\rm{sf}}d\theta\right)^{2}-dr^{2}+\gamma_{\rm{sf}}^{2}r^{2}d\theta^{2},$
(55)

where $w$ is the winding number of the vortex, $\kappa$ is the circulation
quantum and

$\gamma_{\rm{sf}}=\frac{1}{\sqrt{1-\frac{v_{\rm{sf}}^{2}}{c_{\rm{sf}}^{2}}}}$
(56)

is the superfluid Lorentz factor. Writing the space-time line element in this
way allows us to easily work out the tetrads

$e^{0}=c_{\rm{sf}}\gamma_{\rm{sf}}^{-1}dt+\frac{w\kappa}{c_{\rm{sf}}}\gamma_{\rm{sf}}d\theta,\
\ e^{1}=dr\ \ \textrm{and}\ \ e^{2}=\gamma_{\rm{sf}}rd\theta.$ (57)

As already argued by Volovik [48], a quasi-particle propagating past a vortex
will experience a time delay due to the non-vanishing torsion within the
vortex core. The torsion may then be probed by computing the holonomy induced
as a quasi-particle encircles the vortex along a loop $\mathcal{C}$ [49]. In
its most generic form, the geometric phase acquired by the quasi-particle is
given by $e^{i\phi}$ where

$\phi=\oint_{\mathcal{C}}\left(e_{\mu}^{a}P_{a}+\omega^{b}_{a\mu}M^{a}_{b}\right)dx^{\mu},$
(58)

$\omega^{b}_{a\mu}$ is the spin connection and $P_{a}$ and $M^{a}_{b}$ are the
generators of translations and rotations, respectively. Now, since
translations are commutative, we may factor out the component corresponding to
time translation. The phase purely due to torsion can thus be obtained by
integrating the $\theta$ 1-form in $e^{0}$ along $\mathcal{C}$, which gives us

$\phi_{t}=\frac{w\kappa}{c_{\rm{sf}}}\oint_{\mathcal{C}}\gamma_{\rm{sf}}d\theta=\frac{2\pi
w\kappa}{c_{\rm{sf}}}\gamma_{\rm{sf}}.$ (59)

Thus, we may interpret the Lorentz factor as the torsion density over a closed
loop. Note also that the $2\pi$ phase winding is recovered in the long
distance limit as $\lim_{r\longrightarrow\infty}\gamma_{\rm{sf}}(r)=1$. The
relationship between torsion and the tetrad fields is captured by Cartan’s
first structure equation [50]

$T^{a}=de^{a}+\omega^{a}_{b}\wedge e^{b}.$ (60)

The solutions for the spin-connection components have already been worked out
for the metric governing a spinning cosmic string [49], which in the case of a
vortex thus are $\omega^{1}_{\theta 2}=-\omega^{2}_{\theta
1}=\gamma_{\rm{sf}}$, due the the equivalence between the two systems. The
asymmetric connection coefficients, given a tetrad basis, can now be computed
as [43]

$\tilde{\Gamma}^{\lambda}_{\mu\nu}=e^{\lambda}_{a}\partial_{\mu}e^{a}_{\nu}+e^{\lambda}_{a}e^{b}_{\nu}\omega^{a}_{\mu
b}.$ (61)

Contrary to the Levi–Civita connection, this connection takes into account the
effects on parallel transport due to torsion. In the following sections we
shall use this object to couple the field strength to the vortex gravity.

####  III.2.3 Gauss–Einstein law

Going through the same procedure as in the curved case, Gauss’s law becomes

$\displaystyle\nabla_{\mu}\tilde{F}^{\mu
t}=\left(\partial_{r}+\tilde{\Gamma}^{r}_{rr}+\tilde{\Gamma}^{\theta}_{r\theta}+\tilde{\Gamma}^{t}_{tr}\right)\tilde{F}^{rt}+$
$\displaystyle\left(\partial_{\theta}+\tilde{\Gamma}^{r}_{\theta
r}+\tilde{\Gamma}^{\theta}_{\theta\theta}+\tilde{\Gamma}^{t}_{t\theta}\right)\tilde{F}^{\theta
t}+$ (62)
$\displaystyle\left(\tilde{\Gamma}^{t}_{r\theta}-\tilde{\Gamma}^{t}_{\theta
r}\right)\tilde{F}^{r\theta}.$ (63)

Already at this stage something interesting occurs: it seems like the electric
field in Gauss’s law defined on this metric is accompanied by a magnetic field
$\tilde{F}^{r\theta}$, due to the asymmetry of $\tilde{\Gamma}^{t}_{r\theta}$.
However, this should not be too surprising since the current in its
contravariant basis has, due to the non-zero off-diagonal elements in the
metric, mixed components

$J_{\rm{ME}}^{t}=g^{t\alpha}J^{\rm{ME}}_{\alpha}=\rho^{\rm{ME}}-\frac{v_{\rm{sf}}}{c_{\rm{sf}}}j^{\rm{ME}}_{\theta},$
(64)

given an azimuthal flow. This stems from the fact that when the source vortex
is placed in a region in which the torsion is non-negligible, it will start to
precess. In order to find the exact form of Gauss’s law, we need to compute
the asymmetric connection coefficients which yields

$\displaystyle\nabla_{\mu}\tilde{F}^{\mu
t}=\left(\partial_{r}+\beta\partial_{r}\beta\right)\tilde{F}^{rt}+\partial_{\theta}\tilde{F}^{\theta
t}-$
$\displaystyle\left((\partial_{r}-\gamma_{\rm{sf}})\alpha-\alpha\partial_{r}\beta\right)\tilde{F}^{r\theta},$
(65)

where $\alpha(r)=\frac{w\kappa}{c_{\rm{sf}}}\gamma_{\rm{sf}}(r)$ and
$\beta(r)=r\gamma_{\rm{sf}}(r)$. Further, if we go to the rest frame of the
vortex such that $v_{\rm{sf}}=0$, and consequently that
$\alpha(r)=\frac{w\kappa}{c_{\rm{sf}}}$ and $\beta(r)=r$, the magnetic term
vanishes so that Gauss’s law, in the rest frame of the source, becomes (with
unit coupling)

$\left(\partial_{r}+r\right)\tilde{F}^{rt}+\partial_{\theta}\tilde{F}^{\theta
t}=\mu_{0}\rho^{\rm{ME}},$ (66)

which is purely electric.

####  III.2.4 Ampère–Einstein law

Ampère–Einstein’s law is obtained by letting $\nu=r,\theta$ in
$\nabla_{\mu}\tilde{F}^{\mu\nu}$, which yields the components

$\hat{\textbf{e}}_{r}:\ \ \nabla_{\mu}\tilde{F}^{\mu
r}=\partial_{t}\tilde{F}^{tr}+\partial_{\theta}\tilde{F}^{r\theta}$ (67)

and

$\hat{\textbf{e}}_{\theta}:\ \
\nabla_{\mu}\tilde{F}^{\mu\theta}=\partial_{t}\tilde{F}^{t\theta}+\partial_{r}\tilde{F}^{r\theta}.$
(68)

Interestingly, at first sight, it appears that the torsion has no effect on
this equation. This is not the case, however, since the contravariant field
strength components are obtained by applying a non-diagonal metric, causing
mixing to occur between the components, which thus yields

$\tilde{F}^{tr}=\tilde{F}_{tr}+\frac{v_{\theta}}{c_{\rm{sf}}}\tilde{F}_{\theta
r},\ \
\tilde{F}^{t\theta}=\frac{v_{\rm{sf}}^{2}}{c_{\rm{sf}}^{2}}\left(\tilde{F}_{\theta
t}+\tilde{F}_{rt}\right),\ \
\tilde{F}^{r\theta}=\frac{v_{\rm{sf}}^{2}}{c_{\rm{sf}}^{2}}\tilde{F}_{r\theta},$
(69)

where $\tilde{F}_{\mu\nu}$ is defined as in Eq. (III.2.1).

####  III.2.5 Einstein–Faraday and Einstein–Gauss B-law

As already argued, the asymmetry of the torsional connection implies that the
connection terms in the covariant derivative generally do not vanish, as is
the case in the absence of torsion. Nonetheless, the terms containing the
partial derivatives do vanish trivially in $(2+1)$-dimensions. The equation we
are left with for the sourceless part thus is

$\left(\tilde{\Gamma}^{\alpha}_{\mu\gamma}-\tilde{\Gamma}^{\alpha}_{\gamma\mu}\right)\tilde{F}_{\alpha\nu}\\\
+\left(\tilde{\Gamma}^{\beta}_{\nu\gamma}-\tilde{\Gamma}^{\beta}_{\gamma\nu}\right)\tilde{F}_{\beta\mu}+\left(\tilde{\Gamma}^{\delta}_{\nu\mu}-\tilde{\Gamma}^{\delta}_{\mu\nu}\right)\tilde{F}_{\delta\gamma}=0.$
(70)

This equation encompass information that is fundamentally different from that
contained in the conventional homogeneous equations; it says nothing about the
existence of magnetic monopoles, and nor does it assert anything pertaining to
electromagnetic induction. Rather it stipulates how the field strength
components are related to one another via the space-time geometry. We shall
leave the derivation of the exact form of these equations for future work.

##  IV Discussion

A planar superfluid described by a scalar field has been studied. By
generalizing the vortex electrodynamics picture put forth in [40] to systems
with non-constant densities, we have shown that the resulting Maxwell’s
equations can be viewed as a Maxwell theory minimally coupled to a gauge field
$v_{\nabla|\Psi|}$ defined in Eq. (16). A gravity description was then adopted
in which we considered the effective space-time defined by the analogue metric
Eq. (32). By treating this analogue space-time as the background on which
Maxwell’s equations live, we recovered the same set of Maxwell’s equations as
in the hydrodynamics description, given that an appropriate 3-current was
introduced. As such, the core result of this work is the establishment of a
correspondence between the hydrodynamics of a planar superfluid and
Maxwell–Einstein theory, which is summarized in the following proposition:

Proposition (Maxwell–Einstein-superfluidity correspondence)
Let $g_{\mu\nu}(\bf{r})$ be the metric of a $(d+1)$-dimensional conformal
space-time and let $F_{\mu\nu}$ be the field strength tensor in
$(2+1)$-dimensions, then the resulting Maxwell–Einstein theory is equivalent
to the hydrodynamics of a planar superfluid, given that the following
relationships are enforced

$\displaystyle x$ $\displaystyle=\frac{2}{d-1},$ $\displaystyle g_{E}=x+1$
$\displaystyle=\frac{d+1}{d-1}\textrm{ and}$ $\displaystyle g_{M}=x$
$\displaystyle=\frac{2}{d-1}$

between the dimensionless parameter $x$, the dimension $d$ and the
electromagnetic couplings $g_{E}$ and $g_{M}$.

For the sake of generality, we expanded upon this picture by considering the
effective metric in the neighbourhood of a vortex. In this case, the torsion
tensor had non-vanishing components which required us to introduce an
asymmetric connection. Coupling to a torsional connection has, as underscored
in this work, several interesting consequences for the resulting
electrodynamics. Firstly, the field strength tensor takes on a fundamentally
different form since the terms accounting for the non-trivial metric are non-
vanishing, as is the case when only curvature is present. Secondly, the
contravariant components of the field strength tensor are generally formed by
superpositions of various components since the metric has off-diagonal
elements. A third interesting consequence of the presence of torsion is that
the homogeneous equations are no longer insensitive to the metric. If the
connection only has curvature, the resulting homogeneous equations are left
invariant, that is, they are still defined by the Bianchi identity. By
contrast, these equations possess a fundamentally different structure in the
torsional case. In particular, they are reduced from being of a differential-
type to a simple linear-type set of equations, since all of the derivatives
cancel out. As for future work, an interesting direction of study would be to
consider condensates with spin. It has been shown that spinor condensates can
be used to simulate e.g. quantum chromodynamics [51, 52, 53, 54, 55, 56].
Coupling the Yang–Mills field strength tensor to the analogue connection may
thus make possible the study of the effect of a non-vanishing curvature (and
torsion) on analogue quark interactions.

## Acknowledgements

I wish to thank Tapio Simula for many illuminating discussions on the topic of
analogue physics in superfluids. I am also grateful to Mathis Moes for
providing assistance in proof reading the manuscript. This research was
supported by the Australian Research Council Future Fellowship FT180100020,
and was funded by the Australian Government.

## References

  * Albert  _et al._ [1916] E. Albert, W. Perrett, and G. Jeffery, The foundation of the general theory of relativity, Ann. Phys. 354, 769 (1916).
  * Clerk Maxwell [1865] J. Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philos. Trans. R. Soc. 155, 459–512 (1865).
  * Schwinger [1948a] J. Schwinger, On quantum-electrodynamics and the magnetic moment of the electron, Phys. Rev. 73, 416 (1948a).
  * Schwinger [1948b] J. Schwinger, Quantum electrodynamics. I. a covariant formulation, Phys. Rev. 74, 1439 (1948b).
  * Tomonaga [1946] S. Tomonaga, On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields, PTEP 1, 27 (1946).
  * Feynman [1949a] R. P. Feynman, Space-time approach to quantum electrodynamics, Phys. Rev. 76, 769 (1949a).
  * Feynman [1949b] R. P. Feynman, The theory of positrons, Phys. Rev. 76, 749 (1949b).
  * Feynman [1950] R. P. Feynman, Mathematical formulation of the quantum theory of electromagnetic interaction, Phys. Rev. 80, 440 (1950).
  * Yang and Mills [1954] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191 (1954).
  * Jackiw [1980] R. Jackiw, Introduction to the Yang–Mills quantum theory, Rev. Mod. Phys. 52, 661 (1980).
  * Catren [2008] G. Catren, Geometric foundations of classical Yang–-Mills theory, Stud. Hist. Philos. Sci. 39, 511 (2008).
  * Walsh  _et al._ [1979] D. Walsh, R. F. Carswell, and R. J. Weymann, 0957+561 A, B: twin quasistellar objects or gravitational lens?, Nature (London) 279, 381 (1979).
  * Sauer [2008] T. Sauer, Nova Geminorum 1912 and the origin of the idea of gravitational lensing, Arch. Hist. Exact Sci. 62, 1 (2008).
  * Lodge [1919] O. J. Lodge, Gravitation and Light, Nature (London) 104, 372 (1919).
  * Popov [1973] V. Popov, Quantum vortices and phase transitions in Bose systems, Sov. Phys. JETP 37, 2 (1973).
  * Unruh [1981] W. G. Unruh, Experimental black-hole evaporation?, Phys. Rev. Lett. 46, 1351 (1981).
  * Visser [1998] M. Visser, Acoustic black holes: horizons, ergospheres and Hawking radiation, Class. Quantum Gravity 15, 1767 (1998).
  * Garay  _et al._ [2000] L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Sonic Analog of Gravitational Black Holes in Bose–Einstein Condensates, Phys. Rev. Lett. 85, 4643 (2000).
  * Schützhold and Unruh [2002] R. Schützhold and W. G. Unruh, Gravity wave analogues of black holes, Phys. Rev. D 66, 044019 (2002).
  * Weinfurtner [2005] S. Weinfurtner, Analogue model for an expanding universe, Gen. Relativ. Gravit. 37, 1549 (2005).
  * Jain  _et al._ [2007] P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Analog model of a Friedmann-Robertson-Walker universe in Bose–Einstein condensates: Application of the classical field method, Phys. Rev. A 76, 033616 (2007).
  * Hsiang  _et al._ [2013] J.-T. Hsiang, C.-Y. Lin, D.-S. Lee, and R. J. Rivers, The role of causality in tunable Fermi gas condensates, J. Phys. Condens. Matter 25, 404211 (2013).
  * Chapline and Mazur [2014] G. Chapline and P. O. Mazur, Superfluid Picture for Rotating Space-Times, Acta Phys. Pol. B 45, 905 (2014).
  * Furtado  _et al._ [2004] C. Furtado, A. M. d. M. Carvalho, L. C. Garcia de Andrade, and F. Moraes, Holonomy, Aharonov–Bohm effect and phonon scattering in superfluids, arXiv e-prints , gr-qc/0401025 (2004).
  * Volovik [1998a] G. E. Volovik, Gravity of monopole and string and the gravitational constant in 3He-A, JETP Letters 67, 698 (1998a).
  * Gauthier  _et al._ [2016] G. Gauthier, I. Lenton, N. M. Parry, M. Baker, M. Davis, H. Rubinsztein-Dunlop, and T. Neely, Direct imaging of a digital-micromirror device for configurable microscopic optical potentials, Optica 3, 1136 (2016).
  * Einstein [2005] A. Einstein,  _Albert Einstein: Akademie‐Vorträge_ (John Wiley and Sons, Ltd., 2005) pp. 316–321.
  * Pereira [2014] J. G. Pereira, Teleparallelism: A New Insight into Gravity, in  _Springer Handbook of Spacetime_, edited by A. Ashtekar and V. Petkov (Springer, Berlin, 2014) p. 197.
  * Kaluza [2018] T. Kaluza, On the Unification Problem in Physics, Int. J. Mod. Phys. D 27, 1870001 (2018).
  * Klein [1926a] O. Klein, Quantentheorie und fünfdimensionale Relativitätstheorie, Z. Phys. 37, 895 (1926a).
  * Klein [1926b] O. Klein, The Atomicity of Electricity as a Quantum Theory Law, Nature (London) 118, 516 (1926b).
  * Pasini [1988] A. Pasini, A conceptual introduction to the Kaluza–Klein theory, Eur. J. Phys. 9, 289 (1988).
  * Nordström [2007] G. Nordström, Über die möglichkeit, das elektromagnetische feld und das gravitationsfeld zu vereinigen, Z. Phys. 15, 504 (2007).
  * Nordström [1913] G. Nordström, Träge und schwere masse in der relativitätsmechanik, Ann. Phys. 345, 856 (1913).
  * Nordstrom [1913] G. Nordstrom, The theory of the gravitation from the point of view of the relativity principle, Ann. Phys. 42, 533 (1913).
  * Tsagas [2005] C. G. Tsagas, Electromagnetic fields in curved spacetimes, Class. Quantum Gravity 22, 393 (2005).
  * Papapetrou [1974] A. Papapetrou, The Einstein–Maxwell equations, in  _Lectures on General Relativity_ (Springer, Dordrecht, 1974) pp. 134–148.
  * de Felice and Clarke [1992] F. de Felice and C. J. S. Clarke,  _Relativity on curved manifolds_ (Cambridge University Press, 1992).
  * Dalfovo  _et al._ [1999] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose–Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  * Simula [2020] T. Simula, Gravitational vortex mass in a superfluid, Phys. Rev. A 101, 063616 (2020).
  * Misner  _et al._ [2017] C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _Gravitation_ (Princeton University Press, 2017).
  * Barceló  _et al._ [2005] C. Barceló, S. Liberati, and M. Visser, Analogue Gravity, Living Rev. Relativ. 8, 12 (2005).
  * Maluf [2013] J. W. Maluf, The teleparallel equivalent of general relativity, Ann. Phys. 525, 339 (2013).
  * Volovik [1998b] G. E. Volovik, Vortex versus spinning string: Iordanskii force and gravitational Aharonov–Bohm effect, JETP Letters 67, 881 (1998b).
  * Vickers [1987] J. A. G. Vickers, Generalised cosmic strings, Class. Quantum Gravity 4, 1 (1987).
  * Harari and Polychronakos [1988] D. Harari and A. P. Polychronakos, Gravitational time delay due to a spinning string, Phys. Rev. D 38, 3320 (1988).
  * Burges [1985] C. J. C. Burges, Gravitational Aharonov-Bohm effect in three dimensions, Phys. Rev. D 32, 504 (1985).
  * Volovik [1998] G. Volovik, Vortex versus spinning string: Iordanskii force and gravitational Aharonov–Bohm effect, JETP letters 67, 881 (1998).
  * Anandan [1994] J. S. Anandan, Topological and geometrical phases due to gravitational field with curvature and torsion, Physics Letters A 195, 284 (1994).
  * Obukhov [2018] Y. N. Obukhov, Poincaré gauge gravity: An overview, Int. J. Geom. Methods Mod. Phys. 15, 1840005-342 (2018).
  * Vollhardt and Woelfle [2000] D. Vollhardt and P. Woelfle, Superfluid Helium 3: Link Between Condensed Matter Physics and Particle Physics, Acta Phys. Pol. B 31, 2837 (2000).
  * Volovik [1997] G. E. Volovik, Superfluid 3He, Particle Physics and Cosmology, arXiv e-prints , cond-mat/9711031 (1997).
  * Volovik [2001] G. E. Volovik, Superfluid analogies of cosmological phenomena, Phys. Rep. 351, 195 (2001).
  * Eto and Nitta [2018] M. Eto and M. Nitta, Confinement of half-quantized vortices in coherently coupled Bose–Einstein condensates: Simulating quark confinement in a QCD-like theory, Phys. Rev. A 97, 023613 (2018).
  * Tylutki  _et al._ [2016] M. Tylutki, L. P. Pitaevskii, A. Recati, and S. Stringari, Confinement and precession of vortex pairs in coherently coupled Bose–einstein condensates, Phys. Rev. A 93, 043623 (2016).
  * Cirac  _et al._ [2010] J. I. Cirac, P. Maraner, and J. K. Pachos, Cold atom simulation of interacting relativistic quantum field theories, Phys. Rev. Lett. 105, 190403 (2010).


# A small remark on Bernstein’s theorem

Michael Bildhauer
<EMAIL_ADDRESS>Bernhard Farquhar
<EMAIL_ADDRESS>Martin Fuchs
<EMAIL_ADDRESS>
( Saarland University
Department of Mathematics
66123 Saarbrücken, Germany)

###### Abstract

We investigate splitting-type variational problems with some linear growth
conditions. For balanced solutions of the associated Euler-Lagrange equation
we receive a result analogous to Bernstein’s theorem on non-parametric minimal
surfaces.

Without assumptions of this type, Bernstein’s theorem cannot be carried over
to the splitting case, which follows from an elementary counterexample.

We also include some modifications of our main theorem.111AMS Classification
49Q20, 49Q05, 53A10, 35J20, 222Keywords: Bernstein’s theorem, non-parametric
minimal surfaces, two-dimensional variational problems, splitting-type
functionals

August 28, 2024

##  1 Introduction

A famous theorem of Bernstein (see [1]) states that a smooth solution
$u=u(x),x=(x_{1},x_{2})$, of the non-parametric minimal surface equation

$\mathrm{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\right)=0$ (1)

defined on the whole plane must be an affine function. Letting
$f_{0}(P)\vcentcolon=\sqrt{1+|P|^{2}},P\in\mathbb{R}^{2}$, the validity of (1)
on some domain $\Omega\subset\mathbb{R}^{2}$ just expresses the fact that $u$
is a solution of the Euler-Lagrange equation associated to the area functional

$J_{0}[u,\Omega]\vcentcolon=\int_{\Omega}f_{0}(\nabla u)\,\mathrm{d}x.$ (2)

For a general overview on minimal surfaces, variational integrals with linear
growth and for a careful analysis of Bernstein’s theorem the reader is
referred for instance to [2], [3], [4], [5], [6] and [7] and the references
quoted therein.

We ask the following question: does Bernstein’s theorem extend to the case
when the area integrand $f_{0}(\nabla u)$ is replaced by the energy density
$\sqrt{1+(\partial_{1}u)^{2}}+\sqrt{1+(\partial_{2}u)^{2}}$ being also of
linear growth with respect to $|\nabla u|$ but without any obvious geometric
meaning?

More generally we let for $P=(p_{1},p_{2})\in\mathbb{R}^{2}$

$f(P):=f_{1}(p_{1})+f_{2}(p_{2})$ (3)

with functions $f_{i}\in\mathit{C}^{2}(\mathbb{R}),i=1,2$, satisfying

$0<f_{i}^{\prime\prime}(t)\leq
C_{i}(1+t^{2})^{-\frac{\mu_{i}}{2}},t\in\mathbb{R},$ (4)

for numbers $C_{i}>0$ and with exponents

$\mu_{i}>1.$ (5)

Note that (4) implies the strict convexity of $f$ and on account of (5) the
density $f$ is of linear growth in the sense that

$|f_{i}(t)|\leq a|t|+b,\quad t\in\mathbb{R},$

for some constants $a,b>0$. For a discussion of the properties of densities
$f$ satisfying (3)-(5) we refer to [8]. We then replace (1) by the equation

$\mathrm{div}\left(Df(\nabla u)\right)=0$ (6)

and observe that the non-affine function
($\alpha,\beta,\gamma,\delta\in\mathbb{R}$)

$w(x_{1},x_{2}):=\alpha x_{1}x_{2}+\beta x_{1}+\gamma x_{2}+\delta$ (7)

is an entire solution of equation (6), in other words: the classical version
of Bernstein’s theorem does not extend to the splitting case. The behaviour of
the function $w$ defined in (7) is characterized in

######  Definition 1.1.

A function $u\in\mathit{C}^{1}(\mathbb{R}^{2}),u=u(x_{1},x_{2})$, is called
unbalanced, if and only if both of the following conditions hold

$\displaystyle\limsup_{|x|\to\infty}\frac{|\partial_{1}u(x)|}{1+|\partial_{2}u(x)|}$
$\displaystyle=$ $\displaystyle\infty,$ (8)
$\displaystyle\limsup_{|x|\to\infty}\frac{|\partial_{2}u(x)|}{1+|\partial_{1}u(x)|}$
$\displaystyle=$ $\displaystyle\infty.$ (9)

Otherwise we say that $u$ is of balanced form.

######  Remark 1.2.

Condition (8) for example means that there exists a sequence of points
$x_{n}\in\mathbb{R}^{2}$ such that $|x_{n}|\to\infty$ and for which

$\lim_{n\to\infty}\frac{|\partial_{1}u(x_{n})|}{1+|\partial_{2}u(x_{n})|}=\infty.$

######  Remark 1.3.

If for instance (8) is violated, no such sequence exists. Thus we can find
constants $R,M>0$ such that $|\partial_{1}u(x)|\leq
M\left(1+|\partial_{2}u(x)|\right)$ for all $|x|\geq R$. Since $u$ is of class
$\mathit{C}^{1}(\mathbb{R}^{2})$, this just shows

$|\partial_{1}u(x)|\leq m|\partial_{2}u(x)|+M,\quad x\in\mathbb{R}^{2},$ (10)

with suitable new constants $m,M>0$.

Now we can state the appropriate version of Bernstein’s theorem in the above
setting:

######  Theorem 1.4.

Let (3)-(5) hold and let $u\in\mathit{C}^{2}(\mathbb{R}^{2})$ denote a
solution of (6) on the entire plane. Then $u$ is an affine function or of
unbalanced type.

######  Remark 1.5.

We do not know if (7) is the only entire unbalanced solution of (6).

Before proving Theorem 1.4 we formulate some related results: in Theorem 1.7
below we can slightly improve the result of Theorem 1.4 by adjusting the
notation introduced in Definition 1.1 and by taking care of the growth rates
of the second derivatives $f^{\prime\prime}_{i}$ (compare (4)).

######  Definition 1.6.

Let $\mu:=(\mu_{1},\mu_{2})$ with numbers $\mu_{i}>1$, $i=1,2$. A function
$u\in\mathit{C}^{1}(\mathbb{R}^{2})$ is called $\mu$-balanced, if we can find
a positive constant $c$ and a number $\rho>0$ such that at least one of the
following inequalities holds:

$|\partial_{1}u|\leq c(|\partial_{2}u|^{\rho\mu_{2}}+1),$ (11)

$|\partial_{2}u|\leq c(|\partial_{1}u|^{\rho\mu_{1}}+1),$ (12)

where in case (11) we require $\rho\in(1/\mu_{2},1)$, whereas in case (12)
$\rho\in(1/\mu_{1},1)$ must hold.

Note that for example (11) is a weaker condition in comparison to (10).

The extension of Theorem 1.4 reads as follows:

######  Theorem 1.7.

Let (3)-(5) hold and let $u\in\mathit{C}^{2}(\mathbb{R}^{2})$ denote an entire
solution of (6). If the function $u$ is $\mu$-balanced, then it must be
affine.

In Theorem 1.8 we suppose that $|\partial_{1}u|$ is controlled in
$x_{2}$-direction and from this we derive a smallness condition in
$x_{1}$-direction – at least for a suitable sequence satisfying
$|x_{1}|\to\infty$. The idea of proving Theorem 1.8 again is of Bernstein-type
in the sense that the proof follows the ideas of Theorem 1.4 combined with a
splitting structure of the test functions.

######  Theorem 1.8.

Let (3)-(5) hold and let $u\in C^{2}(\mathbb{R}^{2})$ denote a solution of (6)
on the entire plane. Suppose that there exist real numbers $\kappa_{1}>0$,
$0\leq\kappa_{2}<1$ such that

$\mu_{1}>1+\frac{1}{\kappa_{1}}\frac{\kappa_{2}}{1-\kappa_{2}}$ (13)

and such that with a constant $k>0$

$\limsup_{|x_{2}|\to\infty}\frac{|\partial_{1}u(x_{1},x_{2})|}{|x_{2}|^{\kappa_{2}}}\leq
k.$ (14)

Then we have

$\liminf_{|x_{1}|\to\infty}\frac{|\partial_{1}u(x_{1},x_{2})|}{|x_{1}|^{\kappa_{1}}}=0.$
(15)

More precisely, by (13) we choose $\rho$ such that

$\frac{\kappa_{2}}{1-\kappa_{2}}<\rho<\kappa_{1}(\mu_{1}-1).$ (16)

Then the $\limsup$ in (14) is taken in the set

$M_{2,\rho}\vcentcolon=\left\\{(x_{1},x_{2})\colon|x_{1}|\leq
2|x_{2}|^{1/(1+\rho)}\right\\}$

and the $\liminf$ in (15) is taken in the set

$M_{1,\rho}\vcentcolon=\left\\{(x_{1},x_{2})\colon|x_{2}|\leq
2|x_{1}|^{1+\rho}\right\\}.$

Our final Bernstein-type result is given in Theorem 1.9. Here a formulation in
terms of the densities $f_{i}$ is presented without requiring an upper bound
for the second derivatives $f_{i}^{\prime\prime}$ in terms of some negative
powers (see (4)).

######  Theorem 1.9.

Suppose that $f_{i}\in\mathit{C}^{2}(\mathbb{R})$, $i=1,2$, satisfies
$f_{i}^{\prime\prime}(t)>0$ for all $t\in\mathbb{R}$ and $f_{i}^{\prime}\in
L^{\infty}(\mathbb{R})$. Let $u\in\mathit{C}^{2}(\mathbb{R}^{2})$ denote an
entire solution of (6), i.e. it holds

$0=\mathrm{div}\left(Df(\nabla
u)\right)=f_{1}^{\prime\prime}(\partial_{1}u)\partial_{11}u+f_{2}^{\prime\prime}(\partial_{2}u)\partial_{22}u\quad\mbox{on
$\mathbb{R}^{2}$}.$

If

$\Theta:=\frac{f_{2}^{\prime\prime}(\partial_{2}u)}{f_{1}^{\prime\prime}(\partial_{1}u)}\in
L^{\infty}(\mathbb{R}^{2})\quad\mbox{or}\quad\frac{1}{\Theta}\in
L^{\infty}(\mathbb{R}^{2}),$

then $u$ is an affine function.

In the next section we prove our main Theorem 1.4 while in Section 3 the
variants mentioned above are established.

##  2 Proof of Theorem 1.4

Our arguments make essential use of a Caccioppoli-type inequality involving
negative exponents. This result was first introduced in [8].We refer to the
presentation given in Section 6 of [9], where Proposition 6.1 applies to the
situation at hand. Let us assume that the conditions (3)-(5) hold and that $u$
is an entire solution of equation (6) being not necessarily of balanced type.

######  Lemma 2.1 (see [9], Prop.6.1).

Fix $l\in\mathbb{N}$ and suppose that
$\eta\in\mathit{C}_{0}^{\infty}(\Omega)$, $0\leq\eta\leq 1$, where $\Omega$ is
a domain in $\mathbb{R}^{2}$. Then the inequality

$\displaystyle\int_{\Omega}D^{2}f(\nabla
u)(\nabla\partial_{i}u,\nabla\partial_{i}u)\eta^{2l}\Gamma_{i}^{\alpha}\,\mathrm{d}x$
(17) $\displaystyle\leq$ $\displaystyle\int_{\Omega}D^{2}f(\nabla
u)(\nabla\eta,\nabla\eta)\eta^{2l-2}\Gamma_{i}^{\alpha+1}\,\mathrm{d}x,\quad\Gamma_{i}\vcentcolon=1+|\partial_{i}u|^{2},$

holds for any $\alpha>-1/2$ and for any fixed $i=1,2$.

Here and in what follows the letter $c$ denotes finite positive constants
whose value may vary from line to line but being independent of the radius.

Assume next that the solution $u$ is balanced and w.l.o.g. let $u$ satisfy
(10). In order to show that $u$ is affine we return to inequality (17), choose
$i=1$ and fix some function $\eta\in C^{\infty}_{0}(B_{2R}(0))$ according to
$\eta\equiv 1$ on $B_{R}(0)$, $|\nabla\eta|\leq c/R$. Then (17) yields for any
exponent $\alpha\in(-1/2,\infty)$ and with the choice $l=1$
($B_{r}:=B_{r}(0)$, $r>0$)

$\displaystyle\int_{B_{2R}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\eta^{2}\Gamma^{\alpha}_{1}\,\mathrm{d}x$
(18) $\displaystyle\leq$ $\displaystyle c\int_{B_{2R}}D^{2}f(\nabla
u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\alpha+1}\,\mathrm{d}x$
$\displaystyle\overset{\eqref{3}}{=}$ $\displaystyle
c\int_{B_{2R}-B_{R}}\left(f_{1}^{\prime\prime}(\partial_{1}u)|\partial_{1}\eta|^{2}+f_{2}^{\prime\prime}(\partial_{2}u)|\partial_{2}\eta|^{2}\right)\Gamma_{1}^{\alpha+1}\,\mathrm{d}x$
$\displaystyle\leq$ $\displaystyle
cR^{-2}\left(\int_{B_{2R}-B_{R}}f_{1}^{\prime\prime}(\partial_{1}u)\Gamma_{1}^{\alpha+1}\,\mathrm{d}x+\int_{B_{2R}-B_{R}}f_{2}^{\prime\prime}(\partial_{2}u)\Gamma_{1}^{\alpha+1}\,\mathrm{d}x\right)$
$\displaystyle\overset{\eqref{4}}{\leq}$ $\displaystyle
cR^{-2}\left(\int_{B_{2R}-B_{R}}\Gamma_{1}^{\alpha+1-\frac{\mu_{1}}{2}}\,\mathrm{d}x+\int_{B_{2R}-B_{R}}\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\,\mathrm{d}x\right).$

Recall (5) and choose $\alpha$ according to

$\alpha\in\left(-1/2,\mathrm{min}\left\\{-1+\frac{\mu_{1}}{2},-1+\frac{\mu_{2}}{2}\right\\}\right).$
(19)

Here we note that – depending on the values of $\mu_{1}$ and $\mu_{2}$ –
actually a negative exponent $\alpha$ can occur. It follows from (10) that

$cR^{-2}\left(\int_{B_{2R}-B_{R}}\Gamma_{1}^{\alpha+1-\frac{\mu_{1}}{2}}\,\mathrm{d}x+\int_{B_{2R}-B_{R}}\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\,\mathrm{d}x\right)\leq
cR^{-2}\int_{B_{2R}-B_{R}}c\,\mathrm{d}x\leq c<\infty,$ (20)

recalling that $c$ is independent of $R$. Combining (20) and (18) it is
obvious that (by passing to the limit $R\to\infty$)

$\int_{\mathbb{R}^{2}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\Gamma_{1}^{\alpha}\,\mathrm{d}x<\infty$
(21)

for $\alpha$ satisfying (19).

As in the proof of Proposition 6.1 from [9] (with $l=1$) and by applying the
Cauchy-Schwarz inequality we get

$\displaystyle\int_{B_{2R}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\eta^{2}\Gamma_{1}^{\alpha}\,\mathrm{d}x$
(22) $\displaystyle\leq$ $\displaystyle\Biggl{|}\int_{B_{2R}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\eta^{2})\partial_{1}u\Gamma_{1}^{\alpha}\,\mathrm{d}x\Biggr{|}$
$\displaystyle\leq$ $\displaystyle
c\left[\int_{\mathrm{spt}\nabla\eta}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\eta^{2}\Gamma_{1}^{\alpha}\,\mathrm{d}x\right]^{\frac{1}{2}}\left[\int_{\mathrm{spt}\nabla\eta}D^{2}f(\nabla
u)(\nabla\eta,\nabla\eta)\Gamma_{1}^{\alpha+1}\,\mathrm{d}x\right]^{\frac{1}{2}}.\,\,\,\,$

The second integral on the right-hand side is bounded on account of our
previous calculations. Because of the validity of (21) the limit of the first
integral for $R\to\infty$ is $0$. Thus (22) implies

$\nabla\partial_{1}u\equiv 0.$ (23)

In particular (23) guarantees the existence of a number $a\in\mathbb{R}$ such
that

$\partial_{1}u\equiv a.$ (24)

From (24) we obtain

$\displaystyle
u(x_{1},x_{2})-u(0,x_{2})=\int_{0}^{x_{1}}\diff{}{t}u(t,x_{2})\,\mathrm{dt}=\int_{0}^{x_{1}}a\,\mathrm{d}t=a\,x_{1}$

implying

$u(x_{1},x_{2})=u(0,x_{2})+a\,x_{1}.$ (25)

Considering (24) again, equation (6) reduces to

$\partial_{2}\left(f_{2}^{\prime}(\partial_{2}u)\right)=0.$ (26)

We set $\varphi(t)\vcentcolon=u(0,t)$ and interpret the PDE (26) as the ODE

$\diff{}{t}\left(f_{2}^{\prime}(\varphi^{\prime}(t))\right)=0$ (27)

implying

$f_{2}^{\prime}(\varphi^{\prime}(t)))=\mathrm{const.}$

Since $f_{2}^{\prime}$ is strictly monotonically increasing, this just means

$\varphi^{\prime}(t)=b,\quad t\in\mathbb{R},$

for some real number $b$, which consequently gives

$u(x_{1},x_{2})=a\,x_{1}+b\,x_{2}+c,\quad a,b,c\in\mathbb{R}$ (28)

completing our proof. ∎

##  3 Remaining proofs

ad Theorem 1.7. Let the assumptions of Theorem 1.7 hold and assume w.l.o.g.
that we have inequality (11) from Definition 1.6. Consider the mixed term in
the last line of (18) and note that on account of (11) we may estimate

$\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\leq
c\Gamma_{2}^{-\frac{\mu_{2}}{2}(1-2\rho(\alpha+1))}.$ (29)

The validity of $\rho<1$ allows us to choose $\alpha$ sufficiently close to
$-1/2$ such that $1-2\rho(\alpha+1)>0$ which again yields (21) and allows us
to proceed as before giving our claim. ∎

ad Theorem 1.8. Suppose by contradiction that there exists a real number
$\hat{c}>0$ such that w.r.t the set $M_{1,\rho}$

$\hat{c}\leq\liminf_{|x_{1}|\to\infty}\frac{|\partial_{1}u(x_{1},x_{2})|}{|x_{1}|^{\kappa_{1}}}.$
(30)

For intervals $I_{1},I_{2}\subset\mathbb{R}$ we let

$S_{I_{1};I_{2}}\vcentcolon=\\{x\in\mathbb{R}^{2}:|x_{1}|\in I_{1},|x_{2}|\in
I_{2}\\}.$

We fix $0<R_{1}<R_{2}$ and consider

$\eta\in\mathit{C}_{0}^{\infty}(S_{[0,2R_{1});[0,2R_{2})}),\quad 0\leq\eta\leq
1,\quad\eta\equiv 1\quad\text{on}\quad S_{[0,2R_{1});[0,2R_{2})}\,,$
$\text{spt}\partial_{1}\eta\subset
S_{(R_{1},2R_{1});[0,2R_{2})}\,,\quad\text{spt}\partial_{2}\eta\subset
S_{[0,2R_{1});(R_{2},2R_{2})}\,,$ (31) $|\partial_{1}\eta|\leq
c/R_{1}\,,\quad|\partial_{2}\eta|\leq c/R_{2}.$ (32)

Exactly as in (18) one obtains using (31) and (32)

$\displaystyle\int_{S_{[0,2R_{1});[0,2R_{2})}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\eta^{2}\Gamma_{1}^{\alpha}\,\mathrm{d}x$
(33) $\displaystyle\leq$
$\displaystyle\int_{S_{[0,2R_{1});[0,2R_{2})}}D^{2}f(\nabla
u)(\nabla\eta,\nabla\eta)\eta^{2}\Gamma_{1}^{1+\alpha}\,\mathrm{d}x$
$\displaystyle\leq$
$\displaystyle\frac{c}{R_{1}^{2}}\int_{S_{(R_{1},2R_{1});[0,2R_{2})}}\Gamma_{1}^{\alpha+1-\frac{\mu_{1}}{2}}\,\mathrm{d}x+\frac{c}{R_{2}^{2}}\int_{S_{[0,2R_{1});(R_{2},2R_{2})}}\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\,\mathrm{d}x.$

By definition we have

$|S_{[0,2R_{1});(R_{2},2R_{2})}|\leq cR_{1}R_{2}.$

Moreover, our assumption $\kappa_{2}<1$ implies that $\alpha$ can be chosen
such that in the case $\kappa_{1}>0$

$-\frac{1}{2}<\alpha<\frac{1}{2\kappa_{2}}-1.$ (34)

In the case $\kappa_{2}=0$ we do not need an additional condition. We apply
assumption (14), which leads to

$\displaystyle\frac{1}{R_{2}^{2}}\int_{S_{[0,2R_{1});(R_{2},2R_{2})}}\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\,\mathrm{d}x$
$\displaystyle\leq\frac{c}{R_{2}^{2}}\int_{S_{[0,2R_{1});(R_{2},2R_{2})}}|x_{2}|^{2\kappa_{2}(\alpha+1)}\,\mathrm{d}x$
$\displaystyle\leq cR_{1}R_{2}^{2\kappa_{2}(\alpha+1)-1}.$ (35)

Let us consider the first integral on the right-hand side of (33) recalling
that $\alpha+1-\mu_{1}/2<0$. Assumption (30) implies

$\displaystyle\frac{1}{R_{1}^{2}}\int_{S_{(R_{1},2R_{1});[0,2R_{2})}}\Gamma_{1}^{\alpha+1-\frac{\mu_{1}}{2}}\,\mathrm{d}x$
$\displaystyle\leq\frac{c}{R_{1}^{2}}\int_{(R_{1},2R_{1});[0,2R_{2})}|x_{1}|^{2\kappa_{1}(\alpha+1-\frac{\mu_{1}}{2})}\,\mathrm{d}x$
$\displaystyle\leq cR_{2}R_{1}^{-\kappa_{1}(\mu_{1}-2(\alpha+1))-1},$ (36)

where we suppose that $\mu_{1}>2(\alpha+1)$ by choosing $\alpha$ sufficiently
close to $-1/2$. If we further suppose that

$R_{2}=R_{1}^{1+\rho}\quad\text{with a positive real
number}\quad\rho<\kappa_{1}(\mu_{1}-1),$ (37)

then by decreasing $\alpha$, if necessary, still satisfying $\alpha>-1/2$, we
obtain from (36)

$\frac{1}{R_{1}^{2}}\int_{S_{(R_{1},2R_{1});[0,2R_{2})}}\Gamma_{1}^{\alpha+1-\frac{\mu_{1}}{2}}\,\mathrm{d}x\to
0\quad\text{as}\quad R_{1}\to\infty.$ (38)

Using $R_{2}=R_{1}^{1+\rho}$ (recall (37)) we return to (35) recalling that by
the choice (34) we have $2\kappa_{2}(\alpha+1)-1<0$. We calculate

$R_{1}R_{1}^{(1+\rho)(2\kappa_{2}(\alpha+1)-1)}=R_{1}^{2\kappa_{2}(\alpha+1)+\rho(2\kappa_{2}(\alpha+1)-1)}.$
(39)

If we suppose that

$\kappa_{2}+\rho(\kappa_{2}-1)<0,$ (40)

then we may choose $\alpha>-1/2$ sufficiently small such that the exponent on
the right-hand side of (39) is negative, hence together with (35)

$\frac{1}{R_{2}^{2}}\int_{S_{[0,2R_{1});(R_{2},2R_{2})}}\Gamma_{2}^{-\frac{\mu_{2}}{2}}\Gamma_{1}^{\alpha+1}\,\mathrm{d}x\to
0\quad\text{as}\quad R_{1}\to\infty.$ (41)

By (33), (38) and (41) it follows that

$\int_{S_{[0,2R_{1});[0,2R_{2})}}D^{2}f(\nabla
u)(\nabla\partial_{1}u,\nabla\partial_{1}u)\eta^{2}\Gamma_{1}^{\alpha}\,\mathrm{d}x\to
0\quad\text{as}\quad R_{1}\to\infty,$ (42)

provided that we have (37) and (40), i.e. provided that we have (16) which is
a consequence of (13). Hence we have (42) which exactly as in the proof of
Theorem 1.4 shows that $u$ has to be an affine function and this contradicts
(30) which in turn proves Theorem 1.8. ∎

ad Theorem 1.9. W.l.o.g. we suppose that $\Theta\in
L^{\infty}(\mathbb{R}^{2})$ and that $u\in\mathit{C}^{3}(\mathbb{R}^{2})$,
$f_{1}$, $f_{2}\in\mathit{C}^{3}(\mathbb{R})$. Otherwise we argue in a weak
sense. Let

$w_{i}:=f_{i}^{\prime}(\partial_{i}u),\quad i=1,2.$

Then we have

$\partial_{1}w_{1}+\partial_{2}w_{2}=0\quad\mbox{on $\mathbb{R}^{2}$},$ (43)

hence

$\partial_{11}w_{1}+\partial_{1}\partial_{2}w_{2}=\partial_{11}w_{1}+\partial_{2}\partial_{1}w_{2}=0\quad\mbox{on
$\mathbb{R}^{2}$}.$ (44)

A direct calculation shows

$\partial_{1}w_{2}=\partial_{1}\Big{(}f_{2}^{\prime}(\partial_{2}u)\Big{)}=f_{2}^{\prime\prime}(\partial_{2}u)\partial_{1}\partial_{2}u=\Theta
f_{1}^{\prime\prime}(\partial_{1}u)\partial_{2}\partial_{1}u=\Theta\partial_{2}w_{1}$

and the weak form of (44) reads as

$\int_{\mathbb{R}^{2}}\left(\begin{array}[]{c}\partial_{1}w_{1}\\\
\Theta\partial_{2}w_{1}\end{array}\right)\cdot\nabla\varphi\,\mathrm{d}x=0,\quad\varphi\in
C^{1}_{0}(\mathbb{R}^{2}).$ (47)

Inserting $\varphi=\eta^{2}w_{1}$ with suitable $\eta\in C^{1}_{0}(B_{2R})$
such that $0\leq\eta\leq 1$, $\eta\equiv 1$ on $B_{R}$ and $|\nabla\eta|\leq
c/R$, we obtain

$\displaystyle\int_{B_{2R}}|\partial_{1}w_{1}|^{2}\eta^{2}\,\mathrm{d}x+\int_{B_{2R}}\Theta|\partial_{2}w_{1}|^{2}\eta^{2}\,\mathrm{d}x$
(48) $\displaystyle=$
$\displaystyle-2\int_{B_{2R}-B_{R}}\eta\partial_{1}w_{1}\,\partial_{1}\eta\,w_{1}\,\mathrm{d}x-2\int_{B_{2R}-B_{R}}\Theta\,\eta\,\partial_{2}w_{1}\,\partial_{2}\eta\,w_{1}\,\mathrm{d}x.$

Applying Young’s inequality and using $w_{1}$, $\Theta\in
L^{\infty}(\mathbb{R}^{2})$ we obtain that

$\int_{\mathbb{R}^{2}}\Big{(}|\partial_{1}w_{1}|^{2}+\Theta|\partial_{2}w_{1}|^{2}\Big{)}\,\mathrm{d}x<\infty.$
(49)

We then return to (48) and apply the inequality of Cauchy-Schwarz to obtain

$\displaystyle\int_{B_{2R}}|\partial_{1}w_{1}|^{2}\eta^{2}\,\mathrm{d}x+\int_{B_{2R}}\Theta|\partial_{2}w_{1}|^{2}\eta^{2}\,\mathrm{d}x$
(50) $\displaystyle\leq$ $\displaystyle
2\Bigg{[}\int_{B_{2R}-B_{R}}\eta^{2}|\partial_{1}w_{1}|^{2}\,\mathrm{d}x\Bigg{]}^{\frac{1}{2}}\Bigg{[}\int_{B_{2R}-B_{R}}|\partial_{1}\eta|^{2}w_{1}^{2}\,\mathrm{d}x\Bigg{]}^{\frac{1}{2}}$
$\displaystyle+2\Bigg{[}\int_{B_{2R}-B_{R}}\Theta\eta^{2}|\partial_{2}w_{1}|^{2}\,\mathrm{d}x\Bigg{]}^{\frac{1}{2}}\Bigg{[}\int_{B_{2R}-B_{R}}\Theta|\partial_{2}\eta|^{2}w_{1}^{2}\,\mathrm{d}x\Bigg{]}^{\frac{1}{2}}.$

On the right-hand side of (50) we observe that for both parts the first
integral is vanishing when passing to the limit $R\to\infty$ since we have
(49), while the remaining integrals stay uniformly bounded.

This gives $\partial_{1}w_{1}=0$ and $\partial_{2}w_{1}=0$ since we have
$\Theta>0$. Hence we obtain $w_{1}\equiv c_{1}$ for some constant $c_{1}$. The
monotonicity of $f_{1}^{\prime}$ then implies
$\partial_{1}u\equiv\tilde{c}_{1}$ for some different constant
$\tilde{c}_{1}$.

By (43) we then also have $\partial_{2}w_{2}=0$. Since we have already
observed above that $\partial_{1}w_{2}=\Theta\partial_{2}w_{1}$, we deduce
$\partial_{2}u\equiv\tilde{c}_{2}$ for some other real number $\tilde{c}_{2}$
and in conclusion $u$ must be an affine function which completes the proof of
Theorem 1.9. ∎

## References

  * Bernstein [1927] Serge Bernstein.  Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus.  _Math. Z._ , 26(1):551–558, 1927.
  * Dierkes et al. [2010a] Ulrich Dierkes, Stefan Hildebrandt, and Anthony J. Tromba.  _Global analysis of minimal surfaces_ , volume 341 of _Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_.  Springer, Heidelberg, second edition, 2010a.
  * Dierkes et al. [2010b] Ulrich Dierkes, Stefan Hildebrandt, and Anthony J. Tromba.  _Regularity of minimal surfaces_ , volume 340 of _Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_.  Springer, Heidelberg, second edition, 2010b.  With assistance and contributions by A. Küster.
  * Dierkes et al. [2010c] Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny.  _Minimal surfaces_ , volume 339 of _Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]_.  Springer, Heidelberg, second edition, 2010c.  With assistance and contributions by A. Küster and R. Jakob.
  * Nitsche [1989] Johannes C. C. Nitsche.  _Lectures on minimal surfaces. Vol. 1_.  Cambridge University Press, Cambridge, 1989.  Introduction, fundamentals, geometry and basic boundary value problems, Translated from the German by Jerry M. Feinberg,.
  * Nitsche [1957] Johannes C. C. Nitsche.  Elementary proof of Bernstein’s theorem on minimal surfaces.  _Ann. of Math. (2)_ , 66:543–544, 1957.
  * Osserman [1986] Robert Osserman.  _A survey of minimal surfaces_.  Dover Publications, Inc., New York, second edition, 1986.
  * Bildhauer and Fuchs [2020] Michael Bildhauer and Martin Fuchs.  Splitting type variational problems with linear growth conditions.  _J. Math. Sci. (N.Y.) 250(2), 2020. Problems in mathematical analysis_ , 250(105):232–249, 2020.
  * Bildhauer and Fuchs [2022] Michael Bildhauer and Martin Fuchs.  On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting.  _J. Elliptic Parabol. Equ._ , 8(2):853–884, 2022\.