## 1

M. Shokri<EMAIL_ADDRESS>
x_⟂ aainstitutetext: IPM, School of Particles and Accelerators, P.O. Box
19395-5531, Tehran, Iran

##  Generalization of Bantilan-Ishi-Romatschke flow to Magnetohydrodynamics

M. Shokri<EMAIL_ADDRESS>
###### Abstract

We present a generalization of the Bantilan-Ishi-Romatschke (BIR) solution of
relativistic hydrodynamics to relativistic magnetohydrodynamics (RMHD). Using
the symmetries of the boundary of the Kerr-AdS5 black hole, and certain
simplifying assumptions we solve the equations of RMHD on this boundary for a
highly conductive fluid. We then transform the resulting solution to the flat
spacetime. Furthermore, we show that the force-free condition causes the
magnetic field to become singular at particular points and propose a
regularization process for removing the singularities. The regularization
process reveals the importance of non-vanishing electrical current in RMHD.

###### Keywords:

Quark-gluon plasma, Relativistic fluid dynamics, Magnetohydrodynamics,
Evolution of magnetic fields, Analytical solutions, Conformal
Magnetohydrodynamics

††arxiv:

##  2 Introduction

It is expected that high energy collisions of heavy ions produce enormous
electromagnetic fields, probably stronger than any other known electromagnetic
fields in nature Rafelski:1975rf ; Kharzeev-CME . Simulations suggest that the
produced magnetic field is on order of $10^{18}\\!-\\!10^{20}$ Gauss and
linearly proportional to the collision energy skokov2009 ; Voronyuk:2011jd ;
Bzdak:2011yy ; Deng:2012pc . The estimated magnetic characteristic length
Fukushima:2018grm , i.e. $1/\sqrt{eB}$111Here $B$ is the magnitude of the
magnetic field and $e$ is proton’s electric charge., at its peak is comparable
to the typical length scale of strong interactions at low energies, namely
$1/\Lambda_{\text{QCD}}$. This remark has led to dozens of studies on the
probable effects of the magnetic field on the observables of the heavy-ion
collisions Hattori:2016emy ; Ghosh:2018xhh ; Mukherjee:2017dls ;
Sadooghi:2016jyf ; Fukushima:2015wck . Most of the aforementioned works are
based on the assumption of a static long-lasting magnetic field. This
assumption, however, may not be realistic Huang-review  and we can understand
its unrealistic essence by a simple estimation. Consider $eB$ being of the
order of $m_{\pi}^{2}$222$m_{\pi}$ is Pion’s mass. in the timescales that the
QGP is formed. Such an assumption gives rise to an Alfven velocity gedalin1993
of the same order as the speed of sound. If that would be the case, the
electromagnetic forces could compete with the pressure gradient in the
determination of fluid kinematics. Since relativistic hydrodynamics is in a
good agreement with the experimental data, one can infer that this is not the
case and the Alfven speed must be much smaller than the speed of sound. This
rough argument is confirmed by the results of different simulations of the
early time dynamics of the electromagnetic fields that agree on the short
lifetime of magnetic fields skokov2009 ; Voronyuk:2011jd ; Bzdak:2011yy ;
Deng:2012pc . In addition to such effects, the magnetic field is also proposed
as a probe to the chiral properties of the QGP Kharzeev-CME . It is suggested
that the produced QGP may possess a chiral imbalance on an event-by-event
basis that gives rise to a non-dissipative electric current in the direction
of the magnetic field and consequently the separation of opposite charges. The
charge separation produces an electric dipole which induces a difference
between the distribution of final particles of opposite charges. This
phenomenon is called the chiral magnetic effect (CME). The search for the CME
is still a work in progress Kharzeev-Status ; Zhao:2018ixy .

At early times the medium is dominated by the gluons and almost a perfect
insulator Huang-review . In such an insulator the magnetic field decays
quickly. However, as the QGP forms and hydrodynamization happens, the mutual
evolution of the QGP and electromagnetic fields is effectively understood
using relativistic magnetohydrodynamics (RMHD). It has been suggested that
since QGP is highly conductive it may significantly slow down the magnetic
fields decay Tuchin:2013ie . From the RMHD perspective, this is understood by
assuming the ideal MHD (iMHD) limit. In such a limit the electric field is
suppressed and the magnetic field is decayed through the separation of field
lines due to fluid’s expansion beckenstein1978 . The generalization of Bjorken
flow to iMHD confirms that the lifetime of the magnetic field significantly
enhances in such a regime RPRR-Bjorken-iMHD ; RPRR-Bjorken-iMHD-Magnetized .
However, as argued in rotating-trans-mhd  the electric field in the Bjorken
setup is not suppressed unless the electrical conductivity is much larger than
the corresponding proper time, i.e. $\sigma_{e}\tau\gg 1$. Such a condition is
not satisfied by Lattice QCD (LQCD) results LQCD-Sigma  and one may deduce
that the iMHD is a poor approximation for the QGP. Furthermore, it is argued
that the initial configuration of the fields, i.e. rotation in the terminology
of rotating-trans-mhd ; cs-trans-mhd , is dominant over the transport
coefficients of the QGP rotating-trans-mhd . The results of such a simplified
analytical model, which may be crucial in the search for CME, need to be
justified by numerical investigations. Despite significant progress in iMHD
Inghirami:2016iru ; Inghirami:2018ziv ; Inghirami:2019mkc , numerical
computation of resistive RMHD in the QGP context is still missing.

Analytical solutions of RMHD may provide insights into the evolution of
electromagnetic fields in the QGP. As mentioned earlier the generalization of
Bjorken flow to iMHD appeared in RPRR-Bjorken-iMHD . In rotating-trans-mhd ,
the assumption of infinite conductivity is relaxed and in cs-trans-mhd  the
results of the aforementioned work are generalized to the chiral
magnetohydrodynamics. The method of non-conserved charges which was introduced
in rotating-trans-mhd , is also utilized in rotating-trans-mhd  for
investigation of the CME from an RMHD perspective. The generalization of 3+1
self-similar CGHK-cyl-flow  and Gubser Gubser-Conformal ; Gubser-Symmetry
flows to iMHD is studied in cmhd  using a systematic procedure based on
symmetry arguments. In all of these works, a force-free condition is assumed.
Namely, it is assumed that the electromagnetic forces do not disturb the
kinematic of the fluid. This can be regarded as a first approximation if the
Aflven velocity is much smaller than the sound velocity. As it is discussed
earlier, such an assumption is in agreement with the successes of relativistic
hydrodynamics. A different approach is assumed in Accelerated-Pu ;
Accelerated-She , in which the kinematical modifications of the fluid due to
external magnetic fields are studied. In these works, the magnetic fields are
assumed to be external in the sense that they need not satisfy the Maxwell
equations, Recently a novel exact analytical solution to relativistic
hydrodynamics is presented in bir  that breaks rotational and longitudinal
boost symmetries that are assumed by Bjorken and Gubser flows. The
aforementioned solution, which will be abbreviated as BIR333Bantilan-Ishi-
Romatschke flow from now on, is based on the duality of a rotating Kerr-Ads5
black hole with a peripheral heavy-ion collision in the four-dimensional flat
spacetime. The equations of fluid mechanics are solved on the
$S^{3}\times\mathbb{R}$ boundary of the black hole using symmetry arguments,
and then Weyl transformed into the flat spacetime to obtain the solution. We
present a generalization of BIR flow to iMHD in the current paper.

The organization of this paper is as follows: In Sec. 3, we briefly review the
BIR solution. In Sec. 4, we solve the equations of RMHD on the
$S^{3}\times\mathbb{R}$ boundary. After applying the symmetry arguments the
iMHD limit is considered. The resulting equations are fewer than the number of
unknowns. Hence, we make two additional assumptions: The force-free condition
and local electrical neutrality, which help us find the field tensor and
consequently the magnetic field. The section is closed by calculation of the
gauge potential and magnetic helicity. In Sec. 5, we transform the magnetic
field into the flat spacetime and discuss some aspects of its behavior.
Furthermore, we transform the magnetic helicity to the flat spacetime. In Sec.
6, we briefly discuss singularities and provide an approximation to resolve
them and in Sec. 7, we depict some of the results. The paper is concluded in
Sec. 8.

In this paper, the natural units are used in which $\hbar=c=1$. The metric
signature convention is mostly plus. The notations are adapted from cmhd , and
the reader may consult Appendix A of the aforementioned work for more details.
The differential forms are denoted by non-italic bold symbols such as
$\mathbf{A}$, and $\operatorname{\mathbf{d}}$ is the exterior derivative. In
utilizing the language of the differential forms we mainly follow
Markakis:2016udr .

##  3 BIR solution to hydrodynamics

In this section, we briefly review the BIR solution. This solution is obtained
from an assumed duality between a particular solution to Einstein equations in
five-dimensional spacetime, namely the Kerr-AdS5 solution, with the solution
to the equations of fluid mechanics on the four-dimensional boundary. As it is
shown in bir , the line element of Kerr-AdS5 has the following leading
behavior at $r^{\prime}\to\infty$

$\frac{ds_{5}^{2}}{L^{2}}\simeq-(1+r^{\prime 2})dt^{\prime 2}+\frac{dr^{\prime
2}}{1+r^{\prime 2}}+r^{\prime 2}\left(d\theta^{\prime
2}+\sin^{2}\theta^{\prime}\,d\phi^{\prime
2}+\cos^{2}\theta^{\prime}d\chi^{\prime 2}\right)+\mathcal{O}{(1/r^{\prime
2})}.$

Consequently, the line element at the boundary of the spacetime is

$\mathop{\mathrm{d}s^{2}}=L^{2}\left[-\mathop{\mathrm{d}t^{\prime
2}}+\mathop{\mathrm{d}\theta^{\prime
2}}+\sin^{2}\theta^{\prime}\mathop{\mathrm{d}\phi^{\prime
2}}+\cos^{2}\theta^{\prime}\mathop{\mathrm{d}\chi^{\prime 2}}\right]\,,\qquad
x^{a}=(t^{\prime},\theta^{\prime},\phi^{\prime},\chi^{\prime}).$ (1)

The (1) is the line element of a $S^{3}\times\mathbb{R}$ spacetime. To
recognize this, let us use the following coordinate system:

$X=L\sin\theta^{\prime}\cos\phi^{\prime}\,,\quad
Y=L\sin\theta^{\prime}\sin\phi^{\prime}\,,\quad
Z=L\cos\theta^{\prime}\cos\chi^{\prime}\,,\quad
V=L\cos\theta^{\prime}\sin\chi^{\prime}.$ (2)

The above coordinates satisfy the $S^{3}$ equation, namely

$X^{2}+Y^{2}+Z^{2}+V^{2}=L^{2}.$ (3)

The conformal fluid mechanics of such a spacetime is studied in S3xR-Minwalla
and is reviewed here. The fluid energy-momentum tensor is given by Rezzolla

$T^{ab}=\epsilon u^{a}u^{b}+p\Delta^{ab}+T^{ab}_{\text{dissipative}}.$ (4)

Here $u^{a}$ is the fluid’s four-velocity, $p$ and $\epsilon$ are fluid’s
pressure and energy density respectively and $\Delta_{ab}\equiv
g_{ab}+u_{a}u_{b}$ is the projection tensor. At first order S3xR-Minwalla ;
Rezzolla , the dissipative part is

$T^{ab}_{\text{dissipative}}=-\zeta\left(\nabla_{c}u^{c}\right)\Delta^{ab}-2\eta\sigma^{ab}+q^{a}u^{b}+q^{b}u^{a}.$
(5)

Conformal invariance requires that the bulk viscosity, i.e. $\zeta$ vanishes
Gubser-Conformal . In the second term of (5) $\eta$ is the shear viscosity and
$\sigma^{ab}$ is the shear tensor given by Gourgoulhon:2006bn ; Rezzolla

$\sigma_{ab}=\frac{1}{2}\left(\mathrm{\nabla}_{c}{u_{d}}+\mathrm{\nabla}_{d}{u_{c}}\right)\Delta^{c}_{a}\Delta^{d}_{b}-\frac{1}{3}\Delta_{ab}\nabla_{c}u^{c}.$
(6)

In the last two terms of (5) $q^{a}$ is the diffusion current given by S3xR-
Minwalla

$q^{a}=-\kappa\Delta^{ab}\left(\partial_{b}T+a_{b}T\right).$ (7)

Here, $\kappa$ is the thermal conductivity, $T$ is the temperature and $a_{b}$
is the acceleration that reads Rezzolla

$a^{a}=u^{b}\nabla_{b}u^{a}.$ (8)

We assume that the fluid is symmetric under the symmetry group generated by
the following Killing vectors of (1)

$\frac{\partial}{\partial
t^{\prime}}\,,\qquad\frac{\partial}{\partial\phi^{\prime}}\,,\qquad\frac{\partial}{\partial\chi^{\prime}}.$
(9)

The first vector in (9) dictates that the fluid is stationary and thus in
equilibrium. To satisfy this condition any source of dissipation must vanish.
Therefore, the fluid can only experience a rigid motion S3xR-Minwalla ;
Gourgoulhon:2006bn , namely

$u\propto\xi,$

in which $\xi$ is a linear combination of the Killing vectors given in (9). A
choice for $\xi$ is

$\xi=\frac{\partial}{\partial
t^{\prime}}+\omega_{1}\frac{\partial}{\partial\phi^{\prime}}+\omega_{2}\frac{\partial}{\partial\chi^{\prime}}.$

Here $\omega_{1}$ and $\omega_{2}$ are rotation parameters of the Kerr-AdS5
black hole bir . Therefore, using $u^{2}=-1$, the four-velocity is found to be

$u^{a}=\frac{\gamma}{L}\left(1,0,\omega_{1},\omega_{2}\right),\quad\text{with}\quad\gamma=(1-v^{2})^{-1/2},\quad\text{and}\quad
v^{2}=\omega_{1}^{2}\sin^{2}\theta+\omega_{2}^{2}\cos^{2}\theta,$ (10)

It is evident that $\left|\omega_{1}\right|<1$ and
$\left|\omega_{2}\right|<1$. Since $u$ is proportional to $\xi$ it satisfies
the Killing equation Gourgoulhon:2006bn ; Rezzolla ; Zee-GR-Book

$\mathrm{\nabla}_{a}{u_{b}}+\mathrm{\nabla}_{b}{u_{a}}=0,$

and therefore the shear tensor given in (6) vanishes. The only remaining
dissipative part in the energy-momentum tensor is the diffusion current given
in (7). By symmetries, every quantity, including acceleration, must commute
with $u=u^{a}\partial_{a}$. Therefore the only non-vanishing component of the
acceleration reads

$a^{\theta}=\Gamma^{\theta}_{\phi_{1}\phi_{1}}\left(u^{\phi_{1}}\right)^{2}+\Gamma^{\theta}_{\phi_{2}\phi_{2}}\left(u^{\phi_{2}}\right)^{2}=-\cos\theta\sin\theta\frac{\gamma^{2}}{L^{2}}\left(\omega_{1}^{2}-\omega_{2}^{2}\right).$

The following helpful relation is obtained by simple algebra

$\cos\theta\sin\theta\left(\omega_{1}^{2}-\omega_{2}^{2}\right)=\frac{1}{2}\frac{{d}v^{2}}{{d}\theta}=\frac{1}{\gamma^{3}}\frac{{d}\gamma}{{d}\theta},$

that gives rise to

$a^{a}=\frac{1}{L^{2}}\left(0,-\frac{\mathop{{d}}}{\mathop{\mathrm{d}\theta}}\ln\gamma,0,0\right).$

Plugging the above relation in (7) and demanding $q^{a}=0$ one finds

$T=\gamma\mathcal{T},$ (11)

in which $\mathcal{T}$ is a constant. On the other hand, the conformal
invariance implies

$p=A\mathcal{T}^{4}\gamma^{4},\qquad\epsilon=3p.$

For the uncharged fluid $A$ is a constant. Plugging above and (10) in (4) the
energy-momentum tensor is found to be

$T^{ab}=\frac{A\mathcal{T}^{4}\gamma^{6}}{L^{2}}\begin{bmatrix}3+v^{2}&0&4\omega_{1}&4\omega_{2}\\\
0&1-v^{2}&0&0\\\
4\omega_{1}&0&3\omega_{1}^{2}+\csc^{2}\theta-\omega_{2}^{2}\cot^{2}\theta&4\omega_{1}\omega_{2}\\\
4\omega_{2}&0&4\omega_{1}\omega_{2}&3\omega_{2}^{2}+\sec^{2}\theta-\omega_{2}^{2}\tan^{2}\theta\end{bmatrix}.$

Using

$1-v^{2}=\frac{1}{2}\left(2-\omega_{1}^{2}\left(1-\cos
2\theta\right)-\omega_{2}^{2}\left(1-\cos
2\theta\right)\right)=\frac{1}{2}\left[2-\left(\omega_{1}^{2}+\omega_{2}^{2}\right)+(\omega_{1}^{2}-\omega_{2}^{2})\cos
2\theta\right]\,,$

one finds

$\epsilon=u_{a}u_{b}T^{ab}=\frac{3\cdot
4A\mathcal{T}^{4}}{\left[2-\left(\omega_{1}^{2}+\omega_{2}^{2}\right)+(\omega_{1}^{2}-\omega_{2}^{2})\cos
2\theta\right]^{2}}.$

We can rewrite the above relations by defining $A\mathcal{T}^{4}=T_{0}^{4}/3$
as

$\epsilon=\frac{4T_{0}^{4}}{\left[2-\left(\omega_{1}^{2}+\omega_{2}^{2}\right)+(\omega_{1}^{2}-\omega_{2}^{2})\cos
2\theta\right]^{2}}=T_{0}^{4}\gamma^{4}\,,$ (12)

and

$T^{ab}=\frac{T_{0}^{4}\gamma^{6}}{3L^{2}}\begin{bmatrix}3+v^{2}&0&4\omega_{1}&4\omega_{2}\\\
0&1-v^{2}&0&0\\\
4\omega_{1}&0&3\omega_{1}^{2}+\csc^{2}\theta-\omega_{2}^{2}\cot^{2}\theta&4\omega_{1}\omega_{2}\\\
4\omega_{2}&0&4\omega_{1}\omega_{2}&3\omega_{2}^{2}+\sec^{2}\theta-\omega_{2}^{2}\tan^{2}\theta\end{bmatrix}\,.$
(13)

One can check that (13) satisfies the equations of motion, namely

$\nabla_{a}T^{ab}=0.$

At this point, let’s compute the fluid helicity Markakis:2016udr . The one-
form $\mathbf{p}=T\mathbf{u}$ can be assumed as the conjugate momentum of the
fluid and $L=-T$ as its Lagrangian. Therefore, the fluid equations of motion,
in the absence of dissipation, is obtained from

$-\operatorname{\mathbf{d}}T=\mathcal{L}_{u}{\left(T\mathbf{u}\right)}.$

A vorticity two-form is then given by

$\mathbf{\Omega}=\operatorname{\mathbf{d}}\left(T\mathbf{u}\right).$ (14)

The above quantity is called the T-vorticity in the literature
Becattini:2015ska . The T-vorticity vector is found from contracting $u$ with
the Hodge dual of $\mathbf{\Omega}$, namely

$\omega^{a}=u_{b}{{}^{\star}{\Omega}}^{ba}.$

Plugging Eqns. (11) and (10) into (14) one finds the T-vorticity vector as

$\omega^{a}=\left(\frac{8}{3}\right)^{1/4}\frac{T_{0}\gamma^{3}}{L^{2}}\left(\omega_{1}\omega_{2},0,\omega_{2},\omega_{1}\right).$
(15)

The fluid helicity is defined as Markakis:2016udr

$h_{f}^{a}=Tu_{b}{{}^{\star}{\Omega}}^{ba}.$

Putting Eqns. (11), (10), and (14) in above gives rise to

$h^{a}_{f}=\left(\frac{4}{3}\right)^{1/2}\frac{T_{0}^{2}\gamma^{4}}{L^{2}}\left(\omega_{1}\omega_{2},0,\omega_{2},\omega_{1}\right).$
(16)

One can check that the fluid helicity is conserved, i.e. $\nabla.h=0$.

###  3.1 Passing to flat spacetime

Since the Weyl tensor of (1) vanishes it must be locally conformally flat Zee-
GR-Book . Motivated by the well-known conformal transformation of $S^{d}$ to
$\mathbb{R}^{d}$ Zee-GR-Book , it is found that the Weyl factor that relates
$S^{3}\times R$ to Minkowski spacetime reads Bantilan:2012vu

$W=\frac{1}{\cos t^{\prime}+V/L},$ (17)

in which $V$ is defined in (2). To recognize this, let’s consider the ordinary
parametrization of Minkowski spacetime, i.e.

$\mathop{\mathrm{d}s_{\mathbb{R}^{3,1}}^{2}}=-\mathop{\mathrm{d}t^{2}}+\mathop{\mathrm{d}x^{2}_{1}}+\mathop{\mathrm{d}x^{2}_{2}}+\mathop{\mathrm{d}x^{2}_{3}},$
(18)

and use the following coordinate transformation

$t=LW\sin t^{\prime},\quad x_{1}=WX,\quad x_{2}=WY,\quad x_{3}=WZ.$ (19)

Plugging above in (18) and using (3) gives rise to

$\frac{1}{W^{2}}\mathop{\mathrm{d}s_{\mathbb{R}^{3,1}}^{2}}=-L^{2}\mathop{\mathrm{d}t^{\prime
2}}+\mathop{\mathrm{d}X^{2}}+\mathop{\mathrm{d}Z^{2}}+\mathop{\mathrm{d}Y^{2}}+\mathop{\mathrm{d}V^{2}}.$

Instead of (18) we may parameterize the flat spacetime with the spherical
coordinates defined as

$x_{1}=r\sin\theta\cos\phi,\qquad x_{2}=r\sin\theta\sin\phi,\qquad
x_{3}=r\cos\theta.$ (20)

Consequently the conformal relation between $S^{3}\times\mathbb{R}$ and
$M^{4}$ is found as

$\mathop{\mathrm{d}s_{S^{3}\times\mathbb{R}}^{2}}=\frac{1}{W^{2}}\left(-\mathop{\mathrm{d}t^{2}}+\mathop{\mathrm{d}r^{2}}+r^{2}\mathop{\mathrm{d}\theta^{2}}+r^{2}\sin^{2}\theta\mathop{\mathrm{d}\phi^{2}}\right),$
(21)

and the coordinate transformation (19) translates into

$\displaystyle t^{\prime}$ $\displaystyle=$
$\displaystyle\arctan\frac{2Lt}{L^{2}+r^{2}-t^{2}},\qquad\theta^{\prime}=\arctan\frac{r\sin\theta}{\sqrt{L^{2}W^{2}-r^{2}\sin^{2}\theta}},$
$\displaystyle\phi^{\prime}$ $\displaystyle=$
$\displaystyle\phi,\qquad\chi^{\prime}=-\arctan\frac{L^{2}-r^{2}+t^{2}}{2Lr\cos\theta},$
(22)

while the Weyl factor reads

$W^{2}\equiv\frac{L^{4}+(r^{2}-t^{2})^{2}+2L^{2}(r^{2}+t^{2})}{4L^{4}}.$ (23)

For further convenience, we denote the indices of the flat spacetime with
greek letters $\mu,\nu,\cdots$, and the indices of $S^{3}\times\mathbb{R}$
with Latin letters $a,b,\cdots$. At this stage, we can transform any Weyl
covariant quantity using a combination of Weyl and coordinate transformation
from $S^{3}\times\mathbb{R}$ to the flat spacetime Gubser-Symmetry . Following
Gubser-Symmetry  we denote the conformal weight of a quantity with
$\left[\cdots\right]$. Using Gubser-Symmetry

$\Big{[}\epsilon\Big{]}=4\,,$

one finds444The symbol $\gamma$ is used to denote the Lorentz factor in the
$S^{3}\times\mathbb{R}$ spacetime.

$\epsilon_{\mathbb{R}^{3,1}}=\left(\frac{T_{0}\gamma}{W}\right)^{4}\,,$ (24)

or explicitly

$\displaystyle\epsilon_{\mathbb{R}^{3,1}}$ $\displaystyle=$ $\displaystyle
16L^{8}T_{0}^{4}\Big{[}(L^{4}+(r^{2}-t^{2})^{2}+2L^{2}t^{2})(1-\omega_{2}^{2})+2L^{2}r^{2}(1-\omega_{1}^{2})$
(25) $\displaystyle\hskip
142.26378pt+2L^{2}r^{2}(\omega_{1}^{2}-\omega_{2}^{2})\cos
2\theta\Big{]}^{-2}\,.$

On the other hand, the conformal weight of $u_{\mu}$ is $-1$ Gubser-Symmetry ,
therefore, in the language of differential forms

${u_{\mu}}^{\\!\\!\\!\\!\mathbb{R}^{3,1}}\operatorname{\mathbf{d}}\\!{x^{\mu}}=W{u_{a}}^{\\!\\!\\!\\!S^{3}\times\mathbb{R}}\operatorname{\mathbf{d}}\\!{x^{a}}.$

We solve the above equation using (3.1) and find the four-velocity in the flat
spacetime parametrized in spherical coordinates of (20)555We drop the
spacetime labels unless necessary.

$\displaystyle u^{t}$ $\displaystyle=$
$\displaystyle\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{8}T_{0}^{4}}\right]^{1/4}\left(L^{2}+r^{2}+2rt\omega_{2}\cos(\theta)+t^{2}\right)\,,$
$\displaystyle u^{r}$ $\displaystyle=$
$\displaystyle\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{8}T_{0}^{4}}\right]^{1/4}\left(\omega_{2}\cos(\theta)\left(L^{2}+r^{2}+t^{2}\right)+2rt\right)\,,$
$\displaystyle u^{\theta}$ $\displaystyle=$
$\displaystyle-\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{8}T_{0}^{4}}\right]^{1/4}\frac{\omega_{2}\sin(\theta)\left(L^{2}-r^{2}+t^{2}\right)}{r}\,,$
$\displaystyle u^{\phi}$ $\displaystyle=$ $\displaystyle
2\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{8}T_{0}^{4}}\right]^{1/4}L\omega_{1}\,.$
(26)

The fluid T-vorticity and helicity can also be transformed using the same
method. To proceed we notice that $\Big{[}Tu_{\mu}\Big{]}=0$ and therefore
$\Omega_{\mu\nu}$ has also zero conformal weight. Thereby,

$\Big{[}h^{\mu}\Big{]}=\Big{[}Tu_{\nu}{{}^{\star}{\Omega^{\nu\mu}}}\Big{]}=4,\qquad\Big{[}\omega^{\mu}\Big{]}=\Big{[}{{}^{\star}{\Omega^{\nu\mu}u_{\nu}}}\Big{]}=3.$

Therefore, the T-Vorticity found in (15) is transformed into

$\displaystyle\omega^{t}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)^{2}\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{27}\right]^{3/4}\frac{\omega_{1}\left(\omega_{2}\left(L^{2}+r^{2}+t^{2}\right)+2rt\cos(\theta)\right)}{L}\,,$
$\displaystyle\omega^{r}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)^{2}\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{27}\right]^{3/4}\frac{\omega_{1}\left(\cos(\theta)\left(L^{2}+r^{2}+t^{2}\right)+2rt\omega_{2}\right)}{L}\,,$
$\displaystyle\omega^{\theta}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{LT_{0}}\right)^{2}\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{27}\right]^{3/4}\frac{\omega_{1}\sin(\theta)\left(L^{2}-r^{2}+t^{2}\right)}{Lr}\,,$
$\displaystyle\omega^{\phi}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)^{2}\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{27}\right]^{3/4}2\omega_{2}\,,$

while transformation of (16) gives rise to

$\displaystyle h_{f}^{t}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{1}\left(\omega_{2}\left(L^{2}+r^{2}+t^{2}\right)+2rt\cos(\theta)\right)}{L}\,,$
$\displaystyle h_{f}^{r}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{1}\left(\cos(\theta)\left(L^{2}+r^{2}+t^{2}\right)+2rt\omega_{2}\right)}{L}\,,$
$\displaystyle h_{f}^{\theta}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{1}\sin(\theta)\left(L^{2}-r^{2}+t^{2}\right)}{Lr}\,,$
$\displaystyle h_{f}^{\phi}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}2\omega_{2}\,.$

###  3.2 Milne coordinates

The results of Sec. 3.1 can be transformed into Milne coordinates that are
related to the spherical coordinates of (20) by

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\sqrt{t^{2}-r^{2}\sin^{2}\theta\sin^{2}\phi}\,,\quad\eta=\frac{1}{2}\log\frac{t+r\sin\theta\sin\phi}{t-r\sin\theta\sin\phi}\,,$
$\displaystyle x$ $\displaystyle=$ $\displaystyle r\cos\theta\,,\qquad
y=r\sin\theta\cos\phi\,.$ (27)

Using above equations the energy density in Milne coordinates is found to be

$\displaystyle\epsilon_{\mathbb{R}^{3,1}}$ $\displaystyle=$
$\displaystyle\frac{16\gamma^{4}L^{8}T_{0}^{4}}{\left(L^{4}+2L^{2}\left(\tau^{2}\cosh(2\eta)+{\bf
x}_{\perp}^{2}\right)+\left(\tau^{2}-{\bf
x}_{\perp}^{2}\right)^{2}\right)^{2}}\,,$ (28)

in which ${\bf x}_{\perp}^{2}=x^{2}+y^{2}$. Once again we emphasize that we
use $\gamma$ as a shorthand notation for the Lorentz factor in the
$S^{3}\times\mathbb{R}$ spacetime transformed like a scalar. The explicit form
of the above formula is presented in bir . The components of four-velocity are
given by

$\displaystyle u^{\tau}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{4}}\right]^{1/4}\left[(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2})\cosh\eta+2(\tau\omega_{2}x-L\omega_{1}x_{2}\sinh\eta)\right]\,,$
$\displaystyle u^{x}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{4}}\right]^{1/4}\left[\omega_{2}\left(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2}-2y^{2}\right)+2\tau x\cosh(\eta)\right]\,,$ $\displaystyle
u^{y}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{4}}\right]^{1/4}16\left[-L\tau\omega_{1}\sinh(\eta)+y(x\omega_{2}+\tau\cosh\eta)\right]\,,$
$\displaystyle u^{\eta}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{LT_{0}}\right)\left[\frac{\epsilon_{\mathbb{R}^{3,1}}}{16L^{4}}\right]^{1/4}\left[\frac{\sinh(\eta)\left(L^{2}-\tau^{2}+{\bf
x}_{\perp}^{2}\right)-2Ly\omega_{1}\cosh(\eta)}{\tau}\right]\,.$

Finally, we find the fluid helicity in Milne coordinates

$\displaystyle h_{f}^{\tau}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{1}\omega_{2}\cosh(\eta)\left(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2}\right)-2Ly\omega_{2}\sinh(\eta)+2\tau x\omega_{1}}{L}\,,$
$\displaystyle h_{f}^{x}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{1}\left(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2}+2\tau x\omega_{2}\cosh(\eta)-2y^{2}\right)}{L}\,,$
$\displaystyle h_{f}^{y}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\left[\frac{2y\omega_{1}(\tau\omega_{2}\cosh(\eta)+x)}{L}-2\tau\omega_{2}\sinh(\eta)\right]\,,$
$\displaystyle h_{f}^{\eta}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{\mathbb{R}^{3,1}}}{(LT_{0})^{2}\sqrt{3}}\frac{\omega_{2}\left(\omega_{1}\sinh(\eta)\left(L^{2}-\tau^{2}+{\bf
x}_{\perp}^{2}\right)-2Ly\cosh(\eta)\right)}{L\tau}\,.$

##  4 Ideal MHD on the boundary

At this stage, we assume that the fluid on the $S^{3}\times\mathbb{R}$
boundary is coupled with electromagnetic fields and therefore its mutual
evolution with the aforementioned fields is governed by the equation of
relativistic magnetohydrodynamics (RMHD) beckenstein1978 . The electromagnetic
degrees of freedom are encoded in the field tensor, i.e. $F_{ab}$ that
satisfies the homogenous Maxwell equations

$\partial_{a}F_{bc}+\partial_{b}F_{ca}+\partial_{c}F_{ab}=0.$ (1)

Like any other quantity in the system, $F_{ab}$ must respect the symmetries
expressed by (9), namely beckenstein1978 ; cmhd

$\partial_{0}F_{ab}=0\,,\qquad\partial_{2}F_{ab}=0\,,\qquad\partial_{3}F_{ab}=0\,.$

Using above relations in (1) gives rise to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{2}F_{01}+\partial_{1}F_{20,1}+\partial_{0}F_{12}\implies\partial_{1}F_{20}=0\implies
F_{20}=\text{const.}\,,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{3}F_{01}+\partial_{1}F_{30}+\partial_{0}F_{13}\implies\partial_{1}F_{30}=0\implies
F_{30}=\text{const.}\,,$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{3}F_{12}+\partial_{2}F_{31,2}+\partial_{1}F_{23}\implies\partial_{1}F_{23}=0\implies
F_{23}=\text{const.}$

If the gauge potential $\mathbf{A}$ obeys the same spacetime symmetries as
other objects, then the above constants must be zero, because none of the
corresponding components contain a $\theta^{\prime}$-derivative. For example

$F_{20}=\partial_{\phi^{\prime}}A_{t^{\prime}}-\partial_{t^{\prime}}A_{\phi^{\prime}}=0\,.$

Therefore

$F_{03}=0\,,\qquad F_{23}=0\,,\qquad F_{02}=0\,.$

Up to now, we have only utilized the symmetry arguments. To go further, we
assume that the fluid is highly conductive and employ the ideal MHD limit
beckenstein1978 ; cmhd , i.e. $u^{a}F_{ab}=0$, that leads to

$F_{0a}=-\omega_{1}F_{2a}-\omega_{2}F_{3a}\,.$

In particular

$F_{01}=\omega_{1}F_{12}-\omega_{2}F_{31}\,.$

For simplicity we define

$F_{12}=2LT_{0}\Sigma_{0}f(\theta^{\prime})\,,\qquad
F_{31}=2LT_{0}\Sigma_{0}T_{0}^{3}h(\theta^{\prime})\,,$ (2)

in which $\Sigma_{0}$ is an integration constant. The field tensor is now
determined up to two unknown functions

$F_{ab}=2LT_{0}\begin{bmatrix}0&\omega_{1}f(\theta^{\prime})-\omega_{2}h(\theta^{\prime})\hskip
10.00002pt&\hskip 10.00002pt0&0\\\
-\omega_{1}f(\theta^{\prime})+\omega_{2}h(\theta^{\prime})&0&f(\theta^{\prime})\hskip
10.00002pt&-h(\theta^{\prime})\\\ 0&-f(\theta^{\prime})&0&0\\\
0&h(\theta^{\prime})&0&0\end{bmatrix}\,.$ (3)

###  4.1 Force-free condition

Since we assumed the ideal MHD limit, the inhomogeneous Maxwell equations
cannot help us in finding a concrete solution. Therefore the only remaining
set of equations is the energy-momentum conservation that in the presence of
electromagnetic fields is modified as cmhd

$\mathrm{\nabla}_{a}{T^{ab}_{\text{fluid}}}=F^{bc}J_{c}\,.$ (4)

As in relativistic hydrodynamics Gourgoulhon:2006bn , the above equation can
be decomposed into longitudinal and transverse parts with respect to the fluid
velocity. In the longitudinal direction, the RHS vanishes and thus the
electromagnetic degrees of freedom decouple from the energy equation RPRR-
Bjorken-iMHD ; cmhd . To obtain a solution from the transverse direction of
(4), we make the following assumptions

  1. 1.

The fluid kinematics, namely its velocity and acceleration, is not disturbed
by the electromagnetic fields.

  2. 2.

If the fluid is supposed to be highly conductive, then the local charge
density given by $\rho_{e}=-u.J$ is relaxed quickly and the fluid is neutral.

Assumption 1, i.e. the force-free condition, requires that
$\Delta_{cb}\mathrm{\nabla}_{a}{T^{ab}_{\text{fluid}}}=0$, and therefore

$\Delta_{cb}F^{bc}J_{c}=0\,.$ (5)

In the iMHD limit, i.e. $u_{a}F^{ab}=0$, we have beckenstein1978

$F^{ab}=\epsilon^{abcd}B_{c}u_{d}\,,$ (6)

in which $B_{c}$ is the called the magnetic four-vector. Plugging above in (5)
gives rise to

$f^{a}\equiv\epsilon^{abcd}u_{b}J_{c}B_{d}=0\,,$ (7)

which is the covariant counter-part of $\bm{J}\times\bm{B}=0$. The magnetic
four-vector is found from $B^{a}=\frac{1}{2}\epsilon^{abcd}u_{b}F_{cd}$ as

$\displaystyle B^{0}$ $\displaystyle=$
$\displaystyle-\epsilon^{0123}\left(g_{\phi^{\prime}\phi^{\prime}}u^{2}F_{13}-g_{\chi^{\prime}\chi^{\prime}}u^{3}F_{12}\right)=-\frac{\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left(\omega_{1}\sin^{2}\theta^{\prime}F_{13}-\omega_{2}\cos^{2}\theta^{\prime}F_{12}\right)\,,$
$\displaystyle B^{1}$ $\displaystyle=$
$\displaystyle-\epsilon^{0123}\left(g_{t^{\prime}t^{\prime}}u^{0}F_{23}-g_{\phi^{\prime}\phi^{\prime}}u^{2}F_{03}+g_{\chi^{\prime}\chi^{\prime}}u^{3}F_{02}\right)=0\,,$
$\displaystyle B^{2}$ $\displaystyle=$
$\displaystyle\epsilon^{0123}\left(g_{t^{\prime}t^{\prime}}u^{0}F_{13}+g_{\chi^{\prime}\chi^{\prime}}u^{3}F_{01}\right)=\frac{\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left(-F_{13}+\omega_{2}\cos^{2}\theta^{\prime}F_{01}\right)\,,$
$\displaystyle B^{3}$ $\displaystyle=$
$\displaystyle-\epsilon^{0123}\left(g_{t^{\prime}t^{\prime}}u^{0}F_{12}+g_{\phi^{\prime}\phi^{\prime}}u^{2}F_{01}\right)=-\frac{\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left(-F_{12}+\omega_{1}\sin^{2}\theta^{\prime}F_{01}\right)\,.$

Putting (3) in above gives rise to

$\displaystyle B^{0}$ $\displaystyle=$
$\displaystyle\frac{2T_{0}L\Sigma_{0}\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left(\omega_{2}\cos^{2}\theta^{\prime}f(\theta^{\prime})+\omega_{1}\sin^{2}\theta^{\prime}h(\theta^{\prime})\right)\,,$
$\displaystyle B^{1}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
B^{2}$ $\displaystyle=$
$\displaystyle\frac{2T_{0}L\Sigma_{0}\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left[h(\theta^{\prime})\left(1-\omega_{2}^{2}\cos^{2}\theta^{\prime}\right)+\omega_{1}\omega_{2}\cos^{2}\theta^{\prime}f(\theta^{\prime})\right]\,,$
$\displaystyle B^{3}$ $\displaystyle=$
$\displaystyle\frac{2T_{0}L\Sigma_{0}\gamma}{L^{3}\sin\theta^{\prime}\cos\theta^{\prime}}\left[f(\theta^{\prime})\left(1-\omega_{1}^{2}\sin^{2}\theta^{\prime}\right)+\omega_{1}\omega_{2}\sin^{2}\theta^{\prime}h(\theta^{\prime})\right]\,.$
(8)

Generally, the inhomogeneous Maxwell equations, i.e.

$\displaystyle\mathrm{\nabla}_{b}{F^{ab}}=J^{a}\,,$ (9)

are employed to obtain the field tensor. However, in the iMHD limit, the
current is ambiguous and one cannot solve the aforementioned equations to
determine the field tensor. On the other hand, for a given field tensor,
inhomogeneous equations can be utilized to find the current. In the present
case, we have

$J^{a}=\frac{1}{\sqrt{-g}}\partial_{b}\left(F^{ab}\sqrt{-g}\right)=-\left(\tan\theta^{\prime}-\cot\theta^{\prime}\right)F^{a1}+\frac{\partial}{\partial\theta^{\prime}}\left(F^{a1}\right)\,.$
(10)

Here (9) is used. Raising the indices of (3) leads to

$F^{ab}=\frac{2T_{0}\Sigma_{0}}{L^{3}}\begin{bmatrix}0&-\omega_{1}f(\theta^{\prime})+\omega_{2}h(\theta^{\prime})\hskip
10.00002pt&0\hskip 10.00002pt&0\\\
\omega_{1}f(\theta^{\prime})-\omega_{2}h(\theta^{\prime})&0&f(\theta^{\prime})\csc^{2}\theta^{\prime}\hskip
10.00002pt&-h(\theta^{\prime})\sec^{2}\theta^{\prime}\\\
0&-f(\theta^{\prime})\csc^{2}\theta^{\prime}&0&0\\\
0&h(\theta^{\prime})\sec^{2}\theta^{\prime}&0&0\end{bmatrix}\,.$

Plugging above in (10) we find

$\displaystyle J^{0}$ $\displaystyle=$
$\displaystyle\frac{2T_{0}\Sigma_{0}}{L^{3}}\left[\left(\tan\theta^{\prime}-\cot\theta^{\prime}\right)\left(\omega_{1}f(\theta^{\prime})-\omega_{2}h(\theta^{\prime})\right)-\frac{\mathop{{d}}}{\mathop{\mathrm{d}\theta}^{\prime}}\left(\omega_{1}f(\theta^{\prime})-\omega_{2}h(\theta^{\prime})\right)\right]\,,$
$\displaystyle J^{1}$ $\displaystyle=$ $\displaystyle 0\,,$ $\displaystyle
J^{2}$ $\displaystyle=$
$\displaystyle\frac{2T_{0}\Sigma_{0}}{L^{3}}\left[\left(\tan\theta^{\prime}-\cot\theta^{\prime}\right)\frac{f(\theta^{\prime})}{\sin^{2}\theta^{\prime}}-\frac{\mathop{{d}}}{\mathop{\mathrm{d}\theta}^{\prime}}\left(\frac{f(\theta^{\prime})}{\sin^{2}\theta^{\prime}}\right)\right]\,,$
$\displaystyle J^{3}$ $\displaystyle=$
$\displaystyle-\frac{2T_{0}\Sigma_{0}}{L^{3}}\left[\left(\tan\theta^{\prime}-\cot\theta^{\prime}\right)\frac{h(\theta^{\prime})}{\cos^{2}\theta^{\prime}}-\frac{\mathop{{d}}}{\mathop{\mathrm{d}\theta}^{\prime}}\left(\frac{h(\theta^{\prime})}{\cos^{2}\theta^{\prime}}\right)\right]\,.$
(11)

With a bit of algebra, the following relation is obtained from (4.1)

$J^{0}=\omega_{1}\sin^{2}\theta^{\prime}J^{2}+\omega_{2}\cos^{2}\theta^{\prime}J^{3}-2\frac{2T_{0}\Sigma_{0}}{L^{3}}\left[\omega_{1}\cot\theta^{\prime}f(\theta^{\prime})+\omega_{2}\tan\theta^{\prime}h(\theta^{\prime})\right]\,.$
(12)

Using $J^{1}=0$ the components of $f^{a}$ in (7) read

$\displaystyle f_{0}$ $\displaystyle=$ $\displaystyle\sqrt{-g}B^{1}\gamma
L\left(\omega_{1}J^{3}-\omega_{2}J^{2}\right)\,,$ $\displaystyle f_{1}$
$\displaystyle=$ $\displaystyle\sqrt{-g}\gamma
L\left[B^{3}\left(J^{2}-\omega_{1}J^{0}\right)-B^{2}\left(J^{3}-\omega_{2}J^{0}\right)+B^{0}\left(\omega_{1}J^{3}-\omega_{2}J^{2}\right)\right]\,,$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\sqrt{-g}\gamma
LB^{1}\left(J^{3}-\omega_{2}J^{0}\right)\sin^{2}\theta^{\prime}\,,$
$\displaystyle f_{3}$ $\displaystyle=$ $\displaystyle\sqrt{-g}\gamma
LB^{1}\left(\omega_{1}J^{0}-J^{2}\right)\cos^{2}\theta^{\prime}\,.$

Thus, the force-free condition (7) requires that

$\omega_{2}J^{2}=\omega_{1}J^{3}\,,\quad J^{3}=\omega_{2}J^{0}\,,\quad
J^{2}=\omega_{1}J^{0}\,.$ (13)

###  4.2 Current-free solution

At this stage, we employ the assumption of fluid’s neutrality. Using $J^{1}=0$
and (10) in $\rho_{e}=-u.J$, gives rise to

$\rho_{e}=\frac{\gamma}{L^{3}}J^{0}\left(1-\omega_{1}^{2}\csc^{2}\theta^{\prime}-\omega_{2}^{2}\sec^{2}\theta^{\prime}\right)\,.$

Therefore $\rho_{e}=0$ requires that $J^{0}=0$. Putting the latter result in
(13) shows that a vanishing local charge density leads to a free-current
condition, i.e. $J^{a}=0$. According to (4.1), the two azimuthal components of
the four-current vanish if

$f(\theta^{\prime})=C_{1}\tan\theta^{\prime},\qquad
h(\theta^{\prime})=C_{2}\cot\theta^{\prime}\,,$

in which $C_{1}$ and $C_{2}$ are some constants. Plugging above in $J^{0}$
from (4.1) or (12) gives rise to $C_{2}=-\omega_{1}C_{1}/\omega_{2}$. By some
rearrangement, functions $f$ and $h$ can be written as

$f(\theta^{\prime})=\beta\omega_{2}\frac{\tan\theta^{\prime}}{2}\,,\qquad
h(\theta^{\prime})=-\beta\omega_{1}\frac{\cot\theta^{\prime}}{2}\,,$ (14)

in which $\beta$ is a constant. At this stage the equations of RMHD are
completely solved on $S^{3}\times\mathbb{R}$. Plugging (14) in (4.1), we find
the magnetic four-vector

$B^{a}=\frac{T_{0}\gamma}{L^{2}}\Sigma_{0}\Big{(}-(\omega_{1}^{2}-\omega_{2}^{2}),0,-(1-\omega_{2}^{2})\omega_{1}\csc^{2}\theta^{\prime},-(1-\omega_{1}^{2})\omega_{2}\sec^{2}\theta^{\prime}\Big{)}.$
(15)

Here $\beta$ is absorbed into $\Sigma_{0}$. The magnitude of $B^{a}$ reads

$B=\frac{2T_{0}\gamma}{L\sin
2\theta^{\prime}}\Sigma_{0}\sqrt{\omega_{1}^{2}\sin^{2}\theta+\omega_{2}^{2}\cos^{2}\theta-\omega_{1}^{2}\omega_{2}^{2}}.$
(16)

As the above equation suggests, no magnetic field is produced if there is no
rotation. Also, the magnetic field blows up both at the north pole, i.e.
$\theta^{\prime}=\pi/2$ and the equator, i.e. $\theta^{\prime}=0$, if both
angular velocities are non-zero. However, each of the two angular parameters
is related to one of the aforementioned singularities. If
$\omega_{1}$($\omega_{2}$) is absent then the magnetic field becomes regular
at the pole(equator)

$\lim_{\theta^{\prime}\to
0}B\and{\omega_{1}=0}=\frac{T_{0}\Sigma_{0}}{L}\omega_{2}\,,\qquad\lim_{\theta^{\prime}\to\pi/2}B\and{\omega_{2}=0}=\frac{T_{0}\Sigma_{0}}{L}\omega_{1}\,.$

The plasma sigma cmhd , i.e. the ratio of the magnetic energy density to
fluid’s one, reads

$\sigma=\left(\frac{\Sigma_{0}(1-v^{2})}{LT_{0}\sin\theta^{\prime}\cos\theta^{\prime}}\right)^{2}\zeta\,,$
(17)

in which

$2\zeta\equiv(1-v^{2})-(1-\omega_{1}^{2})(1-\omega_{2}^{2})\,.$

Using (17), we find the Alfven speed to be gedalin1993

$v_{A}=\frac{\Sigma_{0}}{LT_{0}}\sqrt{\frac{6\zeta}{\left(\gamma^{2}\sin(2\theta^{\prime})\right)^{2}+6\zeta\left(\Sigma_{0}/(LT_{0})\right)^{2}}}\,.$
(18)

Both at the equator and at the north pole, the Aflven speed $v_{A}$ tends to
unity, which is twice smaller than required by causality gedalin1993 .
Although the causality is guaranteed, one may argue that such a large Aflven
speed is a sign that the force-free condition is a poor approximation close to
the pole and the equator. For angles other than $0$ and $\pi/2$ the force-free
condition is consistent, i.e. $v_{a}\ll c_{s}$, if

$\Sigma_{0}\ll LT_{0}\,.$

One should bear in mind that the above relation is sufficient but not
necessary. For $\Sigma_{0}\ll LT_{0}$, the Alfven speed can be written as

$\displaystyle
v_{A}=\frac{\Sigma_{0}}{LT_{0}}\frac{\sqrt{6\zeta}}{\gamma^{2}\sin(2\theta^{\prime})}+\mathcal{O}{\left(\frac{\Sigma_{0}^{3}}{L^{3}T_{0}^{3}}\right)}\,.$
(19)

The electromagnetic part of the energy-momentum tensor is beckenstein1978

$T^{ab}_{\mbox{\tiny{EM}}}=F^{a}_{~{}c}F^{bc}-\frac{1}{4}g^{ab}F_{cd}F^{cd}\,.$

Plugging the solution (14) in (3) and using the result in above gives rise to

$T^{ab}_{em}=\frac{4T_{0}^{2}\Sigma_{0}^{2}}{L^{4}\sin^{2}2\theta^{\prime}}\left(\begin{array}[]{cccc}\zeta+\omega_{1}^{2}\omega_{2}^{2}&0&\omega_{1}\omega_{2}^{2}&\omega_{1}^{2}\omega_{2}\\\
0&\zeta&0&0\\\
\omega_{1}\omega_{2}^{2}&0&\omega_{2}^{2}-\zeta\csc^{2}(\theta^{\prime})&\omega_{1}\omega_{2}\\\
\omega_{1}^{2}\omega_{2}&0&\omega_{1}\omega_{2}&\omega_{1}^{2}-\zeta\sec^{2}(\theta^{\prime})\\\
\end{array}\right)\,.$

One can check that the divergence of the above energy-momentum tensor vanishes
and therefore our solution satisfies all equations of motion.

###  4.3 Gauge potential and magnetic helicity

We close this section by computing the magnetic helicity which its covariant
definition is Markakis:2016udr

$h_{em}^{a}={{}^{\star}{F^{ab}}}A_{b}\,.$

Here $A_{b}$ is the gauge potential and ${{}^{\star}{F^{ab}}}$ is the dual of
the field tensor. To find $A_{b}$ we need to solve the equation
$\mathbf{F}=\operatorname{\mathbf{d}}\\!\mathbf{A}$. As a subsequent of the
symmetries represented in (9), the $\theta^{\prime}$ component of the gauge
potential is arbitrary and using the gauge freedom we assume it to be zero.
Thus the gauge one-form reads

$\mathbf{A}=A_{0}(\theta^{\prime})\operatorname{\mathbf{d}}\\!{t^{\prime}}+A_{2}(\theta^{\prime})\operatorname{\mathbf{d}}\\!{\phi^{\prime}}+A_{3}(\theta^{\prime})\operatorname{\mathbf{d}}\\!{\chi^{\prime}}\,.$

Plugging above in $\mathbf{F}=\operatorname{\mathbf{d}}\\!\mathbf{A}$ and
using (3) gives rise to

$\mathbf{A}=LT_{0}\Sigma_{0}\left(\omega_{1}\omega_{2}\log\cot\theta^{\prime}\operatorname{\mathbf{d}}\\!{t^{\prime}}-\omega_{2}\log\cos\theta^{\prime}\operatorname{\mathbf{d}}\\!{\phi^{\prime}}+\omega_{1}\log\sin\theta^{\prime}\operatorname{\mathbf{d}}\\!{\chi^{\prime}}\right)\,.$

To find ${\star{\mathbf{F}}}$ we use
${\star{\mathbf{F}}}=-\mathbf{u}\wedge\mathbf{B}$, which is the Hodge star
dual of (6) in the language of differential forms. The result is

${\star{\mathbf{F}}}=-LT_{0}\Sigma_{0}\left(\omega_{1}\operatorname{\mathbf{d}}\\!t^{\prime}\wedge\operatorname{\mathbf{d}}\\!\phi^{\prime}-\omega_{2}\operatorname{\mathbf{d}}\\!t^{\prime}\wedge\operatorname{\mathbf{d}}\\!\chi^{\prime}+\omega_{1}\omega_{2}\operatorname{\mathbf{d}}\\!\phi^{\prime}\wedge\operatorname{\mathbf{d}}\\!\chi^{\prime}\right)\,.$
(20)

Contracting $\mathbf{A}$ with above we find the magnetic helicity to be

${h^{a}}_{em}=\frac{T0^{2}\omega_{1}^{2}\omega_{2}^{2}}{L^{2}}\Sigma_{0}^{2}\left(\frac{\log\cos\theta^{\prime}}{\sin^{2}\theta^{\prime}}+\frac{\log\sin\theta^{\prime}}{\cos^{2}\theta^{\prime}}\right)\left(1,0,\omega_{1},\omega_{2}\right)\,,$

or

$\mathbf{h}=\frac{T0^{3}\omega_{1}^{2}\omega_{2}^{2}}{L}\Sigma_{0}^{2}\left(\frac{\log\cos\theta^{\prime}}{\sin^{2}\theta^{\prime}}+\frac{\log\sin\theta^{\prime}}{\cos^{2}\theta^{\prime}}\right)\,\epsilon^{-1/4}\mathbf{u}\,.$
(21)

As anticipated the magnetic helicity is conserved, i.e. $\nabla.h_{em}=0$.
Also, a non-vanishing magnetic helicity requires both angular parameters to be
non-zero.

##  5 Passing to flat spacetime

Now we are in a situation to find the solution to the equations of RMHD in the
flat spacetime. The procedure is similar to Sec. 3 and is previously examined
in cmhd . As in Sec. 3 and bir , the first step is to perform a combination of
a Weyl and a coordinate transformation to the spherical coordinates given in
(21). The magnetic differential form has a conformal weight of $1$ cmhd  thus

${B_{\mu}}^{\\!\\!\\!\\!\mathbb{R}^{3,1}}\operatorname{\mathbf{d}}\\!{x^{\mu}}=\frac{1}{W}{B_{a}}^{\\!\\!\\!\\!S^{3}\times\mathbb{R}}\operatorname{\mathbf{d}}\\!{x^{a}}\,.$

Using (15) and (24) the above relation is transformed into

${B_{\mu}}^{\\!\\!\\!\\!\mathbb{R}^{3,1}}\operatorname{\mathbf{d}}\\!{x^{\mu}}=\epsilon_{\mathbb{R}^{3,1}}^{1/4}\,\Sigma_{0}\left((\omega_{1}^{2}-\omega_{2}^{2})\operatorname{\mathbf{d}}\\!t^{\prime}-\omega_{1}(1-\omega_{2}^{2})\operatorname{\mathbf{d}}\\!\phi^{\prime}+\omega_{2}(1-\omega_{1}^{2})\operatorname{\mathbf{d}}\\!\chi^{\prime}\right)\,.$

We then use the coordinate transformations of (3.1) in above and solve the
resulting equation to obtain $B^{\mu}$. This process gives rise to

$\displaystyle B^{t}$ $\displaystyle=$
$\displaystyle-\Sigma_{0}\,\left[\frac{\epsilon}{L^{4}}\right]^{1/4}\left[\frac{\left(\omega_{1}^{2}-\omega_{2}^{2}\right)\left(L^{2}+r^{2}+t^{2}\right)}{2L^{2}W^{2}}-\frac{rt\left(1-\omega_{1}^{2}\right)\omega_{2}\cos\theta}{L^{2}W^{2}-r^{2}\sin^{2}(\theta)}\right]\,,$
$\displaystyle B^{r}$ $\displaystyle=$
$\displaystyle\Sigma_{0}\,\left[\frac{\epsilon}{L^{4}}\right]^{1/4}\left[\frac{\left(1-\omega_{1}^{2}\right)\omega_{2}\cos\theta\left(L^{2}+r^{2}+t^{2}\right)}{2\left(L^{2}W^{2}-r^{2}\sin^{2}\theta\right)}-\frac{2rt\left(\omega_{1}^{2}-\omega_{2}^{2}\right)}{2L^{2}W^{2}}\right]\,,$
$\displaystyle B^{\theta}$ $\displaystyle=$
$\displaystyle-\Sigma_{0}\,\left[\frac{\epsilon}{L^{4}}\right]^{1/4}\left[\frac{\left(1-\omega_{1}^{2}\right)\omega_{2}\left(L^{2}-r^{2}+t^{2}\right)}{L^{2}W^{2}-r^{2}\sin^{2}\theta}\right]\frac{\sin(\theta)}{2r}\,,$
$\displaystyle B^{\phi}$ $\displaystyle=$
$\displaystyle-\Sigma_{0}\,\left[\frac{\epsilon}{L^{4}}\right]^{1/4}\omega_{1}\left(1-\omega_{2}^{2}\right)\frac{L}{r\sin\theta}.$

The magnitude of $B$ is also obtained to be

$\displaystyle B$ $\displaystyle=$
$\displaystyle\Sigma_{0}\,\left(\frac{\epsilon^{1/4}}{LW}\right)\Bigg{[}L^{2}W^{2}\left(\frac{\omega_{1}^{2}(1-\omega_{2}^{2})}{r^{2}\sin^{2}\theta}+\frac{\omega_{2}^{2}(1-\omega_{1}^{2})}{L^{2}W^{2}-r^{2}\sin^{2}\theta}\right)-(\omega_{1}^{2}-\omega_{2}^{2})\Bigg{]}.$

At this stage, we transform the results to the Milne coordinates using (3.2).
Applying the ordinary coordinate transformation Zee-GR-Book , the magnetic
four-vector in the Milne coordinates is found to be

$\displaystyle B^{\tau}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{L}\Bigg{[}\frac{y\omega_{1}\left(1-\omega_{2}^{2}\right)L\sinh\eta}{\tau^{2}\sinh^{2}\eta+y^{2}}+\frac{x\tau\left(1-\omega_{1}^{2}\right)\omega_{2}}{L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}}-\frac{\cosh(\eta)\left(\omega_{1}^{2}-\omega_{2}^{2}\right)\left(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2}\right)}{2L^{2}W^{2}}\Bigg{]}\epsilon^{1/4}\,,$ $\displaystyle
B^{x}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{L}\Bigg{[}\frac{\left(1-\omega_{1}^{2}\right)\omega_{2}\left(L^{2}+\tau^{2}+x^{2}-y^{2}\right)}{2\left(L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}\right)}-\frac{x\tau\cosh\eta\left(\omega_{1}^{2}-\omega_{2}^{2}\right)}{L^{2}W^{2}}\Bigg{]}\epsilon^{1/4}\,,$
$\displaystyle B^{y}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{L}\Bigg{[}\frac{xy\left(1-\omega_{1}^{2}\right)\omega_{2}}{L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}}-\frac{y\tau\cosh\eta\left(\omega_{1}^{2}-\omega_{2}^{2}\right)}{L^{2}W^{2}}+\frac{L\tau\omega_{1}\left(1-\omega_{2}^{2}\right)\sinh\eta}{\tau^{2}\sinh^{2}\eta+y^{2}}\Bigg{]}\epsilon^{1/4}\,,$
$\displaystyle B^{\eta}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{L\tau}\Bigg{[}\frac{\sinh(\eta)\left(\omega_{1}^{2}-\omega_{2}^{2}\right)\left(L^{2}-\tau^{2}+{\bf
x}_{\perp}^{2}\right)}{2L^{2}W^{2}}-\frac{Ly\omega_{1}\left(1-\omega_{2}^{2}\right)\cosh(\eta)}{\tau^{2}\sinh^{2}(\eta)+y^{2}}\Bigg{]}\epsilon^{1/4}\,.$

Here $W$ and $\epsilon$ are supposed to be expressed in Milne coordinates. One
can verify that the above results satisfy all equations of motion. The
behavior of the solution at $\bm{x}=0$, i.e. $x=0\,,y=0\,,$ and $\eta=0$ is as
follows: The $\tau$ and $x$ components are nonzero and finite even at
$\tau=0$. The first component vanishes if $\omega_{1}=\omega_{2}$ and the
second if $\omega_{2}=0$. The $y$ component is zero at the aforementioned
point no matter what is the value of $\tau$. The $\eta$ component is not
regular at $y=0$ and behaves as

$\tau B^{\eta}\propto\frac{1}{y/L}\,.$

This singular behavior is translated to $B$ and is related to the singularity
discussed in Sec. 4. The magnitude of the magnetic field in Milne coordinates
is found to be

$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{L}\Bigg{[}\left(\frac{x\tau\cosh\eta\left(\omega_{1}^{2}-\omega_{2}^{2}\right)}{L^{2}W^{2}}-\frac{\left(1-\omega_{1}^{2}\right)\omega_{2}}{2}\frac{L^{2}+\tau^{2}+x^{2}-y^{2}}{L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}}\right)^{2}$
$\displaystyle+\left(\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2}\frac{\left(L^{2}-\tau^{2}+{\bf
x}_{\perp}^{2}\right)\sinh\eta}{L^{2}W^{2}}-\frac{\omega_{1}(1-\omega_{2}^{2})Ly\cosh\eta}{y^{2}+\tau^{2}\sinh^{2}\eta}\right)^{2}$
$\displaystyle+\left(\frac{\omega_{1}^{2}-\omega_{2}^{2}}{2}\frac{\left(L^{2}+\tau^{2}+{\bf
x}_{\perp}^{2}\right)\cosh\eta}{L^{2}W^{2}}-\frac{\omega_{2}(1-\omega_{1}^{2})x\tau}{L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}}-\frac{\omega_{1}(1-\omega_{2}^{2})Ly\sinh^{2}\eta}{\tau^{2}\sinh^{2}\eta+y^{2}}\right)^{2}$
$\displaystyle+\left(\frac{y\tau(\omega_{1}^{2}-\omega_{2}^{2})\cosh\eta}{L^{2}W^{2}}-\frac{\omega_{1}(1-\omega_{2}^{2})L\tau\sinh\eta}{\tau^{2}\sinh^{2}\eta+y^{2}}-\frac{\omega_{2}(1-\omega_{1}^{2})xy}{L^{2}W^{2}-\tau^{2}\sinh^{2}\eta-y^{2}}\right)^{2}\Bigg{]}^{1/2}\epsilon^{1/4}\,.$

As in Sec. 4 there are two singular points related to the two angular
parameters. If $\omega_{1}=0$, then $B$ tends to infinity at $y=L$, but is
regular at $y=0$. On the other hand for $\omega_{1}\neq 0$, the point $y=0$ is
singular, but the magnetic field at $y=L$ becomes finite if one assumes
$\omega_{2}=0$. The leading behavior of $B$ around $y/L=0$ reads

$B\and[\Big{]}{x=0\,,y\ll
L\,,\eta=0}=2LT_{0}\Sigma_{0}\frac{\sqrt{\omega_{1}^{2}(1-\omega_{2}^{2})}}{(L^{2}+\tau^{2})}\frac{1}{y/L}+\mathcal{O}{(y/L)}\,.$
(2)

We come back to the singularity problem in Sec. 6.

Before closing this section, let’s compute the magnetic helicity in the flat
spacetime. The field differential form and consequently the gauge potential
have a conformal weight of zero cmhd . Accordingly, the conformal weight of
the magnetic helicity vector is found from

$\Big{[}h_{em}^{\alpha}\Big{]}=\Big{[}{{}^{\star}{F}}^{\alpha\beta}\Big{]}+\Big{[}A_{\beta}\Big{]}=-\Big{[}\sqrt{-g}\Big{]}+\Big{[}F_{\alpha\beta}\Big{]}=4\,.$

Consequently, the conformal weight of the magnetic helicity one-form is $2$.
As for the magnetic four-vector, we apply a combined Weyl and coordinate
transformation to obtain the magnetic helicity in flat spacetime parameterized
with spherical coordinates of (21). The result is

$\displaystyle h_{em}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{L\Sigma_{0}^{2}T_{0}^{3}\omega_{1}\omega_{2}}{2r^{2}W^{2}\sqrt[4]{\epsilon}\sin^{2}(\theta)}\Bigg{[}\frac{L^{2}W^{2}}{L^{2}W^{2}-r^{2}\sin^{2}\theta}\log\left(\frac{L^{2}W^{2}-r^{2}\sin^{2}(\theta)}{L^{2}W^{2}}\right)$
$\displaystyle\hskip
56.9055pt-\frac{r^{2}\sin^{2}\theta}{L^{2}W^{2}-r^{2}\sin^{2}\theta}\log\left(\frac{L^{2}W^{2}-r^{2}\sin^{2}(\theta)}{r^{2}\sin^{2}(\theta)}\right)\Bigg{]}u^{\mu}\,.$

By transforming above we find the magnetic helicity in Milne coordinates to be

$\displaystyle h_{em}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{L\Sigma_{0}^{2}T_{0}^{3}\omega_{1}\omega_{2}}{2W^{2}\sqrt[4]{\epsilon}\left[L^{2}W^{2}-(\tau^{2}\sinh^{2}\eta+y^{2})\right]}\Bigg{[}\frac{L^{2}W^{2}}{\tau^{2}\sinh^{2}\eta+y^{2}}\log\left(\frac{L^{2}W^{2}-(\tau^{2}\sinh^{2}\eta+y^{2})}{L^{2}W^{2}}\right)$
$\displaystyle\hskip
113.81102pt-\log\left(\frac{L^{2}W^{2}-(\tau^{2}\sinh^{2}\eta+y^{2})}{\tau^{2}\sinh^{2}\eta+y^{2}}\right)\Bigg{]}u^{\mu}\,.$

##  6 Regulaziation with sources

In this section, we comment on the nature of singularities and provide a
physical understanding of their emergence. In Sec. 4, we made two assumptions
in addition to the iMHD limit approximation, i.e. the force-free condition and
the assumption of neutrality. These two assumptions led to a current-free
solution presented in (14). Although it is physically possible for an
infinitely conductive fluid to freeze an initial magnetic field within itself,
a vanishing electrical current is a source for singularities. This can be
revealed by a closer look at (2) in which the singular leading term is similar
to the magnetic field produced by a point-like wire in the z-direction.
Therefore the singularity may be regularized if we put point-like sources of
current at the equator and pole of the $S^{3}\times\mathbb{R}$ spacetime.
Equivalently we can start by changing functions $f$ and $h$ in Eq. (2)

$f(\theta^{\prime})=f_{0}(\theta^{\prime})+\delta
f(\theta^{\prime})\,,h(\theta^{\prime})=h_{0}(\theta^{\prime})+\delta
h(\theta^{\prime})\,.$

Here $f_{0}$ and $h_{0}$ are the functions presented in the current free
solution of (14) with $\beta=1$. Maxwell equations do not put any constraint
on $f$ and $h$. Subsequently adding any differentiable function to $f$ and $h$
provides another solution to Maxwell equations. However, the force-free
condition is not met unless $\delta f=\delta h=0$, and one requires to solve
the energy-momentum conservation to obtain the alterations in the kinematics
and thermodynamics of the fluid. For simplicity, let’s consider that the
disturbance of the fluid by such additions is negligible and $\delta f$ and
$\delta h$ are such that the magnetic field becomes regular at the equator and
north pole of $S^{3}\times\mathbb{R}$. To realize this purpose they should be
in the following form

$\displaystyle\delta f(\theta^{\prime})$ $\displaystyle=$ $\displaystyle
f_{\lambda}(\theta^{\prime})\left(Q_{1}\sin\theta^{\prime}-f_{0}(\theta^{\prime})\right)-h_{\lambda}(\theta^{\prime})f_{0}(\theta^{\prime})\,,$
$\displaystyle\delta h(\theta^{\prime})$ $\displaystyle=$ $\displaystyle
h_{\lambda}(\theta^{\prime})\left(-Q_{2}\cos\theta^{\prime}-h_{0}(\theta^{\prime})\right)-f_{\lambda}(\theta^{\prime})h_{0}(\theta^{\prime})\,.$
(1)

Functions $f_{\lambda}$ and $h_{\lambda}$ in the above Ansatz are used to turn
off $f$ and $h$, as well as the regulators, i.e. $Q_{1}\sin\theta^{\prime}$
and $-Q_{2}\cos\theta^{\prime}$ in the equator and pole. $Q_{1}$ and $Q_{2}$
are numbers to be determined by the initial and boundary conditions. In order
for $B$ to be regular in $\theta^{\prime}=0$ and $\theta^{\prime}=\pi/2$ it is
required that

$\displaystyle f_{\lambda}(0)=1\,,\qquad\lim_{\lambda\to
0}f(\theta^{\prime}>0)=0\,,\qquad
h_{\lambda}(\pi/2)=1\,,\qquad\lim_{\lambda\to 0}f(\theta^{\prime}<\pi/2)=0\,.$
(2)

Here $\lambda$ is a small dimensionless parameter needed for guaranteeing the
above conditions. Assuming the above conditions in disturbing the current free
solution we find that the magnetic field has following finite limits at the
pole and the equator

$\lim_{\theta^{\prime}\to
0}B=\frac{2Q_{1}T_{0}\Sigma_{0}}{L}\,,\qquad\lim_{\theta^{\prime}\to\pi/2}B=\frac{2Q_{2}T_{0}\Sigma_{0}}{L}.$
(3)

However, as anticipated the regulation does not come without a price. The
current is not vanishing anymore and in particular, the following local charge
density appears in the fluid

$\delta{\rho_{e}}^{S^{3}\times\mathbb{R}}=-\frac{4T_{0}\Sigma_{0}}{L^{2}}\left[Q_{1}\omega_{1}\cos\theta^{\prime}f_{\lambda}(\theta^{\prime})-Q_{2}\omega_{2}\sin\theta^{\prime}h_{\lambda}(\theta^{\prime})\right]\gamma.$
(4)

Also the fluid’s acceleration is modified at both $\theta^{\prime}=0$ and
$\theta^{\prime}=\pi/2$. The induced charge by the regularization is small if
we carefully choose the parameters. However, the produced electromagnetic
acceleration at $\theta^{\prime}=0$ and $\theta^{\prime}=\pi/2$ might be still
large. This is because the acceleration at these points is proportional to
$f^{\prime}_{\lambda}(0)$ and $h^{\prime}_{\lambda}(\pi/2)$, which are large
by the conditions of (2). Although the regularization proposed in (6) is ad
hoc, it demonstrates the role of the electrical current in a consistent
solution to the equations of RMHD.

##  7 Numerical results

Although the aim of the present work is mainly theoretical, a numerical
investigation of the results might be fruitful. To set the stage we choose
some benchmark values for $L$ and $T_{0}$. Following Gubser-Conformal , we
first find a relation between the produced entropy and $\Lambda\equiv LT_{0}$.
For a given value of $\Lambda$ the values of $L$ and $T_{0}$ are arbitrary and
depend on the choice of units bir . We then assume the common units in heavy
ion physics to fix values of the aforementioned parameters. The physical
quantities that we use are Inghirami:2019mkc ; Gubser-Conformal

$\epsilon_{0}=55~{}/{\mathop{\rm fm}\nolimits^{3}}=275\mathop{\rm
fm}\nolimits^{-4}\,,\qquad\tau_{0}=0.4\mathop{\rm
fm/c}\nolimits\,,\qquad\frac{{d}S}{{d}\eta}\and[\Big{]}{\eta=0}=5000\,.$ (1)

If the freezeout hypersurface is at $\tau=\tau_{f}$ the entropy per unit
rapidity at mid-rapidity reads Gubser-Conformal ; bir

$\frac{{d}S}{{d}\eta}\and[\Big{]}{\eta=0}=2\pi\tau_{f}(2.43)\int_{0}^{{\bf
x}_{\perp,f}}\\!\\!\\!{\bf
x}_{\perp}\mathop{\mathrm{d}\xbot}\left(\epsilon^{3/4}u^{\tau}\right)\and[\Big{]}{\omega_{i}=0\,,\eta=0\,,\tau=\tau_{f}}$
(2)

At large values of ${\bf x}_{\perp}$ the integrand decays rapidly

${\bf
x}_{\perp}\left(\epsilon^{3/4}u^{\tau}\right)\and[\Big{]}{\omega_{i}=0\,,\eta=0\,,\tau=\tau_{f}}\sim\frac{L^{4}T_{0}^{3}}{{\bf
x}_{\perp}^{5}},\quad\text{for},\quad{\bf x}_{\perp}\gg L\,,$

while it is linear at small values of ${\bf x}_{\perp}$

${\bf
x}_{\perp}\left(\epsilon^{3/4}u^{\tau}\right)\and[\Big{]}{\omega_{i}=0\,,\eta=0\,,\tau=\tau_{f}}\sim\frac{L^{6}T_{0}^{3}}{(L^{2}+\tau^{2})^{2}}{\bf
x}_{\perp},\quad\text{for},\quad{\bf x}_{\perp}\ll L\,.$

Plugging the above approximation for the integrand in (2) and using (1) gives
rise to

$\Lambda\equiv
LT_{0}=0.51\sqrt[3]{\frac{{d}S}{{d}\eta}\and[\Big{]}{\eta=0}}\approx 8.68\,.$
(3)

If we neglect the rotation in (28), then at $\tau_{0}$ we find

$\epsilon_{0}\approx\frac{90823L^{4}}{(L^{2}+\tau_{0}^{2})^{4}}\,.$

Plugging numbers from (1) and using (3) gives rise to

$L=4.22\mathop{\rm fm}\nolimits,\qquad T_{0}=2.05~{}\,.$ (4)

The energy density with the above choice of parameters is depicted in Fig. 1
and Fig. 2.

Figure 1: (color online). Energy density in $y\\!-\\!\eta$ plane from Eq. (28)
using parameters fixed in (4) for $\omega_{1}=0.5\,,\omega_{2}=0.05$ at (a)
$\tau=0.4\mathop{\rm fm/c}\nolimits$ and (b) $\tau=5\mathop{\rm
fm/c}\nolimits$. The dashed line is the contour of $\epsilon=1~{}/\mathop{\rm
fm}\nolimits^{3}$.

Figure 2: (color online). Energy density in the transverse plane from Eq. (28)
using parameters fixed in (4) for $\omega_{1}=0.5\,,\omega_{2}=0.05$ at (a)
$\tau=0.4\mathop{\rm fm/c}\nolimits$ and (b) $\tau=5\mathop{\rm
fm/c}\nolimits$. The dashed line is the contour of $\epsilon=1~{}/\mathop{\rm
fm}\nolimits^{3}$.

The current free solution (5) possesses only one parameter, i.e. $\Sigma_{0}$.
However, due to singularities, it is difficult to choose a value for the
aforementioned parameter in a meaningful fashion. One way is to choose the
value of $B$ such that at $\left\\{\tau=5\mathop{\rm
fm/c}\nolimits\,,x=L\,,y=L\,,\eta=0\right\\}$ it satisfies the experimental
bound proposed by Muller-CME-Bound

$B\leq 1.35\times 10^{-3}\mathop{\rm fm}\nolimits^{-2}\,.$ (5)

The other way which we use, is to first regularize the solution with the
procedure of Sec. 6, fix the parameters and use them in both exact and
regularized solutions. To do so, we need to choose the functions $f_{\lambda}$
and $h_{\lambda}$ in (6). Of course, there are many choices possible for the
aforementioned functions. One appropriate choice could be

$f_{\lambda}(\theta^{\prime})=\frac{2}{1+\exp\Big{[}\cos\theta^{\prime}/\lambda\Big{]}}\,,\qquad
h_{\lambda}(\theta^{\prime})=\frac{2}{1+\exp\Big{[}\sin\theta^{\prime}/\lambda\Big{]}}\,.$
(6)

We adopt $\omega_{1}=0.5$ and $\omega_{2}=0.05$ from bir  and choose
$\lambda=\omega_{1}\omega_{2}$. The value of $Q_{1}$ must be chosen in a way
such that $\Sigma_{0}\ll LT_{0}$ is satisfied. To fix $\Sigma_{0}$ and $Q_{2}$
we then assume that the $S^{3}\times\mathbb{R}$ limits, i.e. (3) for $B$ at
the pole and equator are respectively $10^{-2}m_{\pi}^{2}/e$ and $1.35\times
10^{-3}\mathop{\rm fm}\nolimits^{-2}$. With these considerations in mind, the
following choice of parameters are suitable

$\Big{\\{}Q_{1}\,,Q_{2}\,,\Sigma_{0}\Big{\\}}=\Big{\\{}100\,,-0.60\,,0.016\Big{\\}}\,.$
(7)

Although the resulting regularized solution has finite limits at $y=0$ and
$y=L$, obtaining exact values for $B$ and $\delta\rho_{e}$ at the center of
the transverse plane is not numerically possible, and therefore we assume a
very small value of ${\xbot}_{0}=10^{-4}\mathop{\rm fm}\nolimits$ to fix a
reference point for both $B$ and $\delta\rho_{e}$. The references values for
these quantities are therefore defined as

$\displaystyle\delta\rho_{e,0}$ $\displaystyle=$
$\displaystyle\delta\rho_{e}\left(\tau=2\tau_{0},\,x={\xbot}_{0},\,y=L,\,\eta=0\right)\sim
3\times 10^{-5}\mathop{\rm fm}\nolimits^{-3},\,$ $\displaystyle eB_{0}$
$\displaystyle=$ $\displaystyle
eB\left(\tau=\tau_{0},\,x={\xbot}_{0},\,y={\xbot}_{0},\,\eta=0\right)\sim
4\times 10^{-2}~{}m_{\pi}^{2}\,.$ (8)

Now we are in a position to represent some figures for the solution. In Fig.
3, the magnetic field is depicted in the $y\\!-\\!\eta$ plane at two different
times. In both graphs, there are regions around $y=0$ and $y=L$ in which the
magnetic field is very large. As time goes on the shape of the singular
regions is modified: It is compressed in the $\eta$ direction and extended in
the $y$-direction. Although it is not clear in these pictures, the magnetic
field decays with $\tau$, $y$, and $\eta$. In Fig. 4, the same plane in the
same instances of time is drawn for the regularized solution. Although the
shape of the magnetic field profile becomes regular, the compression and
extension are similar to the exact solution of Fig. 3.

Figure 3: (color online). The magnitude of the magnetic field in
$y\\!-\\!\eta$ plane from current free solution, i.e. Eq. (5) using parameters
fixed in (7) for $\omega_{1}=0.5,\,\omega_{2}=0.05$ at (a)
$\tau=0.4\mathop{\rm fm/c}\nolimits$ and (b) $\tau=5\mathop{\rm
fm/c}\nolimits$. The value of $B_{0}$ depends on the choice of parameters, but
the overall behavior is independent. The white areas are in the neighborhood
of a singularity, and the magnetic field is much larger than other parts.

Figure 4: (color online). The magnitude of the regularized magnetic field in
$y\\!-\\!\eta$ plane using parameters fixed in (7) for
$\omega_{1}=0.5,\,\omega_{2}=0.05$ at (a) $\tau=0.4\mathop{\rm fm/c}\nolimits$
and (b) $\tau=5\mathop{\rm fm/c}\nolimits$.

Fig. 5 represents the magnetic field of the current free solution in the
$x\\!-\\!y$ plane in two instances of time. In contrast to the $y\\!-\\!\eta$
plane, the shape of the magnetic field profile is not significantly modified
by time. This is also true if we consider the regularized magnetic field as in
Fig. 6. In Fig. 7, the values of $\omega_{1}$ and $\omega_{2}$ are exchanged
and the magnetic field’s profile is rotated by $\pi/2$. To represent the
temporal evolution of the magnetic field we depict its profile in the
$t\\!-\\!z$ plane in Fig. 8 as well as a comparison with Bjorken iMHD in Fig.
9. The crucial point in Fig. 8 is that in $y=L$ the early iso-magnetic lines
resemble the Bjorken iMHD ones. However, as Fig. 9 suggests the magnetic field
in BIR flow decays much slower than the Bjorkenian case. One may infer that
the rotation enhances the magnetic field in the earlier case. The enhancement
is more clearly illustrated in Fig. 10 in which the magnetic field at
$\Big{\\{}x=y={\xbot}_{0},\,\eta=0,\,\omega_{1}=0.5\Big{\\}}$ is represented
with respect to the angular parameters. For a fixed value of
$\omega_{1}$($\omega_{2}$) the magnetic field is enhanced by increasing
$\omega_{2}$($\omega_{1}$) and the maximum is on the $\omega_{1}=\omega_{2}$
line.

Figure 5: (color online). The magnitude of the magnetic field in the
transverse plane from current free solution, i.e. Eq. (5) using parameters
fixed in (7) for $\omega_{1}=0.5,\,\omega_{2}=0.05$ at (a)
$\tau=0.4\mathop{\rm fm/c}\nolimits$, and (b) $\tau=5\mathop{\rm
fm/c}\nolimits$. The height of the white areas is decreased with time.

Figure 6: (color online). The magnitude of the regularized magnetic field in
the transverse plane using parameters fixed in (7) for
$\omega_{1}=0.5,\,\omega_{2}=0.05$ at (a) $\tau=0.4\mathop{\rm
fm/c}\nolimits$, and (b) $\tau=5\mathop{\rm fm/c}\nolimits$.

Figure 7: (color online). The magnitude of the magnetic field in the
transverse plane from current free solution, i.e. Eq. (5) using parameters
fixed in (7) for $\omega_{1}=0.05,\,\omega_{2}=0.5$ at (a)
$\tau=0.4\mathop{\rm fm/c}\nolimits$, and (b) $\tau=5\mathop{\rm
fm/c}\nolimits$. The height of the white areas is decreased with time.

Figure 8: (color online). The magnitude of the magnetic field in the
longitudinal ($t\\!-\\!z$) plane from current free solution, i.e. Eq. (5)
using parameters fixed in (7) for $\omega_{1}=0.5,\,\omega_{2}=0.05$ at (a)
$y=0$, (b) $y=L/2$, and (c) $y=L$. At $y=L$ the earliest iso-magnetic lines
resemble the Bjorken iMHD RPRR-Bjorken-iMHD ; rotating-trans-mhd .

Figure 9: (color online). A comparison of the $\tau$-evolution of the current
free solution of Eq. (5) with Bjroken iMHD, i.e
$B=B_{0}\frac{\tau_{0}}{\tau}$. The solid line in the left picture shows the
solution with
$\left\\{\omega_{1}=0,\,\omega_{2}=0.05,\,y=0,\,x=0,\,\eta=0\right\\}$ while
in the right picture it represents the solution with
$\left\\{\omega_{1}=0.5,\,\omega_{2}=0.05,\,y=L,\,x=0,\,\eta=0\right\\}$ is
slightly weaker. Figure 10: (color online). (a) Change in $B(\tau=5\mathop{\rm
fm/c}\nolimits,\,x=y=L,\,\eta=0)$ with respect to $\omega_{1}$ and
$\omega_{2}$. The maximum happens at $\omega_{1}=\omega_{2}$ line. (b)
Temporal evolution of the regularized magnetic field for different values of
$\omega_{2}$. Other parameters are
$\Big{\\{}x=y={\xbot}_{0},\,\eta=0,\,\omega_{1}=0.5\Big{\\}}$

Figure 11: (color online). Temporal evolution of $\delta\rho_{e}$ with respect
to $\delta\rho_{e,0}$ defined in (7) at (a) $y={\xbot}_{0}$, and (b) $y=L$.
Other parameters are
$\Big{\\{}x={\xbot}_{0},\,\eta=0,\,\omega_{1}=0.5,\,\omega_{2}=0.05\Big{\\}}$

Figure 12: (color online). $\delta\rho$ in the $y\\!-\\!\eta$ plane at (a)
$\tau=0$, (b) $\tau=0.4\mathop{\rm fm/c}\nolimits$, and (c) $\tau=5\mathop{\rm
fm/c}\nolimits$. The value of $\delta\rho$ is multiplied by $10^{7}$ to make
the picture more clear. Other parameters are
$\Big{\\{}x={\xbot}_{0},\,\eta=0,\,\omega_{1}=0.5,\,\omega_{2}=0.05\Big{\\}}$

Figure 13: (color online). $\delta\rho$ in the transverse plane at (a)
$\tau=0$, (b) $\tau=0.4\mathop{\rm fm/c}\nolimits$, and (c) $\tau=5\mathop{\rm
fm/c}\nolimits$. The value of $\delta\rho$ is multiplied by $10^{7}$ to make
the picture more clear. Other parameters are
$\Big{\\{}x={\xbot}_{0},\,\eta=0,\,\omega_{1}=0.5,\,\omega_{2}=0.05\Big{\\}}$

As mentioned in Sec. 6, the regularization induces a local charge density at
the boundaries of $S^{3}\times\mathbb{R}$ spacetime given in (4). It worths
that we pause and investigate some of the properties of the induced charge
density in the flat spacetime. The temporal evolution of the local charge
density close to the center of collision, i.e. $y=0$, and at $y=L$ are
depicted in Fig. 11. In both cases, the charge density relaxes as time goes
on. The re-peaking of charge density at $y=L$ is probably a subsequent of our
choice for $f_{\lambda}$ and $h_{\lambda}$. At both points, the charge is
positive which is due to the choice for $Q_{2}$. One should bear in mind that
the value of the reference point, i.e. $\delta\rho_{e,0}$, and therefore the
induced charges are extremely small. Fig. 12 is particularly interesting in
which the charge at the center is quickly squeezed in the $\eta$-direction to
become point-like while the charges at the boundaries are moving outward in
the $y$-direction. If Fig. 13 the charge density is depicted in the transverse
plane at mid-rapidity which displays a similar manner. The charges originally
located at $y=\pm L$ move outward in the $y$-direction and finally disappear.
The central charge fades and spreads through the plane.

We should also comment on the direction of the magnetic field in the current
solution. Essentially due to the singularities, there are regions of spacetime
in which the magnetic field tends to be highly longitudinal, i.e in the
z-direction. Since this is probably the artifact of a vanishing current, we
deeply doubt that such a high tendency of the magnetic field vector to the
longitudinal direction may be likely in any stage of the evolution of the
fireball.

##  8 Concluding remarks

In the current work, we presented a generalization of the BIR flow to RMHD.
For the solution to be manageable, we assumed that the fluid is highly
conductive, electrically neutral and the electromagnetic forces are negligible
in comparison to the pressure gradient. With such assumptions, we solved the
homogeneous Maxwell equation on the boundary of a Kerr-AdS5 black hole by
utilization of symmetries. Solving these equations gave rise to the
electromagnetic field tensor up to two unknown functions of a single
coordinate in the boundary. To determine the aforementioned functions we
employed two other assumptions of electrical neutrality and the force-free
condition. We showed that these assumptions require the electrical current to
be zero. This led to the singularity of the magnetic field at the equator and
pole of the boundary, which then translated into the singularities at $y=0$
and $y=L$ in the flat spacetime. By regularizing the results we explicitly
showed that these singular behaviors are consequences of the vanishing
electrical current. Our solution, albeit being highly theoretical, suggests
that the rotation of the fluid may enhance the lifetime of magnetic fields.
From a purely theoretical point of view, the present work may contribute to
the field of fluid/gravity duality and conformal (magneto)hydrodynamics.

In our point of view, further research on the evolution of the electromagnetic
fields in the QGP requires reflection concerning the simplifying assumptions
that were made in the current work and commonly in the literature. As
mentioned in Sec. 2, the iMHD limit may not be proper for the QGP. For a rough
estimation let’s consider the magnetic Reynolds number, i.e
$R_{m}=Lu\sigma_{e}$. The typical value of $Lu$ for the QGP is of order
$10\mathop{\rm fm}\nolimits$, therefore a high $R_{m}$ regime requires
$\sigma_{e}$ to be larger than $20$ which is not met by LQCD results LQCD-
Sigma . It is also suggested in rotating-trans-mhd  that the ratio of electric
to magnetic field decays as $e^{-\sigma_{e}\tau}$, and thus the LQCD results
do not support the suppression of the electric field. This may signal that the
role of the electric field on an event-by-event basis is highly ignored. It is
also assumed that the QGP is locally neutral essentially because it is highly
conductive. One may argue that the relaxation time for the charge density is
of order $1/\sigma_{e}$, and it is plausible that the local charge density
does not relax in the QGP lifetime. These two considerations may have
significant consequences for our understanding of the evolution of the
electromagnetic fields and more importantly the search for CME signals. The
force-free condition may also become a poor approximation in certain regions
of the fluid, and if that is the case a new window for possible modifications
of some observables of heavy-ion collisions might be uncovered. In particular,
if most of the strength of the magnetic field is lost in the pre-equilibrium
stage then there might be a relation between the hydrodynamization time and
potential effects of magnetic fields on the early-time observables.

We also would like to briefly comment on the BIR solution itself. At the time
of writing, this is the only solution found from the fluid/gravity conjecture
that is exact and also breaks both boost and rotational invariances assumed by
Bjorken and Gubser flows. Also in comparison with the latter flow, the BIR
solution comes from a deeper theoretical framework. However, Gubser flow has a
particular property that is not present in the BIR solution. It has a single
parameter, related to the inverse of fluid transverse size, that when tended
to zero Bjorken flow and its symmetries are reproduced. Unfortunately, the BIR
solution is not manifestly related to the geometrical properties of the
collision and the fluid and cannot reproduce Gubser or Bjorken flow. Looking
for a similar solution with the aforementioned properties may be a possible
direction of research.

###### Acknowledgements.

The author thanks H. Arfaei, H. Bantilan, A. Naseh, L. Rezzolla, N. Sadooghi
and F. Taghinavaz for fruitful discussions. He is also particularly grateful
to F. Elahi for reading the manuscript and providing useful comments.

## References

  * (1) J. Rafelski and B. Muller, Magnetic splitting of quasimolecular electronic states in strong fields, Phys. Rev. Lett. 36, 517 (1976).
  * (2) D. E. Kharzeev, L. D. McLerran and H. J. Warringa, The effects of topological charge change in Heavy Ion Collisions: ’Event by event P and CP violation’, Nucl. Phys. A 803, 227 (2008).
  * (3) V. Skokov, A. Y. Illarionov and V. Toneev, Estimate of the magnetic field strength in Heavy Ion Collisions, Int. J. Mod. Phys. A 24, 5925 (2009).
  * (4) V. Voronyuk, V. D. Toneev, W. Cassing, E. L. Bratkovskaya, V. P. Konchakovski and S. A. Voloshin, (Electro-)Magnetic field evolution in relativistic Heavy Ion Collisions, Phys. Rev. C 83, 054911 (2011).
  * (5) A. Bzdak and V. Skokov, Event-by-event fluctuations of magnetic and electric fields in Heavy Ion Collisions, Phys. Lett. B 710, 171 (2012).
  * (6) W. T. Deng and X. G. Huang, Event-by-event generation of electromagnetic fields in Heavy Ion Collisions, Phys. Rev. C 85, 044907 (2012).
  * (7) K. Fukushima, Extreme matter in electromagnetic fields and rotation, Prog. Part. Nucl. Phys. 107, 167 (2019).
  * (8) K. Hattori and X. G. Huang, Novel quantum phenomena induced by strong magnetic fields in Heavy Ion Collisions, Nucl. Sci. Tech. 28, 26 (2017).
  * (9) S. Ghosh and V. Chandra, Electromagnetic spectral function and dilepton rate in a hot magnetized QCD medium, Phys. Rev. D 98, 076006 (2018).
  * (10) A. Mukherjee, S. Ghosh, M. Mandal, P. Roy and S. Sarkar, Mass modification of hot pions in a magnetized dense medium, Phys. Rev. D 96, 016024 (2017).
  * (11) N. Sadooghi and F. Taghinavaz, Dilepton production rate in a hot and magnetized Quark Gluon Plasma, Annals Phys. 376, 218 (2017).
  * (12) K. Fukushima, K. Hattori, H. U. Yee and Y. Yin, Heavy quark diffusion in strong magnetic fields at weak coupling and implications for elliptic flow, Phys. Rev. D 93, 074028 (2016).
  * (13) X. G. Huang, Electromagnetic fields and anomalous transports in Heavy Ion Collisions — A pedagogical review, Rept. Prog. Phys. 79, 076302 (2016).
  * (14) M. Gedalin, _Linear waves in relativistic anisotropic magnetohydrodynamics_ Phys. Rev. E 47 (1993) 4354.
  * (15) D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions—A status report, Prog. Part. Nucl. Phys. 88, 1 (2016).
  * (16) J. Zhao, Search for the chiral magnetic effect in Relativistic Heavy Ion Collisions, Int. J. Mod. Phys. A 33, 1830010 (2018).
  * (17) E. Stewart and K. Tuchin, Magnetic field in expanding quark-gluon plasma, Phys. Rev. C 97, no. 4, 044906 (2018)
  * (18) K. Tuchin, Particle production in strong electromagnetic fields in relativistic heavy-ion collisions, Adv. High Energy Phys. 2013, 490495 (2013)
  * (19) J. D. Bekenstein and E .Oron, _New conservation laws in general-relativistic magnetohydrodynamics_ , Phys. Rev. D 18, 1809 (1978).
  * (20) J. D. Bjorken, Highly relativistic nucleus-nucleus collisions: The central rapidity region, Phys. Rev. D 27, 140 (1983).
  * (21) V. Roy, S. Pu, L. Rezzolla and D. Rischke, Analytic Bjorken flow in one-dimensional relativistic magnetohydrodynamics, Phys. Lett. B 750, 45 (2015).
  * (22) S. Pu, V. Roy, L. Rezzolla and D. H. Rischke, Bjorken flow in one-dimensional relativistic magnetohydrodynamics with magnetization, Phys. Rev. D 93, 074022 (2016).
  * (23) M. Shokri and N. Sadooghi, _Novel self-similar rotating solutions of nonideal transverse magnetohydrodynamics_ , Phys. Rev. D 96, 116008 (2017).
  * (24) G. Aarts, C. Allton, J. Foley, S. Hands and S. Kim, Spectral functions at small energies and the electrical conductivity in hot, quenched lattice QCD, Phys. Rev. Lett. 99, 022002 (2007). H.-T. Ding, A. Francis, O. Kaczmarek, F. Karsch, E. Laermann and W. Soeldner, Thermal dilepton rate and electrical conductivity: An analysis of vector current correlation functions in quenched lattice QCD, Phys. Rev. D 83, 034504 (2011).
  * (25) N. Sadooghi and M. Shokri, Rotating solutions of nonideal transverse Chern-Simons magnetohydrodynamics, Phys. Rev. D 98, 076011 (2018).
  * (26) G. Inghirami, L. Del Zanna, A. Beraudo, M. H. Moghaddam, F. Becattini and M. Bleicher, Numerical magneto-hydrodynamics for relativistic nuclear collisions, Eur. Phys. J. C 76, no. 12, 659 (2016).
  * (27) G. Inghirami, L. Del Zanna, A. Beraudo, M. Haddadi Moghaddam, F. Becattini and M. Bleicher, Magneto-hydrodynamic simulations of Heavy Ion Collisions with ECHO-QGP, J. Phys. Conf. Ser. 1024, no. 1, 012043 (2018).
  * (28) G. Inghirami, M. Mace, Y. Hirono, L. Del Zanna, D. E. Kharzeev and M. Bleicher, Magnetic fields in heavy ion collisions: flow and charge transport, arXiv:1908.07605 [hep-ph].
  * (29) I. Siddique, R. j. Wang, S. Pu and Q. Wang, “Anomalous magnetohydrodynamics with longitudinal boost invariance and chiral magnetic effect, Phys. Rev. D 99, no. 11, 114029 (2019).
  * (30) T. Csörgö, F. Grassi, Y. Hama and T. Kodama, Simple solutions of relativistic hydrodynamics for longitudinally and cylindrically expanding systems, Phys. Lett. B 565, 107 (2003).
  * (31) S. S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D 82, 085027 (2010).
  * (32) S. S. Gubser and A. Yarom, Conformal hydrodynamics in Minkowski and de Sitter spacetimes, Nucl. Phys. B 846, 469 (2011).
  * (33) M. Shokri and N. Sadooghi, _Evolution of magnetic fields from the 3 + 1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics_ , JHEP 1811, 181 (2018).
  * (34) S. Pu and D. L. Yang, Analytic Solutions of Transverse Magneto-hydrodynamics under Bjorken Expansion, EPJ Web Conf. 137, 13021 (2017).
  * (35) D. She, Z. F. Jiang, D. Hou and C. B. Yang, 1+1 dimensional relativistic magnetohydrodynamics with longitudinal acceleration, arXiv:1907.01250 [hep-ph].
  * (36) H. Bantilan, T. Ishii and P. Romatschke, _Holographic heavy ion collisions: Analytic solutions with longitudinal flow, elliptic flow and vorticity_ , Phys. Lett. B 785, 201 (2018).
  * (37) S. Bhattacharyya, S. Lahiri, R. Loganayagam and S. Minwalla, _Large rotating AdS black holes from fluid mechanics_ , JHEP 0809, 054 (2008).
  * (38) L. Rezzolla and O. Zanotti, “Relativistic Hydrodynamics”, Oxford University Press, Oxford UK, (2013).
  * (39) E. Gourgoulhon, An Introduction to relativistic hydrodynamics, EAS Publ. Ser. 21, 43 (2006).
  * (40) A. Zee, _Einstein gravity in a nutshell_ , Princeton University Press, Princeton, New Jersey, 2013.
  * (41) C. Markakis, K. Uryū, E. Gourgoulhon, J. P. Nicolas, N. Andersson, A. Pouri and V. Witzany, Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows, Phys. Rev. D 96, no. 6, 064019 (2017).
  * (42) F. Becattini et al., A study of vorticity formation in high energy nuclear collisions, Eur. Phys. J. C 75, no. 9, 406 (2015).
  * (43) H. Bantilan, F. Pretorius and S. S. Gubser, Simulation of Asymptotically AdS5 Spacetimes with a Generalized Harmonic Evolution Scheme, Phys. Rev. D 85, 084038 (2012).
  * (44) B. Müller and A. Schäfer, Chiral magnetic effect and an experimental bound on the late time magnetic field strength, Phys. Rev. D 98, 071902 (2018).