\section{Introduction}

In stationary spacetimes, the event horizon of a black hole is a Killing horizon and foliated by surfaces with vanishing outward null expansion:
marginally outer trapped surface (MOTS).
More generally given a Cauchy surface in any spacetime, the boundary of the trapped region is an apparent horizon which is also a MOTS.
Motivated by these facts MOTS are key to many definitions of black hole boundaries including trapping horizons\cite{Hayward:1993wb},
marginally trapped tubes\cite{Ashtekar:2005ez}, isolated and dynamical horizons\cite{Ashtekar:2004cn}, the proposed core of the trapped
region\cite{Bengtsson11} and the very recent future holographic screens\cite{Bousso:2015mqa}.

Apart from being foliated by MOTS the event horizons of the standard stationary black hole solutions (for example Kerr-Newman-deSitter)
have another property: they separate the trapped region from the untrapped region and in particular there are fully trapped surfaces
uniformly close to and  ``just inside'' the MOTS. MOTS with slight variations of this property go by many names including \emph{stable}\cite{Newman87},  \emph{outer trapping}\cite{Hayward:1993wb} or \emph{strictly stably outermost}\cite{Andersson:2005gq}.
MOTS with one of these properties and which foliate a stationary event horizon have been shown to be geometrically rigid against deformations\cite{Andersson:2005gq,Booth07}.

\begin{figure}
\includegraphics[scale=1]{RNdS_Fig}
\caption{One tile of the Penrose-Carter diagram for Reissner-Nordstr\"om-deSitter spacetime. It repeats (subject to possible identifications) across
the dotted lines. The various Killing horizons are labelled: outer and inner black hole, outer and inner white hole, and future and past cosmological
horizons. The null directions are consistently labelled as $L$ and $N$ with the signs of $\theta_{(L)}$ and $\tilde{N}$ in each region being respectively listed as $(\pm,\pm)$. $\theta_{(L)}$ and $\tilde{N}$ vanish on horizons to which they are tangent. Stable horizons are solid lines while unstable  are dashed.
Most of our discussion will focus on the shaded region containing outer and inner black hole horizons along with the past-cosmological horizon.
}
\label{RNdSHorizons}
\end{figure}

However even in stationary spacetimes stable horizons are the (admittedly very important) exception rather than the rule.
Consider, for example, Reissner-Nordstr\"om-deSitter as depicted in Figure \ref{RNdSHorizons}. Focussing on the shaded region,
the inner black hole and past cosmological horizons have trapped surfaces outside rather than inside and so are not stable\footnote{
The terminology of outside versus inside becomes ambiguous in some of these cases but we will return to clarify this in Section \ref{horizons}.
}.

Further, there is at least one case where an unstable horizon \emph{can} be smoothly deformed. Pure deSitter spacetime is homogeneous and isotropic
with a constant positive Ricci curvature $\mathcal{R} = 4\Lambda$ determined by the cosmological constant. However around any point $p$ in
the space it is possible to construct the standard static coordinate patch:
\begin{equation}
ds^2 = - \left(1-\frac{\Lambda}{3}r^2 \right) dt^2 + \frac{dr^2}{1-\frac{\Lambda}{3}r^2} + r^2 d \Omega^2
\end{equation}
from which it is straightforward to show that there is a MOTS at
\begin{equation}
r_{\mbox{\tiny{CH}}} =  l \equiv \sqrt{\frac{3}{\Lambda}} \; .
\end{equation}
Like the cosmological horizons in RNdS, it isn't  stable and it is intuitively clear that in this case we can deform the MOTS. To see this construct
an analogous MOTS around a point $p'$ infinitesimally close to $p$. This represents a deformation of the $p$-MOTS which in this case
is essentially a translation.

So at least in this case the lack of stability corresponds to a freedom to deform the MOTS. In the rest of this paper we will explore unstable
MOTS in more detail. Section \ref{background} reviews basic nomenclature and geometry along with the standard stability result.
Section \ref{InfDef} shows that only particular finely tuned unstable MOTS in RNdS can be infinitesimally deformed. Section \ref{FinDef}
examines whether these infinitesimal deformations can be made finite (and so real!). Section \ref{Discuss} summarizes and discusses our results.
Appendix \ref{triple} reviews some useful identities for Legendre polynomials that are applied in the main text.

\section{Background and general theory}
\label{background}

We begin with a very brief review of the mathematics and geometry of marginally outer trapped surfaces and their deformations.
As general references for the next two subsections, see \cite{Booth07} for more details on the geometry or \cite{Booth:2005qc} for a review of the various types MOTS and their complications.

\subsection{Spacetime and two-surface geometry}
\label{geometry}

Let $(M, g_{ab}, \nabla_a)$ be a $(3+1)$-dimensional time-oriented spacetime and $(S, \tilde{q}_{ab}, d_a)$ be a spacelike
closed and orientable two-surface  embedded in $M$.

The normal space at each $p\in S$ is two-dimensional and timelike and so can be spanned by a pair of null vectors. In particular since $M$ is
time-oriented we may define a pair of future-oriented vector fields $\ell$ and $n$ over $S$ which do this job in each normal space.
Since they are null vector fields they each have one-degree of rescaling freedom. However one of these is removed by requiring
that they be cross-scaled so that $\ell \cdot n = -1$.

Geometric consistency requires that the combined induced metric/projection operator on $S$ satisfies
\begin{equation}
\tilde{q}_{ab} = g_{ab} + \ell_a n_b + n_a \ell_b  \, .  \label{tq}
\end{equation}
This fixes the intrinsic geometry of $S$ while the extrinsic geometry comes from
tangential derivatives of the null-normals. These are the connection on the normal bundle
\begin{equation}
\tilde{\omega}_a = -\tilde{q}_a^c n_b  \nabla_c \ell^b \, .
\end{equation}
and the extrinsic curvatures
\begin{equation}
k^{(\ell)}_{ab} = \tilde{q}_a^c \tilde{q}_b^d \nabla_c \ell_d  \; \; \mbox{and} \; \; k^{(n)}_{ab} =  \tilde{q}_a^c \tilde{q}_b^d  \nabla_c n_d  \, ,
\end{equation}
which may be conveniently decomposed into their trace and trace-free parts
\begin{equation}
k^{(\ell)}_{ab} = \frac{1}{2} \theta_{(\ell)} \tilde{q}_{ab} + \sigma^{(\ell)}_{ab}  \; \; \mbox{and} \; \;  k^{(n)}_{ab} = \frac{1}{2} \theta_{(n)} \tilde{q}_{ab} + \sigma^{(n)}_{ab} \; .
\end{equation}
Respectively these are the expansions and the shears of those vector fields. The reason for these names is clear if we consider their alternate
definition as deformations (also known as variations\cite{Andersson:2005gq}).

Consider a vector field $X^a$ which is normal to $S$ and defined in a neighbourhood of the surface. Hence on $S$
\begin{equation}
X^a = A\ell^a - B n^a \label{X0}
\end{equation}
for some functions $A$ and $B$.\footnote{The negative sign for $B$ is a convention chosen in \cite{Booth07} to simplify calculations when
studying dynamical horizons. Even though they are not considered here, we retain the sign for consistency with that paper from which we draw
almost all of our formulae.} Then that function defines a flow which can be used to evolve and deform $S$.

In coordinate terms, if $x_{S}^\alpha (\theta,\phi)$ is a (local) parameterization of $S$ then infinitesimally the deformation sends:
\begin{equation}
x_{S}^\alpha (\theta,\phi)  \rightarrow x_{S}^\alpha (\theta,\phi) + \varepsilon X^\alpha(\theta,\phi) \, .
\end{equation}
The evolution also identifies points on the original and deformed surfaces along the lines of flow. Thus one may consider the
rate of change of the geometric properties of the surface under the deformation and we denote this differential operator as $\delta_X$. It
is usually referred to as either the \emph{deformation operator} or \emph{variation} with respect to $X$.

If the coordinate system is adapted to $S$ and $X^a$ so that $S$ is a level surface and $X=\partial/\partial \lambda$ a coordinate
vector field then $\delta_X$, the Lie derivative $\Lie_X$ and the partial derivative $\partial/\partial \lambda$ are all the same thing.
This equivalence is often used to simplify discussions of deformations in spherical symmetry. See, for example \cite{Hayward:1993wb}.

It is not hard to show that
\begin{equation}
\delta_X \tilde{q}_{ab} = A k^{(\ell)}_{ab} - B k^{(n)}_{ab} \label{deltaq}
\end{equation}
whence
\begin{equation}
\delta_X \tilde{\epsilon}_{ab} = ( A \theta_{(\ell)} - B \theta_{(n)} ) \tilde{\epsilon}_{ab} \,
\end{equation}
where $\tilde{\epsilon}$ is the area two-form on $S$ (in coordinate form $\tilde{\epsilon} = \sqrt{\tilde{q}} d\theta \wedge d\phi$). Thus
$\theta_{(X)}$ and $\sigma^{(X)}_{CD}$ are respectively the expansion and shear of $S$ as it is evolved by $X^a$. Note too that for these
intrinsic quantities the rates of change are independent of how $X^a$ extends off $S$.

One can also calculate variations of the extrinsic quantities,
however for our purposes we only need
\begin{align}
 \delta_X \theta_{(\ell)}
&  =  \kappa_X \theta_{(\ell)}   - d^2 B + 2 \tilde{\omega}^a d_a B  \label{deltatl} \\
& - B \left(-d_a \tilde{\omega}^a + \| \tilde{\omega} \|^2  - \tilde{K} + G_{ab} \ell^a n^b - \theta_{(\ell)} \theta_{(n)} \right) \nonumber \\
& +A \left(-\|  {\sigma^{(\ell)}} \|^2 - G_{ab} \ell^a \ell^b  - (1/2) \theta_{(\ell)}^2 \right) \, .  \nonumber
\end{align}
Newly appearing quantities are $\kappa_X = - X^a n_b \nabla_a \ell^b$,
 $\tilde{K}$  the Gaussian curvature of $S$  and
$G_{ab}$ the Einstein tensor. Further we  have abbreviated $d^2 = d^a d_a$,
$\| \tilde{\omega} \|^2 = \tilde{\omega}^a \tilde{\omega}_a$ and $\| {\sigma^{(\ell)}} \|^2 = \sigma^{(\ell) ab} \sigma^{(\ell)}_{ab}$.

Unlike (\ref{deltaq}) this variation does depend on derivatives off $S$. This is through the gauge dependent $\kappa_X$ term which
under rescalings $\ell \rightarrow e^f \ell$ and $n \rightarrow e^{-f} n$ of the null vectors transforms as
\begin{equation}
\kappa_X \rightarrow \kappa_X - \Lie_X f \, .
\end{equation}
However, as will now be seen, we are only really interested in situations where $\theta_{(\ell)}$ vanishes and so do not need to worry about this dependence.

\subsection{MOTS: definition, deformation and difficulties}
\label{horizons}

The standard classification of two-surfaces as trapped, untrapped or marginally trapped assumes that one can unambiguously assign
one of the null directions (say $\ell$) as outward pointing and the other ($n$) as inward pointing. Then a closed, spacelike two-surface $S$
is \emph{outer untrapped} if $\theta_{(\ell)}>0$, an \emph{outer trapped} if $\theta_{(\ell)} < 0$ and \emph{marginally outer trapped (MOTS)} if $\theta_{(\ell)} = 0$.
A fully \emph{trapped} surface has both $\theta_{(\ell)}<0$ and $\theta_{(n)}<0$.

A Killing horizon is null and so if it is tangent to the outgoing direction $\ell$, then any two-dimensional slice of that horizon is a MOTS.
Thus for those MOTS
\begin{equation}
\delta_\ell \theta_{(\ell)} = - \|  {\sigma^{(\ell)}} \|^2 - G_{ab} \ell^a \ell^b = 0 \, . \label{dltl}
\end{equation}
Now, intuitively the outer black hole Killing horizon of a stationary spacetime should have outer trapped surfaces ``just inside''.
In terms of deformations, the existence of such surfaces implies that for some inward-oriented spacelike normal vector field
$R = \alpha \ell - \beta n$ ($\alpha \beta < 0$):
\begin{equation}
\delta_R \theta_{(\ell)} = - d^2 \beta + 2 \tilde{\omega}^a d_a \beta  -\beta \delta_n \theta_{(\ell)}  < 0 \, , \label{drtl}
\end{equation}
where
\begin{equation}
\delta_n \theta_{(\ell)} = -d_a \tilde{\omega}^a + \| \tilde{\omega} \|^2  - \tilde{K} + G_{ab} \ell^a n^b \, . \label{dntl}
\end{equation}
The vanishing of $\delta_\ell \theta_{(\ell)}$ renders the value of $\alpha$ irrelevant. Note too that if we rescale the null vectors so that $n \rightarrow \beta n$
(and $\ell \rightarrow \ell/\beta$) this condition becomes $\delta_n \theta_{(\ell)}  < 0$.\footnote{Restricting attention spherical horizons, the correct scaling is
obvious but  for a concrete demonstration of a less trivial situation see the discussion of Kerr in appendix C of \cite{Booth07}.}

Now, a closed MOTS slice $S$ of a Killing horizon with $\delta_n \theta_{(\ell)} < 0$ is guaranteed to be geometrically stable in that it cannot be
smoothly deformed out of the Killing horizon while preserving $\theta_{(\ell)} = 0$. To see this consider variations generated by a vector field
$X$ of the form (\ref{X0}) with $B$ not everywhere vanishing (that would correspond to a variation along the Killing horizon).
Then any such MOTS-preserving variation of $S$ necessarily satisfies
\begin{equation}
\delta_X \theta_{(\ell)} = 0 \;  . \label{dXtL0}
\end{equation}
However as considered above $\delta_\ell \theta_{(\ell)} = 0$ and again $A$ is irrelevant. Thus the variation must satisfy
\begin{equation}
-d^2 B + 2 \tilde{\omega}^a d_a B - B \delta_n \theta_{(\ell)} = 0 \, .  \label{dxtleq}
\end{equation}
For $\delta_n \theta_{(\ell)} < 0$ there are no solutions to this equation and so no MOTS-preserving variation\cite{Andersson:2005gq} (this can also be seen by
a maximum principle argument \cite{Booth07}).

MOTS satisfying versions of this condition have been considered many times over the
years and among other names have been termed \emph{stable}\cite{Newman87},  \emph{outer trapping}\cite{Hayward:1993wb} or
\emph{strictly stably outermost}\cite{Andersson:2005gq}. In this paper we will generally refer to them as stable.

This set-up and labelling is all very well for outer black hole horizons in a spacetime with an unambiguous notion of ingoing and outgoing, however
in a multi-horizon spacetime like that shown in FIG.~\ref{RNdSHorizons}, outward and inward labels are not well-defined. Neither $L$ nor
$N$ is consistently outward-pointing (towards an $r=\infty$) or inward-pointing (towards an $r=0$).

While there are
systems of nomenclature that distinguish between the various types of horizons without reference to ``inner'' and ``outer''\cite{Senovilla:2007zy,Hayward:2009ji}, for this paper we will instead just abuse the name ``MOTS'' and use it to refer to any surface with one
vanishing null expansion. We will always label that  direction $\ell$ (so that $\theta_{(\ell)} = 0$) and the other future null direction $n$. Note that the
geometric stability arguments made in the paragraph surrounding (\ref{dxtleq}) continue to apply regardless of the orientation of $\ell$ and $n$.
Thus we may always test the geometric stability of a MOTS by checking for a scaling of the null vectors such that $\delta_n \theta_{(\ell)} < 0$.

Turning once again to FIG.~\ref{RNdSHorizons} we see that on some horizons $\ell = L$ while on others we will have $\ell = N$. However whatever
the labelling, the outer black and white hole horizons are stable by this measure while all cosmological and inner black hole horizons
are potentially unstable with $\delta_n \theta_{(\ell)} > 0$.

For the rest of this paper we will investigate whether this potential instability translates into finite MOTS-preserving variations.

\section{``Unstable'' MOTS in RNdS}
\label{InfDef}

In the last section we tested stability based on how the null expansions do or don't change signs across a horizon. However to understand
whether the lack of a proof of stability actually corresponds to a real instability we need more calculations and commence with finding
exact expressions for $\delta_n \theta_{(\ell)}$.

First, the RNdS metric in standard form (static for $F(r)>0$) is
\begin{equation}
ds^2 = - F(r) dt^2 + \frac{dr^2}{F(r)} dr^2 + r^2 d \Omega^2
\end{equation}
with
\begin{equation}
F(r) = -\frac{\Lambda}{3}r^2 + 1 - \frac{2m}{r} + \frac{q^2}{r^2} \, . \label{F}
\end{equation}
The horizons are located at roots of $F(r)$.
For black hole solutions like that depicted in FIG.~\ref{Typical} there is one negative, unphysical, root and three positive roots that
correspond to horizons. In increasing order these are the inner black/white hole horizons at $r_{\mbox{\tiny{IH}}}$, outer black/white hole horizons
$r_{\mbox{\tiny{OH}}}$ and future and past cosmological horizons $r_{\mbox{\tiny{CH}}}$. However not all members of this family of solutions are
cosmological black holes. FIG.~\ref{RNdSRange} shows the allowed parameter range. It was produced by examining where the discriminant of
$F$ vanishes (these are double or triple roots and so the boundaries of the ``Regular'' region).

\begin{figure}
\includegraphics[scale=0.5]{DemoRoots}
\caption{$F(r)$ for a typical cosmological black hole solution. In this case $\Lambda \approx 0.1417/m^2$ and $q^2 \approx 0.8496 m^2$. The inner black hole horizon,
outer black hole horizon and cosmological horizon are respectively labelled as IH, OH, and CH.}
\label{Typical}
\end{figure}

\begin{figure}
\includegraphics[scale=1]{RNdS_Allowed}
\caption{Phase space of RNdS spacetimes with $m \neq 0$. $\Lambda$ and $q^2$ are given in units of $m$. Solutions with three horizons
(inner black hole, outer black hole, cosmological) are found in the grey shaded region while naked solutions with a single cosmological horizon form the rest
of the phase space. The only exception is pure RN along the  $\Lambda=0$ line where for $q<m$ there are inner and outer black hole horizons
while $q>m$ is horizon-free.  Along the dashed line outer and cosmological horizons are degenerate as in the Nariai limit of SdS while the solid line
represents extremal black holes. Where the lines meet all three horizons are degenerate.    }
\label{RNdSRange}
\end{figure}

For spherically symmetric $r$=constant surfaces  and a similarly symmetric scaling of the null vectors we have $\tilde{\omega}_a = 0$.  Thus
(\ref{dntl}) becomes
\begin{equation}
\left. \delta_n \theta_{(\ell)}\right|_{\mbox{\tiny{spherical}}} = - \frac{1}{r^2} + G_{ab} \ell^a n^b \, .
\end{equation}
Note that this is invariant with respect to the scaling of the null vectors and in fact we can find it without ever defining them.
By (\ref{tq}),  $G_{ab} \ell^a n^b = \frac{1}{2} G_{ab} \left(\tilde{q}^{ab} - g^{ab} \right)$ so
\begin{equation}
\left. \delta_n \theta_{(\ell)}\right|_{\mbox{\tiny{RNdS}}} =  -  \frac{F'}{r}
\end{equation}
where the prime indicates a derivative with respect to $r$ and we have applied $F(r)=0$. With $\Lambda >0$ the asymptotic behaviour is
fixed and so the requirement that there be three positive roots means that $F(r)$ will generically take a form similar
to FIG.~\ref{Typical}. In particular it is clear that
\begin{equation}
\left. \delta_n \theta_{(\ell)} \right|_{\mbox{\tiny{OH}}} < 0 \; \; \mbox{while} \; \; \left. \delta_n \theta_{(\ell)} \right|_{\mbox{\tiny{IH,CH}}} > 0
\end{equation}
as claimed earlier.

However we now demonstrate that in (at least) the vast majority of cases the inner and cosmological horizons are also stable. To see this we consider
concrete solutions of the stability   equation  (\ref{dxtleq}).
On both horizons this becomes
\begin{equation}
 \nabla^2 B +   (r^2  \delta_n \theta_{(\ell)}) B = 0 \, ,
\end{equation}
where $\nabla^2 B$ is the regular spherical Laplace operator on a unit sphere. Thus potential deformations must satisfy
\begin{align}
& \nabla^2 B = \left( r F' \right) B  \, .  \label{SL}
\end{align}
The only non-diverging solutions of the spherical Laplace eigenvalue equation are spherical harmonics. That is if there
is an integer $l$ such that
\begin{equation}
 -r F' =  l (l+1)
\end{equation}
then (\ref{SL}) has solutions
\begin{equation}
B = P_l (\cos  \theta) (A_m \cos (m\phi) + B_m \sin (m \phi) )
\end{equation}
for integers $ 0 \leq m < l$ and constants $A_m$ and $B_m$.

We can then test for cases where these conditions might be met. First for $m=0$ the only non-naked singularity spacetime is pure deSitter. In that case it is straightforward to
see that $-rF'=2$ on the cosmological horizon and so the MOTS-translation freedom manifests itself as an $l=1$ instability.

Turning to $m \neq 0$, FIG.~\ref{Degree} shows the values $- r F'$ for all horizons in RNdS black hole
spacetimes and so  we can consider them case-by-case.
\begin{figure}
\includegraphics[scale=0.22]{DegreeFigContour.png}
\caption{Values of $r^2 \delta_n \theta_{(\ell)} = - r F'$ for cosmological (blue), outer (grey) and inner (purple) horizons. Potential
instabilities exist when $- r \frac{dF}{dr} = l (l+1)$ for some positive integer $l$. Though it is cut off in the figure, the inner horizon sheet
diverges to infinity. The domain for all horizons is as shown in FIG.~\ref{RNdSRange}.}
\label{Degree}
\end{figure}

First for the outer horizon $-1 \leq -r F' \leq 0$ and so there are no possible solutions. This is not a surprise as we have already twice
concluded that outer black and white hole horizons are stable.

Similarly simple is the inner horizon with $0 \leq -r F' < \infty$. In this case a correct choice of parameter values will allow any possible $l$.
In particular this is even possible for pure RN.

The cosmological horizon is a little more subtle. The figure shows that the only possible case is $-rF' =2 \Leftrightarrow l=1$
however this limit isn't achieved: it is along the $\Lambda = 0$ line where there isn't a cosmological horizon.
So for $m \neq 0$ there are no solutions and the cosmological horizon is stable.

Thus we have now explicitly demonstrated that while the stability condition $\delta_X \theta_{(\ell)} <0$ may be sufficient to exclude deformations it certainly isn't necessary.
Examples are the cosmological horizon in $m \neq 0$ RNdS spacetimes and (at least) all but a finely tuned set of inner
horizons.  In the next section we will further examine those special cases.

\section{Finite Deformations in RNdS}
\label{FinDef}

In this section we develop the formalism necessary to test the higher order stability of the special cases that were found to be first order unstable in the previous section.
It is easiest to do this by moving away from the general formalism of Section \ref{background} and to one  specialized to the RNdS spacetimes.
In all the cases that we check we will see that the apparent first-order instability fails at higher order.

\subsection{MOTS in Painlev\'{e}-Gullstrand coordinates}

We begin by introducing Painlev\`{e}-Gullstrand (PG) coordinates for RNdS spacetimes. Recall that time in these coordinates is measured along
a congruence of infalling timelike geodesics while the spatial slices of constant time are intrinsically flat\cite{Martel:2000rn}.
For RNdS the shaded patch shown in FIG.~\ref{RNdSHorizons} is (almost) covered by coordinates $(T,r,\theta,\phi)$ with metric:
\begin{equation}
ds^2 = - F(r) dT^2 + 2 \sqrt{1-F(r)} dT dr + dr^2 + r^2 d \Omega^2 \, ,
\end{equation}
where $F(r)$ takes its usual form (\ref{F}). The ``almost''  is included in the previous sentence because for $q \neq 0$ there will always be
a region where $1-F < 0$ and so the coordinate system is not well-defined. However, as we shall see, for example in FIG.~\ref{Fin_RN}, this will always be inside
the inner horizon and so not cause us any problems.

We will look for MOTS on hypersurfaces $\Sigma_T$ of constant $T$ and in order to do this it will be
sufficient to know the intrinsic and extrinsic geometry of $\Sigma_T$. The intrinsic geometry on $\Sigma_T$ is given by the Euclidean metric
\begin{equation}
d\Sigma^2 = h_{ab} dx^a dx^b =   dr^2 + r^2 d \Omega^2 \, ,
\end{equation}
while the extrinsic curvature is
\begin{equation}
K =  \left(  \frac{F'}{2 \sqrt{1-F}} \right)  dr^2 -  \left(  r \sqrt{1-F} \right)  d \Omega^2
\end{equation}
which was calculated from the future-oriented  unit timelike normal to  $\Sigma_T$:
\begin{equation}
\hat{u} = \left( \frac{\partial}{\partial T}\right) -  \sqrt{1-F} \left(\frac{\partial}{\partial r} \right) \, .
\end{equation}

Then consider a rotationally symmetric surface $S$ in a $\Sigma_T$ and parameterize it by coordinates
$(\lambda, \phi)$ as
\begin{equation}
(T,R,\theta,\phi) = (T_o,R(\lambda),\Theta(\lambda) , \phi) \,  ,
\end{equation}
for some functions $R(\lambda)$ and $\Theta(\lambda)$. For now we will find it convenient to take $\lambda$ to be the
 arclength parameter as measured from the north pole of $S$ along the constant $\phi$ lines of longitude.
Then the tangent vector
\begin{equation}
\frac{d}{d \lambda} = \dot{R} \left( \frac{\partial}{\partial r}  \right) + \dot{\Theta} \left( \frac{\partial}{\partial \theta} \right)
\end{equation}
is unit length
\begin{equation}
\dot{R}^2 + R^2 \dot{\Theta}^2  = 1 \, , \label{arc}
\end{equation}
where we have marked derivatives with respect to $\lambda$ with dots.

Next the induced two-metric on $S$ is
\begin{equation}
dS^2 = d \lambda^2 + (R^2 \sin^2 \! \Theta) d \phi^2 \, ,
\end{equation}
with inverse:
\begin{equation}
\tilde{q} =    \left( \frac{\partial}{\partial \lambda} \right)   \otimes \left( \frac{\partial}{\partial \lambda} \right)
+ \frac{1}{R^2 \sin^2 \! \Theta} \left( \frac{\partial}{\partial \phi} \right) \otimes  \left( \frac{\partial}{\partial \phi} \right) \nonumber \, .
\end{equation}

The positive-$r$ pointing spacelike normal to $S$ in $\Sigma_T$ is
\begin{equation}
\hat{r} = R \left( \dot{\Theta} \left( \frac{\partial}{\partial r}  \right)
 - \frac{\dot{R}}{r^2} \left( \frac{\partial}{\partial \theta}  \right)  \label{tr}
 \right) \, ,
\end{equation}
whence the trace of the extrinsic curvature of $S$ in $\Sigma_T$ is
\begin{equation}
\theta_{(\hat{r})} \equiv \tilde{q}^{ab} \nabla_a \hat{r}_b =  - \frac{\ddot{R}}{R \dot{\Theta}} + 2 \dot{\Theta} - \frac{\dot{R}}{R} \cot (\Theta) \, ,
\end{equation}
where we have used the arclength condition to somewhat simplify the expression.
Note that no $F$ appears in this expression: $\Sigma_T$ is Euclidean so any geometric calculation intrinsic to $\Sigma_T$ is independent of
$F$.

Next the trace of the extrinsic curvature of $S$ with respect to $\hat{u}$ (and so out of $\Sigma_T$) is:
\begin{equation}
\theta_{(\hat{u}) }   \equiv \tilde{q}^{ab} \nabla_a \hat{u}_b =  K_{ab} h^{ab} -  K_{ab} \hat{r}^a \hat{r}^b \, .
\end{equation}
That is
\begin{equation}
\theta_{(\hat{u}) } = \frac{\left(R F' + 2 (1-F)\right) \dot{R}^2 - 4 (1-F)  }{2 R \sqrt{1-F}}
\end{equation}
Then an outward oriented null vector is $\ell = \hat{u} + \hat{r}$ and if
\begin{equation}
\theta_{(\ell)} = \theta_{\hat{u}} + \theta_{\hat{r}} = 0
\end{equation}
we can combine this with the arclength constraint (\ref{arc}) to get a pair of differential equations for $R$ and $\Theta$
describing a rotationally symmetric MOTS
\begin{align}
\ddot{R} = & \frac{2(1-\dot{R}^2)}{R} - \frac{\dot{R} \sqrt{1-\dot{R}^2}}{R} \cot \Theta\label{ddR}  \\
& + \frac{1}{2R} \sqrt{ \frac{1 - \dot{R}^2}{1-F}} \left( \left(R F' + 2 (1-F)\right) \dot{R}^2 - 4 (1-F)\right) \nonumber
\end{align}
and
\begin{equation}
 \dot{\Theta} = \frac{2 \pi  \sqrt{1-\dot{R}^2}}{R} \, ,  \label{dT}
\end{equation}
where we have assumed that $\dot{\Theta} > 0$ (which turns out to be true for all the situations in which we are interested).

We need initial conditions in order to solve these equations. By the assumed symmetry if we choose $\lambda=0$ at $\theta=0$ (the north pole)
then
\begin{equation}
\dot{R}(0) = 0 \, . \label{IC1}
\end{equation}
Thus given a choice
\begin{equation}
R(0) = R_o \label{IC2}
\end{equation}
for some constant $R_o$ we can find a MOTS candidate. These equations can always be integrated and so by construction will always produce a $\theta_{(\ell)} = 0$
surface. Its closure or lack thereof will determine whether or not it is a MOTS. This shooting method is commonly used for finding axisymmetric apparent horizons in
numerical relativity\cite{Baumgarte:2010:NRS:2019374,Thornburg2007}.

\subsection{Numerical Examples}

\begin{figure}
\includegraphics[scale=0.48]{Fin_RN9o10}
\caption{Axi-symmetric
$\theta_{(\ell)}=0$ surfaces in a $q/m = 0.9$ Reissner-Nordstr\"{o}m spacetime. The inner horizon is purple and the outer horizon is black. Other (open) $\theta_{(\ell)}=0$ surfaces are numerically
solved from initial conditions (\ref{IC1}) and (\ref{IC2}) and coloured blue, grey or dark red  depending on the value of $R_o$. The red circle in the middle is the region that is
not covered by the PG coordinates. The $z$-axis is horizontal with the north pole on the right-hand side. }
\label{Fin_RN}
\end{figure}

\begin{figure}
\includegraphics[scale=0.48]{Fin_SdS}
\caption{Axi-symmetric $\theta_{(\ell)}=0$ surfaces in $\Lambda=0.0768/m^2$ Schwarzschild-deSitter spacetime. The outer black hole and cosmological horizons are respectively black and blue.
Other (open) $\theta_{(\ell)}=0$ surfaces are numerically solved from initial conditions (\ref{IC1}) and (\ref{IC2}) and grey, dark red or green depending on the value of
$R_o$. The $z$-axis is horizontal with the north pole on the right-hand side. }
\label{Fin_SdS}
\end{figure}

Some sample $\theta_{(\ell)} = 0$ surfaces are shown in FIG.~\ref{Fin_RN} and FIG.~\ref{Fin_SdS} which respectively show typical Reissner-Nordstr\"om and Schwazschild-deSitter spacetimes.
Those figures show the system (\ref{ddR})-(\ref{dT}) solved with initial conditions (\ref{IC1}) and (\ref{IC2}). Solutions were obtained using Maple's\cite{maple} built-in routines for
systems of differential equations. Note that while the known horizons certainly show up as solutions there is also a $\theta_{(\ell)} = 0$ surface running through all points on the positive $z$-axis.
The behaviours shown in the figures appear to be generic. Axisymmetric $\theta_{(\ell)} =0$ surfaces that originate from $0 < R_o < r_{\mbox{\tiny{IH}}}$ and
$r_{\mbox{\tiny{OH}}}<R_o<r_{\mbox{\tiny{CH}}}$ ultimately diverge to infinity while those from $r_{\mbox{\tiny{IH}}} < R_o < r_{\mbox{\tiny{OH}}}$ and
$r_{\mbox{\tiny{CH}}}<R_o<\infty$ turn in and disappear into the singularity (or in the case with $q\neq0$ the region where the coordinate system is no longer defined).
Thus those surfaces are not MOTS as they are not smooth and closed.

These divergences can be contrasted with the now familiar pure deSitter case. For that spacetime
\begin{equation}
\theta_{\hat{u}} = - 2 \sqrt{ \frac{3}{\Lambda} }
\end{equation}
and so in a $T=\mbox{constant}$ surface any sphere of radius
\begin{equation}
r_S = \sqrt{\frac{3}{\Lambda}}
\end{equation}
will have $\theta_{(\ell)} = 0$. Examples are shown in FIG.~\ref{Fin_RN} (which despite the preceding analysis were numerically evolved in the same way as the previous examples).
\begin{figure}
\includegraphics[scale=0.45]{Fin_dS}
\caption{Translated cosmological horizons in pure deSitter spacetime. Coordinates are in terms of the deSitter radius. }
\label{Fin_dS}
\end{figure}

Note  that the open $\theta_{(\ell)} =0$ surfaces  shown in FIG.~\ref{Fin_RN} are not leaves of an isolated horizon\cite{Ashtekar:2004cn}. That is, if a particular $\theta_{(\ell)} = 0$ surface that opens up to infinity
(or plunges into the singularity) is extended to a three-surface as the locus of points it traces as $T$ varies, then that three-surface is not null. So
while these are $\theta_{(\ell)} = 0$ surfaces, they should not be viewed as foliating a kind of ``open'' horizon.

As a side note, the ubiquity of $\theta_{(\ell)} = 0$ surfaces seen in  FIG.~\ref{Fin_RN} and FIG.~\ref{Fin_SdS} serves to emphasize the non-local character of MOTS: finding a $\theta_{(\ell)} = 0$ surface is not difficult and in
our examples it is possible to find such a surface through any point. The hard part is finding a $\theta_{(\ell)}=0$ that smoothly closes. Determining whether or not that happens requires an integration to find
the full surface. Hence whether or not a particular section of a $\theta_{(\ell)} =0$ surface is  part of a MOTS may be determined by the detailed geometric properties of a far-away section of spacetime.

\subsection{Higher order stability}

We now return to the first-order unstable cases found in Section \ref{InfDef} to investigate their stability at higher order.

We begin with numerical tests: evolving from initial conditions
$R_o = r_{\mbox{\tiny{IH}}} + \delta R_o$ where $r_{\mbox{\tiny{IH}}}$ is the MOTS of an inner Reissner-Nordstr\"{o}m horizon while $\delta R_o$ is a finite perturbation.
Finite instabilities will manifest as finite deformations (like those in FIG.~\ref{Fin_dS}) while higher order stability will mean that any initially finite deformation will diverge
(like those in FIG.~\ref{Fin_RN} and FIG.~\ref{Fin_SdS}).

The $l=0,1,2,3$ modes are shown in FIG.~\ref{Testing_RN}. For $l=0,2$ the instability appears to fail as the numerical solutions diverge at $\theta = \pi$ however for $l=1,3$
things are not so clear. In those two cases we don't observe any divergences. However,  while suggestive, these observations aren't conclusive as in both cases the
behaviour could change for sufficiently small $\delta R_o$.
\begin{figure*}
\includegraphics[scale=.85]{Fin_Defs_Tuned}
\caption{Numerical solutions of first-order unstable inner Reissner-Nordstr\"om horizons for a) $l=0$, b) $l=1$, c) $l=2$ and d) $l=3$. $l=0,2$ appear to diverge for all sizes of
initial perturbation but no divergence can be seen for $l=1,3$.  As in FIG.~\ref{Fin_RN} the red centre is not covered by the PG coordinates. }
\label{Testing_RN}
\end{figure*}

To better understand what is happening we turn to a higher-order analysis of the equations. Then it is more
convenient to work with a single function and so we
switch to parameterize $R$ with $\theta$. Working from
\begin{equation}
\lambda_\theta = \sqrt{R_\theta^2 + R^2}
\end{equation}
and
\begin{equation}
\dot{R} =\frac{R_\theta}{\lambda_\theta}
\end{equation}
(where derivatives with respect to $\theta$ are indicated with subscripts)
the conversion is straightforward and we get the following differential equation:
\begin{align}
0 =&  R_{\theta \theta}  -  \left(\frac{3}{R} + \frac{\sqrt{R^2 + R_\theta^2}}{2R^2 \sqrt{1-F}} \left(RF'- 2(1-F) \right)\right) R_\theta^2  \nonumber \\
& +  \left(\frac{ R^2 + R_\theta^2  }{R^2}\right)  \cot \! \theta R_\theta  - 2R     + 2 \sqrt{(1-F)( R^2 + R_\theta^2)} \, . \label{Rtt}
\end{align}
Given an inner horizon at $r_{\mbox{\tiny{IH}}}$ we can then look for MOTS of the form:
\begin{equation}
R(\theta) = r_{\mbox{\tiny{IH}}} + m (  \epsilon R_1 (\theta) + \epsilon^2 R_2 (\theta) + \epsilon^3 R_3 (\theta) + \dots )
\end{equation}
where of course $\epsilon$ is the initial perturbation from $r_{\mbox{\tiny{IH}}}$ at $\theta=0$. Then we have initial conditions
\begin{equation}
R_1 (0)  = 1 \; \mbox{ and } R_L(0) = 0 \; \mbox{ for } L>1
\end{equation}
while the first derivative $R_{L\theta} (0) = 0$ for all $L$.

We also expand $F = 1 - \frac{2m}{r} + \frac{q^2}{r^2}$ as a Taylor series around $r_{\mbox{\tiny{IH}}} = m - \sqrt{m^2 - q^2}$ as:
\begin{equation}
F(r) = \sum_{n=1}^{\infty} (-1)^n\big(n l (l+1)  +(n-1) \big) \left(  \frac{r- r_{\mbox{\tiny{IH}}}}{r_{\mbox{\tiny{IH}}}} \right)^n
\end{equation}
where deriving these expansion uses the first-order condition $r_{\mbox{\tiny{IH}}} F_1 = - l(l+1)$.

Then to the first three orders (\ref{Rtt}) expands as
\begin{align}
 0 = & \triangle_l R_1 \label{R1} \\
 0 = & \triangle_l R_2  + \left(  \frac{l^2(l+1)^2 - 4  }{4}   \right) R_{1\theta}^2 \label{R2} \\
 &   - \left(\frac{(l^2 + l + 2)^3}{8} \right) R_1^2 \nonumber  \\
 0 = & \triangle_l R_3  + \left( \frac{(l^2+l+2)^2 \cot \theta}{4} \right) R_{1\theta}^3 \\
  & - \left(\frac{l(l+1)(l^2+l+2)^2(l^2+l+4)}{16}\right) R_1 R_{1 \theta}^2 \nonumber \\
  & + \left(\frac{l^2(l+1)^2 - 4}{2} \right) R_{1 \theta} R_{2 \theta} \nonumber \\
  & - \left(\frac{(l^2+l+2)^3}{4} \right) R_1 R_2 + \left(\frac{(l^2+l+2)^5}{32} \right) R_1^3 \label{R3}
  \nonumber
\end{align}
where $\triangle_l$ is the second-order differential operator that vanishes for first-order unstable perturbations (\ref{SL}):
\begin{equation}
\triangle_l = \frac{d^2}{d\theta^2} + \cot \! \theta \frac{d}{d\theta} + l (l+1)  \, .  \label{trianglel}
\end{equation}
In the following we refer to the non-$\triangle_l R_n$ terms in each equation as $h_{ln}$.

Of course the $R_1$ equation is (\ref{SL}) again. Note that the equations can be solved sequentially. Once we have $R_1$ we can solve for $R_2$ and then both of them
can be used to solve for $R_3$. While the rapidly growing complexity  of the expressions means that it is not practical to show the higher order equations, this pattern continues.

Then we are interested in solutions to equations of the form
\begin{equation}
\triangle_l X + h(\theta) = 0 \, . \label{Prob}
\end{equation}
where $X(\theta):[0,\pi] \rightarrow \mathbb{R}$ satisfies initial conditions $X(0)=X_o \in \mathbb{R}$, $X_\theta (0) = 0$ and
$h(\theta)$ can be expressed as a finite sum of Legendre polynomials:
\begin{equation}
h (\theta) = \sum_{L=0}^{L_{\mbox{\tiny{max}}}} h_{[L]} P_L (\cos  \theta) \, .
\end{equation}

Now, the solution to the homogeneous version of (\ref{Prob}) is the Legendre polynomial $P_l(\cos\theta)$.  Therefore, by the Fredholm Alternative theorem, the inhomogeneous problem
has a solution if and only if $h(\theta)$ is orthogonal to $P_l(\cos\theta)$:
\begin{equation}
h_{[l]} \equiv \frac{2l+1}{2} \int_0^\pi \! \!  \sin  \theta \, h(\theta) P_l (\cos  \theta)  d  \theta = 0 \, . \label{htest}
\end{equation}
Equivalently the Legendre polynomial expansion of $h$ does not contain an $h_l$ term.

If a solution does exist then it is also a finite sum of Legendre polynomials:
\begin{equation}
X (\theta) = \sum_{L=0}^{L_{\mbox{\tiny{max}}}}X_L P_L (\cos  \theta) \label{X}
\end{equation}
where for $L \neq l$
\begin{equation}
X_L = \frac{h_{[L]}}{L(L+1)-l(l+1) } \label{X1}
\end{equation}
and
\begin{equation}
X_l = X_0 - \left. \sum_{L=0}^{L_{\mbox{\tiny{max}}}} \right|_{L \neq l} X_L \, . \label{X2}
\end{equation}

These observation can then be combined as an algorithm to test for solutions to the deformation problem to arbitrary order.
\begin{enumerate}
\item Set $R_1 = \epsilon m  P_l (\cos \theta)$ and $n=2$.
\item \label{s2} Find the $\epsilon^n$ term in the expansion of (\ref{Rtt}).
It will take the form
\begin{equation}
\triangle_l R_n + h_{ln}  = 0 \label{heq}
\end{equation}
where  $h_{ln}$ will always be a sum of terms involving $l$, $\cot \theta$, $R_1$, $R_{1\theta}$, $R_2$, $R_{2\theta}$, \dots,
$R_{n-1}$ and $R_{(n-1) \theta}$.
\item Substitute the known expressions for $R_m$ into $h_{ln} (\theta)$, $m<n$ and
use (\ref{htest}) to test whether (\ref{heq}) has a divergent solution. If $h_{ln[l]} \neq 0$, stop. This case
 cannot be finitely deformed.
\item If the solution of (\ref{heq}) isn't divergent, use (\ref{X})--(\ref{X2}) to generate $R_{n+1}$ and repeat from Step \ref{s2} with $n \rightarrow n+1$.
\end{enumerate}
Appendix \ref{triple} recalls  some results on series expansions of derivatives and products of Legendre polynomials that are used in implementing these steps.

For a finite deformation this algorithm would never terminate. However in all cases that we have checked
we find a divergence at some order and so the horizon cannot be finitely deformed.

Explicitly, the first four even $l$ cases are
\begin{align}
\left. \frac{R}{m} \right|_{l=0} \mspace{-16mu} & \approx 1+ P_0  \epsilon+ \mbox{(divergent term)} \epsilon^2\\
\left. \frac{R}{m} \right|_{l=2} \mspace{-16mu} & \approx  \frac{1}{4} + P_2  \epsilon+ \mbox{(divergent term)} \epsilon^2 \\
\left. \frac{R}{m} \right|_{l=4} \mspace{-16mu} & \approx \frac{1}{11} + P_4   \epsilon+ \mbox{(divergent term)} \epsilon^2 \\
\left. \frac{R}{m} \right|_{l=6} \mspace{-16mu} & \approx \frac{1}{22} + P_6   \epsilon + \mbox{(divergent term)} \epsilon^2  \, ,
\end{align}
while the first three odd cases are
\begin{align}
\left. \frac{R}{m} \right|_{l=1} \mspace{-16mu}  \approx & \frac{1}{2} + P_1 \epsilon+ 4 \left( \frac{P_0-P_2}{3} \right) \epsilon^2  +  16 \left( \frac{-P_1 + P_3 }{5} \right)  \epsilon^3 \nonumber \\
& +  8  \left( - \frac{17}{15} P_0 + \frac{7}{3} P_1 + \frac{1}{21} P_2 - \frac{131}{105} P_4 \right)  \epsilon^4  \\
& + \mbox{(divergent term)} \epsilon^5 \nonumber  \\
\left. \frac{R}{m} \right|_{l=3} \mspace{-16mu} \approx & \frac{1}{7} + P_3  \epsilon + \bigg(  -\frac{11}{12} P_0 + \frac{8}{9} P_2 + \frac{35}{2} P_3 \\
& \mspace{30mu}  - \frac{351}{44}  P_4 - \frac{940}{99} P_6 \bigg) \epsilon^2  + \mbox{(divergent term)} \epsilon^3 \nonumber\\
\left. \frac{R}{m} \right|_{l=5}  \mspace{-16mu} \approx & \frac{1}{16}  + P_5 \epsilon+ \bigg( - \frac{1312}{165} P_0 - \frac{12200}{1287} P_2 - \frac{3456}{715} P_4 \nonumber \\
& + 112 P_5 - \frac{41600}{1683} P_6 - \frac{190400}{8151} P_8 - \frac{9620856}{230945} P_{10} \bigg) \epsilon^2 \nonumber \\
&  + \mbox{(divergent term)} \epsilon^3  \; .
\end{align}
In these expressions the  $\cos \theta$ dependence of the $P_n$ is suppressed.  The pattern  appears to continue for all $l>1$: that is even $l$ diverge at second order while
odd cases diverge at third order. This is demonstrated in FIG.~\ref{EvenOddPlot} where  $h_{l2[l]}$ and $h_{l3[l]}$ are plotted up to $l=31$.
\begin{figure}
\includegraphics[scale=0.5]{EvenOddPlot}
\caption{$-h_{l2[l]}$ (black) and $h_{l3[l]}$ (blue) for even and odd cases respectively for $l=2$ to $l=31$. Both appear to be growing monotonically. }
\label{EvenOddPlot}
\end{figure}

Though the trend in FIG.~\ref{EvenOddPlot} seems clear we were not able to show that the growth continues for all $l$: the expressions, particularly for the odd cases, become
prohibitively complex.

However we can at least see why the odd cases don't diverge at second order. Reading off $h_{2l}$ from (\ref{R2})
and keeping in mind that $R_{1l} = P_l (\cos \theta)$ then applying (\ref{prodEx})  it is straightforward to see that on expanding the  $R_1^2$ term into a Legendre series
we only obtain even terms. Meanwhile the $R_{1\theta}^2$ term can be expanded using (\ref{Pp}) and then (\ref{prodEx}) again to see that it also contains
only even terms. Hence $h_{2l[l]}$ necessarily vanishes for all odd $l$ and any divergence must be at third or higher order.

\section{Discussion}
\label{Discuss}

While a stable MOTS cannot be smoothly deformed, ``instability'' is not sufficient to imply that such deformations are possible. For spherically symmetric MOTS in similarly symmetric spacetimes
we first showed that all but a few finely tuned cases are not deformable. Then on checking those special cases for RN spacetimes we saw that, apart from pure deSitter spacetime, none of them
appeared to be deformable either (though we did not find a completely general proof).

Hence we expect that a much stronger result holds that prevents (virtually all) horizons from being deformed. We expect that only spacetimes with extra symmetries (such as
pure deSitter) can house deformable MOTS and that those deformations will turn out to be translations.

This would not be a particularly shocking result. It would be much more surprising to discover that inner black hole MOTS could be finitely deformed. However as
far as we know there is no extant theorem that proves this.

\acknowledgements

This work was supported by NSERC Discovery Grants 261429-2013 (IB)  and 418537-2012 (HK). AO was also supported by
an NSERC Undergraduate Student Research Award and a Memorial University Student Internship Award.


\section{Introduction}
From their early theoretical framework  to the latest observerence of the blackhole \cite{key-1,key-2}.It has remained the most enigmatic object in the entire cosmos.General Relativity establishes that the space time at the near vicinity of the black hole is so warpped that the photon orbits around the blackhole in an unstable circular orbit. There are plenty of research has been already done on both rotating and non-rotating cases\cite{key-3,key-4,key-5,key-6,key-7,key-8,key-9,key-10,key-11,key-12,key-13}.where the shadow is a perfect circle in later case and distorted in the former due to the frame-dragging effect. A blackhole with presence of charge has also been studied. In addition to this there are black holes with an addisional dark energy quintessenisal field which was first proposed by kesilev. the quintessential field is present to explain the accleratated expansion of our universe \cite{key-14,key-15,key-16}.

In this paper, Instead of assuming point particles,we consider the the blackhole to be reside on on dimensional strings \cite{key-16,key-17,key-18,key-19,key-20,key-21,key-22,key-23,key-24,key-25,key-26}.A static spherically symmetric spacetime metric for this black hole, has been studied  in Ref. In our case we considered a Ressiner nordstrom blackhole in a stingy universe with added quintessential field. using Newman Janis algorithm we have arrived at a metric for rotating charge blackhole with string cloud in pressece of quintessential fieldFin\

This Section 2 we discuss about this Kerr-Newman-Kiselev blackhole with string cloud, Section 3 discusses about the photon orbit and  effective potential,we have plotted the blackhole shadow in section 4 and summarized the paper in section 5.

\section{Kerr - Newman - kiselev BH with string cloud}

The metric of Reissner-Nordstrom surrounded by quintessence and cloud of string\cite{key-28} is given as,
\begin{equation}
\begin{split}
ds^{2}&=-\left(1-b-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{c}{r^{3\omega_q+1}}\right)dt^{2}\\&+\left(1-b-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{c}{r^{3\omega_q+1}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}
\end{split}
\end{equation}
Following newman janis algorithm\cite{key-27}, we have our metric to be,
\begin{equation}
\begin{split}
ds^{2}&=-\left(1-\frac{2\rho r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta( r)}dr^{2}+\Sigma d\theta^{2}\\&-\frac{4 a \rho  r sin^{2}\theta}{\Sigma}{dtd\phi}+sin^{2}\theta\left(r^{2}+a^{2}+\frac{2a^{2}\rho r sin^{2}\theta}{\Sigma}\right)d\phi^{2}
\end{split}
\end{equation}
Where,
\begin{equation}
2\rho r\ =r^{2}+a^{2}-\Delta( r)
\end{equation}

\begin{equation}
\Delta( r)=\ \ r^{2}\ +\ a^{2}\ -\ 2\ M\ r\ -\ q^{2}\ +\ c\ r^{1-3\omega}\ +b\ \ r^{2}
\end{equation}

\begin{equation}
\Sigma = r^{2} + a^{2}\cos^{2}{\theta}
\end{equation}
In the above equations,q being the charge of the blackhole, M and
c term corresponds to mass and quintessence parameter,while a and
b denotes the spin and string cloud parameter. The metric reduces
to kerr newman kiselev blackhole when b = 0 and can be further reducedto
kerr newman with b = c = 0 and to kerr with b=c=q=0 and finally to
schwarschild by a=b=c=q=0.

Solving $\Delta( r)$ = 0 will give us the event horizon
, the plot $\Delta(r)$ vs r below gives us the comparison
for different blackholes.

\begin{figure}[H]
\includegraphics[scale=1]{h1.eps}
\caption{Horizon plots for different blackholes.}
\end{figure}
\section{Photon Orbits}
\subsection{Unstable Circular Orbit}
There is a particular unstable orbit in which a photon revolves around
the blackhole,depending on the conditions of impact parameter the
would leave this orbit to either fell into the blackhole or escape
to infinity. The equation of such orbit can calculated using hamilton-Jacobi
Variable separation method\cite{key-29}. Hamilton - Jacobi equation in its general
form is given as,
\begin{equation}
\frac{\partial S}{\partial\lambda}= - \frac{1}{2}g^{{\mu \nu}}\frac{\partial S}{\partial x^{\mu}}\frac{\partial S}{\partial x^{\nu}}
\end{equation}
Where $\lambda$ is the affine parameter and S is the action thats defined
as,
\begin{equation}
S\ =\frac{1}{2}m^{2}\ \lambda\ -\ Et\ +\ L\varphi\ +\ S_{r}\ (r)\ +\ S_{\theta}\ (\theta)
\end{equation}
where m is the rest mass, E and L are energy and Angular momentum
which are the constants of motion.

Solving the Hamilton jacobi equation will yield,
\begin{equation}
\frac{\partial S_{r}}{\partial r}=\frac{\sqrt{R(r)}}{\Delta(r)}
\end{equation}

\begin{equation}
\frac{\partial S_{\theta}}{\partial\theta}=\sqrt{\Theta(\theta)}
\end{equation}
Where ,\\
 \strut \\
\begin{equation}
R(r)=(aL\ -\ (a^{2}\ +\ r^{2}\ )E)^{2}\ -\ (K\ +\ (aE\ -\ L)^{2}\ )\mathrm{\Delta}
\end{equation}
\begin{equation}
\Theta(\theta)=K\ -\left(\frac{L^{2}}{\text{si}n^{2}\theta}\ -\ a^{2}\text{\ E}\ ^{2}\right)Cos\theta
\end{equation}
where K is the the seperation constant. The trajectory of a photon
is calculated with two impact parameter,

$$\xi=\frac{L}{E}$$
$$\eta=\frac{K}{E^{2}}$$

Rewritting R(r) and $\Theta (\theta)$ in terms of impact parameters
\begin{equation}
R_{p}(r)=(a\ \xi\ -\ (a^{2}\ +\ r^{2}\ ))2-(\eta\ +\ (a-\xi)^{2}
\end{equation}
\begin{equation}
\Theta_{p}(\theta)\ =\ \eta\ - \left( \frac{\xi^{2}}{\text{si}n^{2}\theta}-\ a^{2}\ \right) cos^{2}\theta
\end{equation}
The equation of geodesic motion can be written the form
\begin{equation}
\Sigma\ t'=\frac{((r^{2}+\ a^{2}\ )E\ -\ aL)(r^{2}\ +\ a^{2}\ )}{\Delta(r)}a(aESin^{2}\ (\theta\ )\ -\ L)
\end{equation}
\begin{equation}
\Sigma\ r'\ =\sqrt{R(r)}
\end{equation}
\begin{equation}
\Sigma\ \theta'= \sqrt{\Theta (\theta)}
\end{equation}
\begin{equation}
\Sigma\ \varphi'=\frac{a((r^{2}\ +\ a^{2}\ )E\ -\ aL)}{\Delta(r)}-\frac{\text{aESi}n^{2}\ (\theta\ )\ -\ L}{\text{Si}n^{2}\ (\theta\ )}
\end{equation}
\section{Effective Potential}

The relationship between r' and effective potential is given as,
\begin{equation}
V_{\text{eff}}=\frac{E^{2}-1}{2}-\frac{1}{2}r'
\end{equation}
Effective potential is the maximum potential energy for a photon to
have zero radial velocity and zero radial acceleration and there by
determining the photons's stability.
\begin{figure}[H]
\label{v2}
\includegraphics[scale =1]{v1.eps}
\caption{$ V_{eff}$ vs r for varying q with M = 1; e = 1; k = 4; L = 2.1; a = 1; c = b = 0.1;.}
\end{figure}
\begin{figure}[H]
\label{3}
\includegraphics[scale = 1]{v2.eps}
\caption{$V_{eff}$ vs r for varying
c with M = 1; e = 1; k = 4; L = 2.1; a = 1; b = 0.1;q=0.4.}
\end{figure}

\begin{figure}
\label{v4}
\includegraphics[scale=1]{v3.eps}
\caption{ $V_{eff} $vs r for varying
b with M = 1; e = 1; k = 4; L = 2.1; a = 1; c = 0.1; q=0.4.}
\end{figure}
From figure 2 - 4 , we have plotted effective potential against r
for various profiles such as by varying charge 1, quintessence parameter
c and string parameter b. In fig 2, the maxima of the effective potential
decreases with increase in charge, we can also notice that the peak
is shifted toward the left as charge is being increased. In Fig 3,
the maxima increases with increase in quintessense value and peak
is shifted left. Similarly in fig 4, the maxima increases with string
parameter b and peak shifting towards to left can be seen here too.

\section{Shadows}

For a photon to have spherical orbit,it should have null radial velocity
and null radial acceleration,which means that,

$$R_{p}(r)=\ 0$$
$$\frac{\text{d\ }R_{p}}{\text{d\ r}}\ =\ 0$$

With this we can rewrite the impact parameter as,
\begin{equation}
\xi=\frac{-4r\mathrm{\Delta}(r)\ +\ a^{2}\ \mathrm{\Delta}'(r)\ +\ r^{2}\ \mathrm{\Delta}'(r)}{a\ \mathrm{\Delta}'(r)}
\end{equation}
\begin{equation}
\eta=\frac{r^{2}\ (16a^{2}\ \mathrm{\Delta}(r)\ -\ 16\mathrm{\Delta}^{2}(r)\ +\ 8r\mathrm{\Delta}(r)\ \mathrm{\Delta}'(r)\ -\ r^{2}\ \mathrm{\Delta}'^{2}(r)\ )}{a^{2}\mathrm{\Delta}'^{2}(r)}
\end{equation}
The Shape of the black hole depends on the celestial coordinates $\alpha$
and $\beta$ , which is given as
\begin{equation}
\alpha   = -{r_{0}}^{2}Sin \theta\frac{d\phi}{dr}
\end{equation}
\begin{equation}
 \beta=\ -{{r_{o}}^{2}\frac{d\theta}{\text{dr}}}_{}
\end{equation}
Where $\alpha$ and $\beta$ has limit $r_{o}$ goes to infinity,using
the geodesic equation the celestial coordinates can be written in
terms of impact parameter.
\begin{equation}
\alpha=\frac{-\xi}{sin(\theta)}
\end{equation}

\begin{equation}
\beta=\sqrt{\eta+a^2Cos^2(\theta)-\xi^2 cot^2(\theta)}
\end{equation}

The shadow can we obtained by plotting $\alpha$ vs $\beta$ in the equatorial plane.
\begin{figure*}
\includegraphics[scale=0.8]{s1.eps}
\caption{Shadow plot fot KNK - cloud string black hole for
variation in spin parameter(left) snd charge(right)}
\end{figure*}

\begin{figure*}
\includegraphics[scale=1]{s2.eps}
\caption{ Shadow plot fot KNK - cloud string black hole for
variation in quintessense (left) snd string parameter(right).}
\end{figure*}
\section{conclusion}
In this paper we have took reissner nordstrom blackhole with cloud
of strings and surrounds it with quintessence. we processed the metric
through newman janis algorithm to get its rotating counterpart. the
blackhole in study now is a roating charged blackhole with clouds
of string surrounded by quintessence. we studied its nature of effective
potential and unstable photon orbits. we have observed that the effective
potential maxima is directly proportional to the increase in charge,quintessence
and string parameter and the peak seems to incline towards the left.

Finally, we have plotted the blackhole shadow for various variable
profiles.from figure 5, As expected the shadow shape starts to distort
as we go for higher spin values. it also seems to a distorted circles
for higher values of charge but upon closer observation,the circles
are in deed concentric proving that only size of the shadow decreases
with increase in charge. From figure 6, It is evident that the size
of the shadow decreases with increase in quintessence and string parameters.
\newpage


\section{Introduction}
Knowing the relative importance of gravitational, rotational, turbulent, and magnetic energies is essential to understand the process of star formation,
as all of these effects shape structures and affect disk formation in protostellar sources \citep{McKee07, Li14a}.
Theoretical studies show that if the magnetic field and rotational axis of a dense core are aligned,
magnetic braking is very efficient during the collapse in the ideal magnetohydrodynamic (MHD) condition \citep{Galli06, Allen03, Mellon08}.
As a consequence, the disk formation and growth are suppressed,
and Keplerian disks with sizes larger than 10 au unlikely form around protostars.
When other mechanisms are incorporated, such as the non-ideal MHD effects, misaligned magnetic field and rotational axes in dense cores, or turbulence,
the efficiency of magnetic braking is reduced,
and a large Keplerian disk with a size of tens of au can form around a protostar in numerical simulations \citep{Dapp12,Joos12,Joos13,Santos12,Seifried13,Li13,Krumholz13,Tsukamoto15,Tsukamoto17,Zhao16,Zhao18,Masson16,Matsumoto17,Wurster19a,Wurster19b}.
Keplerian disks with sizes of tens of au have been observed around several young protostars \citep{Murillo13,Lee14,Lee18,Sakai14,Yen17}.
However, it remains unclear as to which mechanisms play more important roles to regulate the disk formation and growth in protostellar sources.

B335 is an isolated Bok globule with an embedded Class 0 protostar at a distance of 100 pc \citep{Keene80, Keene83, Stutz08, Olofsson09}.
B335 is slowly rotating \citep{Saito99,Yen11,Kurono13}, and is associated with a 0.1 pc bipolar molecular outflow as well as several Herbig Haro objects along the east--west direction \citep{Hir88,Gal07}.
The infalling motion has been observed on scales from 3000 au to 10 au in B335 \citep{Zhou93, Zhou95, Choi95, Evans05, Evans15, Saito99, Yen10, Yen11, Yen15, Kurono13, Imai19, Bjerkeli19}.
On the other hand, the rotational motion in the protostellar envelope on scales from 1000 au to 10 au in B335 has been found to be slow \citep{Yen10, Yen15, Imai19, Bjerkeli19},
and no clear sign of Keplerian rotation is observed even on a 10 au scale \citep{Bjerkeli19}.
The James Clerk Maxwell Telescope (JCMT) and Atacama Large Millimeter/submillimeter Array (ALMA) polarimetric observations show that the magnetic field is along the east--west direction, and it is ordered and highly pinched on a 1000 au scale \citep{Maury18,Yen19}.
These results suggest that the magnetic field likely plays an important role in the dynamics in B335.
On the other hand, the presence of the jets and outflows in B335 could hint at the existence of a circumstellar disk \citep[e.g.,][]{Blandford82}.
In addition, a compact continuum component with a size of $\sim$6 au oriented perpendicular to the outflow direction as well as velocity gradients due to the rotational motion on a similar scale have been observed with ALMA \citep{Imai19,Bjerkeli19}.
Thus, a small disk with a size of a few au likely has formed in B335.
Therefore, B335 is an excellent target to study the disk formation under the influence of a dynamically important magnetic field.

To observationally investigate the effects of the magnetic field on the disk formation in B335,
we conducted ALMA polarimetric observations at an angular resolution of 0\farcs2 (20 au) to probe the magnetic field structures on a scale close to the disk forming region.
In the present paper, we describe the details of our observations in Section \ref{observations},
and introduce our observational results in Section \ref{results}.
In addition, we compare our polarization data at 0.87 mm with those at 1.3 mm retrieved from the ALMA archive.
In Section \ref{discussion}, we present the results from our non-ideal MHD simulations for B335,
and discuss the observed magnetic field structures in comparison with the simulations.
Lastly, in Section \ref{summary} we summarize the possible effects of turbulence as well as the misalignment between the magnetic field and rotational axis on the magnetic field structures in B335.

\section{Observations}\label{observations}
The polarimetric observations with ALMA at 0.87 mm toward B335 were conducted during 2016 to 2018, consisting of 13 successful executions (project code: 2015.1.01018.S).
In the observations, 40 to 47 antennae were used in the configurations with baseline lengths ranging from 15 m to 1400 m.
The pointing center was $\alpha({\rm J2000}) = 19^{\rm h}37^{\rm d}00.\!\!^{\rm s}89$, $\delta({\rm J2000}) =  +7\arcdeg34\arcmin09\farcs6$.
The on-source integration time was 7.4 hours.
The observations were conducted with the full polarization mode and at the frequency ranges of 335.5--339.5 GHz and 347.5--351.5 GHz with a total bandwidth of 8 GHz.
In these observations, J1751$-$0939 was observed as the bandpass calibrator, J1938$+$0448 or J1935$+$2021 as the gain calibrators, and
J1924$-$2914 or J2000$-$1748 for polarization calibration.
The flux calibration was performed with the observations of quasars or the asteroid, Pallas.
The data were manually calibrated by the EA ARC node using the Common Astronomy Software Applications (CASA) of the version 5.1.1 \citep{McMullin07}.
We additionally performed self-calibration of the phase using the Stokes {\it I} data.
Then the calibrated visibility data were Fourier-transformed with the Briggs weighting with a robust parameter of +0.5 to generate Stokes {\it IQU} images, and the images were cleaned using the CASA task {\it tclean}.
The achieved synthesized beam is $0\farcs19 \times 0\farcs17$.
The noise level in the Stokes {\it I} image is 40 $\mu$Jy beam$^{-1}$,
and it is 9 $\mu$Jy beam$^{-1}$ in the Stokes {\it Q} and {\it U} images.
When we generated the polarized intensity ($I_{\rm p}$) map, we debiased the polarized intensity ($I_{\rm p}$) with $I_{\rm p} = \sqrt{Q^2 + U^2 - {\sigma_{Q,U}}^2}$,
where $\sigma_{Q,U}$ is the noise level in Stokes {\it Q} and {\it U} \citep{Wardle74,Simmons85}.
To extract polarization detections, we first binned up the Stokes {\it IQU} and $I_{\rm p}$ maps to have a pixel size of 0\farcs1,
which is approximately half of the beam size, and computed polarization orientations and fractions.
Thus, the minimal separation between two polarization detections is 0\farcs1.
The Stokes {\it I} and $I_{\rm p}$ maps with their original pixel size of 0\farcs02 are presented below.
The polarization detections are extracted when the signal-to-noise ratios (S/N) of both Stokes {\it I} and $I_{\rm p}$ are larger than three,
and thus the expected uncertainties in the polarization orientations are $\lesssim$9\arcdeg.

\section{Results}\label{results}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{polmap.eps}
\caption{(a) 0.87 mm continuum map (contours) overlaid on the polarized intensity map (color) obtained with our ALMA observations. The polarized intensity map is in units of mJy Beam$^{-1}$. The contour levels are from 5$\sigma$ in steps of a factor of two, where 1$\sigma$ is 40 $\mu$Jy Beam$^{-1}$.
(b) Magnetic field orientations (orange segments) inferred by rotating the polarization orientations by 90$\arcdeg$. The contours show the Stokes {\it I} intensity, identical to panel (a). The length of all the segments is the same. The minimal separation between two segments is half of the beam size.
The zero offset refers to the pointing center ($19^{\rm h}37^{\rm d}00.\!\!^{\rm s}89 +7\arcdeg34\arcmin09\farcs6$). The filled ellipses at the bottom right corners present the size of the synthesized beam.}\label{polmap}
\end{figure*}

In the 0.87 mm continuum map obtained with our ALMA observations,
there is a compact continuum source at the center with a peak position of $\alpha({\rm J2000}) = 19^{\rm h}37^{\rm d}00.\!\!^{\rm s}9$, $\delta({\rm J2000}) =  +7\arcdeg34\arcmin09\farcs5$ (Fig.~\ref{polmap}a).
This peak position is consistent with the one measured at 1.3 mm at a higher angular resolution of 0\farcs03 \citep{Bjerkeli19}.
The apparent size of this compact continuum source in our observations is 0\farcs2 (20 au).
Thus, it is only marginally resolved, and we could not constrain its orientation.
The peak brightness temperature is measured to be 30 K,
which is more than a factor of two lower than the dust temperature of 67 K at a radius of 25 au estimated by modeling molecular-line and continuum data and the spectral energy distribution of B335 \citep{Shirley11,Evans15}.
Therefore, the continuum emission detected with our observations is most likely optically thin.

The compact source is embedded in a flattened structure with an apparent size of $\sim$2$\arcsec$ and a position angle (PA) of its major axis of 17\arcdeg.
On a larger scale of $\sim$4$\arcsec$ (400 au),
the continuum emission shows a bent structure,
and its distribution is along the outflow cavity wall.
In addition,
the continuum emission on the 400 au scale in B335 lies primarily beyond the edge of the outflow traced by the CO (2--1) emission,
as seen in Fig.~5 in \citet{Bjerkeli19}.
The overall structures in the continuum emission in our observations are consistent with the previous results obtained with ALMA at 1.3 mm and angular resolutions of 0\farcs03 to 0\farcs8 \citep[e.g.,][]{Yen15,Imai16,Maury18,Bjerkeli19}.

The polarized continuum emission at 0.89 mm is primarily detected along the outflow cavity and around the central compact continuum source.
The polarized intensity is highest in the south and north close to the Stokes {\it I} intensity peak but becomes lower at the peak (Fig.~\ref{polmap}a).
As discussed in \citet{Maury18},
more polarized continuum emission detected along the outflow cavity wall could be due to more efficient dust grain alignment caused by stronger illumination from the central star in the outflow cavity \citep{Lazarian07}.
Similar enhancements in polarized continuum emission along outflow cavity walls have also been observed in other protostellar sources \citep[e.g.,][]{Hull17b,Hull19,Gouellec19}.
The orientations of the polarization detections are rotated by 90$\arcdeg$, displaying the orientations of the magnetic field (Fig.~\ref{polmap}b).
As discussed in Section \ref{discussion},
the detected polarized emission is unlikely induced by dust scattering based on the measured polarization fractions \citep{Kataoka15, Yang16, Yang17}.
The magnetic field orientations on a 400 au scale are primarily along the outflow cavity wall.
There is a northern patch of detections with orientations more along the north--south direction.
These features on a scale larger than 100 au are consistent with those observed at 1.3 mm with ALMA at an angular resolution of 0\farcs8 \citep{Maury18,Yen19}.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{bmap.eps}
\caption{Same as Fig.~\ref{polmap} (b) but zooming in on the central 2$\arcsec$~$\times$~2$\arcsec$ region.}\label{bmap}
\end{figure}

With a three times higher resolution,
our observations reveal the magnetic field structures significantly closer to the central source than was possible with the previous, lower-resolution observations \citep{Maury18,Yen19}.
In the present paper, we focus on the features on a small scale within radii of 0\farcs5--1$\arcsec$ (50--100 au).
The magnetic field structures on a larger scale in B335 have been discussed in detail in \citet{Maury18} and \citet{Yen19}.
Figure \ref{bmap} presents the magnetic field orientations in the central 2$\arcsec$ (200 au) region.
At offsets of $\sim$0\farcs5--1$\arcsec$ in the north,
the detected magnetic field orientations are along the wall of the outflow cavity.
In the inner northern region at offsets of $\sim$0--0\farcs5,
the magnetic field is along the northwest--southeast direction with PA of 130\arcdeg--150\arcdeg.
Here, the magnetic field segments appear to connect to the continuum peak,
with a few segments also along the north--south direction with PA of 0\arcdeg--10\arcdeg.
In the south, the magnetic field is primarily along the north--south direction,
and the magnetic field becomes more tilted from the north--south direction by 20\arcdeg--30\arcdeg at outer radii of $\sim$0\farcs5--1\arcsec.
In addition, there are a few magnetic field segments along the northwest--southeast direction detected at $\sim$0\farcs4 southeast from the continuum peak.
In the east, at RA offsets of 0\farcs6--1$\arcsec$, there is a patch of segments with their PA gradually changing from 10$\arcdeg$ to 60$\arcdeg$ toward the center.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{2pol_3p.eps}
\caption{Comparison of (a) the magnetic field orientations and (b) the polarization percentages inferred from the polarization data at 0.87 mm and 1.3 mm. The data were convolved to the same beam size. The dotted lines in the panel (a) show the difference of $\pm$10\arcdeg. (c) presents the polarization percentages at 0.87 mm (orange) and 1.3 mm (black) as a function of projected radius with respect to the continuum peak. Orange solid and black dashed curves show the running means in radial bins of 0\farcs2.}\label{2pol}
\end{figure*}

For comparison, we retrieved the polarization maps at 1.3 mm obtained with ALMA \citep{Maury18,Yen19}.
We convolved our polarization maps at 0.87 mm to have the same beam size as those at 1.3 mm and extracted polarization detections with the same procedure and criteria as those in \citet{Yen19}.
Then we compared the polarization orientations and percentages at 0.87 mm and 1.3 mm at the same positions.
Figure \ref{2pol} presents the comparison of the inferred magnetic field orientations and measured polarization percentages at 0.87 mm and 1.3 mm.
The magnetic field orientations inferred from the polarization data at 0.87 mm and 1.3 mm are consistent within the uncertainties.
The mean difference in the orientations at the two wavelengths is 5$\arcdeg$ with a standard deviation of 4$\arcdeg$.
More than 90\% of the polarization detections show higher polarization percentages at 1.3 mm than at 0.87 mm.
On average, the polarization percentages at 1.3 mm are a factor of 1.7 higher.
We note that the polarization percentages at both wavelengths increase with increasing radii,
and the percentages at 0.87 mm are systematically lower than those at 1.3 mm at all radii (Fig.~\ref{2pol}c).

\section{Discussion}\label{discussion}

\subsection{Origins of the polarized continuum emission}
With the ALMA polarization data at 0.87 mm and 1.3 mm,
we found higher polarization percentages at the longer wavelength.
This is the opposite of the expected trend for the scattering-induced polarization by dust grains with sizes up to $\sim$100 $\mu$m,
where the scattering cross-section drops steeply with wavelength as $\lambda^{-4}$ \citep{Kataoka15, Yang16},
unless the region is optically thick.
In addition, the observed polarization fraction is higher than typically predicted by scattering models.
In B335, the continuum emission on the scale which we observe is optically thin (Section \ref{results}).
Thus, the polarized continuum emission down to a 20 au scale in B335 most likely originates from magnetically aligned grains.
Therefore, the magnetic field structures in B335 can be inferred by rotating the observed polarization orientations by 90\arcdeg.

The polarization percentages as a function of wavelength could depend on the compositions and sizes of dust grains \citep[e.g.,][]{Bethell07,Draine09,Ashton18,Valdivia19}.
Among a few dust models that reproduce the spectral energy distributions and extinction curve of the interstellar medium (ISM),
the one with spheroidal silicate and spherical carbon grains shows increasing polarization percentages from submillimeter to millimeter wavelengths \citep{Draine09},
similar to the trend observed in B335.
Nevertheless, the properties of dust grains and the environments, such as grain sizes and radiation fields, are expected to be different in the ISM and dense cores \citep{Bethell07,Ashton18}.
Consequently, ISM dust models might not be directly applicable to B335.

More specifically,
\citet{Valdivia19} computed polarized radiative transfer in typical protostellar envelopes with the radiative torque theory \citep{Lazarian07} and dust size distributions similar to the Mathis-Rumpl-Nordsieck (MRN) distribution \citep{Mathis77}.
They further studied the dependences of the polarization percentages at 0.8 mm and 1.3 mm on the maximum grain sizes.
They found that the presence of dust grains with sizes larger than 10 $\mu$m in protostellar envelopes on a scale of a few hundred au is needed to explain the observed polarized percentages of a few to more than 10\% in protostellar sources.
In addition, in their model calculations with maximum grain sizes of 30--50 $\mu$m in protostellar envelopes,
the polarization percentages on a scale of a few hundred au are higher at 1.3 mm than at 0.8 mm.
These trends are consistent with our observational results of B335.
Thus, our results may hint at the presence of large dust grains with sizes of a few tens of $\mu$m on a 100 au scale in B335.

\subsection{Magnetic field structures}
The previous JCMT and ALMA polarimetric observations \citep{Maury18, Yen19} show that the magnetic field structures in B335 are ordered and along the outflow (east--west) direction on the scale of the natal dense core of 6000 au, and then become highly pinched on the scale of the infalling envelope of 1000 au.
In addition, close to the center, the magnetic field orientations
are almost along the north--south direction, which is the direction of the disk midplane.
These observed magnetic field structures can be explained with non-ideal MHD simulations of a collapsing dense core with a weak magnetic field of mass-to-flux ratios of $\sim$6--10 aligned with the rotational axis \citep{Maury18, Yen19}.

With our observations at higher angular resolutions,
the magnetic field along the north--south direction is detected within a 100 au scale north and south to the center.
In addition to these field lines along the direction of the disk midplane,
there are also magnetic field lines tilted with respect to the midplane in the 100 au region around the central compact source (Fig.~\ref{bmap}).
Our observations also show that the orientation of its surrounding flattened envelope on a 100 au scale is not aligned with the disk midplane but is tilted eastwards by $\sim$20$\arcdeg$ (Fig.~\ref{bmap}).
Such more complicated magnetic field structures around the central disk and the different orientations between the flattened envelope and the disk are not expected in the non-turbulent simulations of a collapsing dense core with a magnetic field and rotational axis that are aligned \citep[e.g.,][]{Li14b, Masson16, Tsukamoto17, Zhao18, Machida19}.

\subsubsection{Simulation with the aligned magnetic field and rotational axis}
We have compared the magnetic field structures detected with our ALMA observations with those in the non-ideal MHD simulation of a collapsing dense core with the aligned magnetic field and rotational axis.
The synthetic {\it IQU} maps of the simulation results were obtained from \citet{Yen19}.
Among the three non-ideal MHD effects, the simulation only includes ambipolar diffusion, which is the most dominant non-ideal MHD effect in protostellar envelopes on a scale of hundreds of au \citep{Zhao18}.
In this simulation, the dense core has the same mass and angular velocity as the observational estimates of B335, and an initial mass-to-flux ratio of 9.6.
A cosmic ray ionization rate of $5 \times 10^{-17}$ was adopted.
The simulation was stopped when the central stellar mass reached 0.1 $M_\sun$.
At the end of our simulation, the disk size was $<$10 au.
These stellar mass and disk size are comparable to the observational estimates of B335 \citep{Evans15, Yen15, Bjerkeli19, Imai19}.
The synthetic {\it IQU} maps were generated using the radiative transfer code, Simulation Package for Astronomical Radiative Xfer (SPARX; \url{https://sparx.tiara.sinica.edu.tw/}), on the assumption of a constant polarization efficiency meaning that the polarized intensity is simply proportional to the column density.
The details of the simulation are described in \citet{Yen19}.
This simulation can explain observed magnetic field structures on a few hundred au to 6000 au scales in B335.
We then synthetically observed the Stokes {\it IQU} maps output by SPARX using the CASA simulator.
The resulting simulated maps are presented in Fig.~\ref{simob} (a)--(c).

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{simob.eps}
\caption{(a) Density (color; in units of g~cm$^{-3}$) and magnetic field lines (white lines) in our non-ideal MHD simulation of a collapsing dense core with the aligned magnetic field and rotational axis. The outflow direction is along the vertical direction. (b) Synthetic ALMA observational results of our simulation in (a). The synthetic observations are rotated to have an inclination angle of 80$\arcdeg$ and an outflow direction along the east--west direction, the same as in B335. The contours show the distribution of the Stokes {\it I} intensity, and the segments show the magnetic field orientations inferred by rotating the polarization orientations by 90$\arcdeg$. (c) Comparison of the observed (orange) and model (blue) magnetic field orientations. The blue segments are the same as those in (b). Panels (d)--(f) are the same as (a)--(c) but for our simulation of a collapsing dense core with its magnetic field misaligned with the rotational axis by 15\arcdeg. The outflow direction is also along the vertical direction in (d). In (a) and (d), the magnetic field lines were plotted using iso-magnetic flux contours as approximation, and the dashed contours near the polar axis denote the regions with negative magnetic fluxes, where the structures of the magnetic field lines cannot be properly evaluated in two-dimensional plots.}\label{simob}
\end{figure*}

In the simulation with the aligned magnetic field and rotational axis,
the Stokes {\it I} intensity from large to small scale is all elongated along the direction of the disk midplane, which is the north--south direction (Fig.~\ref{simob}b).
The magnetic field inferred from the synthetic polarization maps is perpendicular to the disk midplane within 0\farcs2 in the north and south of the intensity peak.
Then, it becomes almost parallel to the disk midplane at outer radii in the north and the south (Fig.~\ref{simob}b).
Thus, this simulation can explain the magnetic field orientations parallel to the disk midplane observed in the north and the south on a 100 au scale in B335.
Especially, it reproduces most of the observed magnetic field orientations in the south (Fig.~\ref{simob}c).
However, the magnetic field near the intensity peak in the synthetic maps is not along the northwest--southeast direction,
which is different from the observations.
Thus, the observed magnetic field structures within a 100 au scale cannot be fully explained with the simulation of a collapsing dense core with the aligned magnetic field and rotational axis, although this simulation was able to explain the pinched magnetic field observed on scales from 6000 au down to a few hundred au \citep[Fig.~\ref{jcmt_simob} upper; ][]{Yen19}.

\subsubsection{Simulation with the misaligned magnetic field and rotational axis}
On the contrary, in simulations of a collapsing dense core with the misaligned magnetic field and rotational axis or with turbulence,
the magnetic field structures tend to become irregular, different from a simple hourglass shape \citep[e.g.,][]{Li14b, Seifried15, Masson16, Hull17a, Tsukamoto17, Wurster19a}.
Thus, we additionally performed non-ideal MHD simulations of a collapsing dense core with its magnetic field misaligned with its rotational axis by 15\arcdeg.
The mass, angular velocity, and initial mass-to-flux ratio of the dense core are the same as those in our simulation with the aligned magnetic field and rotational axis.
Due to the slight misalignment, the disk is easier to grow in size compared to the simulation without the misalignment.
Thus, in this simulation with the misalignment,
the cosmic-ray ionization rate was increased to be $1\times10^{-16}$~s$^{-1}$, which enhances the coupling between the magnetic field and the matter,
and the disk size was also suppressed to be $<$10 au to match the observational constraint of the disk size in B335.
The simulation was stopped at a similar evolutionary time when the central stellar mass reached 0.1 $M_\sun$.
Thus, the only differences between the initial conditions of these two simulations are the cosmic-ray ionization rates and the angles between the magnetic field and rotational axis.
Finally, we produced synthetic observations using the CASA simulator in the same manner as for the simulations with the aligned magnetic field and rotation axis.

The synthetic maps from our simulation with the misaligned magnetic field and rotational axis are presented in Fig.~\ref{simob}(d)--(f).
Because the rotational axis and magnetic field are misaligned,
the flattened envelope becomes warped on the small scale of 100 au (Fig.~\ref{simob}d),
where the rotational energy becomes also important compared to the gravitational and magnetic energy.
As a result, the elongations of the Stokes {\it I} intensity distributions in the synthetic map change from large to small scales (Fig.~\ref{simob}e).
The magnetic field is dragged by the accretion flows along the warped envelope, resulting in more complicated magnetic field structures.
We note that this simulation with the misalignment has a higher cosmic ray ionization rate.
In the simulations, the magnetic field structures are shaped by the gas motions.
Increasing the cosmic ray ionization rate does not directly affect the magnetic field structures but only strengthens the coupling between the gas motions and the magnetic field \citep{Yen19}.
Thus, the complicated magnetic field structures seen in this simulation is due to the changes in the gas motion caused by the misalignment.

In the synthetic map, the magnetic field is parallel to the disk plane at the intensity peak and is along the northwest--southeast direction within 0\farcs5 around the intensity peak.
Then, the magnetic field becomes more aligned along the north--south direction in the outer regions (Fig.~\ref{simob}e).
These features are similar to what is observed on a 100 au scale in B335,
although the regions showing the magnetic field along the northwest--southeast direction extend further away from the disk midplane in the observations compared to our simulation.
Outside the central 100 au region, the magnetic field structures in the synthetic maps of the simulations with and without misalignment are similar (Fig.~\ref{simob}b \& e).
Thus, our new simulation with the misalignment can also explain the magnetic field structures in the protostellar envelope on a scale of a few hundred au in B335 detected with the previous ALMA observations at a lower spatial resolution of 70--80 au \citep{Maury18,Yen19}.

\subsubsection{Comparison between the aligned and misaligned cases}
In Figure \ref{diff}, we present the number distributions of the angle difference of the magnetic field orientations in the central 100 au region extracted from the observed and synthetic polarization data.
The number of the model segments with their orientations consistent with the observations within 30$\arcdeg$ increases by 50\%,
when we include the misalignment in our simulations.
Thus, the simulation with the misalignment can explain the observed magnetic field orientations better than the simulation without the misalignment.

We have also compared the magnetic field structures on the large scale in this simulation with those observed with JCMT \citep{Yen19}.
The degree of the misalignment between the magnetic field and rotational axis in the initial core in our simulation is only 15$\arcdeg$ in the three-dimensional space.
Thus, after projection on the plane of the sky, the simulation with the misalignment (Fig.~\ref{jcmt_simob} lower panel) can reproduce the observed magnetic field orientations on the 6000 au scale in B335 similar to the simulation with the aligned magnetic field and rotation axis (Fig.~\ref{jcmt_simob} upper panel).
Hence, the critical difference between the aligned and the misaligned cases only manifests itself in the inner 100 au region.

\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{model_diff.eps}
\caption{Number distributions of angle difference between observed and model magnetic field orientations in the central 100 au region. Red solid and blue dashed histograms show the cases for the simulations with the magnetic field misaligned and aligned with the rotational axis, respectively.}\label{diff}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{jcmt_simob.eps}
\caption{Comparison of the magnetic field orientations observed with JCMT (orange segments) and those from our non-ideal MHD simulations (blue segments) of collapsing dense cores with the magnetic field aligned (upper panel) and misaligned (lower panel) with the rotational axes. The contours show the Stokes {\it I} map of B335 obtained with JCMT. The JCMT polarimetric data were retrieved from \citet{Yen19} and were reduced again with the updated software and procedure. The S/N were improved with the new data reduction, and we obtained more detections than \citet{Yen19}.}\label{jcmt_simob}
\end{figure}

We note that the intensity distribution is elongated along the northeast--southwest direction on a 1$\arcsec$ scale and then becomes along the northwest--southeast direction on a 0\farcs5 scale in our synthetic Stokes {\it I} map of the simulation with the misalignment (Fig.~\ref{simob}e).
This change in the intensity distribution is not fully consistent with the observations.
The observed intensity distribution is elongated along the northeast--southwest direction on a 1$\arcsec$ scale,
and then transitions into a roundish morphology on a 0\farcs5 scale.
Our simulations were performed with the barotropic approximation,
which could underestimate the temperature in the inner region of tens of au compared to radiative MHD simulations \citep{Tomida10}.
Consequently, the material could be more concentrated in the midplane in our simulations than in reality.
Because the Stokes {\it I} intensity distribution is sensitive to the density distribution,
this effect could make the warped structures more evident in our synthetic intensity map, compared to the observations.
On the contrary, the orientations of the magnetic field lines are more sensitive to the directions of the gas flows.

In addition,
a constant polarization efficiency was adopted in our radiative transfer calculations,
so our synthetic polarization maps primarily trace the magnetic field in the envelope.
Thus, there are no magnetic field orientations along the outflow cavity wall detected in our synthetic maps,
while such detections are seen in the northeast in the observed maps (Fig.~\ref{bmap}).
We also note that the magnetic field structures and density distributions in our simulations (even for the misaligned case) are more or less axisymmetric.
Thus, the toroidal components of the magnetic field along the line of slight may be canceled out in our radiative transfer calculations,
causing our synthetic polarization maps to show axisymmetric patterns (Fig.~\ref{simob} and \ref{jcmt_simob}).
However, the observations show the magnetic field orientations are different in the north and the south of the center in B335,
which is not seen in our synthetic maps and could suggest the presence of asymmetric structures in the magnetic field or the density distributions along the line of slight.
This observed asymmetry is often seen in numerical simulations with turbulence \citep[e.g.,][]{Li14b, Masson16, Matsumoto17, Wurster19a}.
Thus, in addition to seeing magnetic field perturbations caused by the misalignment of the magnetic field and the rotation axis,
we may also be seeing the effect of gravo-turbulence in B335,
which could cause asymmetries that become more evident on small scales as the core collapses.

Besides B335,
ordered magnetic fields in a pinched hourglass shape have been observed on a 1000 au scale in several protostellar sources \citep{Girart06, Stephens13, Rao14, Cox18, Sadavoy18a, Sadavoy19, Kwon19, Ko19}.
These results hint at an important role of the magnetic field in the dynamics in these protostellar sources.
On the other hand, there are also protostellar sources showing disordered magnetic field structures on a 1000 au scale,
which can be explained better with simulations with turbulence \citep{Hull17a, Hull17b, Cox18, Sadavoy18b, Sadavoy19}.
Our observations together with the previous JCMT and ALMA observations show that the magnetic field from large to small scales changes from the ordered pinched structures to more complicated and asymmetric structures in B335.
A similar trend has also been seen in other protostellar sources, such as IRAS 16293$-$2422 \citep{Rao14,Sadavoy18b}.
These results suggest that the relative importance among magnetic field, rotation, and turbulence changes as a function of scale.
Furthermore, it seems that the influence of rotation and turbulence, which are passed down from larger scales and enhanced by gravitational collapse, may become more significant in the inner envelopes around protostellar disks.

\section{Summary}\label{summary}
We present our observational results of the polarized 0.87 mm continuum emission at an 0\farcs2 resolution in B335 obtained with ALMA.
We compare our results with those from the ALMA polarimetric observations at 1.3 mm with an 0\farcs8 resolution as well as with the synthetic maps generated from our non-ideal MHD simulations of collapsing dense cores.
The main results are summarized below.
\begin{enumerate}
\item{
The polarization orientations at 0.87 mm and 1.3 mm are consistent within the uncertainties.
The polarization percentage is higher at 1.3 mm than 0.87 mm on scales from 1000 au down to tens of au in B335,
suggesting that the polarized emission originates from magnetically aligned dust grains.
In addition, the peak brightness temperature of the 0.87 mm continuum emission on a 20 au scale is 30 K,
more than a factor of two lower than the expected dust temperature.
Thus, the 0.87 mm continuum emission is most likely optically thin down to a 20 au scale in B335,
and the observed polarization orientations can be rotated by 90$\arcdeg$ to infer the magnetic field orientations.}
\item{Our observations show that the magnetic field structures in B335 change from an ordered pinched morphology on a 1000 au scale to more complicated structures on a 100 au scale.
Within a 100 au scale,
in addition to the magnetic field along the equatorial plane in the north and the south to the center,
there are also magnetic field lines along the northwest--southeast direction that are connected to the central disk-forming region.}
\item{We have performed non-ideal MHD simulations of collapsing dense cores with their magnetic field aligned and misaligned with their rotational axes,
and generated synthetic polarization maps to compare with the observations.
The simulation with the aligned magnetic field and rotational axis can explain the observed magnetic field orientations along the equatorial plane on a 100 au scale, but cannot explain those along the northwest--southeast direction connecting the central disk-forming region.
On the contrary,
the simulation with the misaligned magnetic field and rotational axis can reproduce both the magnetic field orientations along the equatorial plane and the northwest--southeast direction observed on a 100 au scale in B355.
The misalignment is 15$\arcdeg$ in our simulation, which is relatively small,
and thus, this simulation can also explain the observed magnetic field orientations on a 6000 au scale with JCMT.}
\item{Our results suggest that the magnetic field and rotational axis in B335 are likely slightly misaligned.
In addition, the observed different magnetic field orientations in the north and south on a scale within tens of au,
which is not seen in our simulation with the misaligned magnetic field and rotation axis, could hint at the contribution of gravo-turbulence in B335.
Therefore, our observational results suggest that the relative importance between magnetic field, rotation, and turbulence changes as a function of scale in protostellar sources.
On the small scale around the disk-forming region in B335,
the rotational energy and influence by turbulence become more significant,
and thus, the protostellar envelope is warped.
The magnetic field lines could be dragged by the accretion flows along the warped envelope, resulting in more complicated and asymmetric structures.
}
\end{enumerate}

\begin{acknowledgements}
We thank Alfonso Trejo-Cruz for his efforts on the manual data calibration.
We thank I-Ta Hsieh and Sheng-Yuan Liu for their assistance and advice on our radiative transfer calculation using SPARX.
This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.1.01018.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
We thank all the ALMA staff supporting this work.
H.-W.Y. acknowledges support from MOST 108-2112-M-001-003-MY2.
P.M.K. acknowledges support from MOST 108-2112-M-001-012 and MOST 107-2119-M-001-023, and from an Academia Sinica Career Development Award.
ZYL is supported in part by NASA 80NSSC18K1095 and NSF AST-1716259 and 1815784.
S.T. acknowledges a grant from JSPS KAKENHI grant No. JP18K03703 in support of this work.
This work was supported by NAOJ ALMA Scientific Research grant No. 2017-04A.
\end{acknowledgements}