# Proof of a Conjecture of Wiegold for nilpotent Lie algebras

Alexander Skutin

###### Key words and phrases:

Lie algebra, Rings and algebras

###### 2010 Mathematics Subject Classification:

17B30

##  1\. Introduction

In this short note we confirm an analog of a conjecture of James Wiegold [1,
4.69] for finite dimensional nilpotent Lie algebras.

######  Conjecture 1.1 (Wiegold’s conjecture).

Let $G$ be a finite $p$-group and let $|G^{\prime}|>p^{n(n-1)/2}$ for some
non-negative integer $n$. Then the group $G$ can be generated by the elements
of breadth at least $n$. The breadth $b(x)$ of an element $x$ of a finite
$p$-group $G$ is defined by the equation $|G:C_{G}(x)|=p^{b(x)}$, where
$C_{G}(x)$ is the centralizer of $x$ in $G$.

An overview of this problem can be found in [3]. In [2] M.R.Vaughan-Lee proved
that for a finite $p$-group $G$ of breadth $b=\max_{g\in G}b(g)$, we have
$|G^{\prime}|\leq p^{b(b-1)/2}$. He also showed that for a finite dimensional
nilpotent Lie algebra $\mathfrak{g}$ of breadth
$b=\max_{g\in\mathfrak{g}}b(g)$, $\dim[\mathfrak{g},\mathfrak{g}]\leq
b(b+1)/2$.

Conjecture 1.1 was proved by the Author in [5]. The goal of this paper is to
prove the Lie algebra analog of this conjecture.

The breadth $b(x)$ of an element $x$ of a finite dimensional Lie algebra
$\mathfrak{g}$ over a field $\mathbb{F}$ is defined by the equation
$\dim\mathfrak{g}-\dim C_{\mathfrak{g}}(x)=b(x)$, where $C_{\mathfrak{g}}(x)$
is the Lie centralizer of $x$ in $\mathfrak{g}$. We prove the following

######  Conjecture 1.2 (Wiegold’s conjecture for nilpotent Lie algebras).

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a field $\mathbb{F}$ and
let $\dim\mathfrak{g}^{\prime}>n(n-1)/2$ for some non-negative integer $n$.
Then the Lie algebra $\mathfrak{g}$ can be generated by the elements of
breadth at least $n$.

In this article we confirm the Conjecture 1.2 and also show that the more
general results are true:

######  Theorem 1.1.

Let $\mathfrak{g}$ be a nilpotent Lie algebra over an infinite field
$\mathbb{F}$ and let $\dim\mathfrak{g}^{\prime}>n(n-1)/2$ for a non-negative
$n$. Then the set of elements of breadth at least $n$ cannot be covered by a
finite number of proper Lie subalgebras in $\mathfrak{g}$.

######  Theorem 1.2.

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a finite field
$\mathbb{F}\not=\mathbb{F}_{2}$ and let $\dim\mathfrak{g}^{\prime}>n(n-1)/2$
for some non-negative integer $n$. Then the set of elements of breadth at
least $n$ cannot be covered by $|\mathbb{F}|-1$ proper Lie subalgebras in
$\mathfrak{g}$.

In the particular case $\mathbb{F}=\mathbb{F}_{2}$ we also get a more general
result:

######  Theorem 1.3.

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a field $\mathbb{F}_{2}$
and let $\dim\mathfrak{g}^{\prime}>n(n-1)/2$ for some non-negative integer
$n$. Then the set of elements of breadth at least $n$ cannot be covered by two
proper Lie subalgebras in $\mathfrak{g}$, one of which has codimension at
least 2 in $\mathfrak{g}$.

##  2\. Formulations and proofs of the main Lemmas

The breadth $b_{\mathfrak{h}}(x)$ of an element $x$ of a finite dimensional
Lie algebra $\mathfrak{g}$ with respect to a subalgebra
$\mathfrak{h}\subseteq\mathfrak{g}$ is defined by the equation
$\dim\mathfrak{h}-\dim C_{\mathfrak{h}}(x)=b_{\mathfrak{h}}(x)$, where
$C_{\mathfrak{h}}(x)=\\{h\in\mathfrak{h}|[x,h]=0\\}$ is the centralizer of $x$
in $\mathfrak{h}$. By definition, $b(x)=b_{\mathfrak{g}}(x)$.

The following two lemmas are the well-known facts in theory of Lie algebras,
so we state them without proof.

######  Lemma 2.1.

Let $\mathfrak{g}$ be a Lie algebra over a finite field $\mathbb{F}$. Then
$\mathfrak{g}$ cannot be covered by $|\mathbb{F}|$ proper subalgebras.
Additionally, if $\mathfrak{g}$ is covered by $|\mathbb{F}|+1$ proper
subalgebras $\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$, $\ldots$,
$\mathfrak{h}_{|\mathbb{F}|+1}$, then each of these $\mathfrak{h}_{i}$ must
have codimension $1$ in $\mathfrak{g}$.

######  Lemma 2.2.

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a field $\mathbb{F}$, such
that there exists a central subalgebra $\mathfrak{f}$ of codimension $2$ in
$\mathfrak{g}$. Then $\dim\mathfrak{g}^{\prime}\leq 1$.

######  Lemma 2.3.

Consider an ideal $\mathfrak{h}$ of a finite dimensional nilpotent Lie algebra
$\mathfrak{g}$. Let $\mathfrak{f}$ be the ideal of $\mathfrak{g}$ generated by
elements $g$ from $\mathfrak{h}$, such that $b_{\mathfrak{h}}(g)=b(g)$. Then

  1. (1)

If $\mathfrak{f}=\mathfrak{h}$, then
$\mathfrak{g}^{\prime}=\mathfrak{h}^{\prime}$;

  2. (2)

If $\mathfrak{f}$ has a codimension at most $1$ in $\mathfrak{h}$, then
$\mathfrak{h}^{\prime}$ has a codimension at most $1$ in
$\mathfrak{g}^{\prime}$.

###### Proof.

Consider the factorization homomorphism
$\pi:\mathfrak{g}\to\mathfrak{g}/\mathfrak{h}^{\prime}$. For each element $g$
in $\mathfrak{h}$, such that $b_{\mathfrak{h}}(g)=b(g)$ we have
$\\{[g,g^{\prime}]|g^{\prime}\in\mathfrak{g}\\}=\\{[g,h]|h\in\mathfrak{h}\\}\subseteq\mathfrak{h}^{\prime}$,
so $\pi(g)$ lies in the center of $\pi(\mathfrak{g})$ and $\pi(\mathfrak{f})$
is the central Lie subalgebra of $\pi(\mathfrak{g})$. If
$\mathfrak{f}=\mathfrak{h}$, then $\pi(\mathfrak{f})=\pi(\mathfrak{h})$ is the
central Lie subalgebra of $\pi(\mathfrak{g})$ of codimension $1$, so
$\pi(\mathfrak{g})$ is abelian and
$\mathfrak{g}^{\prime}=\mathfrak{h}^{\prime}$. If $\mathfrak{f}$ has a
codimension at most $1$ in $\mathfrak{h}$, then $\pi(\mathfrak{f})$ is the
central Lie subalgebra of $\pi(\mathfrak{g})$ of codimension at most $2$, so
from Lemma 2.2 we get $\dim\mathfrak{g}^{\prime}\leq 1$ and
$\mathfrak{h}^{\prime}$ has a codimension at most $1$ in
$\mathfrak{g}^{\prime}$.

∎

######  Lemma 2.4.

Let $\mathfrak{g}$ be a finite dimensional Lie algebra and let $\mathfrak{h}$
be its ideal of codimension $1$ in $\mathfrak{g}$. Then for any element $x$
from the set $\mathfrak{g}\setminus\mathfrak{h}$ we have
$\dim\mathfrak{g}^{\prime}\leq b(x)+\dim\mathfrak{h}^{\prime}$.

###### Proof.

The set $V=\\{[x,h]|h\in\mathfrak{h}\\}$ form a vector space with the
dimension not bigger than $b(x)$. So it is enough to prove that
$\mathfrak{g}^{\prime}=V+\mathfrak{h}^{\prime}$. This follows from the
following properties of the set $V+\mathfrak{h}^{\prime}$ :

  1. (1)

$V+\mathfrak{h}^{\prime}$ is a vector space;

  2. (2)

$V+\mathfrak{h}^{\prime}$ is an ideal in $\mathfrak{g}$ :
$[[x,h],g]+\mathfrak{h}^{\prime}=\overbrace{[x,\underbrace{[h,g]}_{\in\mathfrak{h}}]}^{\in
V}+\overbrace{[h,\underbrace{[x,g]}_{\in\mathfrak{h}}]}^{\in\mathfrak{h}^{\prime}}+\mathfrak{h}^{\prime}$;

  3. (3)

Lie algebra $\mathfrak{g}/(V+\mathfrak{h}^{\prime})$ is abelian.

These facts imply the Lemma 2.4.

∎

##  3\. Proof of Theorem 1.1

Assume the converse. Let the proper subalgebras $\mathfrak{h}_{1}$,
$\mathfrak{h}_{2}$, $\ldots$, $\mathfrak{h}_{k}$ cover all the elements of
breadth at least $n$. We will prove that $\dim\mathfrak{g}^{\prime}\leq
n(n-1)/2$. Considering a sufficiently large finite-dimensional subalgebra in
$\mathfrak{g}$, we can assume that $\mathfrak{g}$ is finite-dimensional. The
proof is by induction on $\dim\mathfrak{g}$. We can assume that
$\mathfrak{h}_{i}$ are the maiximal ideals of codimension $1$ in
$\mathfrak{g}$ (because every proper subalgebra in the nilpotent and finite
dimensional Lie algebra is contained in the ideal of codimension $1$).
Consider any ideal $I\not=\mathfrak{h}_{i}$ of codimension $1$ in
$\mathfrak{g}$. Denote by $J$ the ideal of $\mathfrak{g}$ generated by
elements $g$ from $I$ such that $b_{I}(g)=b(g)$. Consider the case when $J=I$.
Applying the Lemma 2.3 we conclude that $I^{\prime}=\mathfrak{g}^{\prime}$.
The rest follows from the induction hypothesis : the Lie algebra $I$ has
smaller dimension and all its elements of breadth at least $n$ are contained
in the union of proper subalgebras $I\cap\mathfrak{h}_{i}$ (because
$b_{I}(x)\leq b(x)$). Now consider the case $I\not=J$. Notice that the set
$I\setminus\left(J\cup_{i}\mathfrak{h}_{i}\right)$ contain only the elements
of breadth at most $n-2$ (because $J$ is generated by elements $\\{g\in
I|b_{I}(g)=b(g)\\}$). Thus, from the induction hypothesis applied to $I$ and
its $k+1$ proper subalgebras $J$, $I\cap\mathfrak{h}_{1}$,
$I\cap\mathfrak{h}_{2}$, $\ldots$, $I\cap\mathfrak{h}_{k}$ we conclude that
$\dim I^{\prime}\leq(n-1)(n-2)/2$. Finally, consider any element $a$ not lying
in $I\cup_{i}\mathfrak{h}_{i}$, its breadth is less than $n$, so from the
Lemma 2.4 we get $\dim\mathfrak{g}^{\prime}\leq b(a)+\dim
I^{\prime}\leq\frac{n(n-1)}{2}$.

$\Box$

##  4\. Proof of Theorem 1.2

Assume the converse. Let the proper subalgebras $\mathfrak{h}_{1}$,
$\mathfrak{h}_{2}$, $\ldots$, $\mathfrak{h}_{|\mathbb{F}|-1}$ cover all the
elements of breadth at least $n$. We will prove that
$\dim\mathfrak{g}^{\prime}\leq n(n-1)/2$. Considering a sufficiently large
finite-dimensional subalgebra in $\mathfrak{g}$, we can assume that
$\mathfrak{g}$ is finite-dimensional. The proof is by induction on
$\dim\mathfrak{g}$. We can assume that $\mathfrak{h}_{i}$ are the maiximal
ideals of codimension $1$ in $\mathfrak{g}$ (because every proper subalgebra
in the nilpotent and finite dimensional Lie algebra is contained in the ideal
of codimension $1$) and $\mathfrak{h}_{1}\not=\mathfrak{h}_{2}$ (because in
every non-abelian finite dimensional and nilpotent Lie algebra $\mathfrak{g}$,
there are at least two different maximal proper subalgebras). Consider any
ideal $I\not=\mathfrak{h}_{i}$ of codimension $1$ in $\mathfrak{g}$ such that
$I\cap\mathfrak{h}_{1}=I\cap\mathfrak{h}_{2}=\mathfrak{h}_{1}\cap\mathfrak{h}_{2}$.
Denote by $J$ the ideal of $\mathfrak{g}$ generated by elements $g$ from $I$
such that $b_{I}(g)=b(g)$. Consider the case when $J=I$. Applying the Lemma
2.3 we conclude that $I^{\prime}=\mathfrak{g}^{\prime}$. The rest follows from
the induction hypothesis : the Lie algebra $I$ has smaller dimension and all
its elements of breadth at least $n$ are contained in the union of proper
subalgebras $I\cap\mathfrak{h}_{i}$ (because $b_{I}(x)\leq b(x)$). Now
consider the case $I\not=J$. Notice that the set
$I\setminus\left(J\cup_{i}\mathfrak{h}_{i}\right)$ contain only the elements
of breadth at most $n-2$ (because $J$ is generated by elements $\\{g\in
I|b_{I}(g)=b(g)\\}$). Thus, from the induction hypothesis applied to $I$ and
its $|\mathbb{F}|-1$ proper subalgebras $J$,
$I\cap\mathfrak{h}_{1}=I\cap\mathfrak{h}_{2}$, $I\cap\mathfrak{h}_{3}$
$\ldots$, $I\cap\mathfrak{h}_{|\mathbb{F}|-1}$ we conclude that $\dim
I^{\prime}\leq(n-1)(n-2)/2$. Finally, consider any element $a$ not lying in
$I\cup_{i}\mathfrak{h}_{i}$ (such element exists because of the Lemma 2.1),
its breadth is less than $n$, so from the Lemma 2.4 we get
$\dim\mathfrak{g}^{\prime}\leq b(a)+\dim I^{\prime}\leq\frac{n(n-1)}{2}$.
$\Box$

##  5\. Proof of Theorem 1.3

######  Theorem 5.1.

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a finite field
$\mathbb{F}$. Let the next two conditions hold for some integers $n\leq k+1$

  1. (1)

The set of all elements of the breadth at least $n$ can be covered by
$|\mathbb{F}|$ subalgebras of $\mathfrak{g}$;

  2. (2)

The set of elements of the breadth at most $k$ generates $\mathfrak{g}$.

Then $\dim\mathfrak{g}^{\prime}\leq\frac{(n-1)(n-2)}{2}+k$.

###### Proof.

Considering a sufficiently large finite-dimensional subalgebra in
$\mathfrak{g}$, we can assume that $\mathfrak{g}$ is finite-dimensional. The
proof is by induction on $\dim\mathfrak{g}$. Let the proper subalgebras
$\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$, $\ldots$,
$\mathfrak{h}_{|\mathbb{F}|}$ cover all the elements of breadth at least $n$.
We can assume that $\mathfrak{h}_{i}$ are the maiximal ideals of codimension
$1$ in $\mathfrak{g}$ (because every proper subalgebra in the finite
dimensional and nilpotent Lie algebra is contained in the ideal of codimension
$1$) and $\mathfrak{h}_{1}\not=\mathfrak{h}_{2}$ (because in every non-abelian
finite dimensional and nilpotent Lie algebra $\mathfrak{g}$, there are at
least two different maximal proper subalgebras). Consider any ideal
$I\not=\mathfrak{h}_{i}$ of codimension $1$ in $\mathfrak{g}$ such that
$I\cap\mathfrak{h}_{1}=I\cap\mathfrak{h}_{2}=\mathfrak{h}_{1}\cap\mathfrak{h}_{2}$.
Denote by $J$ the ideal of $\mathfrak{g}$ generated by elements $g$ from $I$,
such that $b_{I}(g)=b(g)$. Consider the case when $J=I$. Applying the Lemma
2.3 we conclude that $I^{\prime}=\mathfrak{g}^{\prime}$. The rest follows from
the induction hypothesis : the Lie algebra $I$ has smaller dimension and all
its elements of breadth at least $n$ are contained in the union of proper
subalgebras $I\cap\mathfrak{h}_{i}$ (because $b_{I}(x)\leq b(x)$), also the
set $I\setminus\cup_{i\geq 2}\mathfrak{h}_{i}$ generates $I$ (because of the
Lemma 2.1) and consists only of elements of breadth at most $n-1\leq k$.

So we can assume that $J$ is a proper ideal of $I$. Consider the case when the
codimension of $J$ in $I$ is $1$. Apply the Lemma 2.3 to $I$, we conclude that
$\dim\mathfrak{g}^{\prime}\leq\dim I^{\prime}+1$. Apply the induction
hypothesis to the Lie algebra $I$ and to its proper subalgebras $J$,
$\mathfrak{h}_{1}\cap\mathfrak{h}_{2}$, $\mathfrak{h}_{3}$, $\ldots$,
$\mathfrak{h}_{|\mathbb{F}|}$ so we get $\dim
I^{\prime}\leq\frac{(n-3)(n-2)}{2}+n-1$ (every element from the set
$I\setminus\left((\mathfrak{h}_{1}\cap\mathfrak{h}_{2})\cup\mathfrak{h}_{3}\cup\ldots\cup\mathfrak{h}_{|\mathbb{F}|}\right)$
is of the breadth at most $n-1$ in $I$ and this set generates $I$ (because of
the Lemma 2.1), also for any element $g$ from the set
$I\setminus\left(J\cup(\mathfrak{h}_{1}\cap\mathfrak{h}_{2})\cup\mathfrak{h}_{3}\cup\ldots\cup\mathfrak{h}_{|\mathbb{F}|}\right)$,
we have $b_{I}(g)\leq n-2$). So if $k\geq 2$ we get
$\dim\mathfrak{g}^{\prime}\leq\dim
I^{\prime}+1\leq\frac{(n-3)(n-2)}{2}+n=\frac{(n-2)(n-1)}{2}+2\leq\frac{(n-2)(n-1)}{2}+k$
and the induction step is clear. In the case $k\leq 1$, we get $n=2$ (the case
$n\leq 1$ is trivial) and the set
$I\setminus\left(J\cup(\mathfrak{h}_{1}\cap\mathfrak{h}_{2})\cup\mathfrak{h}_{3}\cup\ldots\cup\mathfrak{h}_{|\mathbb{F}|}\right)$
is contained in the center of $I$. It is clear that the set
$I\setminus\left(J\cup(\mathfrak{h}_{1}\cap\mathfrak{h}_{2})\cup\mathfrak{h}_{3}\cup\ldots\cup\mathfrak{h}_{|\mathbb{F}|}\right)$
generates the central subalgebra of the codimension at most $1$ in $I$ (Lemma
2.1), so $I$ is abelian. From the conditions of Theorem 5.1 there exists an
element $g\notin I$ such that $b(g)\leq k\leq 1$. So the Lie algebra
$C_{\mathfrak{g}}(g)$ has codimension at most $1$ in $\mathfrak{g}$. Also the
subalgebra $C_{\mathfrak{g}}(g)\cap I$ has codimension at most $2$ in
$\mathfrak{g}$ and is central in $\mathfrak{g}$. Apply Lemma 2.2 to the Lie
algebra $\mathfrak{g}$, so we conclude that $\dim\mathfrak{g}^{\prime}\leq
1\leq\frac{(n-1)(n-2)}{2}+k$.

Eventually, we can assume that the codimension of $J$ in $I$ is greater or
equal to $2$. So the set
$I\setminus\left(J\cup(\mathfrak{h}_{1}\cap\mathfrak{h}_{2})\cup\mathfrak{h}_{3}\cup\ldots\cup\mathfrak{h}_{|\mathbb{F}|}\right)$
generates $I$ (Lemma 2.1), thus, $I$ is generated by the elements of the
breadth at most $n-2$ in $I$. And we can apply the induction hypothesis to $I$
and its subalgebras $J$, $\mathfrak{h}_{1}\cap\mathfrak{h}_{2}$,
$\mathfrak{h}_{3}$, $\ldots$, $\mathfrak{h}_{|\mathbb{F}|}$ and conclude that
$\dim I^{\prime}\leq\frac{(n-3)(n-2)}{2}+n-2=\frac{(n-2)(n-1)}{2}$. Also from
the conditions of the Theorem 5.1 there exists an element $a\notin I$ such
that $b(a)\leq k$. Applying the Lemma 2.4 to $I$ and $a$, we conclude that
$\dim\mathfrak{g}^{\prime}\leq b(a)+\dim
I^{\prime}\leq\frac{(n-2)(n-1)}{2}+k$.

∎

###  5.1. Proof of Theorem 1.3

Assume the converse. Let the proper subalgebras $\mathfrak{h}_{1}$ and
$\mathfrak{h}_{2}$ cover all the elements of breadth at least $n$ and
$\mathfrak{h}_{2}$ has codimension bigger or equal to $2$ in $\mathfrak{g}$.
We will prove that $\dim\mathfrak{g}^{\prime}\leq n(n-1)/2$. From Lemma 2.1
the set $\mathfrak{g}\setminus(\mathfrak{h}_{1}\cup\mathfrak{h}_{2})$
generates $\mathfrak{g}$. All the elements from
$\mathfrak{g}\setminus(\mathfrak{h}_{1}\cup\mathfrak{h}_{2})$ have breadth at
most $n-1$ in $\mathfrak{g}$, thus, we can apply Theorem 5.1 to
$\mathfrak{g}$, $\mathfrak{h}_{1}$, $\mathfrak{h}_{2}$, $k=n-1$ and get
$\dim\mathfrak{g}^{\prime}\leq\frac{(n-1)(n-2)}{2}+n-1=\frac{n(n-1)}{2}$.
$\Box$

######  Remark 5.1.

In fact, Theorem 1.2 is also a consequence of Theorem 5.1.

##  6\. Acknowledgements

I am grateful to Anton A. Klyachko for stating the problem.

## References

  * [1] V.D. Mazurov and E.I. Khukhro (Editors), Unsolved problems in group theory - The Kourovka notebook, No. 18, arXiv:1401.0300v10, 8 Sep 2017.
  * [2] M.R. Vaughan-Lee, Breadth and commutator subgroups of p-groups, J. Algebra 32 (1974), 278-285.
  * [3] Wiegold. J, Commutator subalgebras of finite p-groups, J. Australian Math, Soc., 10 (1969), 480-484.
  * [4] Borworn Khuhirun, Kailash C. Misra, Ernie Stitzinger, On nilpotent Lie algebras of small breadth, arXiv:1410.2778, 10 Oct 2014.
  * [5] A. Skutin, Proof of a Conjecture of Wiegold, arXiv:1804.01745, 5 Apr 2018.


# A Distorted Charged Dilaton Black Ring

Shohreh Abdolrahimia,b and Christos C. Tzounisa,c aDepartment of Physics and
Astronomy, California State Polytechnic University, 91768 CA, USA
bKavli Institute for Theoretical Physics, UC Santa Barbara, 93106 CA, USA
cSchmid College of Science and Technology, Chapman University, 92866 CA, USA

(2022 June 9)

###### Abstract

We present the distorted five dimensional static black rings in Einstein-
Maxwell-dilaton (EMd) gravity by using the sigma model structure of the
dimensionally reduced field equations. We investigate how a static and neutral
distribution of external matter distorts the black ring. In the case of a
distorted five dimensional static black rings in EMd gravity the distortion
parameters can be adjusted such that the solution is free of conical
singularities in the region of the validity of the solution.

##  I Introduction

This paper is an extension of our previous results Abdolrahimi2020  to a
charged case. Black holes and black objects interact with external matter and
fields. We can gain a a full understanding of the general theory of relativity
and the properties of black holes only if we study the interaction of black
holes with matter and sources. Many exact solutions of Einstein equations
represent idealized situations, i.e. a black hole in an empty universe. There
are several ways to study interacting black holes. The best way of for
studying interaction of the black holes is the numerical analysis and the
study of the dynamical systems. However, considerable insight can be gained
from studying exact (or approximate) solutions describing a black hole tidally
distorted by external matter. In particular, our goal is to see which
properties of black holes/black objects are more general and remain intact
even for interacting black holes.

Geroch and Hartle GerochH  proposed to consider static (or stationary),
axisymmetric space-times which are not asymptotically flat that model to some
extend the interaction of a black hole with the external matter. We consider a
“local” black hole. Such solution is valid only in the neighborhood of a black
hole and is distorted by external sources, located far away. The external
matter is not included into the solution and “located” at the asymptotic
infinity. In what follows, we use the word distorted black hole in this
context. Analysis of the distorted black holes have the major advantage that
these solutions are valid for broad classes of external matter, although we
may need to impose regularity condition as well as some symmetry restrictions.

In the case of static and vacuum, axisymmetric space-times, the external
metric near distorted black holes is constructed using the Weyl form Weyl .
Four-dimensional, distorted, static, axisymmetric, vacuum black holes were
studied in, e.g., dis1 ; dis2 ; dis3 ; dis4 ; dis5 . Distorted, static,
axisymmetric, electrically charged black holes were studied in, e.g.,
Fairhurst ; Israel1776 ; Abdolrahimi2009 , and distorted, stationary,
axisymmetric, vacuum and electrically charged black holes were studied in,
e.g., Tomimatsu ; Breton1997  and Breton1998 ; Ansorg2008 ; Ansorg2009 ;
Ansorg2009-2 , respectively.

The construction of the distorted black hole solutions to higher dimensions
one relies on generalized Weyl form of the metric GeneralizedWeyl . The first
higher dimensional distorted black hole was constructed in Abdolrahimi2010 .
Distorted five-dimensional electrically charged black holes and distorted
Myers-Perry black holes have been respectively constructed in Abdolrahimi2013
; Abdolrahimi2014 . Distorted and deformed black holes have been constructed
and studied in various other papers SCN3 ; dis2A ; dis4A ; Poisson:2009qj ;
Semerak:2016gfz ; Shoom:2015rda ; Basovnik:2016awa ; Ansorg:2010ru ;
Shoom:2015slu ; Abdolrahimi:2014msa ; Abdolrahimi:2017pmt ; Kunz:2017mfe .

The study of distorted black holes have manifested some strange and remarkable
properties. For sufficiently distorted Schwarzschild solutions, the isolated
horizon can be foliated with neither marginally trapped nor outer trapping
two-dimensional surfaces Pilkington . The distortion can cause the space-time
curvature to become very high at some regions of black hole horizon dis5 ;
Abdolrahimi2009 . In the case of the distorted Myers-Perry black hole, the
ratio of the horizon angular momentum and the mass $J^{2}/M^{3}$ is unbounded,
and can grow arbitrarily large Abdolrahimi2014 . Similarly, in contrast to the
case of the undistorted Kerr black hole, for a distorted Kerr black hole
$J^{2}/M^{4}>1$ for some distortion parameters Abdolrahimi:2015c . In the case
of octupole distortions of a Schwarzschild black hole the local observer may
see two images or eyebrow-like structure from a single black hole
Abdolrahimi:2015a ; Abdolrahimi:2015b ; Grover:2018tbq . These features
illustrate the importance of study of distorted black holes in the
investigation of the properties of black holes and illustrating which
properties of black holes are unique and universal. In higher dimensions,
there exist other black hole solutions that do not have spherical horizons,
such as black rings, black saturns, di-rings, and bicycling black rings, black
helical rings, as well as more general blackfolds ER3 ; Emparan2008 ; ER2 ;
Elvang:2007rd ; Iguchi ; Elvang2008 ; Izumi2008 ; Evslin:2007fv ;
Emparan:2011br . In our perspective, it is necessary to investigate the
features of distorted versions of these black objects to fully be able to know
which properties of these black holes or black objects are universal. Various
studies were devoted to investigating how the properties of 4-dimensional
isolated black holes are influenced if they are distorted by an external
matter field, and which of them remain unaffected, such as the analysis of the
algebraic type of a distorted black hole or the local first law of
thermodynamics for distorted black holes Papadopoulos1984 ; Ashtekar1999 ;
Ashtekar2000 ; Ashtekar2000v2 .

In a recent paper, we constructed a five-dimensional static black ring
distorted by a static and neutral distribution of external matter
Abdolrahimi2020 . We demonstrated that the gravitational field of these
external sources can be adjusted to remove the conical singularity of the
undistorted black ring solution. The properties of the distorted black ring
for the specific cases of dipole and quadrupole distortions was analyzed. In
this paper we construct a charged black ring solution of Einstein-Maxwell-
dilaton (EMd) gravity in 5-dimensions, distorted by a static and neutral
distribution of external matter. This paper is organized as follows: In Sec.
II, we discuss the generating technique for the construction of the solution.
In Sec. III, we first overview the black ring solution, then apply the
generating technique to construct the distorted charged black ring solution.
In Sec. IV we briefly analyze some of the properties of the solution and
summarize our results. In this paper, we use the following convention of
units: $G(5)=c=1$, the spacetime signature is +3, and the sign conventions are
those adopted in MTW  .

##  II Generating Technique

The Einstein-Maxwell-dilaton (EMd) gravity in $D$-dimensional spacetimes is
described by the action Sen ; Gibbons ; Garfinkle

$S={1\over 16\pi}\int
d^{D}x\sqrt{-g}\left(R-2g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-e^{-2\alpha\varphi}F^{\mu\nu}F_{\mu\nu}\right),$
(1)

where $R$ is the Ricci scalar, $\alpha$ is the dilaton coupling constant,
$F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$, is the electromagnetic
field tensor, and $\varphi$ is the dilaton field. The field equations derived
from the action are

$\displaystyle R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\partial_{\mu}\varphi\partial_{\nu}\varphi+2e^{-2\alpha\varphi}\left[F_{\mu\rho}F_{\nu}^{\rho}-{g_{\mu\nu}\over
2(D-2)}F_{\beta\rho}F^{\beta\rho}\right],$ (2)
$\displaystyle\nabla_{\mu}\nabla^{\mu}\varphi$ $\displaystyle=$
$\displaystyle-{\alpha\over 2}e^{-2\alpha\varphi}F_{\nu\rho}F^{\nu\rho},$ (3)
$\displaystyle\nabla_{\mu}$
$\displaystyle\left[e^{-2\alpha\varphi}F^{\mu\nu}\right]=0.$ (4)

Here, $R_{\mu\nu}$ is the Ricci tensor. We consider static spacetimes and
denote the Killing vector by $\xi$ ($\xi={\partial}/{\partial t}$). In this
section, we will restrict ourselves to generating asymptotically flat
solutions. Although, a distorted black hole solution is not asymptotically
flat. The point is that in the absence of the distortion fields the solution
reduces to the asymptotically flat EMd static black ring. We follow the
approach presented in Yazadjiev . Let us consider a static asymptotically flat
solution of $D$-dimensional Einstein equations

$ds_{0}^{2}=-e^{2U_{0}}dt^{2}+e^{-{2U_{0}\over D-3}}h_{ij}dx^{i}dx^{j}.$ (5)

We consider the matrix corresponding to the target space following the
procedure presented in Yazadjiev

$P_{0}=e^{(\alpha_{D}-1)U_{0}}\left(\begin{array}[]{cc}e^{2U_{0}}&0\\\ 0&-1\\\
\end{array}\right),$ (6)

and the metric $h_{ij}$. The action is invariant under the symmetry
transformation which can be used to generate a solution of the $D$-dimensional
EMd gravity given by the matrix

$P=AP_{0}A^{T}.$ (7)

and with the same metric $h_{ij}$, where

$A=\left(\begin{array}[]{cc}\cosh(\gamma)&\sinh(\gamma)\\\
\sinh(\gamma)&\cosh(\gamma)\\\ \end{array}\right).$ (8)

Explicitly, we have, (for details see Yazadjiev )

$\displaystyle e^{U}$ $\displaystyle=$
$\displaystyle{e^{U_{0}}\over\left[\cosh^{2}(\gamma)-e^{2U_{0}}\sinh^{2}(\gamma)\right]^{1\over
1+\alpha^{2}_{D}}},$ (9) $\displaystyle e^{-\varphi_{D}}$ $\displaystyle=$
$\displaystyle\left[\cosh^{2}(\gamma)-e^{2U_{0}}\sinh^{2}(\gamma)\right]^{\alpha_{D}\over
1+\alpha^{2}_{D}},$ (10) $\displaystyle\Phi_{D}$ $\displaystyle=$
$\displaystyle{\tanh(\gamma)\over\sqrt{1+\alpha^{2}_{D}}}{1-e^{2U_{0}}\over
1-e^{2U_{0}}\tanh^{2}(\gamma)}.$ (11)

The static asymptotically flat black ring metric can be presented in a simple
form:

$ds_{0}^{2}=-\frac{F(x)}{F(y)}dt^{2}+\frac{1}{A^{2}(x-y)^{2}}\biggl{[}\biggl{(}\frac{F(y)^{2}}{1-x^{2}}dx^{2}+\frac{F(x)F(y)}{y^{2}-1}dy^{2}\biggl{)}+F(x)(y^{2}-1)d\psi^{2}+\frac{F(y)^{2}}{F(x)}(1-x^{2})d\phi^{2}\biggl{]},$
(12)

where $F(x)=1-\mu x$ and $F(y)=1-\mu y$. The coordinate $y$ is in the range
$y\leq-1$ with the black hole horizon located at $y\rightarrow-\infty$. The
coordinate $x$ is in the range $-1\leq x\leq 1$. The parameters $\mu$ and $A$
will be taken to lie in the range $0\leq\mu\leq 1$, $A>0$. The EMd black ring
solution was generated using SO(1, 1) transformations Yazadjiev . The black
ring is not free of conical singularity. This conical singularity can be
placed either at $x=1$ or at $x=-1$. If the asymptotic metric does not contain
a conical singularity, then the mass of the black ring is

$M=\frac{3\pi\mu(1+\mu)}{4G_{5}A^{2}}\,,$ (13)

where $G_{5}$ is Newton’s constant in five dimensions. For fixed $A$, a change
in $\mu$ changes the black ring’s mass. We can choose, the deficit membrane
extended to infinity. In this case, the mass of the ring is

$M=\frac{3\pi\mu\sqrt{1-\mu^{2}}}{4G_{5}A^{2}}\,.$ (14)

In what follows, the sign $\pm$ corresponds to taking the conical singularity
at $x=\pm 1$, respectively. The area of the horizon is

${\cal A}_{H\pm}=8\pi^{2}\cosh^{3\over
1+\alpha^{2}_{5}}(\gamma){\mu^{2}\sqrt{(1+\mu)(1\pm\mu)}\over A^{3}}$ (15)

The transformations (9-11) were used to generate the following EMd solution
Yazadjiev ,

$\displaystyle ds^{2}=-\left[\cosh^{2}(\gamma)-\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{-2\over 1+\alpha^{2}_{5}}{F(x)\over F(y)}dt^{2}$ (16)
$\displaystyle+{\left[\cosh^{2}(\gamma)-\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{1\over 1+\alpha^{2}_{5}}\over
A^{2}(x-y)^{2}}\left[F(x)(y^{2}-1)d\psi^{2}\right.$
$\displaystyle\left.+{F(x)F(y)\over y^{2}-1}dy^{2}+{F^{2}(y)\over
1-x^{2}}dx^{2}+F^{2}(y){1-x^{2}\over F(x)}d\phi^{2}\right],$ $\displaystyle
e^{-\varphi_{5}}=\left[\cosh^{2}(\gamma)-\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{\alpha_{5}\over 1+\alpha^{2}_{5}},$ (17)
$\displaystyle\Phi_{5}={\tanh(\gamma)\over\sqrt{1+\alpha^{2}_{5}}}{1-{F(x)\over
F(y)}\over 1-\tanh^{2}(\gamma){F(x)\over F(y)}}.$ (18)

Initially, this solution was presented first in KL  without derivation. This
solution was derived by the above generating transformation in Yazadjiev .
Detailed analysis of the solution is given in KL ; Yazadjiev . The solution is
asymptotically flat. The charge and the temperature are given by

$\displaystyle Q_{\pm}=\pi\sqrt{{3\over
1+\alpha^{2}_{5}}}\sinh(\gamma)\cosh(\gamma){\mu\sqrt{(1+\mu)(1\pm\mu)}\over
A^{2}},$ (19) $\displaystyle T={A\over 4\pi\mu\cosh^{3\over
1+\alpha^{2}_{5}}(\gamma)}.$ (20)

##  III Distorted Charged Black Ring

###  III.1 Seed Solution, a Distorted Black Ring

Here, we will consider the distorted black hole ring solution recently
generated in Abdolrahimi2020  given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2(\widehat{U}+\widehat{W})}\frac{F(x)}{F(y)}dt^{2}+\frac{1}{A^{2}(x-y)^{2}}\biggl{[}e^{2(\hat{V}+\widehat{U}+\widehat{W})}\biggl{(}\frac{F(y)^{2}}{1-x^{2}}dx^{2}+\frac{F(x)F(y)}{y^{2}-1}dy^{2}\biggl{)}$
(21) $\displaystyle+$ $\displaystyle
e^{-2\widehat{W}}F(x)(y^{2}-1)d\psi^{2}+e^{-2\widehat{U}}\frac{F(y)^{2}}{F(x)}(1-x^{2})d\phi^{2}\biggl{]}~{}.$

The distortion fields $\widehat{U}$ and $\widehat{W}$ satisfy the three-
dimensional Laplace equation

$\widehat{X}_{,\rho\rho}+\frac{1}{\rho}\widehat{X}{,\rho}+\widehat{X}_{,zz}=0,$
(22)

where $\widehat{X}$ is either $\widehat{U}$, or $\widehat{W}$. Let us consider
the cylindrical coordinates $(\rho,z)$, related to $x$ and $y$ by the
transformation

$\displaystyle\rho=\frac{\alpha}{A^{2}(x-y)^{2}}\sqrt{F(x)F(y)(1-x^{2})(y^{2}-1)}~{},$
(23) $\displaystyle z=\frac{\alpha(1-xy)(F(x)+F(y))}{2A^{2}(x-y)^{2}}~{}.$
(24)

In this case, the solution of Eq. (22) is well known and has the following
form:

$\displaystyle\widehat{X}(\rho,z)=\sum_{n\geq
0}\left[A_{n}\,R^{n}+B_{n}\,R^{-(n+1)}\right]P_{n}(\cos\vartheta)\,,$ (25)

where

$\displaystyle R=\frac{\sqrt{\rho^{2}+z^{2}}}{m}\,,\hskip
5.69046pt\cos\vartheta=z/R\,.$ (26)

Here, $P_{n}(\cos\vartheta)$ are the Legendre polynomials of the first kind.
The coefficients $A_{n}$ and $B_{n}$ in the expansion Eq.(25) correspond to
interior and exterior multipole moments, respectively Multipoles1 ;
Multipoles2 . We consider only the case of a black ring distorted by the
presence of the external sources. In other words, we assume all $B_{n}$’s are
zero. In what follows, for $\widehat{U}$, $A_{n}$’s are named $a_{n}$ and for
$\widehat{W}$, $A_{n}$’s are named $b_{n}$. We shall simply call the $a_{n}$
and $b_{n}$ coefficients multipole moments. One can use the analogy with the
Newtonian picture in order to give an interpretation for multiple moments.
Therefore, we expect that that higher-order multiple moments are expected to
be smaller than the lower-order multiple moments. In section VI, we restrict
our analysis to $n=0...2$. The case with $a_{1}\neq 0$ and $a_{2}=0$ is called
the dipole distortion. The case with with $a_{1}=0$ and $a_{2}\neq 0$ is
called the quadrupole distortion. The case with $a_{1}\neq 0$ and $a_{2}\neq
0$ is called the dipole-quadrupole distortion. If the distortion fields
$\widehat{U}$ and $\widehat{W}$ are known, the function $\widehat{V}$ can be
derived explicitly for multiple moments $n=1..4$. For details see
Abdolrahimi2020

$\displaystyle\hat{V}=\hat{V}_{1}+\hat{V}_{2},$ (27)
$\displaystyle\hat{V}_{1}=\sum_{n,k\geq
1}\frac{nk}{n+k}(a_{n}a_{k}+a_{n}b_{k}+b_{n}b_{k})R^{n+k}[P_{n}P_{k}-P_{n-1}P_{k-1}]\,,$
(28)
$\displaystyle\widehat{V}_{2}=\frac{1}{2z}\biggl{[}a_{1}{(R_{1}-2R_{2}+2R_{3})}-b_{1}{(R_{1}+R_{2}-R_{3})}-3(a_{1}+b_{1})z\biggl{]}RP_{1},~{}~{}~{}~{}~{}n=1,$
(29) $\displaystyle\widehat{V}_{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(a_{2}+b_{2})R^{2}P_{2}+\frac{1}{2m}\biggl{[}\biggl{(}a_{2}(R_{1}-2R_{2}+2R_{3})-b_{2}(R_{1}+R_{2}-R_{3})\biggl{)}$
(30) $\displaystyle+$
$\displaystyle\frac{1}{z}\biggl{(}a_{2}{(R_{1}c_{1}-2R_{2}c_{2}+2R_{3}c_{3})}-b_{2}{(R_{1}c_{1}+R_{2}c_{2}-R_{3}c_{3})}\biggl{)}\biggl{]}RP_{1},~{}~{}~{}~{}~{}n=2$
$\displaystyle\widehat{V}_{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(a_{3}+b_{3})R^{3}P_{3}+\frac{1}{4z^{3}}\biggl{(}-a_{3}(R_{1}^{3}-2R_{2}^{3}+2R_{3}^{3})+b_{3}(R_{1}^{3}+R_{2}^{3}-R_{3}^{3})\biggl{)}R^{3}P_{1}^{3}$
(31) $\displaystyle+$
$\displaystyle\frac{3}{4zm^{2}}\biggl{[}a_{3}\biggl{(}(R_{1}-2R_{2}+2R_{3})z^{2}+R_{1}c_{1}^{2}-2R_{2}c_{2}^{2}+2R_{3}c_{3}^{2}\biggl{)}$
$\displaystyle+$ $\displaystyle
b_{3}\biggl{(}-(R_{1}+R_{2}-R_{3})z^{2}-R_{1}c_{1}^{2}-R_{2}c_{2}^{2}+R_{3}c_{3}^{2}\biggl{)}\biggl{]}RP_{1},~{}~{}~{}~{}~{}~{}n=3$
$\displaystyle\widehat{V}_{2}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(a_{4}+b_{4})R^{4}P_{4}-\frac{1}{z^{3}}\biggl{(}a_{4}(R_{1}^{3}-2R_{2}^{3}+2R_{3}^{3})-b_{4}(R_{1}^{3}+R_{2}^{3}-R_{3}^{3})\biggl{)}R^{4}P_{1}^{4}$
(32) $\displaystyle+$
$\displaystyle\frac{1}{m^{3}}\biggl{[}\left(3a_{4}(\frac{1}{2}R_{1}-R_{2}+R_{3})-\frac{3b_{4}}{2}(R_{1}+R_{2}-R_{3})\right)z^{2}$
$\displaystyle+$ $\displaystyle
3\left(-a_{4}(\frac{1}{2}R_{1}c_{1}-R_{2}c_{2}+R_{3}c_{3})+\frac{b_{4}}{2}(R_{1}c_{1}+R_{2}c_{2}-R_{3}c_{3})\right)z$
$\displaystyle+$
$\displaystyle\frac{a_{4}r^{2}}{4z}\left((R_{1}-2R_{2}+2R_{3})z-R_{1}c_{1}+2R_{2}c_{2}-2R_{3}c_{3}\right)$
$\displaystyle-$
$\displaystyle\frac{b_{4}r^{2}}{4z}\left((R_{1}+R_{2}-R_{3})z-R_{1}c_{1}-R_{2}c_{2}+R_{3}c_{3}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2z}\left(a_{4}(R_{1}c_{1}^{3}-2R_{2}c_{2}^{3}+2R_{3}c_{3}^{3})-b_{4}(R_{1}c_{1}^{3}+R_{2}c_{2}^{3}-R_{3}c_{3}^{3})\right)$
$\displaystyle+$
$\displaystyle\frac{3}{2}\left(a_{4}(R_{1}c_{1}^{2}-2R_{2}c_{2}^{2}+2R_{3}c_{3}^{2})-b_{4}(R_{1}c_{1}^{2}+R_{2}c_{2}^{2}-R_{3}c_{3}^{2})\right)\biggl{]}RP_{1}\,~{}~{}~{}~{}~{}~{}n=4\,.$

Here, $P_{n}\equiv P_{n}(z/R)$, and

$\displaystyle\xi_{i}\equiv z-c_{i}~{},$ (33) $\displaystyle
R_{1}\equiv\sqrt{\rho^{2}+\xi_{1}^{2}}~{},$ (34) $\displaystyle
R_{2}\equiv\sqrt{\rho^{2}+\xi_{2}^{2}}~{},$ (35) $\displaystyle
R_{3}\equiv-\sqrt{\rho^{2}+\xi_{3}^{2}}~{},$ (36) $\displaystyle Y_{ij}\equiv
R_{i}R_{j}+\xi_{i}\xi_{j}+\rho^{2}~{}$ (37) $\displaystyle
c_{1}=1/(2A),~{}~{}~{}c_{2}=\mu/(2A),~{}~{}~{}c_{3}=-\mu/(2A).$ (38)

###  III.2 Einstein-Maxwell-dilaton distorted black ring

For the distorted black ring, we have

$\exp(U_{0})=\exp(\widehat{U}+\widehat{W})\sqrt{\frac{F(x)}{F(y)}}\,$ (39)

Using the transformations Eqs. (9-11) we generate the metric of a static
charged distorted black ring

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2(\widehat{U}+\widehat{W})}\frac{F(x)}{F(y)}\left[\cosh^{2}(\gamma)-\exp[2(\widehat{U}+\widehat{W})]\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{-\frac{2}{1+\alpha^{2}_{5}}}dt^{2}$ (40) $\displaystyle+$
$\displaystyle\frac{1}{A^{2}(x-y)^{2}}{\left[\cosh^{2}(\gamma)-\exp[2(\widehat{U}+\widehat{W})]\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{1\over
1+\alpha^{2}_{5}}}\biggl{[}e^{2(\hat{V}+\widehat{U}+\widehat{W})}\biggl{(}\frac{F(y)^{2}}{1-x^{2}}dx^{2}+\frac{F(x)F(y)}{y^{2}-1}dy^{2}\biggl{)}$
$\displaystyle+$ $\displaystyle
e^{-2\widehat{W}}F(x)(y^{2}-1)d\psi^{2}+e^{-2\widehat{U}}\frac{F(y)^{2}}{F(x)}(1-x^{2})d\phi^{2}\biggl{]}~{}.$

$\displaystyle
e^{-\varphi_{5}}=\left[\cosh^{2}(\gamma)-\exp[2(\widehat{U}+\widehat{W})]\sinh^{2}(\gamma){F(x)\over
F(y)}\right]^{\alpha_{5}\over 1+\alpha^{2}_{5}},$ (41)
$\displaystyle\Phi_{5}=\frac{\tanh(\gamma)}{\sqrt{1+\alpha^{2}_{5}}}\frac{F(y)-F(x)\exp[2(\widehat{U}+\widehat{W})]}{F(y)-F(x)\exp[2(\widehat{U}+\widehat{W})]\tanh^{2}(\gamma)}.$
(42)

For $\gamma=0$ this solution corresponds to the distorted five-dimensional
black ring. For the distortions field $\widehat{U}=\widehat{W}=\widehat{V}=0$
this solution corresponds to asymptotically flat five dimensional black rings
in EMd gravity. The transformation does not affect the distortion fields
$\widehat{U}$, $\widehat{V}$ and $\widehat{W}$. Thus, the transformation
electrically charges the black hole only. In other words, we have a solution
which represents a five-dimensional EMdL black ring distorted by external
electrically neutral sources, such that when the distortion fields
$\widehat{U}$, $\widehat{W}$, and $\widehat{V}$ vanish, the solution
represents a five-dimensional EMdL black ring solution in an empty,
asymptotically flat universe. The metric (40) is not asymptotically flat due
to the presence of sources that distort the black hole.

The first condition that we wish to formulate is the no-conical regularity
condition. The analysis of the no-coniocal singularity of this solution gives
us the same result as the case of the uncharged distorted black ring
Abdolrahimi2020 . For the special case where, $\widehat{W}=-\widehat{U}/2$, we
can remove the conical singularity by considering for the distortion fields to
satisfy the following condition ($n=0..4$)

$\displaystyle\frac{e^{\widehat{V}+2\widehat{U}+\widehat{W}}\rvert_{\theta=\pi}}{e^{\widehat{V}+2\widehat{U}+\widehat{W}}\rvert_{\theta=0}}=\exp\biggl{[}-\frac{3}{2}(1-\mu)\sum_{n=1}^{4}\sum_{k=n}^{4}\frac{a_{k}}{\mu^{n}}\biggl{]}=\frac{\sqrt{1+\mu}}{\sqrt{1-\mu}}.$
(43)

For the dipole distortion ($a_{n>1}=0$) in Eq. (43), we get the following
relation between $a_{1}$ and the parameter $\mu$ ,

$a_{1}=\frac{\mu}{3(1-\mu)}\ln\left(\frac{1-\mu}{1+\mu}\right).$ (44)

For the quadrupole distortion ($a_{1}=0,a_{2}\neq 0$, $a_{n>2}=0$) in Eq.
(43), we get the following relation relation between $a_{2}$ and the parameter
$\mu$:

$a_{2}=\frac{\mu^{2}}{3(1-\mu^{2})}\ln\left(\frac{1-\mu}{1+\mu}\right)\,.$
(45)

This implies that the conical singularity condition in the distorted charged
black ring is removable for special values of multiple moments.

Also consider the $(y,\psi)$ section of the metric (21). We can identify
$\psi$ with period $\Delta_{\psi}=2\pi$, without creating any conical
singularity in our local black hole solution, which is defined in the area
near the horizon (recall that infinity is at $x=y=-1$).

##  IV Analysis and Summary

We consider the space-time near the regular horizon of the black hole far away
from the sources. In this case, the solution represents a local black hole. We
focus on the study of the deformations of the horizon. The distorted black
hole horizon is defined by $y\rightarrow-\infty$. The metric of a constant $t$
slice through the horizon (i.e., horizon surface) is

$ds^{2}=\frac{(\cosh\gamma)^{\frac{2}{1+\alpha_{5}^{2}}}}{A^{2}}\biggl{(}F(x)e^{-2\widehat{W}_{0}}d\psi^{2}+\mu^{2}e^{2(\widehat{V}_{0}+\widehat{U}_{0}+\widehat{W}_{0})}\frac{dx^{2}}{1-x^{2}}+\mu^{2}e^{-2\widehat{U}_{0}}\frac{1-x^{2}}{F(x)}d\phi^{2}\biggl{)}.$
(46)

Here, $\widehat{U}_{0}$, $\widehat{W}_{0}$, and $\widehat{V}_{0}$ are
calculated on the horizon. Boundary values of the distortion functions
$\widehat{U},~{}\widehat{W}$ on the horizon are given by

$\displaystyle\widehat{U}_{0}=\widehat{U}(x)|_{H}=\sum_{n\geq 0}a_{n}x^{n},$
(47) $\displaystyle\widehat{W}_{0}=\widehat{W}(x)|_{H}=\sum_{n\geq
0}b_{n}x^{n}.$ (48)

The boundary value of the distortion function $\widehat{V}$ on the horizon is

$\displaystyle\widehat{V}_{0}=\widehat{V}(x)|_{H}=-3\sum_{n\geq
1}^{4}(a_{n}+b_{n})x^{n}+\sum_{n\geq 1}^{2}(2a_{2n}+b_{2n})-\sum_{n\geq
1}^{4}\frac{(a_{n}-b_{n})}{2\mu^{n}}+C\,.$ (49)

The metric Eq. (46) is conformal to the case of a distorted black ring
presented. One can see that the distortion fields $\widehat{U},\widehat{W}$,
and $\widehat{V}$ are smooth on the black ring horizon. In what follows, we
shall set $C=0$ without loss of generality; solutions with different values of
constant $C$ can be related to this solution by rescaling the periods of
$\psi$, $\phi$ and the parameter $A$. The area of the horizon is

$\mathcal{A}_{H}=\frac{4\pi^{2}\mu^{2}}{A^{3}}\sqrt{1+\mu}\cosh\gamma^{\frac{3}{1+\alpha_{5}^{2}}}e^{(V_{0}+2U_{0}+W_{0})|_{x=-1}}\int^{1}_{-1}e^{V_{0}}dx,$
(50)

Note that in the previous paper this expression had a mistake Abdolrahimi2020
. For $\gamma=0$ this reduces to the correct expression for the area of the
horizon. For the case $\widehat{W}=-\widehat{U}/2$ and for $a_{n\geq 2}=0$ we
have:

$\mathcal{A}_{H}=\frac{8\pi^{2}\mu^{2}}{A^{3}}\sqrt{1+\mu}\cosh\gamma^{\frac{3}{1+\alpha_{5}^{2}}}\frac{e^{3a_{1}}-1}{3a_{1}}e^{\frac{3(a_{0}-a_{1}(1+\mu))}{2\mu}}\,.$
(51)

For the case that $a_{1}\ll 1$, we get

$\mathcal{A}_{H}=\frac{8\pi^{2}\mu^{2}}{A^{3}}\sqrt{1+\mu}\cosh\gamma^{\frac{3}{1+\alpha_{5}^{2}}}e^{\frac{3a_{0}}{2}}\\{1-\frac{3a_{1}}{2\mu}\\}\,.$
(52)

For the case that $a_{1}=0$, $a_{n\geq 3}=0$, and $a_{2}\ll 1$, we have

$\mathcal{A}_{H}=\frac{8\pi^{2}\mu^{2}}{A^{3}}\sqrt{1+\mu}\cosh\gamma^{\frac{3}{1+\alpha_{5}^{2}}}e^{\frac{3a_{0}}{2}}\\{1+\frac{(5\mu^{2}-3)a_{2}}{2\mu^{2}}\\}\,.$
(53)

Here we should clarify that for the undistorted case (i.e. $a_{n}=0=b_{n}$)
the Eq. (50) does not match the Eq. (15) because we consider different period
for the $\psi$ coordinate.

The surface gravity of a distorted charged black ring is given by

$\kappa=\frac{A}{2\mu}\left(\cosh\gamma\right)^{\frac{-3}{\alpha_{5}^{2}+1}}\exp\left[\sum_{n=1}^{2}2^{n-1}\left(\frac{A^{2}}{\alpha\mu}\right)^{n}[b_{n}(c_{1}^{n}+c_{2}^{n}+c_{3}^{n})-a_{n}(c_{1}^{n}-2c_{2}^{n}-2c_{3}^{n})]\right]~{}.$
(54)

Using the no-conical singularity condition we get,

$\kappa=\frac{A}{2\mu}\left(\cosh\gamma\right)^{\frac{-3}{\alpha_{5}^{2}+1}}\exp\left[\sum_{n=1}^{2}\frac{-3a_{n}(\mu^{-n}-(-1)^{n}-1)}{4}\right]~{}.$
(55)

Next, let us consider only the dipole-quadrupole distortions, we get the
following for the rescaled surface gravity,

$\tilde{\kappa}=\frac{\kappa}{\kappa_{iso}}=\exp\left[\frac{3(2\mu^{2}a_{2}-a_{1}\mu-
a_{2})}{4\mu^{2}}\right]\,,$ (56)

where,
$\kappa_{iso}=\frac{A}{2\mu}\left(\cosh\gamma\right)^{\frac{-3}{\alpha_{5}^{2}+1}}$
is the surface gravity of the undistorted charged black ring.

Figure 1: Behavior of $\tilde{\kappa}$ as a function of parameter $\mu$. On
the left we have plotted the behavior of $\tilde{\kappa}$ for dipole multiple
moment with $\mu\in(0..0.656)$. On the right we have plotted the behavior of
$\tilde{\kappa}$ for quadrupole multiple moment with $\mu\in(0..0.771)$. We
choose multiple moments (dipole or quadrupole) to be less than $-1$.

Figure 2: On the left we have plotted the behavior of $\tilde{\kappa}$ as a
function of the dipole multiple moment $a_{1}$ for $\mu=0.5$. On the right we
have plotted the behavior of $\tilde{\kappa}$ as a function of the quadrupole
multiple moment $a_{2}$ for $\mu=0.5$.

In the Fig. 1, we have plotted the behavior of
$\tilde{\kappa}=\kappa/\kappa_{iso}$ as a function of the parameter $\mu$,
where $\mu$ is a parameter of the background undistorted black ring. Relations
(44) and (45) yield a prescription for the multiple moments in terms of $\mu$,
fine-tuned to avoid conical singularity. We consider $|a_{n}|<1$ and
$|b_{n}|<1$. The reason for this is that physically, a small multiple moment
corresponds to distortion of the black object distorted by masses smaller than
the object itself. On the other hand, a large multiple moment would result
from surrounding the black object with masses much larger than the object
itself. In our analysis, we restrict ourselves, to small multiple moment
values, so that the resulting fine tuned multiple moments are likewise small,
i.e., less than $-1$ (large values of $\mu$ imply correspondingly large
multiple moment values using (44) and (45) ). We consider $\mu\in(0,0.656)$
for the dipole multiple moment. As for the quadrupole multiple moment, we
consider $\mu\in(0,0.771)$. In the left diagram of Fig. 1, we consider the
case of dipole multiple moment by setting $a_{2}=0$ and use the Eq. (44) to
related $a_{1}$ with the parameter $\mu$. Note that increasing $\mu$ implies
increasingly negative $a_{1}$.

In the right diagram of Fig. 1, we consider a quadrupole multiple moment by
setting $a_{1}=0$, and use the Eq. (45) to related $a_{2}$ with the parameter
$\mu$. Moreover, if one increases $\mu$ then $a_{2}$ is going to be increased
negatively. We see that an increasingly large dipole moment strengthens the
surface gravity of the distorted black ring. An increasingly large quadrupole
moment strengthens the surface gravity of the distorted black ring. This
happens only up to a certain maximal value. Beyond this value the surface
gravity of the black ring decreases, ultimately becoming less than its
undistorted counterpart as the quadrupole parameter attains its maximal
negative value. For $\mu\rightarrow 1$, the surface gravity $\kappa\rightarrow
0$. However, note that this case corresponds to $|a_{2}|\rightarrow\infty$. We
don’t analyze this case since we consider $|a_{n}|<1$.

Next, let us consider a distorted charged black ring having both dipole and
quadrupole distortions. In Fig. 2, we have plotted the behavior of
$\tilde{\kappa}=\kappa/\kappa_{iso}$ as a function of the multiple moments for
a fixed parameter $\mu=0.5$ in the case of dipole-quadrupole. For the case of
dipole-quadrupole distortion, we have two cases. In the left diagram of Fig.
2, we fine tune $a_{2}$, leaving $a_{1}$ as a free parameter. We plot the
behavior of $\tilde{\kappa}$ as a function of the dipole multiple moment
$a_{1}$ by replacing

$a_{2}=-\frac{\ln\left(\frac{1+\mu}{1-\mu}\right)\mu^{4}-3\mu^{4}a_{1}+3\mu^{3}a_{1}}{3\mu^{2}(1-\mu^{2})}\,,$
(57)

which is derived from the no conical singularity condition (43) for the joint
dipole-quadrupole case. In the right diagram of Fig. 2, we have fine tuned
$a_{1}$ and plotted the behavior of $\tilde{\kappa}$ as a function of the
quadrupole multiple moment $a_{2}$ by inverting (57)

$a_{1}=-\frac{\ln\left(\frac{1+\mu}{1-\mu}\right)\mu^{2}-3\mu^{2}a_{2}+3a_{2}}{3\mu(1-\mu)}\,.$
(58)

We find, regardless of the value of $\mu$, that behavior of $\tilde{\kappa}$
remains the same. Namely, as it is respectively illustrated in the left and
right diagrams of Fig. 2, $\tilde{\kappa}$ monotonically decreases for
positive values of dipole moment $a_{1}$ and monotonically increases for
positive values of quadrupole multiple moment $a_{2}$.

In summary, we have constructed the metric of a distorted five-dimensional
charged static black ring, representing a local charged black ring distorted
by external static and neutral distributions of matter sources. The solution
is free of singularities, with distortion fields $\widehat{U}$, $\widehat{W}$
and $\widehat{V}$ that are regular on the horizon. We observe that by careful
fine-tuning of the external distorting sources we can have a solution that is
free of the conical singularity that is present in the isolated charged black
ring. Similar effect was illustrated for uncharged distorted black ring
Abdolrahimi2020 . We have also analyzed the behavior of the distorted black
ring’s surface gravity. For quadrupole distortions increasingly large
(negative) dipole distortions weaken the surface gravity of the black ring,
and increasingly large (positive) quadrupole distortions strengthen the
surface gravity of the black ring. The surface gravity of the distorted black
hole can be significantly larger than the undistorted charged black ring one.
For further analysis, it would be interesting to study the effects of
distortions on the shape of the horizon.

## Acknowledgments

S.A. appreciated the hospitality of the KITP, as a KITP scholar. This research
was supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958.

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