# Number of bounded distance equivalence classes in hulls of repetitive Delone
sets

Dirk Frettlöh  Technische Fakultät, Bielefeld University
<EMAIL_ADDRESS>,  Alexey Garber  School of Mathematical
& Statistical Sciences, The University of Texas Rio Grande Valley, 1 West
University Blvd., Brownsville, TX 78520, USA<EMAIL_ADDRESS>and
Lorenzo Sadun  Department of Mathematics, University of Texas, 2515 Speedway,
PMA 8.100 Austin, TX 78712, USA<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Two Delone sets are bounded distance equivalent to each other if there is a
bijection between them such that the distance of corresponding points is
uniformly bounded. Bounded distance equivalence is an equivalence relation. We
show that the hull of a repetitive Delone set with finite local complexity has
either one equivalence class or uncountably many.

##  1\. Introduction

Delone sets are central objects of study in the theory of aperiodic order and
give rise to dynamical systems and topological objects with interesting
properties [3]. A _Delone set_ $\Lambda\subset\mathbb{R}^{d}$ is a set that is
both _uniformly discrete_ (that is, there is $r>0$ such that each open ball of
radius $r$ contains at most one point of $\Lambda$) and _relatively dense_
(that is, there is $R>0$ such that each closed ball of radius $R$ contains at
least one point of $\Lambda$). A _point lattice_ is the $\mathbb{Z}$-span
$\langle v_{1},\ldots,v_{d}\rangle_{\mathbb{Z}}$ of $d$ linear independent
vectors in $\mathbb{R}^{d}$. Each point lattice is a Delone set. A Delone set
$\Lambda$ in $\mathbb{R}^{d}$ is _$d$ -periodic_ if the set

$P_{\Lambda}=\\{t\in\mathbb{R}^{d}\mid t+\Lambda=\Lambda\\}$

of its period vectors is a point lattice. Of course for every point lattice
$\Lambda$ we have $\Lambda=P_{\Lambda}$, hence each point lattice is
$d$-periodic. A Delone $\Lambda$ set is _nonperiodic_ if
$P_{\Lambda}=\\{0\\}$.

Two Delone sets $\Lambda,\Lambda^{\prime}$ in $\mathbb{R}^{d}$ are called
_bounded distance equivalent_ ($\Lambda\stackrel{{\scriptstyle\rm
bd}}{{\sim}}\Lambda^{\prime}$) or _bde_ , if there is a bijection
$\varphi\colon\Lambda\to\Lambda^{\prime}$ such that $|x-\varphi(x)|$ is
uniformly bounded. It is easy to see that $\stackrel{{\scriptstyle\rm
bd}}{{\sim}}$ is an equivalence relation for Delone sets.

In the 1990s several authors studied the question of whether a given Delone
set in $\mathbb{R}^{d}$ is bounded distant equivalent to a point lattice [6,
7, 16]. In [7] it was shown that any two point lattices in $\mathbb{R}^{d}$
with the same density are bde. It is a simple consequence that any two
$d$-periodic Delone sets in $\mathbb{R}^{d}$ with the same density are bde.
This leads to considering aperiodic Delone sets, for instance the vertices of
a Penrose tiling. There are two well studied classes of aperiodic Delone sets:
cut-and-project sets and Delone sets arising from substitution tilings. For
details and a precise definition of an aperiodic Delone set, see [3].
Recently, the question of whether an aperiodic Delone set is bde to a point
lattice has gained some interest. For the bd-equivalence of cut-and-project
sets to a point lattice see [8, 10, 13, 14, 15]. The bd-equivalence of Delone
sets from substitution tilings to a point lattice was studied in [1, 10, 11,
16, 23, 24]. Since the bd-equivalence of Delone sets to a point lattice is
well understood, the focus now turns to the question of when two Delone sets
are bde.

This is where hulls of nonperiodic Delone sets come into play. In the theory
of aperiodic order the _hull_ $\mathbb{X}_{\Lambda}$ of a given Delone set
$\Lambda$ is a central object of interest. The hull of $\Lambda$ is the
closure of the orbit of $\Lambda$ under translations in the topology defined
by the “big box” metric (aka the Gromov-Hausdorff topology).111Some authors
work with a different topology, called the “local rubber” or “Chabauty-Fell”
topology. When the Delone set has finite local complexity, defined below, then
the two topologies agree. Specifically, the distance between two
$(r,R)$-Delone sets $\Lambda$ and $\Lambda^{\prime}$ is the infimum over all
$\varepsilon\in(0,1)$ such that there exist $x,x^{\prime}\in\mathbb{R}^{d}$,
each of size $\varepsilon/2$ or less, such that $\Lambda-x$ and
$\Lambda^{\prime}-x^{\prime}$ agree exactly on $B_{1/\varepsilon}$. If no such
$\varepsilon$ exists, then $d(\Lambda,\Lambda^{\prime})=1$.

The hull may be studied as a topological object [2, 17, 21], or as a dynamical
system $(\mathbb{X}_{\Lambda},\mathbb{R}^{d})$ [3], where $\mathbb{R}^{d}$
stands for the action of translations by $x\in\mathbb{R}^{d}$.

This raises the question “How many bde classes does $\mathbb{X}_{\Lambda}$
contain?”. A partial answer was given in [25]: If $\Lambda$ comes from a
substitution tiling meeting some technical conditions, then
$\mathbb{X}_{\Lambda}$ consists of uncountably many bde classes. Here we
provide a generalization that covers all Delone sets that are repetitive and
have finite local complexity (see definitions below). This category includes
the vast majority of the substitution tilings and cut-and-project sets that
have been studied to date.

Two geometric properties of the Delone set $\Lambda$ are closely associated
with dynamical properties of $(\mathbb{X}_{\Lambda},\mathbb{R}^{d})$. A
_patch_ in a Delone set $\Lambda$ is a set $\Lambda\cap K$ for some compact
$K\subset\mathbb{R}^{d}$. A Delone set $\Lambda$ has _finite local complexity_
(FLC) if for any compact set $K\subset\mathbb{R}^{d}$ there are only finitely
many different patches $(\Lambda-x)\cap K$ $(x\in\mathbb{R}^{d})$, up to
translation. A Delone set $\Lambda$ is called _repetitive_ if for each compact
set $K\subset\mathbb{R}^{d}$ such that $K\cap\Lambda$ is non-empty, the set

$\\{x\in\mathbb{R}^{d}\mid(\Lambda-x)\cap K=\Lambda\cap K\\}$

is a Delone set. The uniform density radius $R$ of this Delone set is the
_repetitivity radius_ of the patch $\Lambda\cap K$. The following fact is the
essence of the work of several authors, see [3, 21] for details.

######  Fact 1.

Let $\Lambda\subset\mathbb{R}^{d}$ be a Delone set. Then
$\mathbb{X}_{\Lambda}$ is compact if and only if $\Lambda$ has FLC. If
$\Lambda$ has FLC, then the dynamical system
$(\mathbb{X}_{\Lambda},\mathbb{R}^{d})$ is minimal (that is, each orbit is
dense) if and only if $\Lambda$ is repetitive.

Now we can state our main result.

######  Theorem 1.1.

Let $\Lambda$ be a repetitive Delone set in $\mathbb{R}^{d}$ having FLC and
such that the density of $\Lambda$ exists. Let $\mathbb{X}_{\Lambda}$ be the
hull of $\Lambda$. Then $\mathbb{X}_{\Lambda}$ either consists of a single bde
class or $\mathbb{X}_{\Lambda}$ contains uncountably many bde classes.

In fact we show that the number of bde classes is either 1 or
$2^{\aleph_{0}}$, where $\aleph_{0}$ is the cardinality of $\mathbb{Z}$. It is
clear that the number of bde classes can’t be larger than $2^{\aleph_{0}}$
[12, 22], insofar as there are only $2^{\aleph_{0}}$ elements of
$\mathbb{X}_{\Lambda}$, up to translation.

During the completion of this paper, Smilansky and Solomon released a preprint
[22] in which a stronger version of this result is proven using the language
of dynamical systems. Their theorem, which was developed completely
independently from ours, does not require FLC and considers the orbit closure
of $\Lambda$ in the local rubber topology. They replace our assumption of
repetitivity with one of minimality. They also do not require that the density
of $\Lambda$ exists. The cost of that generality is complexity, in that their
proof is longer and considerably more technical than ours.

An interesting consequence of both results is the following: In [10] it was
shown that a certain one-dimensional collection of Delone sets, namely, the
set of cut-and-project sets using half of the window of the famous Fibonacci
tiling [3], has at least two different bde classes. Theorem 1.1 then yields
the following result.

######  Corollary 1.2.

The hull $\mathbb{X}_{HF}$ of the ’Half-Fibonacci’ cut-and-project tiling
contains uncountably many bde classes.

##  2\. Auxiliary results

Let us fix some more notation. In the sequel, let $\\#M$ denote the
cardinality of a (typically finite) set $M$. We denote the Euclidean norm of
$x\in\mathbb{R}^{d}$ by $\|x\|$. The closed ball of radius $r$ about $x$ is
denoted by $B_{r}(x)$. The $d$-dimensional Lebesgue measure of a set
$A\subset\mathbb{R}^{d}$ is denoted by $\mu(A)$ and all sets in the paper are
compact and measurable unless noted otherwise. Let $A^{+\varepsilon}$ denote
the $\varepsilon$-tube of the boundary of $A$. That is,

$A^{+\varepsilon}=\\{x\in\mathbb{R}^{d}\mid d_{2}(x,\partial
A)\leq\varepsilon\\},$

where $\partial A$ is the topological boundary of $A$ and $d_{2}$ is the
standard Euclidean distance in $\mathbb{R}^{d}$. Hence $\mu(A^{+1})$ is the
Lebesgue measure of the set of all points whose distance to the boundary of
$A$ is one or less.

Any $d$-periodic Delone set has a well defined density, in the sense of
“average number of points per unit volume”. The same holds for cut-and-project
sets and for Delone sets from primitive substitutions. In general, the
definition of the density of a Delone set can be tricky. It is easy to
construct Delone sets having no well defined density (for instance
$-\mathbb{N}\cup 2\mathbb{N}$ in $\mathbb{R}$). There are even repetitive
Delone sets without density, see [19, Thm. 5.1], or [9] for a simpler example.

In order to define the density of an arbitrary Delone sets we need van Hove
sequences. A _van Hove sequence_ is a sequence $(A_{i})_{i}$ of compact
subsets of $\mathbb{R}^{d}$ such that for all $\varepsilon>0$

(1) $\lim_{i\to\infty}\frac{\mu(A_{i}^{+\varepsilon})}{\mu(A_{i})}=0.$

A Delone $\Lambda$ set has density $\operatorname{dens}(\Lambda)$ if for all
van Hove sequences $(A_{i})_{i}$ the limits

$\lim_{i\to\infty}\frac{\\#(A_{i}\cap\Lambda)}{\mu(A_{i})}$

exist and are identical. In that case $\operatorname{dens}(\Lambda)$ is the
value of these limits. An important tool in this context is the following
result. In the context of this paper it translates as follows.

######  Theorem 2.1 ([18]).

Let $\Lambda$ be a Delone set in $\mathbb{R}^{d}$ with density
$\operatorname{dens}(\Lambda)$. $\Lambda$ is bde to some lattice in
$\mathbb{R}^{d}$ if and only if there is $c>0$ such that, for all bounded
measurable sets $E\subset\mathbb{R}^{d}$,

$\big{|}\\#(\Lambda\cap E)-\operatorname{dens}(\Lambda)\mu(E)\big{|}\leq
c\mu(E^{+1})$

The proof of this result relies on the infinite version of the Hall Marriage
Theorem [20]. The same arguments can be used not only to compare a Delone set
$\Lambda$ with $\alpha\mathbb{Z}^{d}$, but to compare two arbitrary Delone
sets $\Lambda,\Lambda^{\prime}$ as well.

######  Theorem 2.2 ([11]).

Let $\Lambda,\Lambda^{\prime}$ be two Delone sets in $\mathbb{R}^{d}$. Suppose
there is a van Hove sequence $(A_{i})_{i}$ such that

(2) $\lim_{n\to\infty}\frac{|\\#(\Lambda\cap A_{i})-\\#(\Lambda^{\prime}\cap
A_{i})|}{\mu(A^{+1}_{i})}=\infty,$

then $\Lambda\not\stackrel{{\scriptstyle\rm bd}}{{\sim}}\Lambda^{\prime}$.

In [11], the result above was formulated using sets $A_{i}$ that are unions of
lattice cubes of appropriate size. Since we deal with Delone sets we can
“approximate” any van Hove sequence $(A_{i})_{i}$ by an appropriate union of
lattice cubes that are small enough that each cube contains either one or zero
points of $\Lambda$. For more details see [12] or [11].

######  Example 2.3.

As a test case for what follows we consider the set
$L=\mathbb{Z}\cup\\{1/2+2^{n}\mid n\in\mathbb{N}_{0}\\}$ and compare $L$ to
the integers $\mathbb{Z}$. Both have density 1, but
$\mathbb{Z}\not\stackrel{{\scriptstyle\rm bd}}{{\sim}}L$. This can be seen by
using Theorem 2.1 above, and observing for each $k>0$ there are intervals
$Q_{i}:=[0,2^{i}+1]$ such that

$\\#(Q_{i}\cap\Lambda)-\\#(Q_{i}\cap\mathbb{Z})>i.$

Note that $L$ contains “rich” regions (i.e. regions where the number of points
is larger than the expected number of points according to the density), but no
“poor” regions.

##  3\. Proof of Theorem 1.1

Note the difference between a patch $E\cap\Lambda$, which is a finite
collection of points, and its support $E$. Also note that we have two
different ways to move patches around: either by translating both its support
and $\Lambda$ to get $(E\cap\Lambda)-x=(E-x)\cap(\Lambda-x)$, or by
translating just the support to get $(E-x)\cap\Lambda$. Both ways yield
patches occurring in some $\Lambda^{\prime}\in\mathbb{X}_{\Lambda}$ and we
will use both.

The strategy of the proof of Theorem 1.1 is as follows. If $\Lambda$ is not
bde to a lattice, then there exist large regions that are “deviant” (meaning
either very rich or very poor in points of $\Lambda$ compared to the expected
number of points according to the density of $\Lambda$) and translates of
those regions that are either “normal” (neither rich nor poor) or have a
discrepancy of the opposite sign of the original region; we will refer to the
latter sets as “normal” anyway. We will recursively define larger and larger
regions $P_{i}$ around the origin and find elements of $\mathbb{X}_{\Lambda}$
for which the regions $P_{i}$ are either deviant or normal. In fact, for each
infinite word in $\\{D,N\\}^{\mathbb{N}}$ we will construct an element of
$\mathbb{X}_{\Lambda}$ where the form of each $P_{i}$ corresponds to the
$i$-th letter in the word. Any two such elements of $\mathbb{X}_{\Lambda}$
with words $u,u^{\prime}$ that differ in infinitely many letters are not bde
to each other by Theorem 2.2. Since there are uncountably many sequences in
$\\{D,N\\}^{\mathbb{N}}$, and since each tail-equivalence class is countable,
there are uncountably many elements of $\mathbb{X}_{\Lambda}$, no two of which
are bde.

We construct deviant and normal patches $D_{i}^{\prime}$ and $N_{i}^{\prime}$
around the origin recursively. Let $\varrho$ be the density of $\Lambda$ and
suppose that $D_{i-1}^{\prime}$ and $N_{i-1}^{\prime}$ have already been
constructed. Since $\Lambda$ is not bde to a lattice, there exist a region
$D_{i}^{\prime}$ such that the ratio

$\frac{|\\#(\Lambda\cap
D_{i}^{\prime})-\varrho\mu(D_{i}^{\prime})|}{\mu(D_{i}^{+1})}$

is arbitrarily large. In other words, for which $D_{i}^{\prime}$ is deviant.
As shown below, there is also a “normal” translate $N_{i}^{\prime}$ of
$D_{i}^{\prime}$ for which the discrepancy
$\\#(N_{1}^{\prime}\cap\Lambda)-\varrho\mu(N_{i}^{\prime})$ has the opposite
sign as the discrepancy of $D_{i}^{\prime}$. Without loss of generality, we
can choose $D_{i}^{\prime}$ so large that there are copies of
$D_{i-1}^{\prime}$ and $N_{i-1}^{\prime}$ near the center of $D_{i}^{\prime}$,
and likewise near the center of $N_{i}^{\prime}$. Note that the construction
so far only involves looking for regions in the fixed Delone set $\Lambda$.

Next we construct Delone sets in $\mathbb{X}_{\Lambda}$ corresponding to each
infinite word in $\\{D,N\\}^{\mathbb{N}}$. Let $u$ be such a word. Place a
copy of $D_{1}^{\prime}$ or $N_{1}^{\prime}$, centered at the origin,
according to whether the first letter of $u$ is $D$ or $N$. Call this copy
$D_{1}$ or $N_{1}$, and let $P_{1}=D_{1}$ or $N_{1}$. Since both
$D_{2}^{\prime}$ and $N_{2}^{\prime}$ contain copies of both $D_{1}^{\prime}$
and $N_{1}^{\prime}$ near their centers, we can extend $P_{1}$ to a copy
$D_{2}$ or $N_{2}$ of $D_{2}^{\prime}$ or $N_{2}^{\prime}$, according to
whether the second letter of $u$ is $D$ or $N$, and we can call this copy
$P_{2}$. Repeat for each index $i\in\mathbb{N}$. Note that each $P_{i}$ is a
patch $N^{\prime}_{i}-x_{i}$ or $D^{\prime}_{i}-x_{i}$ in $\Lambda-x$. The
union of the $P_{i}$’s is a Delone set $\Lambda_{u}\in\mathbb{X}_{\Lambda}$.
Specifically,

$\Lambda_{u}=\bigcup_{i}P_{i}=\lim_{i\to\infty}(\Lambda-x_{i}).$

The tricky point is that, if $u\neq u^{\prime}$, then the $i$-th patch $P_{i}$
of $\Lambda_{u}$ is not perfectly aligned with the $P_{i}$ of
$\Lambda_{u^{\prime}}$, so we cannot directly compare the two $P_{i}$’s in
Theorem 2.2. Instead, we must compare patches defined by small translates
(within $\Lambda_{u}$) of supports of deviant patches $D_{i}$ in $\Lambda_{u}$
to normal patches $N_{i}$ in $\Lambda_{u^{\prime}}$, or vice-versa. The
remainder of the proof is a series of estimates to show that suitable
$D^{\prime}_{i}$’s and $N^{\prime}_{i}$’s exists, such that these small
translates are still sufficiently deviant to apply Theorem 2.2.

The following result is standard, but for completeness we provide a sketch of
the proof.

######  Lemma 3.1.

Let $\Lambda$ be a Delone set in $\mathbb{R}^{d}$ with density $\varrho$ and
let $E$ be a bounded measurable set. Then there exist vectors
$x_{1},x_{2}\in\mathbb{R}^{d}$ such that

$\\#((E+x_{1})\cap\Lambda)\leq\varrho\mu(E)\qquad\text{and}\qquad\\#((E+x_{2})\cap\Lambda)\geq\varrho\mu(E).$

###### Sketch of proof:.

If we average $\\#((E+x)\cap\Lambda)$ over all values of $x\in\mathbb{R}^{d}$
we must get $\varrho\mu(E)$. This means that for every $\varepsilon>0$ there
must a point $x$ for which
$\\#((E+x)\cap\Lambda)\leq\varrho\mu(E)+\varepsilon$. However,
$\\#((E+x)\cap\Lambda)$ is always an integer, and so for small enough
$\varepsilon$ cannot be strictly between $\varrho\mu(E)$ and
$\varrho\mu(E)+\varepsilon$. Thus there must be an $x_{1}$ satisfying the
first inequality. The second is similar. ∎

The following result is then immediate and allows us to construct normal
patches $N^{\prime}_{i}$ from deviant patches $D^{\prime}_{i}$.

######  Lemma 3.2.

Let $\Lambda\subset\mathbb{R}^{d}$ be a Delone set with density $\varrho$ such
that $\Lambda$ is not bde to any lattice in $\mathbb{R}^{d}$. Let $E$ be a
compact subset of $\mathbb{R}^{d}$ such that $\\#(\Lambda\cap
E)-\varrho\mu(E)\geq 0$ (resp. $\leq 0$). Then there is a translation $E-x$ of
$E$ such that $\\#(\Lambda\cap(E-x))-\varrho\mu(E)\leq 0$ (resp. $\geq 0$).

We also need the following elementary identity.

######  Lemma 3.3.

Let $r,\ell>0$, then $(E^{+\ell})^{+r}\subseteq E^{+(\ell+r)}$.

###### Proof.

Let $z\in(E^{+\ell})^{+r}$. By definition there are $y\in\partial E$ and
$y^{\prime}$ such that $\|z-y^{\prime}\|\leq r$ and
$\|y^{\prime}-y\|\leq\ell$, hence

$\|z-y\|\leq\|z-y^{\prime}\|+\|y^{\prime}-y\|\leq\ell+r,$

hence $z\in E^{+(\ell+r)}$. ∎

We also need some result on estimates on the minimal and the maximal number of
points of a Delone set $\Lambda$ within a region $E$. In the sequel we always
assume that $E$ is a bounded measurable set.

######  Lemma 3.4.

There exist constants $\eta,\eta^{\prime}$ such that for any Delone set
$\Lambda$ with parameters $r>0$ and $R>0$ and any bounded region
$E\subset\mathbb{R}^{d}$,

$\\#(E\cap\Lambda)\geq\frac{\eta}{R^{d}}(\mu(E)-\mu(E^{+R}))\quad\mbox{ and
}\quad\\#(E\cap\Lambda)\leq\frac{\eta^{\prime}}{r^{d}}(\mu(E)+\mu(E^{+r}))$

Here, $\eta$ and $\eta^{\prime}$ depend only on $r$, $R$, and the dimension
$d$.

###### Proof.

Consider a periodic packing of $\mathbb{R}^{d}$ by balls of radius $R$. (For
instance, we could tile $\mathbb{R}^{d}$ by cubes of side length $2R$ and
place one ball in each cube.) Averaging over translates of this packing, the
number of balls with center in $E-E^{+R}$ is the packing density (which is a
constant $\eta$ that depends on dimension divided by $R^{d}$) times the volume
of $E-E^{+R}$. Therefore there exists a specific packing where the number of
such balls is at least $\frac{\eta}{R^{d}}\mu(E-E^{+R})$, which in turn is at
least $\frac{\eta}{R^{d}}(\mu(E)-\mu(E^{+R}))$. Each of these disjoint balls
is contained completely in $E$, and by the Delone property each contains at
least one point of $\Lambda$. This establishes the first inequality.

The second inequality is similar, except that we use a covering by balls of
radius $r$ (e.g. by starting with a tiling of $\mathbb{R}^{d}$ by cubes of
side length $2r/\sqrt{d}$ and using circumscribed balls) instead of a packing
by balls of radius $R$. Each point in $E\cap\Lambda$ must lie in a ball whose
center is either in $E$ or in $E^{+r}$. However, each such ball can contain at
most one point in $\Lambda$ and the number of such balls is bounded by
$\frac{\eta^{\prime}}{r^{n}}(\mu(E\cup
E^{+r})\leq\frac{\eta^{\prime}}{r^{n}}(\mu(E)+\mu(E^{+r}))$. ∎

######  Lemma 3.5.

For all $\ell\geq 1$, $\mu(E^{+\ell})\leq\ell^{d}\mu(E^{+1})$.

###### Proof.

First we prove $\mu((\frac{1}{\ell}E)^{+1})\leq\mu(E^{+1})$. Let
$\varepsilon>0$. Let $x_{1},\ldots,x_{n}\in\partial(\frac{1}{\ell}E)$ be
points such that the set of $\varepsilon$-balls centered at $x_{i}$ covers
$\partial(\frac{1}{\ell}E)$.

The balls of radii $1+\varepsilon$ centered at $x_{i}$’s cover
$(\frac{1}{\ell}E)^{+1}$; let $X$ be the union of these balls. Similarly, the
balls of radii $1+\varepsilon$ centered at $\ell x_{i}$’s are contained in
$E^{+(1+\varepsilon)}$; let $Y$ be the union of these balls. Using a variant
of the Kneser-Poulsen conjecture222Despite retaining its original name, this
“conjecture” is actually a proven theorem. for continuous contractions, see
[4, 5], we get that $\mu(X)\leq\mu(Y)$ and therefore

$\mu((\frac{1}{\ell}E)^{+1})\leq\mu(X)\leq\mu(Y)\leq\mu(E^{+(1+\varepsilon)}).$

Taking the limit as $\varepsilon$ goes to $0$,
$\mu((\frac{1}{\ell}E)^{+1})\leq\mu(E^{+1})$.

Now we can prove the lemma. Scaling $E^{+\ell}$ down by $\ell^{-1}$ yields
$(\frac{1}{\ell}E)^{+1}$. Hence

$\mu(E^{+\ell})=\ell^{d}\mu((\frac{1}{\ell}E)^{+1})\leq\ell^{d}\mu(E^{+1}).$

∎

The next lemma ensures that, given a patch of $E\cap\Lambda$, every patch of
$\Lambda$ with translated support $E-x$ has “approximately” the same number of
points, the difference in the number of points being governed by $\mu(E^{+1})$
and the length of the shift.

######  Lemma 3.6.

Let $\Lambda$ be a Delone set with parameters $r>0$ and $R>0$. Let $\ell\geq
r$. Then there is $q>0$ such that, for every $x\in\mathbb{R}^{d}$ with
$\|x\|\leq\ell$ and for every region $E$,

$|\\#\big{(}(E+x)\cap\Lambda\big{)}-\\#\big{(}E\cap\Lambda\big{)}|\leq
q\ell^{d}\mu(E^{+1}).$

Here the constant $q$ depends on the Delone set $\Lambda$ and the dimension
$d$.

###### Proof.

Let $\ell\geq r$ and $\|x\|\leq\ell$.

$\displaystyle|\\#\big{(}(E+x)\cap\Lambda\big{)}-\\#\big{(}E\cap\Lambda\big{)}|$
$\displaystyle\leq\\#\Big{(}\Big{(}\big{(}(E+x)\setminus
E\big{)}\cup\big{(}E\setminus(E+x)\big{)}\Big{)}\cap\Lambda\Big{)}$
$\displaystyle\leq\\#\Big{(}\big{(}(E+x)\setminus
E\big{)}\cap\Lambda\Big{)}+\\#\Big{(}\big{(}E\setminus(E+x)\big{)}\cap\Lambda\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(x\leq\ell)}}{{\leq}}\\#\big{(}(E+x)^{+\ell}\cap\Lambda\big{)}+\\#\big{(}E^{+\ell}\cap\Lambda\big{)}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\scriptsize(Lem. \ref{lem:r-R-
estim})}}}{{\leq}}\frac{\eta^{\prime}}{r^{d}}\Big{(}\mu\big{(}(E+x)^{+\ell}\big{)}+\mu\big{(}((E+x)^{+\ell})^{+r}\big{)}\Big{)}+\frac{\eta^{\prime}}{r^{d}}\Big{(}\mu\big{(}E^{+\ell}\big{)}+\mu\big{(}(E^{+\ell})^{+r}\big{)}\Big{)}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\scriptsize(Lem.
\ref{lem:r+l})}}}{{\leq}}2\frac{\eta^{\prime}}{r^{d}}\big{(}\mu(E^{+\ell})+\mu(E^{+(\ell+r)})\big{)}\leq
2\frac{\eta^{\prime}}{r^{d}}\big{(}\mu(E^{+\ell})+\mu(E^{+(\ell+\ell)})\big{)}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\scriptsize(Lem.
\ref{lem:thickboundary}})}}{{\leq}}2\frac{\eta^{\prime}}{r^{d}}(\ell^{d}+(\ell+\ell)^{d})\mu(E^{+1})\leq
2\frac{\eta^{\prime}}{r^{d}}(1+2^{d})\ell^{d}\mu(E^{+1}).$

With $q=2\frac{\eta^{\prime}}{r^{d}}(1+2^{d})$ and $\eta^{\prime}$ the
constant from Lemma 3.4 the claim follows. ∎

The next lemma is the key to proving Theorem 1.1 and particularly to the
construction of “deviant” patches of $\Lambda$.

######  Lemma 3.7.

Let $\Lambda\subset\mathbb{R}^{d}$ be a repetitive Delone set of FLC with
$\operatorname{dens}(\Lambda)=\varrho$ such that $\Lambda$ is not bde to any
lattice in $\mathbb{R}^{d}$. Let $c>0$ and $\ell>0$ be given. Then there
exists a bounded measurable set $E$ such that for all $x\in\mathbb{R}^{d}$
with $\|x\|\leq\ell$ holds:

$|\\#\big{(}(E-x)\cap\Lambda\big{)}-\varrho\mu(E)|>c\mu(E^{+1})$

###### Proof.

Let $q$ be as in Lemma 3.6. By Theorem 2.1 there is $E$ such that

(3) $|\\#(E\cap\Lambda)-\varrho\mu(E)|>(c+q\ell^{d})\mu(E^{+1}).$

By Lemma 3.6 we have that $|\\#(E\cap\Lambda)-\\#((E-x)\cap\Lambda)|\leq
q\ell^{d}\mu(E^{+1})$. Replacing $\\#(E\cap\Lambda)$ by
$\\#((E-x)\cap\Lambda)$ in (3) changes the left hand side by less than
$q\ell^{d}\mu(E^{+1})$. This yields the claim. ∎

######  Lemma 3.8.

Let $\Lambda$ be a Delone set in $\mathbb{R}^{d}$ with
$\operatorname{dens}(\Lambda)=\varrho$. Let $(E_{i})_{i}$ be a sequence of
bounded measurable subsets of $\mathbb{R}^{d}$ that violate the condition in
Theorem 2.1. That is, let the sequence $(c_{i})_{i}$ be such that
$\lim\limits_{i\to\infty}c_{i}=\infty$, and for each $i$

(4) $|\\#(E_{i}\cap\Lambda)-\varrho\mu(E_{i})|>c_{i}\mu(E^{+1}).$

Then $(E_{i})_{i}$ is a van Hove sequence.

###### Proof.

Inequality (4) is equivalent to

(5)
$\frac{1}{c_{i}}\Big{|}\frac{\\#(E_{i}\cap\Lambda)}{\mu(E_{i})}-\varrho\Big{|}>\frac{\mu(E_{i}^{+1})}{\mu(E_{i})}.$

Using Lemma 3.4 we get

$\frac{\eta}{R^{d}}-\frac{\mu(E_{i}^{+R})}{\mu(E_{i})}\leq\frac{\\#(E_{i}\cap\Lambda)}{\mu(E_{i})}\leq\frac{\eta^{\prime}}{r^{d}}+\frac{\mu(E_{i}^{+r})}{\mu(E_{i})}.$

From Lemma 3.5 we get upper bounds for $\mu(E_{i}^{+r})$ and $\mu(E_{i}^{+R})$
in terms of $\mu(E_{i}^{+1})$ (there is the trivial bound $\mu(E_{i}^{+1})$ if
$r\leq 1$ or $R\leq 1$). Combining these estimates we get the following
inequality for some positive constants $\alpha$ and $\beta$

$\Big{|}\frac{\\#(E_{i}\cap\Lambda)}{\mu(E_{i})}-\varrho\Big{|}\leq\alpha+\beta\frac{\mu(E_{i}^{+1})}{\mu(E_{i})}.$

Once we plug this in inequality (5), we get

$\frac{1}{c_{i}}\left(\alpha+\beta\frac{\mu(E_{i}^{+1})}{\mu(E_{i})}\right)>\frac{\mu(E_{i}^{+1})}{\mu(E_{i})}.$

Considering the limits of both sides yields
$\lim\limits_{i\to\infty}\frac{\mu(E^{+1})}{\mu(E_{i})}=0$. In order to show
$\lim\limits_{i\to\infty}\frac{\mu(E^{+\varepsilon})}{\mu(E_{i})}=0$ for all
$\varepsilon>0$, we consider two cases. If $\varepsilon\leq 1$, then

$\lim\limits_{i\to\infty}\frac{\mu(E_{i}^{+\varepsilon})}{\mu(E_{i})}\leq\lim\limits_{i\to\infty}\frac{\mu(E_{i}^{+1})}{\mu(E_{i})}=0.$

If $\varepsilon>1$, then we use the estimate from Lemma 3.5 to get

$\lim\limits_{i\to\infty}\frac{\mu(E_{i}^{+\varepsilon})}{\mu(E_{i})}\leq\lim\limits_{i\to\infty}\frac{\varepsilon^{d}\mu(E_{i}^{+1})}{\mu(E_{i})}=0.$

∎

Frequently a van Hove sequence $(E_{i})_{i}$ is defined by requiring another
property in addition to (1): namely, that the $E_{i}$ exhaust the entire
space. For instance, the sequence $(B_{i}(x_{i}))_{i}$, with
$x_{i}=(i,0,\cdots,0)^{T}$, fulfills (1), but the (closure of the) union of
the balls is only the half-space $\mathbb{R}^{\geq 0}\times\mathbb{R}^{d-1}$.
The next result ensures that we do not need this further requirement in our
context.

######  Proposition 3.9.

Let $(E_{i})_{i}$ be a van Hove sequence. Then for each $R>0$ there are $i\geq
0,t_{i}\in\mathbb{R}^{d}$ such that $E_{i}$ contains a ball $B_{R}(t_{i})$.

###### Proof.

Assume the contrary; that is, suppose there is $R>0$ such that for all
$t\in\mathbb{R}^{d},i\geq 0$ holds: $B_{R}(t_{i})\not\subseteq E_{i}$. This
implies that for all $t,i$ we have that the distance between $t$ and $\partial
E_{i}$ is not greater than $R$. Consequently, for all $i$ we have
$E_{i}\setminus(E_{i})^{+R}=\varnothing$. Since $E_{i}\neq\varnothing$ this
implies that for all $i>0$ holds $\mu(E_{i})<\mu((E_{i})^{+R})$, contradicting
the van Hove property (1). ∎

Now we can state the proof of our main result.

###### Proof of Theorem 1.1.

Let $\Lambda$ be a Delone set in $\mathbb{R}^{d}$ with
$\operatorname{dens}(\Lambda)=\varrho$ such that $\Lambda$ is not bde to any
lattice in $\mathbb{R}^{d}$. We choose an infinite word $u=u_{1}u_{2}\cdots$
over the alphabet $\\{D,N\\}$, and $(c_{i})_{i}$ such that
$\lim\limits_{i\to\infty}c_{i}=\infty$.

For $c>0$, we call a bounded measurable set $E$ $c$-_deviant_ if

$|\\#(E\cap\Lambda)-\varrho\mu(E)|>c\mu(E^{+1}),$

otherwise we call $E$ $c$-_normal_.

Let $\ell_{1}$ be arbitrary. By Theorem 2.1 there is a compact set
$D_{1}^{\prime}$ that is $c_{1}$-deviant. By Lemma 3.2 there is set
$N_{1}^{\prime}:=D_{1}^{\prime}-y_{1}$ such that the discrepancy
$\\#(N_{1}^{\prime}\cap\Lambda)-\varrho\mu(N_{1}^{\prime})$ of
$N_{1}^{\prime}$ is zero or has the opposite sign as the discrepancy of
$D_{1}^{\prime}$. ($N_{1}^{\prime}$ may or may not be $c$-normal for any
prescribed $c$, but in all cases the discrepancy of $N_{1}^{\prime}$ differs
greatly from that of $D_{1}^{\prime}$, which is what we actually need.) Pick a
point $x_{1}\in D_{1}^{\prime}$. If the first letter $u_{1}$ of $u$ is $D$
then let
$P_{1}:=(D^{\prime}_{1}\cap\Lambda)-x_{1}=(D_{1}^{\prime}-x_{1})\cap(\Lambda-
x_{1})$, which is a patch in $\Lambda-x_{1}$. Otherwise let
$P_{1}:=(N_{1}^{\prime}\cap\Lambda)+y_{1}-x_{1}=(N^{\prime}_{1}+y_{1}-x_{1})\cap(\Lambda+y_{1}-x_{1})$,
which is a patch in $\Lambda+y_{1}-x_{1}$. The shape of support of $P_{1}$ is
the same in both cases, but the underlying patch of the Delone set is
different.

Choose $\ell_{2}$ such that
$\ell_{2}>2\max\\{R_{rep}(D_{1}^{\prime}\cap\Lambda),R_{rep}(N_{1}^{\prime}\cap\Lambda)\\}$,
where $R_{rep}(P)$ denotes the repetitivity radius of the patch $P$ (compare
Section 1). By Lemma 3.7 there is $D_{2}^{\prime}$ such that
$D_{2}^{\prime}-x$ is $c_{2}$-deviant for all $x$ with $\|x\|\leq\ell_{2}$. By
Lemma 3.2 there is $y$ such that $N_{2}^{\prime}=D_{2}^{\prime}-y$ has the
opposite discrepancy as $D_{2}^{\prime}$ (or this discrepancy is 0). Let
$B_{s}(t_{1})$ be the largest ball contained in $D_{2}^{\prime}$.

By repetitivity, both $D_{2}^{\prime}\cap\Lambda$ and
$N_{2}^{\prime}\cap\Lambda$ contain translates of both
$D_{1}^{\prime}\cap\Lambda$ and $N_{1}^{\prime}\cap\Lambda$ within distance
$\frac{\ell_{2}}{2}$ of the center $t_{1}$ of $B_{s}(t_{1})$, say with the
points $x_{1}\in D_{1}^{\prime}$ and $y_{1}\in N_{1}^{\prime}$ corresponding
to $x_{D2}$ and $x_{N2}$ in $D_{2}^{\prime}$. If $u_{1}=D$ we take
$x_{2}=x_{D2}$ and if $u_{1}=N$ we take $x_{2}=x_{N2}$. Either way, define
$D_{2}=D_{2}^{\prime}-x_{2}$, viewed as a pattern in $\Lambda-x_{2}$. That is,
$D_{2}$ is a translate (of both support and Delone set) of $D_{2}^{\prime}$
with a copy of $P_{1}$ close to the center. We similarly create $N_{2}$ as a
translate of $N_{2}^{\prime}$ that likewise extends $P_{1}$. Finally, we pick
$P_{2}$ to be either $D_{2}$ or $N_{2}$, depending on whether $u_{2}$ is $D$
or $N$.

The supports of the two possible choices of $P_{2}$ are not identical, since
the relative positions of copies of $P_{1}$ in $D_{2}^{\prime}$ and
$N_{2}^{\prime}$ are not the same. However, their supports differ by
translation by less than $\ell_{2}$. A translate of $D_{2}$ that has the same
support as $N_{2}$ still has a discrepancy greater in magnitude than
$c_{2}\mu(D_{2}^{+1})$, while the discrepancy of $N_{2}$ has a discrepancy of
the opposite sign.

Now iterate: let $\ell_{i+1}$ be such that
$\ell_{i+1}>2\max\\{r_{rep}(D_{i}^{\prime}\cap\Lambda),r_{rep}(N_{i}^{\prime}\cap\Lambda)\\}$
and continue as above, finding regions $D_{i+1}^{\prime}$ and
$N_{i+1}^{\prime}$ in $\Lambda$ such that $D_{i+1}^{\prime}$ and all its
translations by at most $\ell_{i+1}$ are $c_{i+1}$-deviant, $N_{i+1}^{\prime}$
has a discrepancy of the opposite sign as $D_{i+1}^{\prime}$, and such that
both $D_{i+1}^{\prime}$ and $N_{i+1}^{\prime}$ contain copies of
$D_{i}^{\prime}$ and $N_{i}^{\prime}$ within a distance $\ell_{i+1}/2$ of the
center of a large sphere (as in Proposition 3.9). We then defined translates
$D_{i+1}$ and $N_{i+1}$ of $D_{i+1}^{\prime}$ and $N_{i+1}^{\prime}$, viewed
as regions in translates of $\Lambda$, such that these pattern extend $P_{i}$.
Finally we pick $P_{i+1}$ to be $D_{i+1}$ or $N_{i+1}$ depending on the
$(i+1)$st letter of $u$.

For any word $u\in\\{D,N\\}^{\mathbb{N}}$, this procedure gives a nested
sequence of patches $(P_{i})_{i}$ (i.e. $P_{i}\subset P_{i+1}$). By Lemma 3.8
the supports of the patches $P_{i}$ are a van Hove sequence. By Proposition
3.9 these van Hove sequences contain arbitrary large balls. Since we have
chosen the balls $B_{s}(t_{i})$ in each step to be the largest possible, their
diameters $s$ tend to infinity. By the closedness of $\mathbb{X}_{\Lambda}$
under translations we can assume without loss of generality that these balls
are centered at 0. The union of the patches $P_{i}$ is then a Delone set
$\Lambda_{u}\in\mathbb{X}_{\Lambda}$. If
$u,u^{\prime}\in\\{D,N\\}^{\mathbb{N}}$ differ in infinitely many letters,
then we look at the points of disagreement and compare $N_{i}$ in one Delone
set to a translate by less than $\ell_{i}$ of $D_{i}$ in the other. Applying
Theorem 2.2, we see that $\Lambda_{u}\not\stackrel{{\scriptstyle\rm
bd}}{{\sim}}\Lambda_{u^{\prime}}$ because the translate by at most $\ell_{i}$
of $D_{i}$ is still $c_{i}$-deviant. There are $2^{\aleph_{0}}$ elements of
$\\{D,N\\}^{\mathbb{N}}$ and only countably many elements of each bde class,
so there are $2^{\aleph_{0}}$ distinct bde classes among the
$\\{\Lambda_{u}\\}$ and there are at least $2^{\aleph_{0}}$ bde classes in
$\mathbb{X}_{\Lambda}$.

However, Delone sets that differ only by a translation are bde and
$\mathbb{X}_{\Lambda}$ consists of only $2^{\aleph_{0}}$ translational orbits,
so $\mathbb{X}_{\Lambda}$ can have at most $2^{\aleph_{0}}$ bde classes. Thus
the cardinality of the bde classes in $\mathbb{X}_{\Lambda}$ is exactly
$2^{\aleph_{0}}$. ∎

######  Remark.

In order to maintain a strict separation between our results and those of
Smilansky and Solomon [22], we stated and proved Theorem 1.1 in the form in
which we originally developed it, including the assumption that an overall
density exists. However, it is extremely easy to drop that assumption. If
there is no overall density, then there is an upper density $\varrho_{+}$ and
a lower density $\varrho_{-}$. Instead of looking for regions that are rich or
poor relative to “the” overall density, we can look instead for regions that
are rich or poor relative to an arbitrary intermediate density
$\varrho\in(\varrho_{-},\varrho_{+})$. For instance, we can look for Euclidean
balls of large radius whose densities are greater than
$(\varrho_{+}+\varrho)/2$ or are less than $(\varrho_{-}+\varrho)/2$. The rest
of the proof that there exist uncountably many bde classes proceeds exactly as
before.

## Acknowledgements

It is a pleasure to thank Philipp Gohlke and Yasushi Nagai for providing the
proof of Proposition 3.9 (actually, two proofs).

The authors are also extremely thankful to Yaar Solomon and Alan Haynes for
useful comments.

## References

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# Extracting spinning wormhole energy via magnetic reconnection

Xu Ye, Chao-Hui Wang, Shao-Wen Wei 111Corresponding author. E-mail:
<EMAIL_ADDRESS>1 Key Laboratory of Quantum Theory and Applications of MoE,
Lanzhou University, Lanzhou 730000, China
2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, People’s Republic of China,
3Institute of Theoretical Physics $\&$ Research Center of Gravitation, Lanzhou
University, Lanzhou 730000, People’s Republic of China

###### Abstract

Magnetic reconnection has been extensively shown to be a promising approach to
extract spinning black hole energy. In this paper, we focus on extracting
spinning wormhole energy via such mechanism. The study shows that it is indeed
possible to extract rotating energy from a spinning wormhole with small
regularization parameter $\ell$ of the central singularity. The efficiency and
power of the energy extraction are also evaluated. Quite different from the
Kerr black hole, the spin of the wormhole can take arbitrarily large value.
However, the increasing of wormhole spin not always improves the efficiency
and power of energy extraction. By further comparing with the Kerr black hole,
we find the wormhole is more efficient when the magnetic reconnection happens
within radial distance $r/M<1$. These studies reveal the features of
extracting spinning wormhole energy, and more underlying properties are
expected to be disclosed for the horizonless objects.

Wormhole, black hole mimicker, energy extraction, magnetic reconnection

###### pacs:

04.70.Bw, 52.27.Ny, 52.30.Cv

##  I Introduction

In astrophysics, a key aspect of highly energetic phenomena is accompanied
with energy release, such as active galactic nuclei McKinney2004 ; Hawley2005
, gamma-ray bursts SwiftScience2004 ; Woosley2006 , and ultraluminous X-ray
binaries Remillard2006 ; Bachetti2014 . In these phenomena, it is widely
believed that energy released originates from black holes whose energy
consists of gravitational potential of matter, electromagnetic field energy,
and black hole itself. With the energy conservation, black hole loses its mass
during the energy-releasing process, accordingly. Nevertheless, the black hole
mass can not completely vanish for the existence of the irreducible mass
Christo1970 . Taking Kerr black hole as an example, its total mass $M$ can be
decomposed into the reducible part $M_{re}$ and irreducible part $M_{ir}$
Christo1970

$M=M_{re}+M_{ir},$ (1)

where, the irreducible mass is

$M_{ir}=M\sqrt{\frac{1}{2}\left(1+\sqrt{1-\frac{a^{2}}{M^{2}}}\right)}.$ (2)

Black hole spin $a=J/M$ measures the angular momentum $J$ per unit mass.
Obviously, this formula shows that the irreducible mass decreases with the
black hole spin $a$, and tends to the total mass $M$ for a nonrotating black
hole. On the other hand, the black hole entropy can be expressed in terms of
the irreducible mass Bekenstein1972 ; Bekenstein1973 ; Hawking1974

$\displaystyle S=4\pi M_{ir}^{2}.$ (3)

The entropy always increases if one extracts black hole energy by decreasing
its spin. As a result, the second law of black hole thermodynamics holds as
expected. Meanwhile, the extractable energy is given by

$E_{re}=M-M_{ir}=\left(1-\sqrt{\frac{1}{2}\left(1+\sqrt{1-\frac{a^{2}}{M^{2}}}\right)}\right)M.$
(4)

For an extremal Kerr black hole, the amount of the energy can grow up to 29%.

In Penrose’s landmark paper Penrose1971 , he originally proposed a thought
experiment in which a particle fission process ($A\to B+C$) happens in ergo-
sphere of rotating black holes. The subtlety of this process lies that the
particle can have negative energy to the infinity observer. Provided that
particle B with negative energy is swallowed by the spinning black hole and
particle C with positive energy escapes to infinity, more energy will be
carried out by particle C. Accordingly, the black hole spin reduces after
absorbing a negative energy particle. This provides a first process to extract
black hole energy. However, a major problem is that these two newborn
particles B and C should be separated with a relative high velocity, even to
the speed of light Wald1974 . This requirement gives rise to an extremely low
expected rate of particle fission. Hence it is pretty inefficient to extract
black hole rotational energy via such process.

Soon afterwards, such issue was improved, and several new mechanisms were put
forward such as the superradiant scattering Teukolsky1974 , collisional
Penrose process Piran1975 , Blandford-Znajek (BZ) process Blandford1977 , and
the magnetohydrodynamic (MHD) Penrose process Takahashi1990 . A significant
result on BZ process was that the strongly magnetized BZ mechanism is more
effective than the non-magnetized neutrino annihilation process to power
gamma-ray-burst jets Liu2017 . This reveals that the magnetic field is a key
to enhance extraction efficiency. Actually, around an astrophysical black
hole, magnetic field can be generated from the charged matter of the accretion
disk, and is believed to extensively exist. Such fact was also confirmed by
EHT collaborations on observing the image of M87* via the polarized emission
EHT2021 .

Recently, in high energy astrophysics, magnetic reconnection associated with
the acceleration of energetic particles has gained tremendous significance
Medina2023 . It is a strong candidate to generate ultra-high energy cosmic
rays in magnetically dominated regions. It was noted by Koide and Arai
Koide2008  that fast magnetic reconnection redistributes the energy of the
plasma. This could possibly lead to that the falling particle with negative
energy enters the horizon, whereas other plasmas escaping from the ergo-region
take more energy away. As a result, black hole energy is extracted. Such idea
was supported by Parfrey Parfrey2018 , who confirmed that magnetic
reconnection releases magnetic energy and converts it into the kinetic energy
of plasmas according to a relativistic kinetic simulation. Very recently, in
Ref. Comisso2020 , Comisso and Asenjo further improved such magnetic
reconnection mechanism and calculated the efficiency and power of the energy
extraction. The result indicates that it is feasible only for a rapidly
spinning Kerr black hole. Furthermore, some other relevant studies have
attempted to deal with the issue in various black hole models Wei2022 ;
Wang2022 ; Khodadi2022 ; Carleo2022 ; Liu38 ; Yuan31 ; Yuan53 .

Wormhole is one potential alternative to the black hole both in theory and
astronomical observation. The wormhole with non-trivial topology is one kind
exact solution of Einstein field equation. It can connect either two distant
regions of the same universe or two different universes. The first solution
was given by Einstein and Rosen Einstein1935 . While the transverse wormhole
was built by Thorne and Morris Morris1988 . Nevertheless, the biggest
challenge is that the matter constructing the wormhole must be exotic leading
to the violation of the null energy condition. Until recently, it was shown
that the transverse wormhole can be well constructed with ordinary matter
Blazquez ; Konoplya ; Wang50 . Another significant difference from the black
hole is that the wormhole is singularity free object. Especially, the
gravitational waves and images, respectively, observed by LIGO and EHT, could
be well interpreted by the wormholes Hopper ; Vincent . Furthermore, it is
worth to investigate other phenomena and disclose whether the wormhole can be
used to replace black hole.

The aim of this paper is to study the energy extraction from a spinning
wormhole by using the magnetic reconnection mechanism. First, we show that,
like the black hole, it is feasible for extracting wormhole energy via such
process. Then after calculating the efficiency and power, the results imply
that, in certain parameter regions, the wormhole has great advantages than the
black hole. The obtained findings uncover possible features of the wormholes
on energy extraction. And wormhole can be treated as an alternative of the
black hole to extract energy via magnetic reconnection.

This paper is structured as follows. In Sec. II, we briefly review the
spinning wormhole and the corresponding spacetime structure. In Sec. III, we
focus on the magnetic reconnection process and give an analytic expression of
energy-at-infinity for the accelerated/decelerated plasmas. Furthermore, we
explore the conditions to achieve the energy extraction. The efficiency and
power of mechanism for the wormhole are studied in detail in Sec. IV. The last
Sec. V devotes to the conclusion and discussion of our findings.

##  II Rotating wormhole model and geodesics

In this section, we will briefly introduce the wormhole metric proposed by
Mazza, Franzin, and Liberati in Ref. Mazza2021 , and then show the equation of
motion of the test particles.

The wormhole model we considered is a rotating counterpart of the static,
spherically symmetric Simpson-Visser (SV) one described by the following line
element Simpson2018

$ds^{2}=-\left(1-\frac{2M}{\sqrt{r^{2}+\ell^{2}}}\right)dt^{2}+\left(1-\frac{2M}{\sqrt{r^{2}+\ell^{2}}}\right)^{-1}dr^{2}+(r^{2}+\ell^{2})(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (5)

where $M$ represents the ADM mass and non-negative $\ell$ is a parameter
responsible for the regularization of the central singularity. Note that the
radial coordinate $r\in(-\infty,+\infty)$ under this wormhole model. Moreover,
SV metric is a minimal modification of the Schwarzschild solution that can be
restored by taking $\ell=0$. Since the spacetime structure is completely
governed by the parameter $\ell$, one can acquire a wormhole or a black hole
by selecting $\ell>2M$ or $\ell<2M$. Obviously, the positive parameter $\ell$
eliminates the central singularity of the spacetime, and thus we can call it
regularization parameter.

Making use of the Newman-Janis procedure Newman1965a ; Newman1965b , a
spinning wormhole can be obtained Mazza2021

$ds^{2}=-(1-\frac{2M\sqrt{r^{2}+l^{2}}}{\Sigma})dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}-\frac{4Ma\sin^{2}\theta\sqrt{r^{2}+l^{2}}}{\Sigma}dtd\phi+\frac{A\sin^{2}\theta}{\Sigma}d\phi^{2},$
(6)

with

$\displaystyle\Sigma=r^{2}+\ell^{2}+a^{2}\cos^{2}\theta,\quad\Delta=r^{2}+\ell^{2}+a^{2}-2M\sqrt{r^{2}+\ell^{2}},$
(7) $\displaystyle A=(r^{2}+\ell^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta,$
(8)

where $a$ is the angular momentum of wormhole per unit mass. Taking limits
$a\to 0$ or $\ell\to 0$, respectively, it shall reduce to the SV metric (5) or
Kerr metric. The metric (6) is symmetric under the reflection transformation
$r\to-r$ due to the fact that the position $r=0$ is a throat of the wormhole
that connects two distinct universes.

More importantly, the spacetime structure is characterized by the
regularization parameter $\ell$ and wormhole spin $a$. From $\Delta=0$, the
corresponding horizons are calculated as

$r_{\pm}=\sqrt{(\rho_{\pm})^{2}-\ell^{2}},$ (9)

here $\rho_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$. For a black hole, there exists at
least one horizon located at $r_{+}$, which requires $\ell<\rho_{+}$. If we
further impose $\ell<\rho_{-}$, the black hole shall possess two horizons. On
the other hand, a wormhole has no horizon outside its throat, and which leads
to $a>M$ or $\ell>\rho_{+}$.

For the sake of clarity, we show the black hole and wormhole regions in ($a$,
$\ell$) plane in Fig. 1. The left lower corner marked with light blue color is
for the black hole region with at least one horizon, and the maximal spin
bound is $a/M=1$. The remaining other regions are for the wormhole. Obviously
they can be slowly spinning $a/M<1$ or rapidly spinning $a/M>1$ marked with
light green or light purple color. In these two regions, we will show their
structures have significant difference and shall have influence on the energy
extraction.

Figure 1: Rapidly spinning Wormholes (RS-WH), slowly spinning Wormholes (SS-
WH), and black hole (BH) regions in the ($a/M$, $\ell/M$) plane.

Since the energy extraction via magnetic reconnection happens within the ergo-
region, it is necessary to examine the structure of the wormholes. The outer
and inner ergo surfaces are determined by $g_{tt}=0$, which gives

$r^{\pm}_{erg}=\sqrt{(\rho_{\,erg}^{\pm})^{2}-\ell^{2}},$ (10)

with $\rho_{erg}^{\pm}=M\pm\sqrt{M^{2}-a^{2}\cos^{2}\theta}$. Energy
extraction requires the existence of the ergo surface, and thus we must have
$\rho_{erg}^{+}>\ell$ for slowly spinning wormhole and $\rho_{erg}^{\pm}>\ell$
for rapidly spinning wormhole.

In order to avoid the misunderstanding, we exhibit the structure of the
ergoregion in ($\rho$, $\theta$, $\phi$) plane instead of the ($r$, $\theta$,
$\phi$) plane in Fig. 2. In Fig. 2, the structure of the ergoregion is shown
for the slowly spinning wormhole case $a/M<1$. Although there is a horizon, it
hides behind the wormhole throat. It is clear that the ergoregion between the
outer ergo surface and wormhole throat is very narrow. While for the rapidly
spinning wormhole, the structure is quite different, see Fig. 2. The
ergoregion bounded by the outer, inner ergo surfaces, and the throat is broad.
Such feature potentially indicates high efficiency and power for the energy
extraction. It is worth emphasizing that the ergo-region structures for both
the wormhole cases are significantly different from that of Kerr black holes.

Figure 2: The structure of ergo-region in ($\rho$, $\theta$, $\phi$) plane.
(a) Slowly spinning wormhole with $a/M<1$. (b) Rapidly spinning wormhole with
$a/M>1$. $E_{out}$ and $E_{in}$ denote the outer and inner ergo-surfaces,
respectively.

After examining the ergoregion for the wormhole, we turn to the motion of the
test particle, which is another key for the magnetic reconnection mechanism.
Adopting the Hamilton-Jacobi method, one can easily obtain the geodesic
equations in a wormhole background Mazza2021

$\displaystyle\Sigma\frac{dt}{d\tau}$
$\displaystyle=a(\mathcal{L}-a\mathcal{E}\sin^{2}\theta)+\frac{\rho^{2}+a^{2}}{\Delta}[\mathcal{E}(\rho^{2}+a^{2})-\mathcal{L}a],$
(11) $\displaystyle\Sigma\frac{dr}{d\tau}$ $\displaystyle=\pm\sqrt{R},$ (12)
$\displaystyle\Sigma\frac{d\theta}{d\tau}$ $\displaystyle=\pm\sqrt{\Theta},$
(13) $\displaystyle\Sigma\frac{d\phi}{d\tau}$
$\displaystyle=\frac{\mathcal{L}}{\sin^{2}\theta}-a\mathcal{E}+\frac{a}{\Delta}[\mathcal{E}(\rho^{2}+a^{2})-\mathcal{L}a],$
(14)

where

$\displaystyle\mathcal{R}$
$\displaystyle=[\mathcal{E}(\rho^{2}+a^{2})-\mathcal{L}a]^{2}-\Delta[\mu^{2}\rho^{2}+(\mathcal{L}-a\mathcal{E})^{2}+\mathcal{Q}],$
(15) $\displaystyle\Theta$
$\displaystyle=\mathcal{Q}-\cos^{2}\theta[a^{2}(\mu^{2}-\mathcal{E}^{2})+\frac{\mathcal{L}^{2}}{\sin^{2}\theta}],$
(16) $\displaystyle\mathcal{Q}$
$\displaystyle=u^{2}_{\theta}+\cos^{2}\theta[a^{2}(1-\mathcal{E})^{2}-\frac{\mathcal{L}}{\sin^{2}\theta}].$
(17)

Here $\mu^{2}=0$ and $1$ for the null and timelike geodesics, respectively.
Constants $\mathcal{E}$, $\mathcal{L}$, and $\mathcal{Q}$ are the energy,
angular momentum, and Carter constant for the test particle along each
geodesic.

There are two characterized circular orbits playing important role in magnetic
reconnection. One is the circular photon orbit with $\mu^{2}=0$ satisfying the
following conditions

$\mathcal{R}(r_{ph})=0,\quad\frac{d\mathcal{R}(r)}{dr}\bigg{|}_{r=r_{ph}}=0.$
(18)

Substituting (15) into (18), we solve

$r_{ph}=\sqrt{4M^{2}\left[1+\cos\left(\frac{2}{3}\arccos(\mp\frac{a}{M})\right)\right]^{2}-\ell^{2}}.$
(19)

Another is the innermost stable circular orbit for the test particle with
$\mu^{2}=1$. The corresponding conditions are

$\mathcal{R}(r_{ISCO})=0,\quad\frac{d\mathcal{R}(r)}{dr}\bigg{|}_{r=r_{ISCO}}=0,\quad\frac{d^{2}\mathcal{R}(r)}{dr^{2}}\bigg{|}_{r=r_{ISCO}}=0.$
(20)

Solving them, one has

$r_{ISCO}=\sqrt{{\left[3M+Z_{2}M\mp\left[(3-Z_{1})(3+Z_{1}+2Z_{2})\right]^{1/2}M\right]^{2}}-\ell^{2}},$
(21)

with

$\displaystyle Z_{1}$
$\displaystyle=1+(1-\frac{a^{2}}{M^{2}})^{1/3}\left[(1+\frac{a}{M})^{1/3}+(1-\frac{a}{M})^{1/3}\right],$
(22) $\displaystyle Z_{2}$
$\displaystyle=(3\frac{a^{2}}{M^{2}}+Z_{1}^{2})^{1/2}.$ (23)

Moreover, the Keplerian angular velocity of circular orbits on the equatorial
plane is $\Omega_{K}\equiv\frac{d\phi}{d\tau}/\frac{dt}{d\tau}$. By making use
of Eqs. (11) and (14), we have

$\frac{dg_{\phi\phi}}{dr}\Omega_{K}^{2}+2\frac{dg_{t\phi}}{dr}\Omega_{K}+\frac{dg_{tt}}{dr}=0.$
(24)

Solving it, the Keplerian angular velocity reads

$\Omega_{K\pm}=-\frac{aM\mp\sqrt{M\left(r^{2}+\ell^{2}\right)^{3/2}}}{\left(r^{2}+\ell^{2}\right)^{3/2}-a^{2}M}.$
(25)

The upper and lower signs correspond to the corotating and counterrotating
orbits. Taking the limit $\ell\to 0$, this result can recover that of the Kerr
black hole.

##  III Energy extraction regions

In the magnetic reconnection process, a current sheet appears owing to the
direction change of the magnetic field perpendicular to the equatorial plane.
When this current reaches a critical aspect ratio, it would be destroyed by
the plasmoid instability Comisso2016 ; Uzdensky2014 ; Comisso2017 . Then, the
formation of plasmoids/flux ropes drives fast magnetic reconnection and
converts available magnetic energy into the kinetic of plasma Daughton2009 ;
Bhattacharjee2009 . It is interesting to imagine that, after the occurrence of
the magnetic reconnection, one part of the plasmas is accelerated and another
part is decelerated. If the decelerated part with negative energy-at-infinity
is swallowed by the black hole, while the accelerated part with positive
energy escapes to radial infinity, net energy will be extracted from the black
hole. Thus, such magnetic reconnection process provides us with a potential
mechanism to extract black hole energy. In this section, we shall consider the
parameter range that can implement this process.

To analyze the process of magnetic reconnection, we begin with exploring the
energy-at-infinity of accelerated and decelerated plasma. For the purpose, we
adopt the locally nonrotating frame, the zero-angular-momentum-observer (ZAMO)
frame, under which the observers do not feel any angular velocity related to
the spinning wormhole.

Denoting $(\hat{t},\hat{x}^{1},\hat{x}^{2},\hat{x}^{3})$ and
$(t,x^{1},x^{2},x^{3})$ as the coordinates in ZAMO frame and Boyer-Lindquist
frame, the transformation relations between them will be

$d\hat{t}=\alpha dt,\quad d\hat{x}^{i}=\sqrt{g_{ii}}dx^{i}-\alpha\beta^{i}dt,$
(26)

where $\alpha$ and $\beta^{i}$ are, respectively, the lapse function and shift
vector. In terms of the metric components, they are expressed as

$\alpha=\sqrt{-g_{tt}+\frac{g_{\phi
t}^{2}}{g_{\phi\phi}}},\quad\beta^{\phi}=\frac{\sqrt{g_{\phi\phi}}}{\alpha}\omega^{\phi}.$
(27)

Here $\omega^{\phi}=-g_{\phi t}/g_{\phi\phi}$ is the angular velocity of the
frame dragging.

Employing (26), the line element (6) in ZAMO frame becomes

$ds^{2}=-d\hat{t}^{2}+\sum_{i=1}^{3}(d\hat{x}^{i})^{2}=\eta_{\mu\nu}d\hat{x}^{\mu}\hat{x}^{\nu}.$
(28)

It is clear that the spacetime in this ZAMO frame is locally Minkowskian for
observers. Furthermore, it is convenient to derive the relation of the vector
$\psi$ in the ZAMO frame
$(\hat{\psi}^{0},\hat{\psi}^{1},\hat{\psi}^{2},\hat{\psi}^{3})$ and Boyer-
Lindquist coordinate $(\psi^{0},\psi^{1},\psi^{2},\psi^{3})$

$\displaystyle\hat{\psi}^{0}$ $\displaystyle=\alpha\psi^{0},$
$\displaystyle\hat{\psi}^{i}=\sqrt{g_{ii}}\psi^{i}-\alpha\beta^{i}\psi^{0},$
(29) $\displaystyle\hat{\psi}_{0}$
$\displaystyle=\frac{\psi_{0}}{\alpha}+\sum_{i=1}^{3}\frac{\beta^{i}}{\sqrt{g_{ii}}}\psi_{i},$
$\displaystyle\hat{\psi}_{i}=\frac{\psi_{i}}{\sqrt{g_{ii}}}.$ (30)

Next, we would like to evaluate the capability of magnetic reconnection to
extract wormhole energy. This can be achieved by examining the formation of
negative and positive energy of decelerated and accelerated plasmas at
infinity. Under the one-fluid approximation, the stress-energy tensor can be
formulated as

$T^{\mu\nu}=pg^{\mu\nu}+\omega
U^{\mu}U^{\nu}+F^{\mu}_{\,\sigma}F^{\nu\sigma}-\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta},$
(31)

where $p$, $\omega$, $U^{\mu}$, and $F^{\mu\nu}$ are the pressure, enthalpy
density, four-velocity, and electromagnetic field tensor of the plasma.

The energy-at-infinity is

$e^{\infty}=-\alpha g_{\mu 0}T^{\mu
0}=\alpha\hat{e}+\alpha\beta^{\phi}\hat{P}^{\phi},$ (32)

with the total energy density $\hat{e}$ and the azimuthal component of the
momentum density $\hat{P}^{\phi}$ given by

$\hat{e}=\omega\hat{\gamma}^{2}-p+\frac{\hat{B}^{2}+\hat{E}^{2}}{2},\quad\hat{P}^{\phi}=\omega\hat{\gamma}^{2}\hat{v}^{\phi}+(\hat{B}\times\hat{E})^{\phi}.$
(33)

The Lorentz factor is
$\hat{\gamma}=\hat{U}^{0}=1/\sqrt{1-\sum_{i=1}^{3}(\hat{v}^{i})^{2}}$. The
components of magnetic and electric fields
$\hat{B}^{i}=\epsilon^{ijk}\hat{F}_{jk}/2$ and
$\hat{E}^{i}=\eta^{ij}\hat{F}_{j0}=\hat{F}_{i0}$. The sign $\hat{v}^{\phi}$
denotes the azimuthal component of the out flow velocity of plasma for a ZAMO
observer.

Further, the energy-at-infinity can be divided into the hydrodynamic and
electromagnetic components such that
$e^{\infty}=e^{\infty}_{hyd}+e^{\infty}_{em}$ with

$e^{\infty}_{hyd}=\alpha\hat{e}_{hyd}+\alpha\beta^{\phi}\omega\hat{\gamma}^{2}\hat{v}^{\phi},\quad
e^{\infty}_{em}=\alpha\hat{e}_{em}+\alpha\beta^{\phi}(\hat{B}\times\hat{E})_{\phi},$
(34)

where $\hat{e}_{hyd}=\omega\hat{\gamma}^{2}-p$ and
$\hat{e}_{em}=(\hat{E}^{2}+\hat{B}^{2})/2$ denote the hydrodynamic and
electromagnetic energy densities observed in the ZAMO frame. Since the
electromagnetic energy $e^{\infty}_{em}$ is converted into the kinetic energy
of the plasma, the remained electromagnetic energy can be negligible and
$e^{\infty}=e^{\infty}_{hyd}$. Supposing that the plasma is incompressible and
adiabatic, the energy-at-infinity density takes the form Koide2008

$e^{\infty}=\alpha\omega\hat{\gamma}(1+\beta^{\phi}\hat{v}^{\phi})-\frac{\alpha
p}{\hat{\gamma}}.$ (35)

By introducing the local rest frame for the bulk plasmas
$x^{\mu\prime}=(x^{0\prime},x^{1\prime},x^{2\prime},x^{3\prime})$, which
rotates with Keplerian angular velocity $\Omega_{k}$ in the equatorial plane,
the localized reconnection process can be assessed. For convenience, one can
choose the directions of $x^{1\prime}$ and $x^{3\prime}$ such that they are
parallel to radial direction $x^{1}=r$ and azimuthal direction $x^{3}=\phi$,
respectively.

Using Eq. (29), the corotating Keplerian velocity in the ZAMO frame can be
expressed as

$\hat{v}_{K}=\frac{d\hat{x}^{\phi}}{d\hat{x}^{t}}=\frac{\sqrt{g_{\phi\phi}}dx^{\phi}-\alpha\beta^{\phi}dx^{t}}{\alpha
dx^{t}}=\frac{\sqrt{g_{\phi\phi}}}{\alpha}\Omega_{K}-\beta^{\phi}.$ (36)

Here $\Omega_{K}$ is the Keplerian angular velocity in Boyer-Lindquist
coordinate given in Eq. (25). From the previous formula, the Lorentz factor
$\hat{\gamma}_{K}=1/\sqrt{1-\hat{v}_{K}^{2}}$ in ZAMO frame is obtained.

According to “relativistic adiabatic incompressible ball method”, the
hydrodynamic energy associated with accelerated/decelerated plasma at the
infinity per enthalpy is derived as follows Comisso2020

$\epsilon_{\pm}^{\infty}=\alpha\hat{\gamma}_{K}\left((1+\beta^{\phi}\hat{v}_{K})\sqrt{1+\sigma_{0}}\pm\cos\xi(\beta^{\phi}+\hat{v}_{K})\sqrt{\sigma_{0}}-\frac{\sqrt{1+\sigma_{0}}\mp\cos\xi\hat{v}_{K}\sqrt{\sigma_{0}}}{4\hat{\gamma}_{K}^{2}(1+\sigma_{0}-\cos^{2}\xi\hat{v}_{K}^{2}\sigma_{0})}\right),$
(37)

where $\pm$ corresponds to the accelerated and decelerated plasma. Parameters
$\sigma_{0}$ and $\xi$ are, respectively, the plasma magnetization and
orientation angle between the magnetic field lines and the azimuthal direction
in the equatorial plane. From Eq. (37), it is clear that the energy-at-
infinity per enthalpy of the accelerated/decelerated plasma relies on wormhole
parameters $(a,M,\ell)$ and magnetic reconnection configuration ($\sigma_{0}$,
$\xi$, $r_{X}$), where $r_{X}$ refers to the position of the reconnection
point and we denote it as the X-point.

If the decelerated plasmas acquire negative energy as measured at infinity,
then the accelerated plasmas will have more positive energy compared to their
rest mass and thermal energy. As a result, the rotating energy of the wormhole
shall be extracted via magnetic reconnection process. Considering it, the
corresponding conditions for the allowed energy extraction are

$\epsilon^{\infty}_{-}<0,\quad\text{and}\quad\Delta\epsilon_{+}^{\infty}=\epsilon_{+}^{\infty}-\left(1-\frac{\Gamma}{\Gamma-1}\frac{p}{\omega}\right)=\epsilon^{\infty}_{+}>0,$
(38)

where the polytropic index is $\Gamma=4/3$ for the relativistically hot
plasmas.

Now, we concern on exploring the above energy extraction conditions (38) for
the spinning wormhole. For the low spinning case ($a=0.7M$), we show
$\epsilon^{\infty}_{+}$ and $\epsilon^{\infty}_{-}$ as a function of the
regularization parameter $\ell$ in Figs. 3(a) and 3(b), where they present
significantly different behaviors. The green, red, blue, black, and purple
curves are for the X-point positions $r_{X}/M=$0, 0.5, 1.0, 1.5, and 2.0,
respectively. Obviously, $\epsilon^{\infty}_{+}$ shown in Fig. 3(a) always
gradually decreases with the regularization parameter $\ell$, and importantly,
the condition $\epsilon^{\infty}_{+}>0$ is satisfied for the accelerated
plasmas. In order to extract more rotating energy, one needs to tune the
suitable regularization parameter $\ell$ for a high $\epsilon^{\infty}_{+}$,
which can be well achieved by reducing $\ell$ and shifting the magnetic
reconnection location near the throat of wormhole. Nevertheless, there always
exists a lower bound for $\ell$, where $\epsilon^{\infty}_{+}$ diverges.
$\epsilon^{\infty}_{-}$ is another important factor in this process, and we
plot it in Fig. 3(b). Similar to $\epsilon^{\infty}_{+}$,
$\epsilon^{\infty}_{-}$ is also bounded by $\ell$, while with lower values.
Quite differently, each curve exhibits a local minimum, and which is shifted
towards to small $\ell$ by $r_{X}$. Despite these behaviors, we find that
$\epsilon^{\infty}_{-}$ is always positive for arbitrary values of $\ell$ and
$r_{X}$ leading to the violation of condition (38). So it is impossible to
extract the energy for a slow spinning wormhole. This result also holds for a
slow spinning Kerr black hole Comisso2020 . One notable difference of them is
that the Kerr black hole is bounded by a maximal spin $a/M=1$, while it is
unbounded for the wormhole. So it is worthwhile to consider such process for a
rapidly spinning wormhole.

For the purpose, we take high-spin situation $a/M=1.5$ and plot
$\epsilon^{\infty}_{\pm}$ in Figs. 3(c) and 3(d). It is clear that
$\epsilon^{\infty}_{+}$ still decreases with $l$. While the local minimum of
$\epsilon^{\infty}_{-}$ disappears. Significantly, $\epsilon^{\infty}_{-}$
could be negative at small $\ell$ and $r_{X}$, which strongly indicates that
it is feasible to extract energy from a spinning wormhole via the magnetic
reconnection. Simultaneously, $\epsilon^{\infty}_{+}$ takes a larger positive
value for small $\ell$ and $r_{X}$, making the energy extraction more easily
to occur.

(a) $a$=0.7$M$

(b) $a$=0.7$M$

(c) $a$=1.5$M$

(d) $a$=1.5$M$

Figure 3: Energy $\epsilon_{\pm}^{\infty}$ as a function of the regularization
parameter $\ell/M$ for $\xi=0$ and $\sigma_{0}=5$. (a) $\epsilon_{+}^{\infty}$
with $a=0.7M$. $r_{X}/M=$0, 0.5, 1.0, 1.5, 2.0 from right to left. (b)
$\epsilon_{-}^{\infty}$ with $a=0.7M$. $r_{X}/M=$0, 0.5, 1.0, 1.5, 2.0 from
right to left. (c) $\epsilon_{+}^{\infty}$ with $a=1.5M$. $r_{X}/M=$0, 0.7,
1.0, 1.3, 2.0 from top to bottom. (d) $\epsilon_{-}^{\infty}$ with $a=1.5M$.
$r_{X}/M=$0, 0.7, 1.0, 1.3, 2.0 from bottom to top.

(a) $r_{X}=0.5M$ and $\xi=\pi/12$

(b) $r_{X}=0.5M$ and $\sigma_{0}=100$

(c) $r_{X}=1.5M$ and $\xi=\pi/12$

(d) $r_{X}=1.5M$ and $\sigma_{0}=100$

Figure 4: Parameter regions for negative $\epsilon_{-}^{\infty}$ in the
($a/M$, $\ell/M$) plane. (a) $r_{X}=0.5M$ and $\xi=\pi/12$. $\sigma_{0}=$1, 5,
10, 30 from bottom to top. (b) $r_{X}=0.5M$ and $\sigma_{0}=100$. $\xi=\pi/6$,
$\pi/8$, $\pi/12$, $\pi/20$ from bottom to top. (c) $r_{X}=1.5M$ and
$\xi=\pi/12$. $\sigma_{0}=$1, 5, 10, 30 from bottom to top. (d) $r_{X}=1.5M$
and $\sigma_{0}=100$. $\xi=\pi/6$, $\pi/8$, $\pi/12$, $\pi/20$ from bottom to
top.

Next, we turn to consider the possible parameter regions of $a$ and $\ell$
allowed for the energy extraction. Noting that $\epsilon^{\infty}_{+}$ is
always positive, we only focus on the condition with $\epsilon^{\infty}_{-}<0$
by varying $\sigma_{0}$, $\xi$, and $r_{X}$.

After a simple algebra, we show the shaded negative energy regions in Fig. 4
with these curves denoting $\epsilon^{\infty}_{-}=0$. The magnetic
reconnection locations are set to $r_{X}=0.5M$ in Figs. 4(a) and 4(b), and
$r_{X}=1.5M$ in Figs. 4(c) and 4(d). It is quite obvious that the shaded
regions shrink with $r_{X}$, for example, the maximal $\ell$ decreases from
1.9$M$ to 1.2$M$. By setting $\xi=\pi/12$, we observe that the shaded regions
enlarge with the magnetization $\sigma_{0}$ from Figs. 4(a) and 4(c). On the
other hand, when taking $\sigma_{0}=100$, the negative energy regions shrink
with $\xi$ as shown in Figs. 4(b) and 4(d). In summary, the parameter region
allowed for the energy extraction enlarges with $\sigma_{0}$ while shrinks
with $\xi$ and $r_{X}$.

Another thing worth emphasizing is that the spin of the wormhole is always
bounded within a certain range, see Fig. 4. Thus, both high and low spin shall
make it impossible to extract energy, quite different from the result of Kerr
black hole, where high spin always promotes the process.

##  IV Efficiency and Power

We now proceed to the efficiency of the magnetic reconnection from a spinning
wormhole. As mentioned above, after the magnetic reconnection process, the
magnetic field energy inside the ergo-region of wormholes is redistributed.
The decelerated plasmas with negative energy-at-infinity are swallowed by
wormholes, while the positive energy plasmas escape to infinity. Consequently,
the proportion of positive energy-at-infinity after redistribution can be used
to scale the efficiency of energy extraction. Following Ref. Comisso2020 , the
efficiency of the plasma energization process via magnetic reconnection is
defined by

$\eta=\frac{\epsilon_{+}^{\infty}}{\epsilon^{\infty}_{+}+\epsilon^{\infty}_{-}}.$
(39)

To achieve energy extraction, we must have $\eta>1$. The lower bound of the
efficiency $\eta=1$ corresponds to the vanishing negative energy-at-infinity,
i.e. $\epsilon^{\infty}_{-}=0$.

With fixing $\sigma_{0}=100$ and $\xi=\pi/20$, the efficiency $\eta$ is
plotted as a function of $\ell$ in Fig. 5 for $r_{X}/M=$0, 0.4, 0.8, and 1.2.
The spin parameter $a/M$ is set to 1.15, 1.18, 1.22, and 1.28 from top to
bottom. It is notable that $\eta$ decreases with the black hole spin,
indicating that high spin does not necessarily have an advantage for energy
extraction. For small $r_{X}/M=0$ and 0.4, respectively, shown in Figs. 5(a)
and 5(b), a peak emerges for each $a/M$. Its maximum value is significantly
reduced by the spin of wormhole from 6 to 2.5 when $a/M$ increases from 1.15
to 1.28. Meanwhile, the location of the peak is shifted towards to small
$\ell/M$. Further increasing $r_{X}/M$ such that $r_{X}/M$=0.8 and 1.2 in
Figs. 5(c) and 5(d), the peak tends to disappear and $\eta$ will be only a
monotonically decreasing function of $\ell/M$. The maximum $\eta$ continues to
decrease below 1.6 for $r_{X}/M$=1.2, and such value will approach a lower
value with the increase of $r_{X}/M$. As a result, in order to reach a high
efficiency of energy extraction, increasing the spin is not necessarily a good
choice. Instead, reducing $r_{X}/M$ is an alternative.

(a) $r_{X}=0M$

(b) $r_{X}=0.4M$

(c) $r_{X}=0.8M$

(d) $r_{X}=1.2M$

Figure 5: Efficiency $\eta$ of magnetic reconnection mechanism. (a) $r_{X}=0M$
and $a/M=$1.15, 1.18, 1.22, 1.28 from top to bottom. (b) $r_{X}=0.4M$ and
$a/M=$1.15, 1.18, 1.22, 1.28 from top to bottom. (c) $r_{X}=0.8M$ and
$a/M=$1.15, 1.18, 1.22, 1.28 from top to bottom. (d) $r_{X}=1.2M$ and
$a/M=$1.15, 1.18, 1.22, 1.28 from top to bottom. Here $\sigma_{0}=100$ and
$\xi=\pi/20$.

Besides the efficiency, another important key to measure the magnetic
reconnection mechanism is the energy absorbing power, which depends on the
negative energy-at-infinity of decelerated plasmas swallowed by wormholes in
unit time. According to the energy conservation, a high absorbing power
potentially results in a high energy extraction rate by escaping plasmas.

The power of energy extraction per enthalpy via magnetic reconnection from
wormholes can also be defined by Comisso2020

$P_{extr}=-\epsilon^{\infty}_{-}A_{in}U_{in},$ (40)

where $U_{in}=\mathcal{O}(10^{-1})$ and $\mathcal{O}(10^{-2})$ for the
collisionless and collisional regimes Comisso2016b , respectively. For
spinning wormholes, $A_{in}$ is the cross-sectional area of the inflowing
plasma and is estimated as

$A_{in}\sim\begin{cases}(r^{+}_{erg})^{2}-r_{ph}^{2},\quad&\text{with photon
orbit},\\\ (r^{+}_{erg})^{2},\quad&\text{without photon orbit}.\end{cases}$
(41)

Here $r^{+}_{reg}$ and $r_{ph}$ are the radii of the ergo surface and photon
orbit of wormhole, seen previously in Eq. (10) and (19).

We first consider the power for the case with spin $a/M<1$. As a typical
example, we take $a=0.93M$ and $0.96M$. The former possesses a photon orbit
while the latter does not. Such significant difference shall produce different
scenarios of the power, see Figs. 6(a) and 6(b). Although the ending point is
at $r_{erg}^{+}$, the most obvious difference is that the starting point
locates at $r_{ph}$ for $a=0.93M$, while at the throat of wormhole for another
case without photon orbit. Nevertheless, we observe that the enhancement of
magnetization $\sigma_{0}$ raises the extracting power for both scenarios as
expected. For each given $\sigma_{0}$, the maximum value of the power almost
keeps the same for $a=0.93M$ and $0.96M$.

(a) $a$=0.93$M$

(b) $a$=0.96$M$

Figure 6: Power of magnetic reconnection mechanism with $\ell/M=1.4$. (a)
Spinning wormhole with photon orbit. The spin $a=0.93M$ and magnetization
$\sigma_{0}$=10, $10^{2}$, $10^{3}$, $10^{4}$, and $10^{5}$ from bottom to
top. The left and right dashed vertical lines stand for the position of the
photon orbit and outer ergo-surface, respectively. (b) Spinning wormhole
without photon orbit. The spin $a=0.96M$ and magnetization $\sigma_{0}$=10,
$10^{2}$, $10^{3}$, $10^{4}$, and $10^{5}$ from bottom to top. Here we take
$\xi=0$ and $U_{in}=0.1$ for the collisionless regime.

(a) $r_{X}$=0$M$

(b) $r_{X}$=0.4$M$

(c) $r_{X}$=0.8$M$

(d) $r_{X}$=1.2$M$

Figure 7: Power $P_{extr}$ as a function of the regularization parameter
$\ell/M$ with $\sigma_{0}=10^{4}$ and $\xi=\pi/12$. (a) $r_{X}=0M$. (b)
$r_{X}=0.4M$. (c) $r_{X}=0.8M$. (d) $r_{X}=1.2M$. Wormhole spin $a/M=$1.1,
1.3, 1.5, 1.8 from top to bottom.

Another interesting case is that the wormhole spin is beyond one exceeding the
maximal bound of Kerr black hole, which presents a unique feature for the
wormhole. For the purpose, we take $a/M=$1.1, 1.3, 1.5, and 1.8, and show the
power as a function of $\ell$ in Fig. 7. For different $r_{X}/M=0$, 0.4, 0.8,
and 1.2, we see that the power $P_{extr}$ decreases with $\ell$. The increase
of the wormhole spin will also reduce the power. Thus we can conclude that the
farther away of the X-point from a rapidly spinning wormhole throat, the lower
the power. It is worth noting that the power is not well-defined by
simultaneously taking $r_{X}/M=0$ and $\ell\to 0$, see the divergent behaviors
shown in Fig. 7(a).

(a) Kerr BH

(b) $\ell$=0.5$M$

(c) Kerr BH

(d) $\ell$=0.5$M$

Figure 8: (a) Energy extracting efficiency for the Kerr black holes. Black
hole spin $a/M=$0.93, 0.96, 0.99, 1.0 from bottom to top. (b) Efficiency ratio
of wormhole with respect to the maximal Kerr black hole efficiency. Spin
$a/M=$1.15, 1.18, 1.22, 1.28 from top to bottom. (c) Energy extracting power
for the Kerr black hole. Black hole spin $a/M=$0.93, 0.96, 0.99, 1.0 from
bottom to top. (d) Power ratio of wormhole with respect to the maximal Kerr
black hole power. Spin $a/M=$1.1, 1.3, 1.5, 1.8 from top to bottom. Other
parameters are set to $U_{in}=0.1$, $\sigma_{0}=100$, and $\xi=\pi/12$.

Before ending this section, we would like to make a comparison between the
Kerr black hole and wormhole. By setting $\ell\to 0$, the wormhole model will
recover the Kerr black hole. Let us examine the efficiency and power for the
Kerr black hole, which are shown in Figs. 8(a) and 8(c) by taking
$U_{in}=0.1$, $\sigma_{0}=100$, and $\xi=\pi/12$. A significant result is that
both the efficiency and power reach their maximal values for the extremal
black hole with spin $a/M=1$. In order to examine whether the wormhole has
advantage than the Kerr black hole, we define two ratios of the efficiency and
power

$R_{\eta}=\frac{\eta}{\lim\limits_{a,\;r_{X}\to M}\eta_{Kerr}},\quad
R_{p}=\frac{P_{extr}}{\lim\limits_{a,\;r_{X}\to M}P_{Kerr}},$ (42)

where the denominators denote the efficiency and power of extremal Kerr black
holes, respectively. If they are above one, the wormhole would be more
efficient than the Kerr black hole to extract energy via the magnetic
reconnection mechanism. Otherwise, the black hole will dominate.

These two ratios are plotted as a function of $r_{X}/M$ in Figs. 8(b) and 8(d)
with $\ell/M=0.5$. The ratio $R_{\eta}$ of the efficiency exhibits a non-
monotonic behavior by varying the wormhole spin $a/M$ from 1.15 to 1.28. These
peaks are near $r_{X}/M=0.4$. However, for the ratio $R_{p}$, it gradually
decreases with $r_{X}/M$ for $a/M=$1.1-1.8. We also observe that the increase
of the wormhole spin reduces both ratio $R_{\eta}$ and $R_{p}$, which is
consistent with our above result that continuously increasing spin is not a
better approach to extract energy. Moreover, from Figs. 8(b) and 8(d), one can
see that both the ratios go beyond one when $r_{X}/M$ is smaller than some
certain values approximately near $r_{X}/M=1$. Therefore, wormhole has
advantages on extracting energy via magnetic reconnection only at small
X-point.

##  V conclusions

In this paper, we mainly focus on the energy extraction via the magnetic
reconnection mechanism from spinning wormholes characterized by spin $a/M$ and
regularization parameter $\ell$.

At first, we pointed out the wormhole region we concerned in the ($a$, $\ell$)
plane. Such region must satisfy $a/M>1$ or $\ell>\rho_{+}$. Considering that a
Kerr black hole is always bounded by its maximal spin $a/M=1$, the spinning
wormholes are divided into two cases, the slowly spinning one with $a/M<1$ and
the rapidly spinning one with $a/M>1$. These two cases have different
spacetime structures. For example, they present distinguished ergo-regions,
see Fig. 2. The former wormhole only has one ergo surface outside the throat,
while the latter has two.

We also showed that for some slowly spinning wormholes, both
$\Delta\epsilon_{+}^{\infty}$ and $\epsilon_{-}^{\infty}$ of accelerated and
decelerated plasmas are positive, making the energy extraction impossible.
Such result is quite similar to the Kerr black hole with lower spin. However,
for a rapidly spinning wormhole, the conditions $\epsilon_{+}^{\infty}>0$ and
$\epsilon_{-}^{\infty}<0$ can be satisfied simultaneously for small $\ell/M$.
Hence, this fact indicates that the spinning wormhole can act as a source for
energy extraction via the magnetic reconnection process.

In order to implement the energy extraction, we exhibited the possible
parameter region in ($a$, $\ell$) plane. It is obvious that both too large and
too small wormhole spins are unfavorable for energy extraction. The
regularization parameter $\ell$ is also bounded. On the contrary, the
parameter region shrinks with the location of X-point and orientation angle
$\xi$, while enlarges with the magnetization $\sigma_{0}$ as expected.

After examining the possibility of energy extraction, we calculated the
efficiency $\eta$ and power of the magnetic reconnection process. If such
process occurs near the wormhole throat, the efficiency always has a peak for
each given wormhole spin. The peak shall disappear when the magnetic
reconnection process occurs far away from the throat. And the maximal
efficiency is shifted to $\ell/M=0$. The power behaves quite differently for
the slowly and rapidly spinning wormhole for the reason that the cross-
sectional area takes different forms with or without circular photon orbits.
The numerical result indicates that the farther away of the X-point from a
rapidly spinning wormhole throat, the lower the power.

We also compared our results with that of an extremal Kerr black hole by
defining the efficiency and power ratios. We observed peaks for the efficiency
ratio. But the power ratio is monotonically decreasing with the location of
X-point. Both the results show that the wormhole will dominate the energy
extraction for the X-point location $r_{X}/M<1$. This presents the most
advantage of spinning wormhole than the Kerr black hole.

In summary, our study confirms the feasibility of extracting energy from the
wormhole via magnetic reconnection under the appropriate parameters $a/M$ and
$\ell/M$. The study of the efficiency and power also reveals that wormhole
indeed has advantages, especially, when the location of X-point is close to
the wormhole throat. These results shed light into the energy extraction via
magnetic reconnection from horizonless objects.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grants No. 12075103, No. 12247101).

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