# Thermodynamics of EiBI-AdS black holes with global monopole

A. M. Kusuma  B. N. Jayawiguna  H. S. Ramadhan  Departemen Fisika, FMIPA,
Universitas Indonesia, Depok, 16424, Indonesia.

###### Abstract

It is well-known that black hole can be endowed with topological charge coming
as a result of phase transition(s) in the early universe. It has also recently
been revealed that such object might exist in one of the modified theory of
gravity called the Eddington-inspired-Born-Infeld (EiBI) theory. Here we shall
investigate the the possibility of phase transitions, and in general
thermodynamic phenomena, of EiBI-anti-de Sitter (AdS) black hole with
(topologically-charged) global monopole. This is the first work that applies
the full Euclidean action formalism in this model. We provide a counterterm to
cancel inifinities and argue that it is the most suitable among other
possibilities. Our investigation reveals that the state variables obtained are
found to obey the first law of the black hole mechanics and the Smarr’s law
for black holes with $\Lambda\neq 0$. Related to the second feature, we
obtained the forbidden range of parameter space for EiBI AdS black hole with
global monopole, which corresponds to
$\frac{-(1-\Delta)}{\Lambda}\leq\kappa\leq-\frac{1}{\Lambda}$. The
dependencies on $\Lambda$, the EiBI constant $\kappa$ and the global monopole
charge $\eta$ of the state variables and state functions obtained manifests in
a Schwarzchild AdS-like phase transition for black holes with parameters below
the lower bound of the forbidden range, and could also manifest in a
Schwarzschild-flat like phase behavior for black holes with parameters above
the higher bound of the forbidden range.

##  I Introduction

A series of phase transitions in the early universe might have produced global
monopole(s) when global $SO(3)$ symmetry is spontaneously broken Vilenkin .
These objects, along with other types of topological defects, are of great
interest in cosmology as their existence might reveal information about the
early history of our cosmos and the unification of fundamental interactions.
The gravitational field of this object was first studied by Barriola and
Vilenkin in 1989 Barriola:1989hx  that shows that, although the monopole
exerts no gravitational force, nonetheless the metric is not asymptotically
flat but suffers a deficit solid angle. When the monopole core is much smaller
than the Schwarszchild radius, the solution describes a black hole eating up a
global monopole Dadhich:1997mh . Ever since then, numerous studies have been
devoted to the study of classical properties as well as cosmological
signatures of global monopole black holes in various modifications. Black
holes with non-canonical global monopole within the framework of General
Relativity (GR) were studied, for example, in Jin:2007fz ; Liu:2009eh ;
Prasetyo:2017rij . Extensive works have also been done to investigating
canonical black holes within the modified gravity models, for example in
Carames:2011uu  and the references therein. Recently, gravitational field of
global monopole was studied in the framework of Eddington-inspired-Born-Infeld
(EiBI) gravity Lambaga:2018yzv , a modified gravity model that revives an old
proposal by Eddington and was proposed to solve singularity problems within GR
Banados:2010ix ; Delsate:2012ky .

Semiclassically, black holes radiate Bekenstein:1973ur ; Bardeen:1973gs  and,
as a consequence, may undergo phase transition Roychowdhury:2014cva . The
study of black hole phase transition began when Hut and, independently, Davies
pointed out such possibility HutP1977 ; Davies:1978zz . Interests in this
field of study then has been steadily increasing ever since Hawking and Page’s
seminal finding in 1983 that in AdS space, Schwarzschild black holes display
phase behavior and thus undergo phase transitions Hawking:1982dh . Adopting
the Euclidean formulation, they found in Schwarzschild-AdS there are two kinds
of black holes that can exist, the thermodynamically-stable large and the
thermodynamically-unstable small black holes. By treating the AdS space as the
background metric, it has also been found that at certain critical temperature
there would be phase transition between the thermal AdS space and the stable
large Schwarzschild AdS black hole state. From then on numerous mechanisms to
explain the thermodynamics and critical behaviour of AdS black holes have been
proposed. One of the most interesting finding between all the proposed
discourses is the fact that the cosmological constant in black holes could be
thought of as a varying parameter Kastor:2009wy . Once the cosmological
constant variation is taken into account, there shall be a modification in the
first law of black hole mechanics so that its form would be consistent with
the Smarr’s relation. The cosmological constant variation itself then can be
done in its associated form, namely as the variation of spacetime’s negative
pressure $P$. It has been established that for $4d$ black holes, the
cosmological constant in terms of the natural units is defined as

$\Lambda=-8\pi P=-\frac{3}{l^{2}}.$ (1)

Note that the $l$ above is the AdS radius constant. The existence of pressure
indicates the presence of its conjugate form, thermodynamic volume $V$
Sekiwa:2006qj ; Dolan:2010ha . The introduction of $P$ and $V$ in the first
law of black hole mechanics leads to a more comprehensive study of
thermodynamic quantities of black holes, one of its most enterprising branch
of study is now widely dubbed as black hole chemistry Kubiznak:2012wp ;
Kubiznak:2016qmn .

In contrast to the extensive studies on thermodynamics and phase transitions
of vacuum or electrically-charged black holes, the study of them charged with
global monopole is relatively rare, and even much less for the case of
modified gravity. The thermodynamics of global monopole black holes in $4d$
were investigated in Jing:1993np ; Yu:1994fy ; Jensen:1995fz ; Chen:2001jn ;
Deng:2018wrd ; Kumara:2019xgt ; Soroushfar:2020wch , while their higher-
dimensional generalization was discussed, for example, in Ramadhan:2018gbf .
Semiclassical analysis of black holes in EiBI gravity were studied in
He:2016yuc ; Ozen:2017uoe ; Jayawiguna:2018tba . It is therefore interesting
to investigate the effect of global monopole to the critical behavior of AdS
black holes with global monopole in EiBI gravity. That is the purpose of this
work. To achieve that, this paper proceeds as follows. In the next section we
review the static black hole with global monopole in EiBI gravity. In Sec. III
we present the Euclidean action formalism of EiBI AdS black hole with global
monopole from which the corresponding partition function can be obtained. In
Sec. IV we calculate the black hole’s state variables and in Sec. V we present
the local and global stability of this model by investigating its phase
structure. Sec. VI is devoted to the conclusion and discussion. Note that
throughout this work we have set $\hbar=c=G=1$, unless otherwise stated.

##  II EiBI-AdS global monopole (EiBI-GM-AdS)

It is instructive to briefly review the EiBI global monopole solutions laid
out in Lambaga:2018yzv . The action is

$S=\frac{1}{8\pi\kappa}\int d^{4}\tilde{x}\left(\sqrt{-|g_{\mu\nu}+\kappa
R_{\mu\nu}(\Gamma)|}-\lambda\sqrt{-g}\right)+S_{M}[g,\Phi_{M}].$ (2)

The matter Lagrangian is given by the global SO(3) Higgs field,

$\mathcal{L}=\frac{1}{2}(\nabla\Phi^{a})^{2}-\frac{\sigma}{4}(\Phi^{a}\Phi^{a}-\eta^{2})^{2}.$
(3)

Employing the Palatini formalism, the gravitational field equations are

$\sqrt{-q}q^{\mu\nu}=\lambda\sqrt{-g}g^{\mu\nu}-8\pi\kappa\sqrt{-g}T^{\mu\nu},$
(4)

and

$q_{\mu\nu}=g_{\mu\nu}+\kappa R_{\mu\nu},$ (5)

for $g_{\mu\nu}$ and $q_{\mu\nu}$, respectively.

Assuming static, spherically symmetric configuration, we choose ansatz

$\displaystyle ds^{2}_{g}$ $\displaystyle=$
$\displaystyle-\psi^{2}(\tilde{r})f(\tilde{r})d\tilde{t}^{2}+\frac{1}{f(\tilde{r})}d\tilde{r}^{2}+\tilde{r}^{2}d\Omega^{2}_{2},$
(6) $\displaystyle ds^{2}_{q}$ $\displaystyle=$
$\displaystyle-G^{2}(\tilde{r})F(\tilde{r})d\tilde{t}^{2}+\frac{1}{F(\tilde{r})}d\tilde{r}^{2}+H^{2}(\tilde{r})d\Omega^{2}_{2},$
(7)

and

$\Phi^{a}=\phi(r)\eta\frac{x^{a}}{r},$ (8)

where $ds^{2}_{g}$ and $ds^{2}_{q}$ are the physical and auxiliary metrics,
respectively. Outside the monopole core, the exterior energy momentum tensors
are ($\phi\approx 1$)
$T^{\mu}_{\nu}=\eta^{2}/\tilde{r}^{2}\leavevmode\nobreak\ diag(-1,-1,0,0)$.
Substituting it to Eq. (4), we get

$\displaystyle
H^{2}=\lambda\tilde{r}^{2}+8\pi\kappa\eta^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ G=\lambda\psi,\leavevmode\nobreak\ \leavevmode\nobreak\
F=\frac{f}{\lambda}.$ (9)

On the other hand, Eq. (5) read

$\displaystyle\frac{2}{\kappa F}\left(\frac{\psi^{2}f}{G^{2}F}-1\right)$
$\displaystyle=$
$\displaystyle\frac{F^{\prime\prime}}{F}+\frac{2G^{\prime\prime}}{G}+\frac{3G^{\prime}F^{\prime}}{GF}+\frac{2F^{\prime}H^{\prime}}{FH}+\frac{4G^{\prime}H^{\prime}}{GH},$
$\displaystyle\frac{2}{\kappa F}\left(\frac{F}{f}-1\right)$ $\displaystyle=$
$\displaystyle\frac{F^{\prime\prime}}{F}+\frac{2G^{\prime\prime}}{G}+\frac{4H^{\prime\prime}}{H}+\frac{2F^{\prime}H^{\prime}}{FH}+\frac{3F^{\prime}G^{\prime}}{FG},$
$\displaystyle\frac{1}{\kappa F}\left(\frac{r^{2}}{H^{2}}-1\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{H^{2}F}+\frac{F^{\prime}H^{\prime}}{FH}+\frac{H^{\prime
2}}{H^{2}}+\frac{H^{\prime\prime}}{H}+\frac{H^{\prime}G^{\prime}}{HG}.$

Solving Eqs. (LABEL:4) simultaneously, the solutions are

$\psi(\tilde{r})=\frac{\sqrt{\lambda}\tilde{r}}{\sqrt{\lambda\tilde{r}^{2}+8\pi\kappa\eta^{2}}},$
(11)

and

Figure 1: Plot of $\psi(r)$ for $\kappa=1$ and $\eta=0.5$.

$f(\tilde{r})=1-\frac{2\tilde{m}\sqrt{\lambda}\sqrt{\lambda\tilde{r}^{2}+\Delta\kappa}}{\lambda\tilde{r}^{2}}-\frac{\Lambda}{3}\tilde{r}^{2}-\left[\frac{2\Lambda}{3}+\frac{(\tilde{r}^{2}-1)}{\tilde{r}^{2}}\right]\frac{\Delta}{\lambda}-\left(1+\frac{\Lambda}{3}\right)\frac{\Delta^{2}}{\lambda^{2}\tilde{r}^{2}},$
(12)

with $\Delta\equiv 8\pi\eta^{2}$. Defining $\tilde{\Delta}=\Delta/\lambda$ and
applying the following rescaling prescriptions,

$\displaystyle\tilde{t}\rightarrow\frac{t}{\sqrt{1-\tilde{\Delta}}},$
$\displaystyle\tilde{r}\rightarrow r\sqrt{1-\tilde{\Delta}},$
$\displaystyle\tilde{\Delta}\kappa\rightarrow(1-\tilde{\Delta})\kappa,$
$\displaystyle\tilde{m}\rightarrow m(1-\tilde{\Delta})^{3/2},$ (13)

the line elements from Eqs. (6)-(7) are thus obtained to be

$\displaystyle ds^{2}_{g}$ $\displaystyle=$
$\displaystyle-\left[1-\frac{2m}{\sqrt{r^{2}+\kappa}}-\frac{1}{3}\Lambda(\kappa+r^{2})\right]dt^{2}+\frac{r^{2}}{r^{2}+\kappa}\left[1-\frac{2m}{\sqrt{r^{2}+\kappa}}-\frac{1}{3}\Lambda(\kappa+r^{2})\right]^{-1}dr^{2}$
(14) $\displaystyle+r^{2}(1-\tilde{\Delta})d\Omega^{2}_{2}.$ $\displaystyle
ds^{2}_{q}$ $\displaystyle=$
$\displaystyle-\lambda\left[1-\frac{2m}{\sqrt{\kappa+r^{2}}}-\frac{1}{3}\Lambda\left(\kappa+r^{2}\right)\right]dt^{2}+\frac{\lambda
r^{2}}{\kappa+r^{2}}\left[1-\frac{2m}{\sqrt{\kappa+r^{2}}}-\frac{1}{3}\Lambda\left(\kappa+r^{2}\right)\right]^{-1}dr^{2}$
(15) $\displaystyle+\lambda(r^{2}+\kappa)(1-\tilde{\Delta})d\Omega^{2}_{2}.$

The profiles of $\psi(r)$ and $f(r)$ are shown in In Figs. 1-2.

|
---|---
Figure 2: Profiles of $f(r)$ for several values of $\kappa$. In this plot, we
set $m=1$.

##  III Euclidean action formulation

The complexified action integrals may serve as a starting point to obtain the
state variables of black holes, as what has been first established by Gibbons
and Hawking in 1977 Gibbons:1976ue . For convenience, we shall relabel some
components in Eqs. (14)-(15) as follows

$\displaystyle A(r)$ $\displaystyle\equiv$ $\displaystyle
1-\frac{2m}{\sqrt{r^{2}+\kappa}}-\frac{1}{3}\Lambda\left(r^{2}+\kappa\right),$
$\displaystyle B^{2}(r)$ $\displaystyle\equiv$
$\displaystyle\frac{r^{2}}{(r^{2}+\kappa)},$ $\displaystyle C^{2}(r)$
$\displaystyle\equiv$ $\displaystyle(r^{2}+\kappa).$ (16)

Applying Wick rotation $(t\rightarrow i\tau)$ to the previously defined line
elements, we obtain

$\displaystyle ds^{2}_{g}$ $\displaystyle=$ $\displaystyle
A(r)d\tau^{2}+B^{2}(r)A^{-1}(r)dr^{2}+r^{2}(1-\tilde{\Delta})d\Omega^{2}_{2},$
(17) $\displaystyle ds^{2}_{q}$ $\displaystyle=$ $\displaystyle\lambda
A(r)d\tau^{2}+\lambda B^{2}(r)A^{-1}(r)dr^{2}+\lambda
C^{2}(r)(1-\tilde{\Delta})d\Omega^{2}_{2}.$ (18)

To construct the thermal states, we utilize the relation between the
geometrical period of spacetime with the inverse temperature in Euclidean time
Hawking:1982dh . In our case, the Hawking temperature reads Visser:1992qh

$\displaystyle T_{H}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2\pi}=\frac{1}{4\pi}\left[\frac{\partial_{r}(g_{00})}{\sqrt{g_{00}g_{11}}}\right]$
(19) $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left[\frac{1}{\sqrt{r^{2}_{+}+\kappa}}-\Lambda\sqrt{r^{2}_{+}+\kappa}\right].$

We should point out that in our case, the Hawking temperature obtained from
the physical and auxiliary metric are exactly alike with each other. Thus,
based on the result above the period of the black hole is given by

$\beta=4\pi\left[\frac{\sqrt{r^{2}_{+}+\kappa}}{1-\Lambda(r^{2}_{+}+\kappa)}\right].$
(20)

The total Euclidean action considered would be of the following form

$I=I_{b}+I_{GHY}+I_{ct},$ (21)

where $I_{b}$ is the bulk action, $I_{GHY}$ is the Gibbons-Hawking-York
boundary term and $I_{ct}$ is the counterterm. The bulk action is the Wick
rotated form of Eq. (2), while $I_{GHY}$ and $I_{ct}$ serves to regulate the
divergences that arise in the bulk action evaluation. First we evaluate the
bulk action with the line elements given in Eqs. (17)-(18), which gives

$\displaystyle I_{b}$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi\kappa}\int
d^{4}x\left(-\sqrt{q}+\lambda\sqrt{g}-8\pi\kappa\mathcal{L}_{m}\sqrt{g}\right)$
(22) $\displaystyle=$
$\displaystyle\frac{\beta\lambda}{2}\left[\frac{\Lambda}{3}(r_{+}^{2}+\kappa)^{3/2}-\frac{\Lambda}{3}(r_{b}^{2}+\kappa)^{3/2}\right](1-\tilde{\Delta})$

Note that the result above can also be obtained by working with the unrescaled
line elements given in Eqs. (6)-(7) on the bulk action evaluation, as long as
we conduct appropriate rescaling prescription previously outlined in Eq. (13)
before the $t$ and $r$ elements integration.

Before we construct the $I_{GHY}$ for EiBI gravity, first let us note that the
action given in Eq. 2 could also be written in its alternative form
Delsate:2012ky ,

$\displaystyle I_{b}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi\kappa}\int
d^{4}x\sqrt{-q}\left[\kappa
R-2+\left(q^{\mu\nu}g_{\mu\nu}-2\lambda\sqrt{\frac{(-g)}{(-q)}}\right)\right]+S_{m},$
(23) $\displaystyle=$ $\displaystyle\frac{1}{16\pi}\int
d^{4}x\sqrt{-q}\left(R-\frac{2\Lambda}{\lambda}\right).$

We can see that expression has similar structure to the usual Einstein-Hilbert
action. We then argue that the appropriate boundary term is the usual Gibbons-
Hawking-York (GHY) form Gibbons:1976ue ; York:1986it ; Arroja:2016ffm . Using
the constructed trace of the extrinsic curvature from Eq. (18) it yields

$\displaystyle I_{GHY}$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi}\int
d^{3}x\sqrt{h}K,$ (24) $\displaystyle=$
$\displaystyle\frac{\beta\lambda}{2}\left(3m+\Lambda(r_{b}^{2}+\kappa)^{3/2}-2\sqrt{r_{b}^{2}+\kappa}\right)(1-\tilde{\Delta}).$

Besides the GHY term, we also need to subtract the contribution of the black
hole’s background space. It might be tempting to immediately construct the
counterterm in such a way that it comes from 3D form of Eq (23), but such
approach would not yield an effective counterterm, as we show in the Appendix
A. Instead, we shall take the advantage of how the alternative form of the
EiBI GM AdS action is similar to the usual Einstein AdS action. Consider the
second term of (23). Using the rescaled line element given in Eq. (18), we can
see that the AdS background in EiBI gravity could be written in the following
form

$\displaystyle I_{EiBIAdS}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi\lambda}\int
d^{4}x\sqrt{\lambda^{4}A(r)B^{2}(r)A^{-1}(r)C^{4}(r)(1-\tilde{\Delta})^{2}\sin^{2}\theta}\left(2\Lambda\right),$
(25) $\displaystyle=$
$\displaystyle\frac{\lambda}{16\pi}(1-\tilde{\Delta})\int
d^{4}x\sqrt{g}\left(2\Lambda\right).$

This looks similar, up to some overall constant, to the usual AdS term found
in Einstein-Hilbert (EH) action,

$I_{EHAdS}=\frac{1}{16\pi}\int d^{4}x\sqrt{g}\left(2\Lambda\right).$ (26)

Due to the similarity of the form above with the usual EH action, it is
justifiable to use the same counterterm prescription as used in AdS black
holes with GR framework. In the search for the counterterm prescription, we
found that the most appropriate and effective approach to compute the
counterterm action is by utilizing the generalized form of the
Balasubramanian-Kraus counterterm (BK) action Emparan:1999pm ;
Balasubramanian:1999re . For $4D$ AdS black holes the counterterm is given as
follows

$I_{ct}=\frac{1}{8\pi}\int
d^{3}x\sqrt{h^{o}}\left[\frac{2}{l}+\frac{l}{2}R-\frac{l^{3}}{2}\left(R_{\mu\nu}R^{\mu\nu}-\frac{3}{8}R^{2}\right)\right].$
(27)

Thus, we will use the counterterm prescription for AdS black holes given in
Eq. (27) by treating the $\lambda(1-\tilde{\Delta})$ as a constant. The Ricci
scalar and tensor shall be constructed from the following boundary metric

$h^{o}_{ij}=\left\\{A_{o}(r_{b}),C^{2}(r_{b}),C^{2}(r_{b})\sin^{2}\theta\right\\},$
(28)

with $A_{o}$ is the modified AdS metric in EiBI gravity and we will define it
as follows

$A_{o}(r)=1-\frac{\Lambda}{3}(r^{2}+\kappa).$ (29)

Using the quantities above, the counterterm gives

$\displaystyle I_{ct}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{8\pi}(1-\tilde{\Delta})\int
d^{3}x_{o}\sqrt{h_{o}}\left[\frac{2}{l}+\frac{l}{2}R-\frac{l^{3}}{2}\left(R_{\mu\nu}R^{\mu\nu}-\frac{3}{8}R^{2}\right)\right],$
(30) $\displaystyle=$
$\displaystyle\frac{\lambda\beta_{o}\sqrt{A_{o}(r_{b})}}{2}\left\\{-\frac{l^{3}}{(r_{b}^{2}+\kappa)}+\frac{3l^{3}}{4(r_{b}^{2}+\kappa)}+\frac{2(r_{b}^{2}+\kappa)}{l}+l\right\\}(1-\tilde{\Delta}).$

Rescale the period of the AdS background so that it matches with that of the
black hole,

$\beta_{o}\sqrt{A_{o}(r_{b})}=\beta\sqrt{A(r_{b})},$ (31)

and with some approximation we shall obtain

$\beta_{o}=\beta\left[-\frac{l^{3}\left(\sqrt{r_{b}^{2}+\kappa}-2m\right)^{2}}{8\left(r_{b}^{2}+\kappa\right)^{5/2}}+\frac{l\left(\sqrt{r_{b}^{2}+\kappa}-2m\right)}{2\left(r_{b}^{2}+\kappa\right)}+\frac{\sqrt{r_{b}^{2}+\kappa}}{l}\right].$
(32)

Substituting the results above into Eq. (30) and omitting the null terms
(terms that becomes zero as $r_{b}\rightarrow\infty$) as we revert the $l$
constant back into $\Lambda$, leave us with the following results

$I_{ct}=\frac{\beta\lambda}{2}\left[-2m+2\sqrt{r_{b}^{2}+\kappa}-\frac{2\Lambda}{3}\left(r_{b}^{2}+\kappa\right)^{3/2}\right](1-\tilde{\Delta}).$
(33)

Combining the results that we have obtained so far from Eqs. (22), (24) and
(33) yields the total Euclidean action as follows

$I=\frac{\beta\lambda}{2}\left[\frac{\Lambda}{3}(r_{+}^{2}+\kappa)^{3/2}+m\right](1-\tilde{\Delta}).$
(34)

The value of $m$ can be obtained from $f(r_{+})=0$ and gives

$m=-\frac{1}{6}\sqrt{r_{+}^{2}+\kappa}\left[\Lambda(r_{+}^{2}+\kappa)-3\right].$
(35)

Thus, the total Euclidean action can be written again as follows

$I=\frac{\beta\lambda}{2}\left[\frac{\Lambda}{6}(r_{+}^{2}+\kappa)^{3/2}+\frac{1}{2}\sqrt{r_{+}^{2}+\kappa}\right](1-\tilde{\Delta}).$
(36)

##  IV State Variables, First law and Smarr’s formula

Using the total Euclidean action, we could compute state variables and state
functions of the black hole. First we shall evaluate the entropy of the
system, which gives

$\displaystyle S$ $\displaystyle=$ $\displaystyle\beta\left(\frac{\partial
I}{\partial\beta}\right)_{A}-I,$ (37) $\displaystyle=$
$\displaystyle\beta\left(\frac{\partial I}{\partial
r_{+}}\right)_{A}\Big{/}\left(\frac{\partial\beta}{\partial
r_{+}}\right)_{A}-I,$ $\displaystyle=$
$\displaystyle\lambda\pi(r_{+}^{2}+\kappa)(1-\tilde{\Delta}).$

We can see that the entropy for our case is different from that of the
ordinary Einstein-Hilbert gravity, due to the existence of the $\lambda$,
$\kappa$ and $\Delta$. Nevertheless, as $\kappa\rightarrow 0$ we can see that
the result reduces to that of the Bekenstein-Hawking entropy of a black hole
that is endowed with global monopole Deng:2018wrd . Note that the form given
in Eq. (37) can also be obtained from the known relation between the black
hole’s entropy and the event horizon area calculated from the auxiliary metric

$S=\frac{A}{4}=\frac{1}{4}\int\sqrt{q_{\theta\theta}q_{\phi\phi}}d\theta
d\phi.$ (38)

The Generalized Second Law (GSL) of black hole mechanics states that the
entropy of a black hole must always increase, $S\geq 0$ Bekenstein:1974ax ;
Wald:1999vt . For the entropy obtained in Eq. (37), it seems certain value of
$\kappa$ would yield decreasing amount of entropy, which clearly violates GSL.
Thus we proceed to find the allowed set of parameters for $\kappa$ by applying
the GSL requirement to the entropy expression, which yields the following
result

$\displaystyle 1-\tilde{\Delta}\geq 0,$ $\displaystyle
1-\frac{\Delta}{1+\kappa\Lambda}\geq 0.$ (39)

Setting the conditions that $0\leq\Delta<1$ (based on the critical value for
monopole charge Barriola:1989hx ; Prasetyo:2015bga ) and $\Lambda<0$, we
obtain the following range that should be considered as the requirement that
the hole is thermodynamically feasible

$\displaystyle\kappa\leq\frac{-(1-\Delta)}{\Lambda}$ or
$\displaystyle\kappa>-\frac{1}{\Lambda}.$ (40)

We can see the first condition on the equation above may serve as some sort of
the lower bound while the other one acts as the upper bound. The region
between these two boundaries
$\left(\frac{-(1-\Delta)}{\Lambda}\leq\kappa\leq-\frac{1}{\Lambda}\right)$
will be considered as the forbidden region for our thermodynamic analysis.

The ADM mass from the euclidean action is computed as follows

$\displaystyle M$ $\displaystyle=$ $\displaystyle\left(\frac{\partial
I}{\partial\beta}\right),$ (41) $\displaystyle=$
$\displaystyle-\frac{\lambda(1-\tilde{\Delta})}{6}\sqrt{r_{+}^{2}+\kappa}\left[\Lambda(r_{+}^{2}+\kappa)-3\right].$

Comparing the result above with Eq. (35), we can see that the ADM mass can
also be expressed in the following form

$M=\lambda(1-\tilde{\Delta})m.$ (42)

We can also see that once again there is an explicit dependence on $\lambda$,
and consequently $\kappa$, on the ADM mass. As $\kappa\rightarrow 0$ the
result would still reduces to the known value in EH gravity Deng:2018wrd .

In 2008, Kastor Kastor:2008xb  constructed a new Komar integral for black
holes with $\Lambda\neq 0$ by defining an antisymmetric 2-form potential
$\omega^{\mu\nu}$ such that

$M=\int_{\partial\Sigma}dS_{\mu\nu}\left(\nabla^{\mu}\xi^{\nu}+\Lambda\omega^{\mu\nu}\right),$
(43)

where $\omega^{\mu\nu}$ satisfies $\nabla_{\mu}\omega^{\mu\nu}=0$ and
$\xi^{\mu}\equiv\\{1,0,0,0\\}$ is the time-invariant Killing vector. The
equation above can also be used to obtain the ADM mass given in Eq. (42). In
the following year Kastor et al found that there is a consistency between the
derivation of Smarr’s formula for AdS black holes from the newly defined Komar
integral and the scaling argument (the method which was actually used by Smarr
in his original proposal), if the variation of $\Lambda$ is also taken into
account Kastor:2009wy . The modified Smarr’s formula could be expressed as
follows

$M=2TS-2VP.$ (44)

The volume term $V$ above is defined by the relation between $P$ and $\Lambda$
given in Eq. (1), and is given by

$V=\left(\frac{\partial M}{\partial P}\right)_{A}.$ (45)

The first law of black hole mechanics as the variation of $\Lambda$ is taken
into account can then be defined as follows

$dM=TdS+VdP.$ (46)

Employing Eqs (1) and (42) to the equation above, the conjugate volume is
obtained to be

$V=\frac{4\pi}{3}(1-\tilde{\Delta})(r^{2}_{+}+\kappa)^{3/2}.$ (47)

It can be verified that the state variables that we have obtained so far for
our model, namely Eqs. (19), (37), (42) and (47), along with the definition of
$P$ given in Eq. (1) would obey (44) and (46).

##  V The thermodynamic stability of an AdS EiBI black hole with global
monopole

After obtaining the relevant state variables of the EiBI-GM-AdS black holes,
the thermodynamical stability can be investigated. We start with the local
stability, which can be done by analyzing the Hawking temperature and the
specific heat of the configuration. The plot of Hawking temperature $T_{H}$ vs
the event horizon $r_{+}$ is shown in Fig. 3. There is a discontinuous
transition from lower ($\kappa=0$) to higher $(\kappa>0)$ EiBI parameter. In
the Schwarzschild case ($\kappa=0$ and $\Lambda=0$), the Hawking temperature
depends only on its mass or event-horizon radius and slowly decreases as the
radius gets larger. When $\kappa=0$ (with nonvanisihing cosmological
constant), the Hawking temperature becomes similar to the well known
Schwarzschild-AdS. In this case, the black hole has minimum temperature at
$r_{0}=i/\sqrt{\Lambda}$. This condition forces
$T_{0}=\frac{-i\sqrt{\Lambda}}{2\pi}$, where the cosmological constant must
have a negative value (AdS), $\Lambda=-|\Lambda|$. From this plot, we can also
infer that for $T<T_{0}$ the spacetime is filled with pure radiation and hence
there is no AdS black hole. For $T>T_{0},$ AdS black holes tend to have two
different conditions. When EiBI parameter $\kappa$ gets larger, we still have
a same behavior as Schwarzschild-AdS but with the nonzero Hawking temperature
at the origin. We shall discuss in detail about this conditions and also the
stability on specific heat.

Figure 3: A typical plot of Hawking temperature $T_{H}$ versus radius $r_{+}$.
Figure 4: The heat capacity $C$ vs $r_{+}$ for $\Lambda=-0.03$ and
$\Delta=0.4$.

The Hawking temperature in Eq. (19) indicates that there is a possible
transition from small to large black hole. In detail, we can use the specific
heat $C$ and free energy $F$ to investigate the stability and phase
transitions. Combined with the entropy (37), the specific heat $C$ can be
obtained:

$\displaystyle C=T_{H}\leavevmode\nobreak\ \frac{\partial S}{\partial
T_{H}}=-\frac{2\pi\lambda(r_{+}^{2}+\kappa)(-1+\Lambda
r_{+}^{2}+\kappa\Lambda)}{(1+\Lambda
r_{+}^{2}+\kappa\Lambda)}(1-\tilde{\Delta}).$ (48)

Fig. 4 shows the specific heat described in Eq. (48), compared to those of
Schwarzschild black holes in AdS and asymptotically flat spacetimes. Their
stability depend on the value of $C$, where positive sign ($C>0$) indicates
the black holes are stable and $C<0$ means unstable.

In general, the specific heat plot of an EiBI-GM-AdS black hole that follows
the first parameter space given in Eq. (40) would have diverging points
separating the two distinct phases, illustrated with the blue-colored plot in
Fig. 4. The form is completely similar to that of the Schwarzschild-AdS black
hole (plotted as the green-colored plot in Fig. 4), which tells us that the
nature of the phase transition taking place between these two configurations
black holes is the same; there would be a phase transition taking place
between the unstable small black hole (SBH) and the stable large black hole
(LBH). Meanwhile, for EiBI GM AdS black holes that follow the second parameter
space in Eq. (40), we observed the specific heat would always be completely
negative, reminiscent with the usual flat Schwarzschild black hole
configuration. The main difference between the two configurations is the
existence of unstable equilibrium for the EiBI-GM-AdS, while for flat
Schwarzschild black hole there would be none (due to the quadratic nature of
the function) whereas for EiBI GM AdS configuration there is one visible.

With the results obtained so far, we have been able to analyze the local
stability of the holes. Now we proceed to extend the analysis by determining
the global stability of the configuration in order to study the phase of a
system that corresponds to the global maximum of the system’s total entropy,
in other words phase that minimize the free energy used would be the preferred
one Monteiro:2010cq . The free energy in black hole mechanics is defined as
follows,

$F=\frac{I}{\beta}.$ (49)

Using the previously obtained results for the thermodynamic variables, the
free energy as a function of the black hole’s event horizon is obtained to be

$F=\frac{\lambda}{2}\left[\frac{\Lambda}{6}(r_{+}^{2}+\kappa)^{3/2}+\frac{1}{2}\sqrt{r_{+}^{2}+\kappa}\right](1-\tilde{\Delta}).$
(50)

Note that the results above agree with $F=M-TS$, using expressions previously
obtained in Eqs (19), (37) and (41).

Figure 5: Figure that represents the on-shell free energy of AdS holes as a
function of event horizon with different configurations. In this plot, we have
set $\Lambda=-0.03$ and $\Delta=0.4$.

The plots the free energy as a function of event-horizon are shown in Fig. 5.
Echoing the earlier results presented in the specific heat results, we
observed similar features between the EiBI-GM-AdS with $\kappa$ that follows
the first parameter space given in Eq. (40) with the Schwarzschild AdS black
holes. The main difference between these two configurations are the fact that
EiBI-GM-AdS starts with non zero free energy. At certain point, increasing the
amount of $\kappa$ eventually renders the free energy configuration to be
completely positive-valued. This drastic change of the free energy profile
starts after the lower bound of the ”forbidden region” of the EiBI parameter
obtained previously from the applied GSL requirement to the entropy expression
of the holes. Above this value, $F$ would abruptly change its profile into an
ever-increasing value of free energy with no critical points, which leads us
to consider any configuration above this particular value would not be
thermodynamically stable.

##  VI Conclusion

To summarize, we consider the 4d (A)dS black hole with a global monopole
Lambaga:2018yzv  as a test case to apply Euclidean formalism and derive the
corresponding thermodynamical properties in EiBi gravity. To the best of our
knowledge this is the first approach to do so in this type of modified gravity
model. In non-canonical GR one is faced with ambiguity in choosing the
appropriate boundary and counter terms to render the Euclidean action finite.
In this work we argue that the most suitable respective terms are the GHY and
the generalized KS forms.

Once the finite Euclidean action is obtained, it is easy to verify that the
state variables satisfy the black hole first law and the Smarr formula. We
obtained the parameter space in terms of $\kappa$, $\eta$ and the cosmological
constant $\Lambda$ that would allow a thermodynamically feasible
configuration. The phase structure of EiBI AdS with global monopole is found
to be quite similar to that of Schwarzchild-AdS black hole endowed with global
monopole, as long as $\kappa\leq\frac{-(1-\Delta)}{\Lambda}$. On the other
hand, once $\kappa>-\frac{1}{\Lambda}$, a Schwarzschild-flat like phase
behavior shall be observed in our configuration.

All in all, we have been able to examine the nature of the phase transition
taking place in the EiBI AdS black hole endowed with global monopole.
Naturally, further studies on the nature of the phase transition in the
electrically-charged black hole endowed with global monopole in EiBI gravity
would seem to be an interesting inquiry to pursue.

###### Acknowledgements.

We thank Ihsan Fauzi for the help with the preliminary manuscript. We also
thank the anonymous referee for the constructive suggestions. This work is
funded by PUTI Q2 grant under the contract No. NKB-1654/UN2.RST/HKP.05.00/2020
and Hibah Riset PPI Q1 No. NKB-583/UN2. RST/HKP.05.00/2021.

## Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

##  Appendix A Alternative Form of the Counterterm Action

One of the pivotal steps of Euclidean action calculation is the counterterm
action evaluation. There are many approaches that can be done to calculate
counterterm actions Gibbons:1976ue ; York:1986it ; Balasubramanian:1999re ;
Mann:1999pc , and in this work we obtain finite result by utilizing the
relatively simple form of counterterm action for AdS black holes
Emparan:1999pm . This is not a unique choice due to the ambiguity of choosing
it in non-canonical gravity. In this section we shall show several attempts in
constructing the counterterm action, and why they fail111We thank the
anonymous PRD referee for suggesting these models for us to consider.. The
first approach is motivated by the known results in AdS black holes
thermodynamics with GR structure, in which $I_{GHY}$ would eventually give
vanishing contribution due to the nature of AdS. Thus, to normalize the
divergent term arising from the bulk action one could construct a counter term
from the Euclideanized form of the pure AdS geometry. In this approach, the
integrand is the physical metric tensor, not the induced metric. Looking back
at Eq. (23), since there is similarity between EiBI and EH action, it is
tempting to construct the counterterm in such a way that the form becomes as
follows

$I_{ct_{1}}=\frac{1}{16\pi\lambda}\int d^{4}x\sqrt{q}(2\lambda).$ (51)

Using the known result in AdS thermodynamics that for even-dimensional black
holes in GR framework $r_{+}=0$ Gibbons:2004ai , we then obtain

$\displaystyle I_{ct_{1}}$ $\displaystyle=$
$\displaystyle\frac{\omega\beta_{0}\Lambda\lambda}{8\pi}\left\\{\int_{0}^{r_{b}}dr\left[r\sqrt{r^{2}+\kappa}\right]\right\\}(1-\tilde{\Delta}),$
(52) $\displaystyle=$
$\displaystyle\frac{\lambda\beta_{0}\Lambda}{2}\left\\{\frac{1}{3}\left(r^{2}+\kappa\right)^{3/2}\Big{|}_{0}^{r_{b}}\right\\}(1-\tilde{\Delta}).$

Rescaling the period to the metric results in the following form

$\beta_{0}=\beta\left[1+\frac{l^{2}\left(\sqrt{r_{b}^{2}+\kappa}-2m\right)}{2\left(r_{b}^{2}+\kappa\right)^{3/2}}\right].$
(53)

Plugging the result above into Eq. (52) gives

$I_{ct}=\frac{\beta\lambda}{2}\left\\{m-\frac{(r_{b}^{2}+\kappa)}{2}+\frac{\Lambda}{3}(r_{b}^{2}+\kappa)^{5/2}-\frac{\kappa^{3/2}}{3}\right\\}(1-\tilde{\Delta}).$
(54)

The result above actually reduces the divergent term that arise in the bulk
action, yet we can see there is an extra divergent term that comes up in Eq
(54), $-\left(r_{b}^{2}+\kappa\right)/2$. Thus the regular AdS counterterm
approach does not seem to work completely in our case.

Next, we attempt to slightly modify the first approach that we have taken. If
the counterterm was to be calculated as the rescaled to the AdS space version
of the bulk action, then we would have

$\displaystyle I_{ct_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\kappa}\int d^{4}x\left(\sqrt{g_{\mu\nu}+\kappa
R_{\mu\nu}}-\lambda\sqrt{g_{\mu\nu}}\right),$ (55) $\displaystyle=$
$\displaystyle\frac{1}{8\pi\kappa}\int
d^{4}x\left(\sqrt{q_{\mu\nu}}-\lambda\sqrt{g_{\mu\nu}}\right).$

Similar to the previous approach, by taking advantage of $r_{+}=0$ in even
dimensional AdS counterterm calculation, we obtain

$\displaystyle I_{ct_{2}}$ $\displaystyle=$
$\displaystyle\frac{\omega\beta_{0}\lambda}{8\pi\kappa}\int_{0}^{r^{b}}dr\left[\lambda
B(r)C^{2}(r)-B(r)r^{2}\right](1-\tilde{\Delta}),$ (56) $\displaystyle=$
$\displaystyle\frac{\beta_{0}\lambda}{2\kappa}\left\\{\kappa\left[\sqrt{r^{2}+\kappa}+\frac{\Lambda}{3}(r^{2}+\kappa)^{3/2}\right]\Bigg{|}_{0}^{r_{b}}\right\\}(1-\tilde{\Delta}),$
$\displaystyle=$
$\displaystyle\frac{\beta_{0}\lambda}{2}\left[\sqrt{r_{b}^{2}+\kappa}+\frac{\Lambda}{3}(r_{b}^{2}+\kappa)^{3/2}-\sqrt{\kappa}-\frac{\Lambda}{3}\kappa^{3/2}\right](1-\tilde{\Delta}).$

Using the $\beta_{0}$ rescaling obtained in Eq. (53) and substituting it into
the form above would yield the following form (after the null terms are
eliminated $r_{b}\rightarrow\infty$)

$\displaystyle I_{ct_{2}}$ $\displaystyle=$
$\displaystyle\frac{\beta\lambda}{2}\left[m-\frac{\sqrt{\kappa}}{3}(\lambda+2)+\frac{\sqrt{r_{b}^{2}+\kappa}}{2}+\frac{\Lambda}{3}(r_{b}^{2}+\kappa)^{3/2}\right](1-\tilde{\Delta}).$
(57)

Once again we see that the regular AdS counterterm approach does not
completely normalizes the Euclidean action, instead we are left with more
divergent terms. This is one of the reason why we choose to use the
generalized BK boundary counterterm action, since they wonderfully work well
in eliminating the divergence.

Our last attempt is by considering another approach, namely by reinterpreting
the 3D Dirac-Born-Infeld action as the counterterm Ozen:2017uoe ; Gullu:2010pc
. This is motivated by the similarity of EiBI gravity with the new massive
gravity BeltranJimenez:2017doy . The 3D Dirac-Born-Infeld action itself is
given by

$I_{DBI}=-\frac{m^{2}}{4\pi
G_{3}}\int_{\mathcal{M}}d^{3}x\left[\sqrt{-\Big{|}g_{\mu\nu}+\frac{\sigma}{m^{2}}\mathcal{G}_{\mu\nu}\Big{|}}-\left(1-\frac{\lambda_{0}}{2}\right)\sqrt{-|g_{\mu\nu}|}\right].$
(58)

In the form above $m$ is a spin-related mass paramater. $G_{3}$ is the
gravitational constant and $\sigma$ is term related to the helicity of
Einstein-Hilbert term Bergshoeff:2009aq . $\mathcal{G}_{\mu\nu}$ is the usual
Einstein tensor $(\mathcal{G}_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)$. In Dirac-Born-Infeld gravity, the
effective cosmological constant is expressed as

$\Lambda=\sigma m^{2}\lambda_{0}\left(1-\frac{\lambda_{0}}{2}\right).$ (59)

Our goal is to reinterpret Eq. (58) with appropriate terms in EiBI gravity, so
that we may evaluate it as the counterterm action for our model. First we can
see that the $m^{2}$ term in (58) is dimensionally similar to $\kappa$ in EiBI
gravity, in which both of them are inversely proportional to the cosmological
constant. Based on this observation if we were to reinterpret the form given
in Eq (58), it is tempting to translate $m^{2}\rightarrow\kappa$. Next we may
set $\sigma=-1$ since we are using the regular form of EH action. $G_{3}$ can
be set to 1 so that eventually we obtain the following form

$\displaystyle I_{ct_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\kappa}\int
d^{3}x\left[\sqrt{|g_{\mu\nu}+\kappa\mathcal{G}_{\mu\nu}|}+\lambda\sqrt{|g_{\mu\nu}|}\right],$
(60) $\displaystyle=$ $\displaystyle\frac{1}{8\pi\kappa}\int
d^{3}x\left[\sqrt{\Big{|}g_{\mu\nu}+\kappa\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)\Big{|}}+\lambda\sqrt{|g_{\mu\nu}|}\right],$
$\displaystyle=$ $\displaystyle\frac{1}{8\pi\kappa}\int
d^{3}x\left[\sqrt{\Big{|}q_{\mu\nu}-\frac{\kappa}{2}g_{\mu\nu}R\Big{|}}+\lambda\sqrt{|g_{\mu\nu}|}\right].$

The Ricci scalar from 3-D $q_{\mu\nu}$ could be constructed as

$R=\frac{2}{(1-\tilde{\Delta})\lambda\left(\kappa+r^{2}\right)}.$ (61)

Note that the form above is different from the one used in Eqs (27) and (30),
since in this current approach we could not take advantage of the similarity
between EiBI and EH action ($\lambda$ and $1-\tilde{\Delta}$ could not be
treated as mere constants in the counterterm evaluation anymore). Evaluating
Eq (60) gives

$\displaystyle I_{ct}$ $\displaystyle=$
$\displaystyle\frac{\omega\beta_{0}}{8\pi\kappa}\left\\{\sqrt{\frac{\left(6m+\sqrt{\kappa+r_{b}^{2}}\left(\kappa\Lambda+\Lambda
r_{b}^{2}-3\right)\right)\left(\kappa\left(1-(1-\tilde{\Delta})\lambda^{2}\right)-(1-\tilde{\Delta})\lambda^{2}r_{b}^{2}\right)}{3(1-\tilde{\Delta})\lambda\left(\kappa+r_{b}^{2}\right)^{3/2}}}\right.$
$\displaystyle\left.\times\left[\kappa\lambda+r_{b}^{2}\left(\lambda-\frac{\kappa}{(1-\tilde{\Delta})\lambda\left(\kappa+r_{b}^{2}\right)}\right)\right]-\lambda\sqrt{1-\frac{2m}{\sqrt{\kappa+r_{b}^{2}}}-\frac{1}{3}\Lambda\left(\kappa+r_{b}^{2}\right)}(r_{b}^{2}+\kappa)\right\\}$
$\displaystyle\times(1-\tilde{\Delta}).$

Using the rescaled form of $\beta_{0}$ previously given in Eq. (53) as the
form above is expanded, we obtain the following results

$\displaystyle I_{ct}$ $\displaystyle=$
$\displaystyle\frac{\beta_{0}}{2\kappa}\bigg{[}-\frac{r^{10}\left(l^{5}+12\kappa
l^{3}-72\kappa^{2}l\right)}{1728\kappa^{3}\left(\kappa+r^{2}\right)^{9/2}}-\frac{r^{8}\left(-72\kappa^{2}+l^{4}+12\kappa
l^{2}\right)}{144\kappa^{2}l\left(\kappa+r^{2}\right)^{7/2}}+\frac{r^{6}\left(-72\kappa^{2}+l^{4}+12\kappa
l^{2}\right)}{24\kappa l^{3}\left(\kappa+r^{2}\right)^{5/2}}$ (63)
$\displaystyle+\cdots\bigg{]}(1-\tilde{\Delta}).$

The latest approach that we attempt to conduct does not seem to normalize the
bulk action, instead, more non-linear divergent terms appear in the
counterterm action evaluation. This leads us to conclude that the said
approach is not effective, and should not be considered as the procedure taken
for the counterterm computation.

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###### Abstract

We investigate the relation between the convergence of a sequence of lattices
and the set-theoretic convergence of their corresponding Voronoi cells
sequence. We prove that if a sequence of full rank lattices converges to a
full rank lattice, then the closures of the limit infimum and limit supremum
of the Voronoi cells converges to the corresponding Voronoi cell. It remains
an open question if the converse is also true.

Keywords: $lattices$, $Voronoi$ $cells$, $convergence$.

††The publisher does not claim any copyright for the technical reports. The
author keeps the full copyright for the paper, and is thus free to transfer
the copyright to a publisher if the paper is accepted for publication
elsewhere.

Faculty of Computer Science, Alexandru Ioan Cuza University Iaşi

Technical Report TR 19-01, March 2019

A note on sequences of lattices

Emanuel Florentin OLARIU111Faculty of Computer Science, Alexandru Ioan Cuza
University Iaşi, General Berthelot 16, 700483 Iasi̧, Romania, Email:
<EMAIL_ADDRESS>
##  1 Introduction

A lattice is a discrete additive group of an euclidean space. Lattices are of
great interest for discrete optimization, for example ${\mathbb{Z}}^{n}$, the
lattice of points of integral coordinates, is extensively used in integer
programming. We analyze the relation between the convergence of a sequence of
lattices and the set theoretic convergence of the corresponding Voronoi cells.

We prove that if $\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ is a
sequence of full rank lattices from ${\mathbb{R}}^{n}$ and
$\displaystyle\lim_{k\to\infty}\Lambda_{k}=\Lambda$ \- a full rank lattice
also, then

$\overline{\liminf_{k\to\infty}V(\Lambda_{k})}=\overline{\limsup_{k\to\infty}V(\Lambda_{k})}=V(\Lambda),$

where, for a given sequence $\displaystyle\left(A_{k}\right)_{k\geq 1}$ of
sets from ${\mathbb{R}}^{n}$

$\liminf_{k\to\infty}V(A_{k})=\bigcup_{h\geq 1}\bigcap_{j\geq
h}V(A_{j}),\liminf_{k\to\infty}V(A_{k})=\bigcap_{h\geq 1}\bigcup_{j\geq
h}V(A_{j})$

are the set theoretic limit inferior and limit superior.

##  2 Notations and definitions

For the following definitions one can consult [1] and [2].

######  Definition 1.

A lattice $\Lambda\subset{\mathbb{R}}^{n}$ is a discrete additive group of
${\mathbb{R}}^{n}$, i. e., ${\mathbf{x}}-{\mathbf{y}}\in\Lambda$,
$\forall\>{\mathbf{x}},{\mathbf{y}}\in\Lambda$ and there exists an
$\epsilon>0$, such that $B({\mathbf{0}};\epsilon)\cap\Lambda=\varnothing$. Its
rank, $rank(\Lambda)$, is the dimension of its spanned subspace,
$\dim(\operatorname{span}(\Lambda))$.

For every lattice there exist $m$ independent vectors
${\mathbf{v}}_{1},{\mathbf{v}}_{2},\ldots,{\mathbf{v}}_{m}$ in $\Lambda$, such
that

$\Lambda=\left\\{\sum_{i=1}^{m}\alpha_{i}{\mathbf{v}}_{i}\>:\>\alpha_{1},\ldots,\alpha_{m}\in{\mathbb{Z}}\right\\},$

where $m=rank(\Lambda)$. This a basis of $\Lambda$. For a given base, if $A$
is the matrix whose columns are the basis vectors, the determinant of
$\Lambda$ is

$\operatorname{det}(\Lambda)=\sqrt{\operatorname{det}(A^{T}A)}.$

When the lattice has full rank, the determinant is the volume of the
fundamental parallelepiped:

$\Pi=\left\\{\sum_{i=1}^{m}a_{i}{\mathbf{v}}_{i}\>:\>a_{i}\in[0,1),\forall
i=\overline{1,n}\right\\}.$

The fundamental parallelipiped has the following property: every vector
${\mathbf{x}}$ in $lin(\Lambda)$ can be written uniquely as
${\mathbf{x}}={\mathbf{v}}+{\mathbf{y}}$, where ${\mathbf{v}}\in\Lambda$ and
${\mathbf{y}}\in\Pi$. That is the spanned subspace can be tiled with copies of
the fundamental parallelepiped centered in the vectors of the lattice. The
fundamental parallelepiped depends on a particular basis but there is a
similar polytope which can be uniquely associated with a lattice:

######  Definition 2.

Let $\Lambda\subseteq{\mathbb{R}}^{n}$ be a full rank lattice. Its Voronoi
cell is
$V(\Lambda)=\\{{\mathbf{x}}\in{\mathbb{R}}^{n}\>:\>\lVert{\mathbf{x}}\rVert\leq\lVert{\mathbf{x}}-{\mathbf{v}}\rVert,\forall\>{\mathbf{v}}\in\Lambda\\}$.

The Voronoi cell has a similar property of $\operatorname{span}(\Lambda)$
tessellation:

$\operatorname{span}(\Lambda)=\Lambda+V(\Lambda),$

where the intersection of different tiles (cells) occurs only on their
boundaries (frontiers) and
$\operatorname{det}(\Lambda)=\operatorname{vol}(V(\Lambda))$.

The following two lattice parameters (see [2], [3]) illustrates the importance
of the Voronoi cell. If $\Lambda$ is a lattice, the packing radius of
$\Lambda$, $\rho(\Lambda)$, is half the length of the shortest non-zero vector
of $\Lambda$, i. e.

$\rho(\Lambda)=(\min_{{\mathbf{v}}\in\Lambda}{\lVert{\mathbf{v}}\rVert})/2=\operatorname{inradius}\left[V(\Lambda)\right].$

The covering radius of $\Lambda$ is the smallest $\mu$ such that spheres of
radius $\mu$ centered around all the points in $\Lambda$ cover the entire
$lin(\Lambda)$:

$\mu(\Lambda)=\max_{{\mathbf{x}}\in
lin(\Lambda)}{d({\mathbf{x}},\Lambda)}=\operatorname{circumradius}\left[V(\Lambda)\right].$

######  Definition 3.

Let $\Lambda$ and $\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ be full
rank lattices in ${\mathbb{R}}^{n}$, we say that
$\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ converges to $\Lambda$,
$\displaystyle\lim_{k\to\infty}\Lambda_{k}=\Lambda$ if, for each $k\geq 1$,
there exists a basis
${\mathbf{v}}_{k1},{\mathbf{v}}_{k2},\ldots,{\mathbf{v}}_{kn}$ of
$\Lambda_{k}$ such that there exist the limits

$\lim_{k\to+\infty}{\mathbf{v}}_{ki}={\mathbf{v}}_{i},\forall 1\leq i\leq n,$

and ${\mathbf{v}}_{1},{\mathbf{v}}_{2},\ldots,{\mathbf{v}}_{n}$ is a basis of
$\Lambda$.

##  3 Main result

######  Lemma 1.

Let $\Lambda\subseteq{\mathbb{R}}^{n}$ be a full rank lattice. There exists
$R\in{\mathbb{R}}^{*}_{+}$ such that
$V(\Lambda)=V_{R}(\Lambda)=\\{{\mathbf{x}}\in{\mathbb{R}}^{n}\>:\>\lVert{\mathbf{x}}\rVert\leq\lVert{\mathbf{x}}-{\mathbf{v}}\rVert|,\forall\>{\mathbf{v}}\in\Lambda,\lVert{\mathbf{v}}\rVert\leq
R\\}=\\{{\mathbf{x}}\in{\mathbb{R}}^{n}\>:\>2{\mathbf{v}}^{T}{\mathbf{x}}\leq\lVert{\mathbf{v}}\rVert^{2},\forall\>{\mathbf{v}}\in\Lambda,\lVert{\mathbf{v}}\rVert\leq
R\\}$ and $V(\Lambda)$ is a bounded set. (Consequence: $V(\Lambda)$ is a
polytope.)

proof: We prove first that $V(\Lambda)$ is bounded; let
$({\mathbf{u}}_{i})_{1\leq i\leq n}$ be a basis of $\Lambda$ and
$\displaystyle r=\max_{1\leq i\leq n}{\lVert{\mathbf{u}}_{i}\rVert}$. If
${\mathbf{x}}\in V(\Lambda)$, then
$\displaystyle{\mathbf{x}}=\sum_{i=1}^{n}\beta_{i}{\mathbf{u}}_{i}$, for some
$\beta_{i}\in{\mathbb{R}}$, $\forall 1\leq i\leq n$ and

$\lVert{\mathbf{x}}\rVert=\left\lVert\sum_{i=1}^{n}[\beta_{i}]{\mathbf{u}}_{i}+\sum_{i=1}^{n}\\{\beta_{i}\\}{\mathbf{u}}_{i}\right\rVert\leq\left\lVert\sum_{i=1}^{n}{\beta_{i}}{\mathbf{u}}_{i}\right\rVert\leq\sum_{i=1}^{n}\lVert{\mathbf{u}}_{i}\rVert\leq
n\cdot\max_{1\leq i\leq n}{\lVert{\mathbf{u}}_{i}\rVert}=n\cdot r.$

Define $R=2rn$; obviously, $V(\Lambda)\subseteq V_{R}(\Lambda)$. Let
${\mathbf{x}}\in V_{R}(\Lambda)$ and ${\mathbf{v}}\in\Lambda$, with
$\lVert{\mathbf{v}}\rVert\geq R$. We have
$\lVert{\mathbf{x}}-{\mathbf{v}}\rVert\geq\lVert{\mathbf{v}}\rVert-\lVert{\mathbf{x}}\rVert\geq
R-R/2=R/2\geq\lVert{\mathbf{x}}\rVert$.  $\square$

######  Lemma 2.

Let $\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ be a sequence of full
rank lattices from ${\mathbb{R}}^{n}$. If
$\displaystyle\lim_{k\to\infty}\Lambda_{k}=\Lambda$, a full rank lattice from
${\mathbb{R}}^{n}$, then there exists $R^{\prime}>0$ such that
$V(\Lambda)=V_{R^{\prime}}(\Lambda)$ and
$V(\Lambda_{k})=V_{R^{\prime}}(\Lambda_{k})$, $\forall k\geq 1$.

proof: Define $\displaystyle R_{k}=2n\cdot\max_{1\leq i\leq
n}{\lVert{\mathbf{u}}_{ki}\rVert}$ and $\displaystyle R=2n\cdot\max_{1\leq
i\leq n}{\lVert{\mathbf{u}}_{i}\rVert}$; we have
$\displaystyle\lim_{k\to\infty}R_{k}=R$. There exists $k_{0}\in{\mathbb{N}}$
such that $R_{k}\leq 2R$, $\forall k\geq k_{0}$. Hence by choosing
$R^{\prime}\geq 2R$ and $R^{\prime}\geq R_{k}$, $\forall 1\leq k<k_{0}$ we
have the desired property.  $\square$

######  Theorem 1.

([2], Theorem 1, V.3) A necessary and sufficient condition that
$\displaystyle\lim_{k\to+\infty}\Lambda_{k}=\Lambda$ is that the following two
conditions be both satisfied

  * (i)

if ${\mathbf{u}}\in\Lambda$, there are points
${\mathbf{v}}_{k}\in\Lambda_{k}$, for $k=1,2,\ldots$ such that

$\lim_{k\to+\infty}{\mathbf{v}}_{k}={\mathbf{u}}.$

  * (ii)

if ${\mathbf{x}}\notin\Lambda$, there is a number $\epsilon_{0}>0$ and an
integer $k_{0}$, both depending on $x$, such that

$\lVert{\mathbf{u}}-{\mathbf{x}}\rVert>\epsilon_{0},\forall\>{\mathbf{u}}\in\Lambda_{k},\forall
k\geq k_{0}.$

Where $\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ and $\Lambda$ are full
rank lattices from ${\mathbb{R}}^{n}$.

######  Theorem 2.

Let $\displaystyle\left(\Lambda_{k}\right)_{k\geq 1}$ be a sequence of full
rank lattices from ${\mathbb{R}}^{n}$. If
$\displaystyle\lim_{k\to\infty}\Lambda_{k}=\Lambda$, a full rank lattice from
${\mathbb{R}}^{n}$, then
$\displaystyle\overline{\liminf_{k\to\infty}V(\Lambda_{k})}=\overline{\limsup_{k\to\infty}V(\Lambda_{k})}=V(\Lambda)$.

proof: Suppose that $({\mathbf{u}}_{ki})_{1\leq i\leq n}$ is a basis of
$\Lambda_{k}$, $\forall k\geq 1$, and $({\mathbf{u}}_{i})_{1\leq i\leq n}$ is
a basis of $\Lambda$ such that
$\displaystyle\lim_{j\to\infty}{\mathbf{u}}_{ji}={\mathbf{u}}_{i}$, $\forall
1\leq i\leq n$. For every $\epsilon>0$, there exists $k_{\epsilon}$ such that
$\lVert{\mathbf{u}}_{ki}-{\mathbf{u}}_{i}\rVert\leq\epsilon$, $\forall k\geq
k_{\epsilon}$ and $\forall 1\leq i\leq n$.

Let
$\displaystyle{\mathbf{x}}\in\liminf_{k\to\infty}V(\Lambda_{k})\subseteq\limsup_{k\to\infty}V(\Lambda_{k})=\bigcap_{k\geq
1}\bigcup_{j\geq k}V(\Lambda_{j})$, i. e., $\forall k\geq 1$ there exists
$j=j_{k}\geq k$ such that ${\mathbf{x}}\in V(\Lambda_{j_{k}})$. Choose
$\displaystyle\left(\alpha_{i}\right)_{1\leq i\leq
n}\subseteq{\mathbb{Z}}^{n}$ and
$\displaystyle{\mathbf{u}}=\sum_{i=1}^{n}\alpha_{i}{\mathbf{u}}_{i}\in\Lambda$;
we have
$\displaystyle{\mathbf{u}}_{j_{k}}=\sum_{i=1}^{n}\alpha_{i}{\mathbf{u}}_{j_{k}i}\in\Lambda_{j_{k}}$
and

$\lVert{\mathbf{x}}\rVert\leq\left\lVert{\mathbf{x}}-\sum_{i=1}^{n}\alpha_{i}{\mathbf{u}}_{j_{k}i}\right\rVert,\forall\>k\geq
1\stackrel{{\scriptstyle
k\to\infty}}{{\Longrightarrow}}\lVert{\mathbf{x}}\rVert\leq\left\lVert{\mathbf{x}}-\sum_{i=1}^{n}\alpha_{i}{\mathbf{u}}_{i}\right\rVert=\lVert{\mathbf{x}}-{\mathbf{u}}\rVert,$

hence ${\mathbf{x}}\in V(\Lambda)$.

We proved that
$\displaystyle\liminf_{k\to\infty}V(\Lambda_{k})\subseteq\limsup_{k\to\infty}V(\Lambda_{k})\subseteq
V(\Lambda)$. Now we prove that
$V(\Lambda)\subseteq\displaystyle\overline{\liminf_{k\to\infty}V(\Lambda_{k})}$.
Suppose not and let ${\mathbf{x}}_{0}\in
V(\Lambda)\setminus\displaystyle\overline{\liminf_{k\to\infty}V(\Lambda_{k})}$.
It means that there exists an $\epsilon_{0}>0$ such that

$\displaystyle
B({\mathbf{x}}_{0};\epsilon_{0})\cap\displaystyle\liminf_{k\to\infty}V(\Lambda_{k})=\varnothing\Leftrightarrow
B({\mathbf{x}}_{0};\epsilon_{0})\cap\left(\bigcap_{j\geq
k}V(\Lambda_{j})\right)=\varnothing,\forall\>k\geq 1.$

Since ${\mathbf{x}}_{0}\in V_{R^{\prime}}(\Lambda)$, we have
$\lVert{\mathbf{x}}_{0}-{\mathbf{u}}\rVert\geq\lVert{\mathbf{x}}_{0}\rVert$,
$\forall\>{\mathbf{u}}\in\Lambda$ with $\lVert{\mathbf{u}}\rVert\leq
R^{\prime}$. By perturbing ${\mathbf{x}}_{0}$ and decreasing $\epsilon_{0}$
(if necessary), we can suppose that

${\mathbf{x}}_{0}\notin\bigcap_{j\geq k}V(\Lambda_{j}),\forall\>k\geq 1\mbox{
and
}\lVert{\mathbf{x}}_{0}-{\mathbf{u}}\rVert>\lVert{\mathbf{x}}_{0}\rVert,\forall{\mathbf{u}}\in\Lambda,\lVert{\mathbf{u}}\rVert\leq
R^{\prime}.$

Using this property we can define an increasing sequence
$\displaystyle(j_{k})_{k\geq 1}$ such that ${\mathbf{x}}_{0}\notin
V(\Lambda_{j_{k}})$, for any $k\geq 1$; hence there exist the points
${\mathbf{u}}_{j_{k}}\in\Lambda_{j_{k}}$, such that
$\lVert{\mathbf{x}}_{0}-{\mathbf{u}}_{j_{k}}\rVert<\lVert{\mathbf{x}}_{0}\rVert$,
$\forall k\geq 1$. Since $({\mathbf{u}}_{j_{k}})_{k\geq 1}\subseteq
B({\mathbf{0}};R^{\prime})$, we can extract a convergent subsequence from
$({\mathbf{u}}_{j_{k}})_{k\geq 1}$, which, for the sake of simplicity, will be
named in the same way:
$\displaystyle\lim_{k\to\infty}{\mathbf{u}}_{j_{k}}={\mathbf{u}}^{\prime}$.
Obviously
$\lVert{\mathbf{x}}_{0}-{\mathbf{u}}^{\prime}\rVert\leq\lVert{\mathbf{x}}_{0}\rVert$,
$\lVert{\mathbf{u}}^{\prime}\rVert\leq R^{\prime}$, and, by (ii) from Theorem
1, ${\mathbf{u}}^{\prime}\in\Lambda$ \- a contradiction with the above
property.  $\square$

##  4 Conclusions

The converse of our main result it remains an open problem for now, but it can
be proved (which is left for another technical report) that, if the extremal
points of $V(\Lambda)$ are contained in
$\displaystyle\liminf_{k\to\infty}V(\Lambda_{k})$ (when this happens,
$V(\Lambda)\subseteq V(\Lambda_{k})$, starting from a certain $k_{0}$), then
$\displaystyle\lim_{k\to\infty}\Lambda_{k}=\Lambda$. It remains to see what
happens when some of the extremal points of $V(\Lambda)$ belong on the
boundary of $\displaystyle\liminf_{k\to\infty}V(\Lambda_{k})$.

## References

  * [1] A. Barvinok.  Combinatorics, geometry and complexity of integer points. _electronic lecture notes_.  http://www.math.lsa.umich.edu/~barvinok/latticenotes669.pdf.
  * [2] J. W. S. Cassels  An introduction to the geometry of numbers.  Springer Verlag, New York, 1997.
  * [3] M. D. Sikiric, A. Tarlecki.  Complexity and algorithms for computing Voronoi cells of lattices, _Mathematics of Computation_ 78(267), 1713–1731, 2009.  doi:10.1090/S0025-5718-09-02224-8.