# The generalized uncertainty principle effect in acoustic black holes

M. A. Anacleto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  G. C. Luna
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de
Campina Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil
E. Passos<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil

###### Abstract

We obtain an effective acoustic metric with quantum corrections that are
provided by a minimum length implemented by the generalized Heisenberg
uncertainty principle (GUP) in the Abelian Higgs model. The effective acoustic
metric now depends on the contribution of scalar and vector potentials. We
also explore the Hawking radiation and entropy by considering the effective
canonical acoustic black hole and find that the modified Hawking temperature
leads to logarithm corrections to the entropy. Finally, we investigate the
dispersion relations of the model to establish the relationships among the
deviations of the group velocity, frequency and temperature due to the GUP.

##  I Introduction

Black holes are fascinating objects in our universe and an astrophysical
observation of these objects is a subject of great interest in General
Relativity. Recently, the LIGO-VIRGO collaboration LIGOScientific:2016aoc ;
LIGOScientific:2017vwq  with the remarkable observation of GW150914 made its
first direct detection of gravitational waves that emerge from the merger of a
binary black hole system. In addition, the Event Horizon Telescope captured
the image of a supermassive black hole at the center of galaxy M87
event2019firstI ; event2019firstVI . Thus, with these two observations that
evidence the existence of black holes, a wide range appears to deepen studies
in the development of black hole physics.

In recent decades great advances have been made in the physics of black holes
in the investigation of properties near the event horizon. One of the
characteristics of black holes is Hawking radiation. From an observational
point of view, the detection of Hawking radiation is difficult due to
technical limitations. To overcome this difficulty, Unruh Unruh:1980cg ;
Unruh:1994je  in 1981 introduced the acoustic black hole model that makes
possible to detect Hawking radiation from the acoustic black hole in the
laboratory. Thus, several works have been done to analyze analogous Hawking
radiation in different physical scenarios Visser:1997ux (see Barcelo:2005fc
for review). In Vieira:2014rva  the analytical solutions of the massless
scalar field and the analogous Hawking radiation were addressed.

Furthermore, starting with the Abelian Higgs model, relativistic metrics of
acoustic black holes have been constructed Ge:2010wx ; Anacleto:2010cr ;
Anacleto:2011bv ; Anacleto:2013esa  (see also Bilic:1999sq ; Fagnocchi:2010sn
; Visser:2010xv ). In Ge:2019our , by considering relativistic Gross-
Pitaevskii theory and Yang-Mills theory, the acoustic black hole in curved
geometry has been obtained. In addition, based on a holographic approach, the
acoustic black hole constructed from black D3-brane has been discussed in
Yu:2017bnu . Moreover, by applying these effective metrics, studies have been
carried out to examine the phenomena of superradiance Basak:2002aw ;
Richartz:2009mi ; Anacleto:2011tr ; Zhang:2011zzh ; Ge:2010eu , entropy
Zhao:2012zz ; Anacleto:2014apa ; Anacleto:2015awa ; Anacleto:2016qll ;
Anacleto:2019rfn , quasinormal modes Cardoso:2004fi ; Nakano:2004ha ;
Berti:2004ju ; Chen:2006zy ; Guo:2020blq ; Ling:2021vgk  and the analogous
Aharonov-Bohm effect Dolan:2011zza ; Anacleto:2012ba ; Anacleto:2012du ;
Anacleto:2015mta ; Anacleto:2016ukc ; Anacleto:2018acl  as well as quantum
stochastic motion Anacleto:2020kxj ; Anacleto:2021wmv . In Qiao:2021trw  the
gravitational bending of acoustic Schwarzschild black hole was investigated.
Recently, great progress has been made in detecting Hawking radiation in the
laboratory MunozdeNova:2018fxv ; Isoard:2019buh . A first experimental
realization of the acoustic black hole in the Bose-Einstein condensate (BEC)
has been reported in Lahav:2009wx . In addition, experimental studies have
been carried out on optical systems Steinhauer:2014dra ; Drori:2018ivu ;
Rosenberg:2020jde  and also on other physical systems Guo:2019tmr ;
Bera:2020doh ; Blencowe:2020ygo .

The aim of this paper is to find a metric with quantum corrections for the
acoustic black hole. For this purpose, we start with the Abelian Higgs model
and apply the same procedure done in Ge:2010wx ; Anacleto:2010cr ;
Anacleto:2011bv ; Anacleto:2013esa . It has been obtained in Quesne:2006fs
generalized commutation relation that arises due to the presence of a minimum
length given by

$\displaystyle[X^{\mu},P^{\nu}]=-i[(1-\beta
p^{\sigma}p_{\sigma})g^{\mu\nu}-2\beta p^{\mu}p_{\nu}],$ (1)

where $\beta$ is the deformation parameter with minimum length of the order of
$\sqrt{\beta}$. This naturally leads to the generalized Heisenberg uncertainty
principle (GUP) Garay:1994en ; AmelinoCamelia:2000ge ; Ali:2009zq ;
Das:2008kaa ; Das:2009hs ; Ali:2011fa ; Scardigli:1999jh ; Capozziello:1999wx
; Scardigli:2003kr ; Tawfik:2014zca ; Dahab:2015 ; Dutta:2015 ; KMM ;
Gangopadhyay:2013ofa . As shown in Quesne:2006fs ; Dias:2016lkg ;
HoffDaSilva:2020uov  the above commutation relation holds in the configuration
space for $P^{\mu}=(1-\beta p^{\sigma}p_{\sigma})p^{\mu}$, and then a
correction for the partial derivative operator is obtained, i.e.,
$\partial_{\mu}\rightarrow(1+\beta\Box)\partial_{\mu}$. Thus, in order to
generate the effective acoustic metric, we incorporate the contribution of the
minimum length with this prescription into the Lagrangian. In recent years,
the exploration of models considering the GUP effect has been a subject of
great interest in the literature Yang:2018wlb ; Feng:2017gms ; Feng:2015jlj ;
Javed:2019btd ; Li:2016mwq ; Chen:2016ftz ; Maluf:2018lyu ; Gomes:2018oyd ;
Pourhassan:2018chj ; Sadeghi:2016xym ; Scardigli:2014qka ; Casadio:2020rsj ;
Scardigli:2016pjs ; Scardigli:2019pme ; Anacleto:2021qoe . In particular, the
analysis of the effect of the GUP on Hawking radiation has been an issue that
has been widely explored by many authors Sakalli:2016mnk ; Gecim:2017zid ;
Alonso-Serrano:2018ycq ; Anacleto:2015mma ; Silva:2012mt ; Anacleto:2015rlz ;
Anacleto:2015kca ; Haldar:2018zyv ; Haldar:2019fcz ; Iorio:2019wtn ;
Scardigli:2008jn ; Anacleto:2020efy ; Anacleto:2020zfh ; Nozari:2005ah ;
Nozari:2008gp ; Nozari:2008rc ; Nozari:2006ka . Therefore, the obtained metric
presents quantum corrections that are identified by the deformation parameter.
As a result, we show that terms associated with scalar and vector potentials
appear in the effective metric of the acoustic black hole due to the
deformation parameter $\beta$.

The paper is organized as follows. In Sec. II we derive the acoustic metric
under the GUP. In Sec. III we investigate the effect of the GUP in the
dispersion relations to establish relationships among deviations on the group
velocity, frequency and Hawking temperature. In the Sec. IV we make our final
comments.

##  II GUP-Corrected Acoustic Metric

In this section, we consider the Abelian Higgs model that describes high
energy physics in order to derive a GUP-corrected acoustic black hole metric.
This study is motivated by the fact that acoustic black holes could be formed
in high energy physical processes, such as quark matters, neutron stars and
quark gluon plasma. Thus, it seems to be natural to look for acoustic black
holes in a quark gluon plasma fluid with GUP correction in this regime. In
addition acoustic phenomena in quark gluon plasma matter was investigated in
Casalderrey-Solana:2004fdk  and acoustic black holes in a plasma fluid was
explored in GarciadeAndrade:2008kz . Thus, we start with the following
Lagrangian of the Abelian Higgs model:

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left|\left(\partial_{\mu}-ieA_{\mu}\right)\phi\right|^{2}+m^{2}|\phi|^{2}-b|\phi|^{4},$
(2)

where $A_{\mu}$ is the gauge field and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
tensor. The model deals with a charged fluid since the Lagrangian is invariant
under $U(1)$ gauge symmetry which allows a minimal coupling between the
complex scalar field $\phi$ and the gauge field $A_{\mu}$ via Noether current
and coupling constant $e$.

Let us first consider only the free scalar field sector given by

$\displaystyle\mathcal{L}=\left|\partial_{\mu}\phi\right|^{2}+m^{2}|\phi|^{2}.$
(3)

In order to obtain an effective acoustic metric, we introduce the contribution
of the minimum length in the above Lagrangian by means of the following
prescription Quesne:2006fs ; Dias:2016lkg ; HoffDaSilva:2020uov :

$\displaystyle\partial_{\mu}\rightarrow\left(1+\beta\square\right)\partial_{\mu},$
(4)

where $\beta$ is the deformation parameter. By substituting (4) into (3), we
have

$\displaystyle\mathcal{L}=\left|\partial_{\mu}\phi\right|^{2}+2\beta\square\left|\partial_{\mu}\phi\right|^{2}+m^{2}|\phi|^{2}.$
(5)

Here we keep the terms up to first order in $\beta$. Thus, we obtain the
modified Klein-Gordon equation given by

$\displaystyle\square\phi+2\beta\square^{2}\phi-m^{2}\phi=0.$ (6)

Now isolating the term $\square\phi$ in (6) and replacing it in (4), we find

$\displaystyle\mathcal{L}=\left|\left(1+\beta
m^{2}\right)\partial_{\mu}\phi\right|^{2}+m^{2}|\phi|^{2}.$ (7)

So the new contribution of the derivative is
$\partial_{\mu}\rightarrow\left(1+\beta m^{2}\right)\partial_{\mu}$. Now we
replace this new contribution of the derivative in the Lagrangian (2) and so
we get

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}(1+2\beta
m^{2})+\left|\left[\left(1+\beta
m^{2}\right)\partial_{\mu}-ieA_{\mu}\right]\phi\right|^{2}+m^{2}|\phi|^{2}-b|\phi|^{4}.$
(8)

Using the Madelung representation for the scalar field
$\phi=\sqrt{\rho\left(\vec{x},t\right)}e^{iS\left(\vec{x},t\right)}$, the
Lagrangian becomes

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}(1+2\beta
m^{2})+\frac{1}{4\rho}\partial_{\mu}\rho\partial^{\mu}\rho\left(1+2\beta
m^{2}\right)+\left(1+2\beta m^{2}\right)\rho\partial_{\mu}S\partial^{\mu}S$
(9) $\displaystyle-$ $\displaystyle 2\left(1+\beta m^{2}\right)e\rho
A^{\mu}\partial_{\mu}S+e^{2}A_{\mu}A^{\mu}\rho+m^{2}\rho-b\rho^{2},$

and the equations of motion for $S$ and $\rho$ are

$\displaystyle\partial_{\mu}\left[\rho(\partial^{\mu}S-eA^{\mu})\left(1+2\beta
m^{2}\right)+\beta m^{2}e\rho A^{\mu}\right]=0,$ (10)

and

$\displaystyle\frac{1}{\sqrt{\rho}}\partial_{\mu}\partial^{\mu}\sqrt{\rho}\,(1+2\beta
m^{2})-\left(\partial_{\mu}S-eA_{\mu}\right)^{2}(1+2\beta m^{2})-2\beta
m^{2}\left(\partial_{\mu}S-eA_{\mu}\right)eA^{\mu}-m^{2}+2b\rho=0.$ (11)

The Eq. (10) is the continuity equation and Eq. (11) is an equation describing
a hydrodynamical fluid with a quantum potential term
$\frac{1}{\sqrt{\rho}}\partial_{\mu}\partial^{\mu}\sqrt{\rho}\,(1+2\beta
m^{2})$ which can be negligible in the hydrodynamic region. However, in this
regime the model still describes relativistic fluids. On the other hand, one
can also apply the Madelung representation of the condensate wave function in
Abelian Higgs model and in relativistic Bose-Einstein condensate as has been
argued in Ge:2010wx  and Fagnocchi:2010sn ; Giacomelli:2017eze , respectively.
Thus, just as in the Schroedinger equation, this approach yields the
continuity and the Euler equation applied to an irrotational and inviscid
fluid in the absence of the quantum potential.

For the gauge field $A_{\mu}$, we find

$\displaystyle\partial_{\mu}F^{\mu\nu}(1+2\beta m^{2})=2e\rho(1+\beta
m^{2})u^{\nu}+2\beta m^{2}e^{2}\rho A^{\nu},$ (12)

or

$\displaystyle\partial_{\mu}F^{\mu\nu}=2e\rho(1-\beta m^{2})u^{\nu}+2\beta
m^{2}e^{2}\rho A^{\nu},$ (13)

where $u^{\mu}=\partial^{\mu}S-eA^{\mu}=(-\omega,-v^{i})$.

Next we make the following perturbation in the equations of motion (10) and
(11):

$\displaystyle S=S_{0}+\epsilon S_{1}+\mathcal{O}(\epsilon^{2}),$ (14)
$\displaystyle\rho=\rho_{0}+\epsilon\rho_{1}+\mathcal{O}(\epsilon^{2}),$ (15)

so that we obtain

$\displaystyle\partial_{\mu}\left\\{\rho_{1}(1+2\alpha)u_{0}^{\mu}+\rho_{1}\alpha
eA^{\mu}+\rho_{0}\left(1+2\alpha\right)\partial^{\mu}S_{1}\right\\}=0,$ (16)

and

$\displaystyle(1+2\alpha)u_{0}^{\mu}\partial_{\mu}S_{1}+\alpha
eA^{\mu}\partial_{\mu}S_{1}-b\rho_{1}=0.$ (17)

For simplicity we have defined $u_{0}^{\mu}=\partial^{\mu}S_{0}-eA^{\mu}$ and
$\alpha=\beta m^{2}$ is the dimensionless GUP parameter. Now, solving (17) for
$\rho_{1}$ and substituting into (16), we find

$\displaystyle\partial_{\mu}\left[u^{\mu}_{0}u^{\nu}_{0}(1+4\alpha)+\alpha
eu^{\mu}_{0}A^{\nu}+\alpha
eA^{\mu}u^{\nu}_{0}+b\rho_{0}\left(1+2\alpha\right)g^{\mu\nu}\right]\partial_{\nu}S_{1}=0.$
(18)

We can also write the above equation as follows:

$\displaystyle\partial_{t}\left\\{\tilde{\omega}^{2}_{0}\left[-1-4\alpha+\frac{2\alpha
eA_{t}}{\tilde{\omega}_{0}}-\frac{b\rho_{0}}{\tilde{\omega}^{2}_{0}}\left(1+2\alpha\right)\right]\partial_{t}S_{1}+\tilde{\omega}^{2}_{0}\left[-\frac{v^{i}_{0}}{\tilde{\omega}_{0}}\left(1+4\alpha-\frac{eA_{t}}{\tilde{\omega}_{0}}\right)+\alpha\frac{eA^{i}}{\tilde{\omega}_{0}}\right]\partial_{i}S_{1}\right\\}$
$\displaystyle+\partial_{i}\left\\{\tilde{\omega}^{2}_{0}\left[-\frac{v^{i}_{0}}{\tilde{\omega}_{0}}\left(1+4\alpha-\frac{eA_{t}}{\tilde{\omega}_{0}}\right)+\alpha\frac{eA^{i}}{\tilde{\omega}_{0}}\right]\partial_{t}S_{1}\right.$
$\displaystyle+\left.\tilde{\omega}^{2}_{0}\left[-(1+4\alpha)\frac{v^{i}_{0}v^{j}_{0}}{\tilde{\omega}^{2}_{0}}+\alpha\left(v^{i}\frac{eA^{j}}{\tilde{\omega}_{0}}+v^{j}\frac{eA^{i}}{\tilde{\omega}_{0}}\right)+\frac{b\rho_{0}}{\tilde{\omega}^{2}_{0}}\left(1+2\alpha\right)\delta^{ij}\right]\partial_{j}S_{1}\right\\}=0,$
(19)

where $\tilde{\omega}_{0}=eA_{t}+\omega_{0}$, $\omega_{0}=-\partial^{t}S_{0}$
and $v_{0}^{i}=\partial_{i}S_{0}+eA^{i}$ (the local velocity field). In
addition, at the weak field limit, we define
$c^{2}_{s}=b\rho_{0}/\omega^{2}_{0}$, $v^{i}=v^{i}_{0}/\omega_{0}$,
$\Phi=A_{t}/\omega_{0}$ and $\Lambda^{i}=A^{i}/\omega_{0}$. In order to
identify an acoustic metric, the last three quantities should be recognized as
the vector velocity, electric scalar potential and magnetic vector potential,
respectively. So the equation (II) becomes

$\displaystyle\partial_{t}\left\\{\frac{b\rho_{0}}{c_{s}^{2}}\left[-1-4\alpha+2\alpha
e\Phi-
c_{s}^{2}\left(1+2\alpha\right)\right]\partial_{t}S_{1}+\frac{b\rho_{0}}{c_{s}^{2}}\left[-(1+4\alpha-\alpha
e\Phi)v^{i}+\alpha e\Lambda^{i}\right]\partial_{i}S_{1}\right\\}$
$\displaystyle+\,\partial_{i}\left\\{\frac{b\rho_{0}}{c_{s}^{2}}\left[-(1+4\alpha-\alpha
e\Phi)v^{i}+\alpha e\Lambda^{i}\right]\partial_{t}S_{1}\right.$
$\displaystyle+\left.\frac{b\rho_{0}}{c_{s}^{2}}\left[-(1+4\alpha)v^{i}v^{j}+\alpha
e\Lambda^{j}v^{i}+\alpha
v^{j}e\Lambda^{i}+c_{s}^{2}\left(1+2\alpha\right)\delta^{ij}\right]\partial_{j}S_{1}\right\\}=0.$
(20)

We can now write the above equation as the equation of the fluctuations in a
(3+1) dimensional curved space as follows:

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\right)\psi=0,$
(21)

where

$\displaystyle\sqrt{-g}g^{\mu\nu}=\frac{b\rho_{0}}{c^{2}_{s}}\left(\begin{array}[]{ccc}-1-c^{2}_{s}(1+2\alpha)-2\alpha(2-e\Phi)&\vdots&-v^{i}(1+4\alpha-\alpha
e\Phi)+\alpha e\Lambda^{i}\\\ \cdots\cdots&\cdot&\cdots\cdots\\\
-v^{j}(1+4\alpha-\alpha e\Phi)+\alpha
e\Lambda^{j}&\vdots&c^{2}_{s}(1+2\alpha)\delta^{ij}-v^{i}v^{j}(1+4\alpha)+\alpha
v^{i}e\Lambda^{j}+\alpha e\Lambda^{i}v^{j}\end{array}\right).$ (25)

Therefore, the effective relativistic acoustic metric is given by

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}\sqrt{\mathcal{Q}}}\left(\begin{array}[]{ccc}-c^{2}_{s}(1+2\alpha)+v^{2}(1+4\alpha)-2\alpha
e\vec{v}\cdot\vec{\Lambda}&\vdots&-v^{i}(1+4\alpha-\alpha e\Phi)+\alpha
e\Lambda^{i}\\\ \cdots\cdots&\cdot&\cdots\cdots\\\ -v^{j}(1+4\alpha-\alpha
e\Phi)+\alpha
e\Lambda^{j}&\vdots&[1+c^{2}_{s}(1+2\alpha)-2\alpha(e\Phi-2)]\delta^{ij}\end{array}\right),$
(29)

where

$\displaystyle\mathcal{Q}=(1+c^{2}_{s}-v^{2})+2\alpha(e\vec{v}\cdot\vec{\Lambda}-3v^{2}-e\Phi+3)+4\alpha
c^{2}_{s}.$ (30)

Note that we have obtained a modified relativistic acoustic metric with new
terms emerging. Especially the terms generated due to the scalar potential
$\Phi$ and the vector potential $\vec{\Lambda}$. Furthermore, in the absence
of potentials, i.e., by taking $e=0$, we find the relativistic acoustic metric
with GUP corrections, given by

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}\sqrt{\mathcal{Q}}}\left(\begin{array}[]{ccc}-c^{2}_{s}(1+2\alpha)+v^{2}(1+4\alpha)&\vdots&-v^{i}(1+4\alpha)\\\
\cdots\cdots&\cdot&\cdots\cdots\\\
-v^{j}(1+4\alpha)&\vdots&[1+4\alpha+c^{2}_{s}(1+2\alpha)]\delta^{ij}\end{array}\right),$
(34)

where

$\displaystyle\mathcal{Q}=(1+c^{2}_{s}-v^{2})+2\alpha(3+2c^{2}_{s}-3v^{2}).$
(35)

In addition, for $\alpha=0$, we have

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}\sqrt{1+c^{2}_{s}-v^{2}}}\left(\begin{array}[]{ccc}-c^{2}_{s}+v^{2}&\vdots&-v^{i}\\\
\cdots\cdots&\cdot&\cdots\cdots\\\
-v^{j}&\vdots&(1+c^{2}_{s})\delta^{ij}\end{array}\right),$ (39)

such that the relativistic acoustic metric obtained in Ge:2010wx  is
recovered. However, in the non-relativistic limit the metric (29) becomes

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}\sqrt{f}}\left(\begin{array}[]{ccc}-c^{2}_{s}(1+2\alpha)+v^{2}(1+4\alpha)-2\alpha
e\vec{v}\cdot\vec{\Lambda}&\vdots&-v^{i}(1+4\alpha-\alpha e\Phi)+\alpha
e\Lambda^{i}\\\ \cdots\cdots&\cdot&\cdots\cdots\\\ -v^{j}(1+4\alpha-\alpha
e\Phi)+\alpha
e\Lambda^{j}&\vdots&[1-2\alpha(e\Phi-2)]\delta^{ij}\end{array}\right),$ (43)

where

$\displaystyle f=1+6\alpha+2\alpha(e\vec{v}\cdot\vec{\Lambda}-e\Phi).$ (44)

So for $e=0$ in the metric above, we find

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}\sqrt{1+6\alpha}}\left(\begin{array}[]{ccc}-c^{2}_{s}(1+2\alpha)+v^{2}(1+4\alpha)&\vdots&-v^{i}(1+4\alpha)\\\
\cdots\cdots&\cdot&\cdots\cdots\\\
-v^{j}(1+4\alpha)&\vdots&\left(1+4\alpha\right)\delta^{ij}\end{array}\right),$
(48)

which is the metric of the acoustic black hole with GUP corrections without
the effect of potentials.

Now, for $\alpha=0$, we obtain the metric found by Unruh Unruh:1980cg  up to
an overall factor

$\displaystyle
g_{\mu\nu}=\frac{b\rho_{0}}{c_{s}}\left(\begin{array}[]{ccc}-(c^{2}_{s}-v^{2})&\vdots&-v^{i}\\\
\cdots\cdots&\cdot&\cdots\cdots\\\
-v^{j}&\vdots&\delta^{ij}\end{array}\right).$ (52)

We can also write the effective acoustic line element for the metric (43) as
follows:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{b\rho_{0}}{c_{s}\sqrt{f}}\left[-F(v)\,dt^{2}-2\vec{V}\cdot
d\vec{x}dt+[1-2\alpha(e\Phi-2)]dx^{2}\right],$ (53)

where

$\displaystyle F(v)=c^{2}_{s}(1+2\alpha)-v^{2}(1+4\alpha)+2\alpha
e\vec{v}\cdot\vec{\Lambda},$ (54) $\displaystyle\vec{V}=(1+4\alpha-\alpha
e\Phi)\vec{v}-e\alpha\vec{\Lambda}.$ (55)

Now, by performing a coordinate change

$\displaystyle d\tau=dt+\frac{\vec{V}\cdot d\vec{x}}{F(v)},$ (56)

the line element can be written in stationary form

$\displaystyle
ds^{2}=\frac{b\rho_{0}}{c_{s}\sqrt{f}}\left[-F(v)\,d\tau^{2}+\left(\frac{V^{i}V^{j}}{F(v)}+\left[1-2\alpha(e\Phi-2)\right]\delta^{ij}\right)dx^{i}dx^{j}\right].$
(57)

At this point, to address the Hawking temperature issue, we will consider the
configuration of an incompressible fluid with spherical symmetry. For this
situation the density $\rho$ is a position independent quantity, the sound
speed is also a constant and the continuity equation implies that $v\sim
1/r^{2}$. Thus, disregarding position-independent factors, the line element
(57) written in the Schwarzschild-type form is given by

$\displaystyle
ds^{2}=-\mathcal{F}(v_{r})d\tau^{2}+\frac{c^{2}_{s}\left(1+6\alpha-2\alpha
e\Phi\right)}{f(v_{r})\mathcal{F}(v_{r})}dr^{2}+\frac{\left[1-2\alpha(e\Phi-2)\right]}{\sqrt{f(v_{r})}}r^{2}d\Omega^{2},$
(58)

where

$\displaystyle\mathcal{F}(v_{r})=\frac{F(v_{r})}{\sqrt{f(v_{r})}}\approx
c^{2}_{s}[1-\alpha(1-e\Phi+ev_{r}\Lambda_{r})]-v^{2}_{r}[1-\alpha(ev_{r}\Lambda_{r}-e\Phi-1)]+2e\alpha
v_{r}\Lambda_{r}.$ (59)

Next, by continuity equation $v_{r}=c_{s}r^{2}_{h}/r^{2}$, where $r_{h}$ is
the radius of the event horizon, we have that the modified canonical acoustic
black hole reads

$\displaystyle\mathcal{F}(r)=c^{2}_{s}\left[1-\alpha\left(1-e\Phi(r)+\frac{ec_{s}r^{2}_{h}\Lambda_{r}(r)}{r^{2}}\right)\right]-\frac{c^{2}_{s}r^{4}_{h}}{r^{4}}\left[1-\alpha\left(\frac{ec_{s}r^{2}_{h}\Lambda_{r}(r)}{r^{2}}-e\Phi(r)-1\right)\right]+\frac{2e\alpha
c_{s}r^{2}_{h}\Lambda_{r}(r)}{r^{2}}.$ (60)

Notice that when $e=0$ and $c_{s}=1$ the metric function above is given by

$\displaystyle\mathcal{F}(r)=1-\frac{r^{4}_{h}}{r^{4}},$ (61)

which is the canonical acoustic black hole metric found by Visser
Visser:1997ux . This metric is different from the well-known four-dimensional
black hole geometries normally found in general relativity. To obtain a result
conforming to the Schwarzschild geometry, but not identical to it, we can
consider from the continuity equation that $\rho v_{r}\propto 1/r^{2}$ with
$v_{r}=\sqrt{2GM/r}$ and $\rho\propto r^{-3/2}$, such a way for the metric
function, we find that $\mathcal{F}(r)=1-2GM/r$ — see Visser:1997ux  for
further details.

Therefore, the Hawking temperature in this case is given by

$\displaystyle\tilde{T}_{H}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{F}^{\prime}(r_{h})}{4\pi}$ (62) $\displaystyle=$
$\displaystyle\frac{c^{2}_{s}(1+\alpha)+e\alpha
c^{2}_{s}\left[\Phi(r_{h})-c_{s}\Lambda_{r}(r_{h})\right]}{\pi
r_{h}}+\frac{2e\alpha
c_{s}}{4\pi}\left(\Lambda^{\prime}_{r}(r_{h})-\frac{2\Lambda_{r}(r_{h})}{r_{h}}\right).$
(63)

Note that the Hawking temperature of the modified canonical acoustic black
hole is affected by the GUP deformation parameter $\alpha$ and also by the
contribution of the potentials $\Phi$ and $\Lambda_{r}$. In particular, for
$\Lambda_{r}=0$ the Hawking temperature in this case is reduced to

$\displaystyle\tilde{T}_{H}=\frac{c^{2}_{s}[1+\alpha+e\alpha\Phi(r_{h})]}{\pi
r_{h}}.$ (64)

Furthermore, assuming $c_{s}=1$ and a scalar potential of the type
$\Phi=r_{0}/r$, the temperature becomes

$\displaystyle\tilde{T}_{H}=T_{H}\left(1+\alpha+\frac{\alpha
er_{0}}{r_{h}}\right),$ (65)

where $T_{H}=1/\pi r_{h}$ is the Hawking temperature of the canonical acoustic
black hole for $\alpha=0$ and $r_{0}$ is a parameter with length dimension.
Therefore, in the absence of the scalar potential ($r_{0}=0$ or $e=0$), we
have

$\displaystyle\tilde{T}_{H}=T_{H}\left(1+\alpha\right),$ (66)

and so the Hawking temperature has its value increased when we vary the
parameter $\alpha$. On the other hand, when the scalar potential is considered
the temperature is modified and a new correction term appears.

Furthermore, with the result obtained for the temperature in (65) we can
determine the entropy (entanglement entropy Anacleto:2019rfn ) of the modified
canonical acoustic black hole by applying the first law of thermodynamics such
that

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\frac{dE}{dT}=\int\frac{\kappa\,dA}{8\pi\tilde{T}_{H}}=\int\frac{dA}{4}\left(1-\alpha^{2}-\frac{2\sqrt{\pi}\alpha^{2}er_{0}}{\sqrt{A}}-\frac{4\pi\alpha^{2}e^{2}r^{2}_{0}}{A}\right),$
(67) $\displaystyle=$
$\displaystyle\frac{(1-\alpha^{2})A}{4}-\frac{4\sqrt{\pi}\alpha^{2}er_{0}\sqrt{A}}{4}-\frac{4\pi\alpha^{2}e^{2}r^{2}_{0}}{4}\ln\left(\frac{A}{4r^{2}_{0}}\right),$
(68)

where $A=4\pi r^{2}_{h}$ is the horizon area of the canonical acoustic black
hole. Note that we have found a logarithmic correction term for entropy. This
correction term is generated due to the GUP and the scalar potential. The
presence of the logarithmic correction term is related to the appearance of
black hole remnants in gravitation Nozari:2005ah .

The metric (58) can also be written, for $\Lambda_{r}=0$, as follows

$\displaystyle
ds^{2}=-\mathcal{F}(v_{r})d\tau^{2}+\frac{c^{2}_{s}}{\mathcal{F}(v_{r})}dr^{2}+\left(1+\alpha-\alpha
e\Phi\right)r^{2}d\Omega^{2},$ (69)

where

$\displaystyle\mathcal{F}(v_{r})=c^{2}_{s}[1-\alpha(1-e\Phi)]-v^{2}_{r}[1+\alpha(1+e\Phi)].$
(70)

Now, recalling that $v_{r}=c_{s}r^{2}_{h}/r^{2}$ and $\Phi=r_{0}/r$ and making
$c_{s}=1$, we find

$\displaystyle
ds^{2}=-\mathcal{F}(r)d\tau^{2}+\frac{dr^{2}}{\mathcal{F}(r)}+\left(1+\alpha-\frac{\alpha
er_{0}}{r}\right)r^{2}d\Omega^{2},$ (71)

being

$\displaystyle\mathcal{F}(r)=1-\alpha+\frac{\alpha
er_{0}}{r}-\frac{r^{4}_{h}}{r^{4}}\left[1+\alpha\left(1+\frac{er_{0}}{r}\right)\right].$
(72)

At the limit of $r\gg r_{h}\equiv\alpha er_{0}$ the metric becomes

$\displaystyle
ds^{2}=-\mathcal{F}(r)d\tau^{2}+\frac{dr^{2}}{\mathcal{F}(r)}+r^{2}d\Omega^{2},$
(73)

that in this case the metric function $\mathcal{F}(r)$ is now given by

$\displaystyle\mathcal{F}(r)=1-\alpha^{2}+\frac{r_{h}}{r},$ (74)

which is an asymptotically global monopole metric. So this result is a
consequence of the scalar potential effect on the effective acoustic metric.

Now, for the case $\Phi=0$ and $\Lambda=\lambda/r$, being $\lambda$ a
parameter with a length dimension. In this case the Hawking temperature in
(63) becomes

$\displaystyle\tilde{T}_{H}=T_{H}\left(1+\alpha-\frac{5\alpha
e\lambda}{2r_{h}}\right),$ (75)

and for entropy we find

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{(1-\alpha^{2})A}{4}+\frac{20\sqrt{\pi}\alpha^{2}e\lambda\sqrt{A}}{4}-\frac{25\pi\alpha^{2}e^{2}\lambda^{2}}{4}\ln\left(\frac{A}{4\lambda^{2}}\right).$
(76)

Again a logarithmic correction is also generated for the pure vector potential
contribution whose pre-factor with respect to the pure scalar potential case
is $(5\lambda/2r_{0})^{2}$, which mimics logarithm corrections due to species
with different spins.

Here it is worth mentioning that considering the contributions of
$\Phi=r_{0}/r$ and $\Lambda_{r}=\lambda/r$ the metric function (60) becomes

$\displaystyle\mathcal{F}(r)=1-\alpha\left(1-\frac{er_{0}}{r}-\frac{e\lambda
r^{2}_{h}}{r^{3}}\right)-\frac{r^{4}_{h}}{r^{4}}\left[1+\alpha\left(1+\frac{er_{0}}{r}-\frac{e\lambda
r^{2}_{h}}{r^{3}}\right)\right].$ (77)

From (63), by making $c_{s}=1$, $\Phi=r_{0}/r$ and $\Lambda_{r}=\lambda/r$, we
obtain the Hawking temperature which is given by

$\displaystyle\tilde{T}_{H}=T_{H}\left(1+\alpha+\frac{e\alpha
r_{0}}{r_{h}}-\frac{5e\alpha\lambda}{2r_{h}}\right).$ (78)

On the other hand, if $\Lambda_{r}=\lambda_{0}$, where $\lambda_{0}$ is a
dimensionless constant, then we have

$\displaystyle\mathcal{F}(r)=1-\alpha\left(1-\frac{er_{0}}{r}-\frac{e\lambda_{0}r^{2}_{h}}{r^{2}}\right)-\frac{r^{4}_{h}}{r^{4}}\left[1+\alpha\left(1+\frac{er_{0}}{r}-\frac{e\lambda_{0}r^{2}_{h}}{r^{2}}\right)\right].$
(79)

Our result shows similarity with the metric obtained in Ling:2021vgk  with the
identification of effective charge and mass as $Q^{2}=\alpha
e\lambda_{0}r^{2}_{h}$ and $2M=\alpha er_{0}$ for $r_{0}<0$, respectively.
Therefore, we highlight another new result that arises due to the GUP effect,
that is, the contributions of $\Phi$ and $\Lambda$ can generate Reissner-
Nordström-like contributions for the metric. In this case the Hawking
temperature is given by equation (63) making $c_{s}=1$, $\Phi=r_{0}/r$ and
$\Lambda_{r}=\lambda_{0}$. So we have

$\displaystyle\tilde{T}_{H}=T_{H}\left(1+\alpha-2e\alpha\lambda_{0}+\frac{e\alpha
r_{0}}{r_{h}}\right).$ (80)

Hence, when $e=0$, that is, $Q=M=0$, we have the temperature given by equation
(66).

##  III The Dispersion Relation

In order to study further effects from the GUP we shall now investigate the
dispersion relation. For this we will use the following notation:

$\displaystyle
S_{1}\sim\mbox{Re}\left[e^{i\tilde{\omega}t-i\vec{k}\cdot\vec{x}}\right],\qquad\tilde{\omega}=\frac{\partial
S_{1}}{\partial t},\qquad\vec{k}=\nabla S_{1}.$ (81)

In this way the Klein-Gordon equation (II) in terms of momenta and frequency
becomes

$\displaystyle a\,\tilde{\omega}^{2}+b\,\tilde{\omega}+c=0,$ (82)

where

$\displaystyle a=1-2\alpha(e\Phi-2)+c_{s}^{2}\left(1+2\alpha\right),$ (83)
$\displaystyle b=2(1+4\alpha-\alpha
e\Phi)(\vec{v}\cdot\vec{k})-2\alpha(\vec{\Lambda}\cdot\vec{k}),$ (84)
$\displaystyle c=-\left[-(1+4\alpha)v^{2}k^{2}+2\alpha
e(\vec{\Lambda}\cdot\vec{k})(\vec{v}\cdot\vec{k})+c_{s}^{2}\left(1+2\alpha\right)k^{2}\right].$
(85)

Now, by considering $k^{i}=\delta^{i1}$, we have

$\displaystyle\tilde{\omega}=\frac{-2\left[(1+4\alpha+\alpha
e\Phi)v_{1}-\alpha e\Lambda_{1}\right]k\pm\sqrt{\Delta}}{2a},$ (86)

with

$\displaystyle\sqrt{\Delta}=2c_{s}k\sqrt{1+2\alpha-2\alpha(e\Phi-2)+4\alpha
e\Phi\frac{v_{1}^{2}}{c^{2}_{s}}-{2\alpha\frac{e\Lambda_{1}v_{1}}{c^{2}_{s}}}+\left[(1+4\alpha)c^{2}_{s}-(1+6\alpha)v^{2}_{1}+{2\alpha
e\Lambda_{1}v_{1}}\right]}.$ (87)

In particular for $\Lambda_{1}=0$ and in the non-relativistic limit, we obtain

$\displaystyle\tilde{\omega}\approx\pm\frac{c_{s}\sqrt{1+6\alpha-2\alpha
e\Phi}}{1+4\alpha-2\alpha e\Phi}k=\eta\,\omega\left(1-\alpha+\alpha
e\Phi\right),$ (88)

being $\omega=c_{s}k$ (the dispersion relation for $\alpha=0$) and $\eta=\pm
1$ the polarizations. Now, to explore the scalar potential effect recall that
$\Phi=r_{0}/r$. Hence, focusing on the dispersion relation in the vicinity of
the event horizon, $r\rightarrow r_{h}$, becomes

$\displaystyle\tilde{\omega}=\eta\,\omega\left(1-\alpha+\frac{er_{0}\alpha}{r_{h}}\right)=\eta(1-\alpha)\omega\left(1+\frac{er_{0}\alpha}{r_{h}}\right)+\mathcal{O}(\alpha^{2}).$
(89)

Next, we redefine $\bar{\omega}=\tilde{\omega}/(1-\alpha)$ and apply
$k\sim\Delta k\geq 1/\Delta x=1/r_{h}$. Thus, the dispersion relation above is
rewritten in the form

$\displaystyle\bar{\omega}=\eta\,\omega\left(1+\alpha er_{0}k\right).$ (90)

So we note that the scalar potential contribution generated by the GUP is
associated with a viscosity effect in the fluid. Similar behavior has been
analyzed in Anacleto:2019rfn  in the background with Lorentz violation.

We can also write an expression in terms of the frequency difference given by

$\displaystyle\frac{\Delta\omega}{\omega}=\frac{\bar{\omega}-\eta\omega}{\omega}=\eta\alpha
er_{0}k.$ (91)

Now we can get from the dispersion relation (90) the phase velocity ($v_{p}$)
and the group velocity ($v_{g}$) as follows:

$\displaystyle v_{p}=\frac{\bar{\omega}}{k}=\eta c_{s}(1+\alpha er_{0}k),$
(92)

and

$\displaystyle v_{g}=\frac{d\bar{\omega}}{dk}=\eta c_{s}(1+2\alpha er_{0}k).$
(93)

Using the Rayleigh’s formula

$\displaystyle v_{g}=v_{p}+k\frac{dv_{p}}{dk},$ (94)

we obtain an expression for the velocity difference given by

$\displaystyle\frac{v_{g}-v_{p}}{v_{p}}=\alpha er_{0}k.$ (95)

In this case, we have a supersonic behavior ($v_{g}>v_{p}$) as in BEC systems
whose excitations follow ‘superluminal’ dispersion relations Robertson:2012ku
.

Furthermore, from equation (65), for the Hawking temperature we can also
obtain a relationship for the temperature difference. So we have

$\displaystyle\mathcal{T}_{H}=T_{H}(1+\alpha er_{0}k).$ (96)

Here we have redefined $\mathcal{T}_{H}=\tilde{T}_{H}/(1+\alpha)$. Hence, we
find the following relationship for the Hawking temperature variation due to
the potential effect

$\displaystyle\frac{\Delta
T_{H}}{T_{H}}=\frac{\mathcal{T}_{H}-T_{H}}{T_{H}}=\alpha er_{0}k.$ (97)

Therefore, we find the following relationship:

$\displaystyle\frac{v_{g}-v_{p}}{v_{p}}=\eta\frac{\Delta\omega}{\omega}=\frac{\Delta
T_{H}}{T_{H}}.$ (98)

Moreover, we emphasize that the modification of the dispersion relation
associated with $\delta=\alpha er_{0}k$ is analogous to Lorentz-violating
parameter in BEC physics whose deviation is $\delta\sim 10^{-5}$ — see
Anacleto:2019rfn  and references therein. As a result, by recalling $k\sim
1/r_{h}$, such that $\delta\sim\alpha e\Phi(r_{h})$, we obtain

$\displaystyle\frac{\Delta T_{H}}{T_{H}}\sim\Phi(r_{h})\sim 10^{-5}.$ (99)

Interestingly enough, this result is also analogous to the small temperature
fluctuations in the anisotropy of the cosmic microwave background (CMB), with
the scalar field evaluated at horizon corresponding to fluctuations of the
gravitational potential in the cosmological scenario.

##  IV Conclusions

We obtained an effective acoustic metric for acoustic black holes with quantum
corrections implemented by the generalized Heisenberg uncertainty principle
(GUP) in the Abelian Higgs model. Thus, contrary to the usual acoustic
metrics, the obtained effective acoustic metric now depends on the
contribution of scalar and vector potentials that emerge as an effect of the
GUP. We have examined the Hawking temperature and the entropy for the modified
canonical acoustic black hole. As a result, the Hawking temperature that
changes due to the effect of the GUP provides logarithmic corrections to the
entropy. We address the modification of the Hawking temperature by considering
different effects by turning on scalar and vector potential contributions
separately and also turning on them together. Moreover, by analyzing the
dispersion relation of the model we have obtained a relationship between the
group and phase velocity difference due to the GUP effect, as well as
expressions for the frequency and temperature deviations. Finally, we have
made a few comments on how they can be related to BEC physics with supersonic
dispersion relations and small temperature fluctuations in the anisotropy of
the cosmic microwave background (CMB).

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant nos.
165/2018 and 015/2019), for partial financial support. MAA, FAB and EP
acknowledge support from CNPq (Grant nos. 306962/2018-7 and 433980/2018-4,
312104/2018-9, 304852/2017-1).

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