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Spinning black hole in a fluid
Surojit Dalui,1,* Arpan Krishna Mitra,2,†Deeshani Mitra ,3,‡and Subir Ghosh 3,§
1Department of Physics, Shanghai University, 99 Shangda Road, Baoshan District,
Shanghai 200444, People Republic of China
2Aryabhatta Research Institute of Observational Sciences (ARIES),
Manora Peak Nainital—263001, Uttarakhand, India
3Indian Statistical Institute, 203, Barrackpore Trunk Road, Kolkata 700108, India
(Received 15 August 2023; accepted 21 February 2024; published 18 March 2024)
In this paper, we propose a new analog gravity example—a spinning (or Kerr) black hole in an
extended fluid model; the latter was derived in an earlier work [A. K. Mitra and S. Ghosh, Divergence
anomaly and Schwinger terms: Towards a consistent theory of anomalous classical fluids, Phys.Rev.D
106, L041702 (2022).] by two of the present authors. The fluid model receives Berry curvature
contributions and applies to electron dynamics in condensed matter lattice systems in the hydrodynamic
limit. We construct the acoustic metric for sonic fluctuations that obey a structurally relativistic
wave equation in an effective curved background. In a novel approach of dimensional analysis, we
have derived explicit expressions for effective mass and angular momentum per unit mass in the
acoustic metric (in terms of fluid parameters), to identify with corresponding parameters of the Kerr
metric. The spin is a manifestation of the Berry curvature-induced effective noncommutative structure in
the fluid. Finally we put the Kerr black hole analogy in a robust setting by revealing explicitly the
presence of horizon and ergoregion for a specific background fluid velocity profile. We also show that
near horizon behavior of the phase-space trajectory of a probe particle agrees with Kerr black hole
analogy. In a fluid dynamics perspective, the presence of a horizon signifies the wave blocking
phenomenon.
DOI: 10.1103/PhysRevD.109.064055
I. INTRODUCTION
Analog gravity [1] started with the work of Unruh [2],
who showed that first-order fluctuations in irrotational,
nonviscous, barotropic flow obey a structurally relativistic
massless scalar wave equation in an effectively curved
background, with an acoustic metric (AM), comprising of
fluid flow parameters (for diverse models, see [3,4]). AM
reveals black/white holelike features in velocity space,
known as wave blocking in fluid dynamics [5].
In this paper, we construct a new AM, (in the framework
of [2]), in an extended fluid model with the Berry curva-
ture effects, derived in [6]. This phase space describes
semiclassical electron dynamics in a magnetic Bloch band,
with periodic potential in an external magnetic field and
Berry curvature [7]. This fluid dynamics is relevant in
electron hydrodynamics in condensed matter, where elec-
tron flow obeys hydrodynamic laws instead of Ohmic [8].
Generically electrons in metals act as nearly-free Fermi gas
with a large mean free path for electron-electron collision.
Recently hydrodynamic regime was achieved in extremely
pure, high quality, electronic materials—especially gra-
phene [9], layered materials with very high electrical
conductivity such as metallic delafossites PdCoO2and
PtCoO2[10].
The salient feature in our work is that the AM after a
coordinate transformation [11] is similar to Kerr metric
[12] in Eddington-Finkelstein (EF) coordinates [13].
Recently, there have been several attempts to construct
analog models of BHs other than the nonrotating ones
[14–17]. The fluid, with the presence of a vortex in it, has
been considered as a system to construct an analog of a
rotating BH [14].In[15], authors reasoned that in a
shallow water system, with a varying background flow
velocity, metric analogs of Kerr metric can be constructed.
Later, the presence of superradiance was found in it [16]
andalsointheBose-Einsteincondensate[17].Inarecent
work in this direction but exploiting the optical vortex
is [18], the authors have used Laguerre-Gaussian-type
beams, bearing phase singularities. These types of beams
have transverse intensity profiles comprising all character-
istics of a vortex. The fluctuations in the amplitude and the
phase of the electric field have been shown to satisfy a
*surojitdalui@shu.edu.cn
†arpankmitra@aries.res.in
‡deeshani1997@gmail.com
§subirghosh20@gmail.com
PHYSICAL REVIEW D 109, 064055 (2024)
2470-0010=2024=109(6)=064055(7) 064055-1 © 2024 American Physical Society
massless scalar field equation on a curved background,
similar to the Kerr metric. However, this present paper is
possibly the first instance of an analog Kerr metric in the
fluid subjected to an external magnetic field and Berry
curvature. However, it is not unexpected since a spinlike
feature appears in Berry curvature-modified particle
dynamics [19].ThephysicsbehindthisAMisrevealed
through explicit construction of Kerr-like parameters,
such as effective mass meff and angular momentum per
unit mass aeff , out of fluid composites via dimensional
analysis. More interestingly, using a specific form of
nonuniform background fluid velocity we explicitly pro-
vide spatial positions of the ergoregion and horizon,
characteristic of the Kerr metric.1Recently, multiple
articles have shown studies on the trajectories of Weyl
fermions in curved spacetimes [20–22].Oneofthem
presented the trajectory of the massless Weyl particles
around an analog Schwarzschild black hole [22].Herewe
depict the phase space trajectories of a probe particle
around the (Berry curvature-induced) analog Kerr metric
that we have found and point out that the location of the
analog horizon in this analog Kerr metric is the same as
that of one in the Kerr metric in general relativity.
II. AM WITH BERRY CURVATURE EFFECTS
We consider a fluid with pressure PðρÞ.eis electronic
charge, Bexternal magnetic field and ΩðkÞis Berry
curvature in momentum ð¯
kÞspace. For small ΩðkÞthe
extended fluid model [with full expressions [6] in the
Supplemental Material Eqs. (1)–(3)]is½Aðx;kÞ¼
1þeBðxÞ·ΩðkÞ
˙
ρ¼−∇ρv
A;ð1Þ
˙
vþðv·∇Þv
A¼−
∇P
ρA:ð2Þ
Irrotational v¼−∇ψis written by a velocity potential ψ.
The velocity csof sonic disturbance in the medium and the
system enthalpy hare cs¼ffiffiffiffiffi
dP
dρ
q;∇h¼∇P=ρand (2)
becomes
−∇˙
ψþ∇ð∇ψÞ2
2A¼−∇h
A
→
˙
ψ−ð∇ψÞ2
2A¼h
A:ð3Þ
With the fluid variables as background þfluctuation [2],
ρ¼ρ0þϵρ1;P¼P0þϵc2
sρ1;
vi¼v0iþϵv1i¼∂iψ0þϵ∂iψ1;∇h1¼c2
s∇ρ1=ρ0:ð4Þ
the first-order perturbation terms are
ρ1¼ρ0
˙
ψ1
c2
sþρ0v0
!·∇ψ1
c2
sA;ð5Þ
˙
ρ1¼−
1
A∇:ðρ1v0
!−ρ0∇ψ1Þ:ð6Þ
Taking the time derivative of (5) and comparing it with (6),
(keeping external parameters and csfixed), we arrive at the
wave equation of massless relativistic scalar in a curved
spacetime,
∂μðfμν∂νψ1Þ¼0;
fμν ¼ρ0
c2
s
0
B
B
B
B
B
B
@
Avxvyvz
vx
v2
x−c2
s
A
vxvy
A
vxvz
A
vy
vxvy
A
v2
y−c2
s
A
vyvz
A
vz
vxvz
A
vyvz
A
v2
z−c2
s
A
1
C
C
C
C
C
C
A
:ð7Þ
Note that fμν depends on the background velocity
v0iwhich we write as vi. The effective background metric
gμν is fμν ¼ffiffiffiffiffiffi
−g
pgμν with the determinant of fμν given by
jfμνj¼ð ffiffiffiffiffiffi
−g
pÞ41
g¼g¼−ρ4
0
c2
sA2. The AM is constructed
out of background fluid velocity and inherits symmetries
of the latter. The AM is stationary as flow is stationary
(or steady in fluid dynamics terminology). Thus,
cherished AM gμν, one of our major results, in polar
form is
gμν ¼ρ0
Acs
0
B
B
B
B
B
@
c2
s−ðv2
rþv2
θþv2
ϕÞ
Avrrvθrsin θvϕ
vr−A00
rvθ0−Ar20
rsin θvϕ00−Ar2sin2θ
1
C
C
C
C
C
A
:
It is important to remember that fluid particles see the flat
Minkowski metric (for fluid velocity ≪velocity of the
electromagnetic field in vacuum) whereas acoustic fluctu-
ations feel only the AM; some basic properties of the latter
carry a legacy of the former. From the above AM it is
clear [23] that the regions of supersonic flow are ergo-
regions where gtt changes sign, gtt ¼0→vr¼cscorre-
sponds to the event horizon (wave-blocking zone in fluid
dynamics); the boundary that null geodesics (or phonons)
cannot escape. In fact, here the ergosphere coincides with
the event horizon. Other notions such as trapped surface,
surface gravity, etc also exist for AM [23]. Spatial positions
of the analog horizon in the fluid will appear indirectly
from csðrÞ;v
rðrÞ.
1We thank the anonymous referee for the suggestion.
DALUI, MITRA, MITRA, and GHOSH PHYS. REV. D 109, 064055 (2024)
064055-2
III. meff,aeff IN AM-KERR ANALOGY
For matching with Kerr, we convert the acoustic path-
length dimension to jds2j¼ðlengthÞ2¼½L2. In GR, the
metrics have dimensional parameters such as Newton’s
constant Gand velocity of light c, among others. Similarly,
AM can depend on cs, background fluid density ρ0(both
not constant in general), etc. Another fluid parameter
is the dynamic (or absolute) viscosity μof dimension of
jμj¼½M½L−1½T−1(with kinematic viscosity being μ=ρ0).
A length scale l(∼spatial dimension of the fluid system)
enters our acoustic model. k-dependence in ΩðkÞrefers to
the quasimomentum of a single band (in the crystalline
solid) of the Bloch electron, comprising the electron fluid
in the hydrodynamic limit, where the AM is constructed.
For the present work, kis just a label and is treated as a
constant. For a uniform BðrÞ¼B,Ais effectively a
constant. The resulting acoustic path has jds2
AMj¼
ðlengthÞ2dimension,
ds2
AM ¼cslρ0
μAðc2
s−v2Þ
Adt2þ2vrdtdrþ2rvθdtdθ
þ2rsinθvϕdtdϕ−Afdr2þr2dθ2þr2sin2θdϕ2g;
ð8Þ
where v2¼v2
rþv2
θþv2
ϕ. Now we perform a coordinate
transformation
dt →dt þdr
csþdθ
ωsþdϕ
Ωs
;
ωs¼angular frequency;
Ωs¼azimuthal frequency of sonic disturbance;ð9Þ
on the acoustic path to obtain,
ds2
AM ¼cslρ0
μAðc2
s−v2Þ
Adt2þðc2
s−v2Þ
Ac2
sþ2vr
c2
s
−Adr2þ2ðc2
s−v2Þ
Acsþvrdtdr þ2ðc2
s−v2Þ
AΩsþrsin θvϕdtdϕ
þ2ðc2
s−v2Þ
Aωsþrvθdtdθþ2ðc2
s−v2Þ
Acsþvr
ωsþrvθ
csdrdθþ2ðc2
s−v2Þ
AΩsωsþrsin θvϕ
ωsþrvθ
Ωsdθdϕ
þ2ðc2
s−v2Þ
AcsΩsþvr
Ωsþrsin θvϕ
csdrdϕþðc2
s−v2Þ
AΩ2
s
−Ar2sin2θþ2rsin θ
Ωs
vϕdϕ2
þðc2
s−v2Þ
Aω2
s
−Ar2þ2r
ωs
vθdθ2:ð10Þ
Our major observation is that in the equatorial plane (i.e., θ¼π=2hypersurface) and with vθ¼0, the acoustic path,
ds2
AM ¼cslρ0
μAðc2
s−v2Þ
Adt2þðc2
s−v2Þ
Ac2
sþ2vr
c2
s
−Adr2þ2ðc2
s−v2Þ
Acsþvrdtdr þ2ðc2
s−v2Þ
AΩsþrvϕdtdϕ
þ2ðc2
s−v2Þ
AcsΩsþvr
Ωsþrvϕ
csdrdϕþðc2
s−v2Þ
AΩ2
s
−Ar2þ2r
Ωs
vϕdϕ2;ð11Þ
is structurally equivalent to Kerr metric path length in Eddington-Finklestein coordinates [full expressions in Supplemental
Material Eqs. (5)–(10)],
ds2
Kerr ¼1−
2Gm
rc2c2dt2−
4Gm
rc dtdr þ4Gma
rc2dtdϕ−1þ2Gm
rc2dr2þ2a
c1þ2Gm
rc2drdϕ
−r2þa2
c2−
2Gma2
rc4dϕ2:ð12Þ
We exploit the dimensional equality ds2
AM ¼ds2
Kerr ¼ðlengthÞ2to construct effective mass and spin parameters for AM,
in analogy with mass (m) and angular momentum per unit mass a¼J=m of Kerr black hole [with details in Supplemental
Material Eqs. (11)–(17)]:
(i) Comparison of the dimensions of gtt gives
meff ≡l3ρ0v2
A2c2
sð13Þ
and
SPINNING BLACK HOLE IN A FLUID PHYS. REV. D 109, 064055 (2024)
064055-3
(ii) Comparison of the dimensions of gtϕgives
aeff ≡lρ0c5
s
μA2Ωsv2:ð14Þ
This constitutes another set of important results since these effective parameters are, in principle, measurable.
In terms of meff and aeff the same metric (11) turns out to be
ds2
AM ¼cslρ0
μA2−meff
cs
μl2c2
sdt2þ2meff
c2
s
μl2c2
s
v2−1þcsvrA
v2dtdr þ2meff aeff A2
l3ρ01−v2þΩsA
c2
s
rvϕdtdϕ
þcslρ0
μA21þ2vr
c2
s
A−A2−meff
cs
μl2dr2þ2aeff
v2
r
c3
s1−meff
A2
l3ρ0csþAvr
csþAΩs
c2
s
rvϕdrdϕ
þμA2v4
c7
slρ0
a2
eff 1−meff
A2
l3ρ0−cslρ0
μr2þ2cslρ0
μAΩs
rvϕdϕ2:ð15Þ
Remarkably, our entirely algebraic methodology for
implementing coordinate transformations (10) and pre-
scription of identifying fluid mass and spin parameters
have resulted in an AM (15), which can be compared term
by term with the Kerr metric (see Supplemental Material
[24]). Notice that meff ;a
eff in AM (15) occupy identical
positions as m,ain Kerr metric (see Supplemental Material
[24]). This provides a mathematical consistency of our
framework and reveals the physics behind AM.
IV. PHASE SPACE PROBE TRAJECTORY
The AM at θ¼π=2has a timelike Killing vector
χa¼ð1;0;0;0Þwith conserved energy of a particle given
by E¼−χapa¼−pt, where pa¼ðpt;p
r;p
θ;p
ϕÞ. Using
the AM in the dispersion relation gabpapb¼−M2for a
particle of mass Min AM (with pθ¼0), particle energy E
is obtained in terms of the other momentum components,
as the positive energy root. Hamilton’s equations of
motion are
˙
r¼∂E
∂pr
;˙
pr¼−
∂E
∂r;
˙
ϕ¼∂E
∂pϕ
;˙
pϕ¼−
∂E
∂ϕ:ð16Þ
Let us consider a particular background fluid profile known
as “draining bathtub”flow (for details see [23])
v¼Aˆ
rþB
ˆ
ϕ
r:ð17Þ
with constant A,B. In this idealized model, background
fluid flow is planar until it reaches a linear sink along
perpendicular direction. The background fluid density ρis
taken to be constant throughout the flow. Furthermore, for
the barotropic fluid considered here, (2) and specific
enthalpy (h) indicate that the background pressure P
and the speed of sonic disturbance csare also constant.
The equation of continuity (1), in cylindrical coordinates
with a sink along z-direction, reduces to
1
r∂
∂rðrvrÞþ∂vϕ
∂ϕþ∂
∂zðrvzÞ¼0;ð18Þ
and clearly the profile (17) (with vz¼0on the plane just
away from the sink and planar distance rmeasured from
z-axis) is a solution of (18) (for more details, see Sec. 2.4.3
of [23]).
In this model the acoustic ergosphere and event horizon
form at
rergosphere ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2þB2
p
cs
;r
horizon ¼jAj
cs
:ð19Þ
However, one important thing needs be to mentioned here
is that the distinction between ergosphere and the acoustic
horizon is critical for this model [23]. Therefore, keeping
that in mind, we proceed here to solve these coupled
differential equations numerically [Eqs. (16)].
After the numerical solutions we have plotted the phase-
space plot between the radial coordinate (r) and the
corresponding radial momentum of the particle ðprÞ.
Depending on the sign of A(þand −) we plot two cases.
The values of the other parameters are as follows: A¼5,
cs¼100,Ωs¼1.0,Γ¼cslρ0
μA¼100. In the first figure, i.e.,
Fig. 1the amplitudes of the velocity components (i.e., vr
and vϕ) are respectively A¼B¼100;000. In Fig. 1,we
can see, that the phase-space trajectory of the particle starts
with some lower-momentum value for a larger value of r,
but as rdecreases, the corresponding radial momentum
value (pr) increases and at r¼1000 the momentum
reaches its maximum value. This nature of the graph
depicts that as the particle moves near to r¼1000 the
particle experiences a “sudden change”in its trajectory.
DALUI, MITRA, MITRA, and GHOSH PHYS. REV. D 109, 064055 (2024)
064055-4
Moreover, according to the “draining bathtub”model the
acoustic horizon should appear at r¼1000 [see Eq. (19)]
which exactly happens in our case based on our specified
parameter values. Consequently, from this occurrence, we
can identify the position of the horizon, which exactly
matches the theoretical value of the horizon, i.e., at
jAj=cs¼1000. In this context, it is worth mentioning that
in some near-horizon contexts [13,25,26] it has been shown
that in the near-horizon region, a particle experiences this
kind of “sudden change”or ”instability”in its phase-space
trajectory.
Similarly, in Fig. 2we have chosen A¼−100;000 and
B¼500 keeping the other parameters the same and we
find that the radial momentum value of the massless
particle does not change much until reaching r≃2000.
After r≃1000 the momentum value falls abruptly which
suggests that the particle is sucked inside the horizon which
is situated at r¼1000. This characteristic is exactly similar
to an ingoing massless particle in the near-horizon region of
a SSS BH [see Eq. (26) and Page 8 of [26] ] and a Kerr BH
(see Page 6 of Ref. [13]).
In a nutshell, we can say that with the radial-dependent
background flow of fluid, we can construct an analog
metric which mimics the exact structure of Kerr BH, and by
studying the particle dynamics in this background we can
pinpoint the exact location of the horizon for particular
values. Future works will involve a more rigorous analysis
with the full anomalous fluid dynamics and an arbitrary
fluid flow. Attempts of laboratory demonstrations of this
new analog black hole model will be worthwhile.
FIG. 1. Phase-space diagram for a massless particle (for A¼B¼1;000;000). From the figure we can see as the particle moves near
to r¼1000 its radial momentum princreases exponentially reaching its peak as rattains the value of 1000. This occurrence enables us
to pinpoint the horizon’s position, which precisely corresponds to the theoretical expectation at jAj=cs¼1000.
FIG. 2. Similarly, for A¼−100;000 and B¼500 we plot again plot the phase-space trajectory of the massless particle remaining
other parameter values the same. We see that until around r≃2000 the radial momentum of the particle does not change much, but near
to r≃1000 the momentum of the particle suddenly falls down which suggests that the massless particle falls inside the horizon.
SPINNING BLACK HOLE IN A FLUID PHYS. REV. D 109, 064055 (2024)
064055-5
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