# Hydrodynamical duals of the gravitational axial anomaly and the cosmological
constant

R. V. Khakimov  Joint Institute for Nuclear Research, Joliot-Curie str. 6,
Dubna 141980, Russia  NRC Kurchatov Institute, Moscow, Akademika Kurchatova
pl. 1, Moscow, 123098, Russia  Physics Department, Lomonosov Moscow State
University, 1-2 Leninskie Gory, Moscow 119991, Russia  G. Yu. Prokhorov  Joint
Institute for Nuclear Research, Joliot-Curie str. 6, Dubna 141980, Russia  NRC
Kurchatov Institute, Moscow, Akademika Kurchatova pl. 1, Moscow, 123098,
Russia  O. V. Teryaev  Joint Institute for Nuclear Research, Joliot-Curie str.
6, Dubna 141980, Russia  NRC Kurchatov Institute, Moscow, Akademika Kurchatova
pl. 1, Moscow, 123098, Russia  V. I. Zakharov  Joint Institute for Nuclear
Research, Joliot-Curie str. 6, Dubna 141980, Russia  NRC Kurchatov Institute,
Moscow, Akademika Kurchatova pl. 1, Moscow, 123098, Russia  Pacific Quantum
Center, Far Eastern Federal University, 10 Ajax Bay, Russky Island,
Vladivostok 690950, Russia

## Abstract

We construct the hydrodynamic expansion for a rotating and accelerated medium
in a curved space-time, and establish a duality between the currents related
to the cosmological constant and the acceleration. Then we consider the more
general case with a non-zero Weyl tensor, and show the relationship between
the current in flat space-time and the gravitational axial anomaly. This
generalizes the previous derivation to the case with a non-zero Ricci tensor.

##  1 Introduction

Since the discovery of the chiral anomaly [1, 2] its study has been of great
interest. To date, there is a large number of different effects associated
with it found in many different physical fields [3, 4, 5, 6, 7]. Relativistic
fluid effects of the anomalies [3, 8, 4, 5, 9, 10, 11] are of particular
interest at the present time, indicating the nontrivial interplay between the
infrared and the ultraviolet theories. Besides undoubted theoretical interest,
these effects can be applied to describe processes in heavy-ion-collisions
[12, 13, 14].

Hydrodynamics, which describes physics at large spatial and temporal
distances, is based on the conservation relations, such as conservation of the
stress-energy tensor and current [15]

$\displaystyle\displaystyle\partial_{\mu}T^{\mu\nu}=0\,,\quad\partial_{\mu}j^{\mu}=0\,.$
(1)

In particular, in the case of a viscous and heat-conducting fluid the
equations (1) allow us to describe linear dissipative effects [15]. A
significant step was made in the work [8], where an additional linear
correction associated with vorticity (and magnetic field) was obtained. It was
shown that the magnitude of the corresponding vorticity effect (more
precisely, the part with chemical potential), so-called chiral vortical effect
(CVE), in the axial current

$\displaystyle\displaystyle
j^{\nu}_{A,\text{CVE}}=\left(\sigma_{(T)}T^{2}+\sigma_{(\mu)}\mu^{2}\right)\omega^{\nu}+\mathcal{O}(\partial^{3}u)\,,$
(2)

is determined by the quantum axial anomaly in the electromagnetic field
$\displaystyle F_{\mu\nu}$ [1]

$\displaystyle\displaystyle\partial_{\mu}j^{\mu}_{A}=C\,\varepsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\,,$
(3)

which, being included, modifies the basic hydrodynamic equations (1). In (2)
$\displaystyle T$ is the proper temperature, $\displaystyle\mu$ is the
chemical potential,
$\displaystyle\omega^{\mu}={\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}u_{\nu}\partial_{\alpha}u_{\beta}}$
is the vorticity and $\displaystyle\mathcal{O}(\partial^{3}u)$ denotes terms
cubic in gradients of 4-velocity $\displaystyle u_{\mu}$. The relationship is
fixed by the equation, connecting conductivity $\displaystyle\sigma_{(\mu)}$
with a numerical factor from the anomaly $\displaystyle C$

$\displaystyle\displaystyle\sigma_{(\mu)}=-8C\,,$ (4)

which was obtained in [8] from the second law of thermodynamics for entropy
current. In particular, for Dirac field current (2) has the form [3, 16, 11]

$\displaystyle\displaystyle
j^{\nu}_{A,\text{CVE}}=\left(\frac{T^{2}}{6}+\frac{\mu^{2}}{2\pi^{2}}\right)\omega^{\nu}+\mathcal{O}(\partial^{3}u)\,,$
(5)

which exactly satisfies the well known anomaly for the Dirac field
$\displaystyle C=-1/(16\pi^{2})$. In the same way the correspondence (4) was
demonstrated for spin 3/2 [17] and spin 1 [18].

Recently, it was shown that in a similar way the same effects can be obtained
without the use of the entropic current [19, 20], taking advantage of the so-
called global thermodynamic equilibrium [21]. This simplifies the
calculations, which made it possible, in particular, to analyze the second-
order gradient effects [19].

Note that the anomalous transport associated with the electromagnetic chiral
anomaly was mainly analyzed (3). However, there are also a number of works
[22, 23, 24, 25] that consider the transport associated with the gravitational
axial anomaly [2]

$\displaystyle\displaystyle\nabla_{\mu}j_{A}^{\mu}=\mathcal{N}\epsilon^{\alpha\beta\mu\nu}R_{\mu\nu\lambda\rho}R^{\quad\lambda\rho}_{\alpha\beta}\,,$
(6)

where
$\displaystyle\epsilon^{\alpha\beta\mu\nu}=\frac{1}{\sqrt{-g}}\varepsilon^{\alpha\beta\mu\nu}$
is Levi-Civita tensor in a curved space-time and $\displaystyle\mathcal{N}$ is
a numerical constant.

However, there was a problem of direct generalization of the method of [8, 19]
to the gravitational chiral anomaly. Unlike the case with an electromagnetic
anomaly (associated with a linear order in gradients), it is necessary to
consider higher orders of the gradient expansion. This, as well as the
relative complexity of gravity itself, makes a generalization of [8, 19]
somewhat nontrivial and massive.

The solution was recently found in [9], where it was shown that the imprints
of the gravitational chiral anomaly (6) arise in the third order of the
gradient expansion for the axial current. Interestingly, this effect exists
even in flat space-time, when the corresponding current includes only
acceleration and vorticity

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\left(\lambda_{1}\omega^{2}+\lambda_{2}a^{2}\right)\omega_{\mu}\,,$
(7)

where $\displaystyle a_{\mu}=u^{\nu}\partial_{\nu}u_{\mu}$ is acceleration and
$\displaystyle\lambda_{2}$ and $\displaystyle\lambda_{1}$ are dimensionless
conductivities. The connection of the current (7) with the gravitational
chiral anomaly (6) is fixed by the equation

$\displaystyle\displaystyle\frac{\lambda_{1}-\lambda_{2}}{32}=\mathcal{N}\,.$
(8)

In particular, for Dirac fields conductivities in (7) were found directly
using the density operator and has the form [26] (see also [27, 28])

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\left(-\frac{\omega^{2}}{24\pi^{2}}-\frac{a^{2}}{8\pi^{2}}\right)\omega_{\mu}\,.$
(9)

At the same time, the gravitational chiral anomaly for Dirac fields is well
known [2]

$\displaystyle\displaystyle\nabla_{\mu}j_{A}^{\mu}=\frac{1}{384\pi^{2}}\epsilon^{\alpha\beta\mu\nu}R_{\mu\nu\lambda\rho}R^{\quad\lambda\rho}_{\alpha\beta}\,.$
(10)

It’s easy to see that (8) is satisfied. Similarly, this relation was verified
in [29] for the Rarita-Schwinger-Adler model [30] containing fields with spin
3/2 (one mode) and 1/2 (two modes). The current was obtained from the quantum-
statistical correlators and has the form

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\left(-\frac{53}{24\pi^{2}}\omega^{2}-\frac{5}{8\pi^{2}}a^{2}\right)\omega_{\mu}\,.$
(11)

At the same time, a gravitational chiral anomaly was found in [31]

$\displaystyle\displaystyle\nabla_{\mu}j_{A}^{\mu}=-\frac{19}{384\pi^{2}}\epsilon^{\alpha\beta\mu\nu}R_{\mu\nu\lambda\rho}R^{\quad\lambda\rho}_{\alpha\beta}\,,$
(12)

Comparing (11) and (12), we see that the general expression (8) is satisfied
again. Thus, there is a new anomalous transport effect in a vortical and
accelerated medium (7), called the kinematic vortical effect (KVE).

However, the derivation in [9] was not the most general, since it used an
assumption about the properties of the manifold. Namely, Ricci-flat space-
times were considered

$\displaystyle\displaystyle R_{\mu\nu}=0\,.$ (13)

And the question is, will the obtained key relation (8) remain valid beyond
this approximation? To answer this question, in this work we will consider a
more general case when the Ricci tensor is not equal to zero, but is
proportional to the cosmological constant

$\displaystyle\displaystyle R_{\mu\nu}=\Lambda g_{\mu\nu}\,,$ (14)

which, on the one hand, (if we set the Weyl tensor equal to zero) corresponds
to the very important case of (anti-)de Sitter spaces, while from a technical
point of view will simplify the derivation, since it leads to additional
equations such as $\displaystyle\nabla_{\rho}R_{\mu\nu}=0$.

Similar hydrodynamic expansion of the stress-energy tensor for the medium in
the curved space with (14) was recently considered in [32] for particular case
with a zero Weyl tensor, where the anomaly (6) is absent, and accelerated
(non-rotating) medium. In this case, an alternative confirmation for the Unruh
effect in curved space-time was obtained, with a temperature depending
simultaneously on curvature and acceleration [33]. Below we will show that the
gradient expansion of the axial current also indicates the generalized Unruh
effect in curved space.

As a warm-up, we start by constructing a hydrodynamic expansion for the axial
current in a vortical and accelerated medium in the case when the Weyl tensor
is equal to zero.

The paper organized as follows. We begin in Section 2 by considering the
simplest case of curved space-time with a zero Weyl tensor. The relationship
between the KVE and the anomaly cannot be fixed in this case, however, the
relation between the cosmological constant and acceleration will be obtained.
In Section 3 we consider the more general case of Einstein manifolds with
nonzero Weyl tensor. In this case we obtain the KVE relation, and confirm the
duality between the cosmological constant and acceleration obtained in Section
2. In Section 4 we discuss some aspects and implications of our work and give
a brief qualitative description of the obtained effects. In the Conclusion we
summarize the results obtained.

##  2 Duality of the cosmological constant and acceleration

###  2.1 Riemann tensor and gradient expansion

We consider an uncharged non-dissipative fluid of massless fermions, moving
with four-velocity $\displaystyle u_{\mu}(x)$ and characterized by proper
temperature $\displaystyle T(x)$, placed in an external gravitational field
with the metric $\displaystyle g_{\mu\nu}(x)$. The system is assumed to be in
a state of global thermodynamic equilibrium [21], for which the inverse
temperature vector $\displaystyle\beta_{\mu}=\frac{u_{\mu}}{T}$ satisfies the
Killing equation

$\displaystyle\displaystyle\nabla_{\mu}\beta_{\nu}+\nabla_{\nu}\beta_{\mu}=0\,,$
(15)

which means that we work with the hydrodynamic $\displaystyle\beta$-frame.
Note that the condition (15) is not some artificial condition and is very
close to the known criteria of thermal equilibrium; we will consider this in
more detail in the Discussion section. In particular, due to (15), we obtain
for the second-order covariant derivative

$\displaystyle\displaystyle\nabla_{\mu}\nabla_{\nu}\beta_{\alpha}=-R^{\rho}_{\>\>\mu\nu\alpha}\beta_{\rho}\,.$
(16)

The anti-symmetric combination of covariant derivatives forms the thermal
vorticity tensor [34]

$\displaystyle\displaystyle\varpi_{\mu\nu}=-\frac{1}{2}\left(\nabla_{\mu}\beta_{\nu}-\nabla_{\nu}\beta_{\mu}\right)\,,$
(17)

which has one vector and one pseudo-vector component, corresponding to
(‘‘thermal’’) acceleration $\displaystyle\alpha_{\mu}$ and vorticity
$\displaystyle w_{\mu}$

$\displaystyle\displaystyle\alpha_{\mu}=\varpi_{\mu\nu}u^{\nu}\,,\quad
w_{\mu}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}u^{\nu}\varpi^{\alpha\beta}\,,\quad\varpi_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}w^{\alpha}u^{\beta}+\alpha_{\mu}u_{\nu}-\alpha_{\nu}u_{\mu}\,.$
(18)

Kinematic vorticity $\displaystyle\omega_{\mu}$ is proportional to the
‘‘thermal’’ vorticity $\displaystyle w_{\mu}$, and the kinematic acceleration
$\displaystyle a_{\mu}$ is proportional to the ‘‘thermal’’ acceleration in the
state of global equilibrium (15)

$\displaystyle\displaystyle
w_{\mu}=\frac{\omega_{\mu}}{T}\,,\quad\alpha_{\mu}=\frac{a_{\mu}}{T}\,.$ (19)

In the general case the Riemann tensor has the form

$\displaystyle\displaystyle
R_{\alpha\mu\beta\nu}=C_{\alpha\mu\beta\nu}-\frac{R}{6}\left(g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\nu}g_{\mu\beta}\right)+\frac{1}{2}\left(R_{\alpha\beta}g_{\mu\nu}+R_{\mu\nu}g_{\alpha\beta}-R_{\alpha\nu}g_{\beta\mu}-R_{\beta\mu}g_{\alpha\nu}\right)\,,$
(20)

where $\displaystyle C_{\alpha\mu\beta\nu}$ is the Weyl tensor.

Let us start with the simplest case when the Weyl tensor is zero and the Ricci
tensor is given by the cosmological constant

$\displaystyle\displaystyle C_{\alpha\mu\beta\nu}=0\,,\,R_{\mu\nu}=\Lambda
g_{\mu\nu}\,.$ (21)

Then the Riemann tensor can be decomposed into the components

$\displaystyle\displaystyle
R_{\alpha\mu\beta\nu}=\frac{\Lambda}{3}\left(g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\nu}g_{\mu\beta}\right)\,,$
(22)

which means that we consider de Sitter space-time ($\displaystyle\Lambda>0$)
or anti-de Sitter space-time ($\displaystyle\Lambda<0$) [35]. Our further
analysis takes place in a state of global equilibrium (15), so we choose
static patch of dS space-time, therefore having a time-like Killing vector.

Now we construct the hydrodynamic gradient expansion for the current. The only
possibility to add space-time curvature to the current expression is the
scalar curvature $\displaystyle R$, which is of the second order in gradients.
Then to make this term pseudovector one has to multiply it by the axial
quantity $\displaystyle w_{\mu}$, which also contains a derivative and
therefore the axial current will be of the third order. Thus, the curvature
effects arise only in the third order in gradients. Since the Ricci tensor is
proportional to the cosmological constant, this term will have the form
$\displaystyle\Lambda w_{\mu}$

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\xi_{1}(T)w^{2}w_{\mu}+\xi_{2}(T)\alpha^{2}w_{\mu}+\xi_{3}(T)(\alpha
w)\alpha_{\mu}+\xi_{\Lambda}(T)\Lambda w_{\mu}\,.$ (23)

We will denote the coefficient before the term with the cosmological constant
as $\displaystyle\xi_{\Lambda}(T)$. The unknown coefficients
$\displaystyle\xi_{n}(T)$ and $\displaystyle\xi_{\Lambda}(T)$ depend on the
proper temperature $\displaystyle T$.

###  2.2 Axial current divergence

Due to the symmetry of the chosen Riemann tensor (the Weyl tensor is zero),
there is no gravitational anomaly, and the axial current in this case is
conserved

$\displaystyle\displaystyle\nabla_{\mu}j^{\mu}_{A}=0\,.$ (24)

The covariant derivatives of the kinematic variables can be found from (15),
(16) and the condition $\displaystyle u^{\mu}u_{\mu}=1$

$\displaystyle\displaystyle\begin{cases}\nabla_{\mu}T=T^{2}\alpha_{\mu}\,,\\\
\nabla_{\mu}w_{\nu}=T\left(-g_{\mu\nu}(\alpha
w)+\alpha_{\mu}w_{\nu}\right)\,,\\\
\nabla_{\mu}\alpha_{\nu}=T\left(w^{2}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)-\alpha^{2}u_{\mu}u_{\nu}-w_{\mu}w_{\nu}-u_{\mu}\eta_{\nu}-u_{\nu}\eta_{\mu}\right)-\frac{1}{T}\frac{\Lambda}{3}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)\,,\end{cases}$
(25)

where
$\displaystyle\eta_{\mu}=\epsilon_{\mu\nu\rho\sigma}u^{\nu}w^{\rho}\alpha^{\sigma}$.

An important consequence of considering the system in global thermodynamic
equilibrium is that hydrodynamics is defined only in terms of the first
gradients of velocity (acceleration and vorticity), and all higher derivatives
are expressed through the lower ones. This is clearly seen from the system
(25). Now, substituting (23) into (24), we obtain an expression for the axial
current divergence

$\displaystyle\displaystyle\nabla_{\mu}j^{\mu(3)}_{A}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle(\alpha
w)w^{2}\left(T^{2}\xi_{1}^{\prime}-3T\xi_{1}+2T\xi_{3}\right)+(\alpha
w)\alpha^{2}\left(T^{2}\xi_{2}^{\prime}-3T\xi_{2}+T^{2}\xi_{3}^{\prime}-T\xi_{3}\right)$
(26) $\displaystyle\displaystyle+\Lambda(\alpha
w)\left(T^{2}\xi_{\Lambda}^{\prime}-3T\xi_{\Lambda}-\frac{4}{3T}\xi_{3}-\frac{2}{3T}\xi_{2}\right)=0\,.$

The pseudo-scalar terms before each bracket in this expression are
independent, so we can equate all expressions in brackets to zero and obtain a
system of differential equations

$\displaystyle\displaystyle\begin{cases}T^{2}\xi_{1}^{\prime}-3T\xi_{1}+2T\xi_{3}=0\,,\\\
T^{2}\xi_{2}^{\prime}-3T\xi_{2}+T^{2}\xi_{3}^{\prime}-T\xi_{3}=0\,,\\\
T^{2}\xi_{\Lambda}^{\prime}-3T\xi_{\Lambda}-\frac{4}{3T}\xi_{3}-\frac{2}{3T}\xi_{2}=0\,.\end{cases}$
(27)

Based on dimensional analysis (in a massless theory without boundary
conditions, etc., temperature $\displaystyle T$ is the only dimensional
parameter), we move on to the dimensionless coefficients
$\displaystyle\lambda$

$\displaystyle\displaystyle\xi_{1}=T^{3}\lambda_{1},\quad\xi_{2}=T^{3}\lambda_{2},\quad\xi_{3}=T^{3}\lambda_{3},\quad\xi_{\Lambda}=T\lambda_{\Lambda}\,.$
(28)

And left with the system ($\displaystyle\lambda_{1}$ remains arbitrary)

$\begin{cases}\lambda_{3}=0\,,\\\
\lambda_{\Lambda}=-\frac{\lambda_{2}}{3}\,.\end{cases}$ (29)

The first equality in (29) provides the conservation of the axial current in
flat space-time in the absence of external fields [26].

The phenomena analyzed in this paper, from the point of view of conservation
laws, can also be derived directly from the equilibrium perturbation theory
[34, 26], where they are determined by the effective interaction with boost
operator (in the case of acceleration) and with operator of angular momentum
(in the case of vorticity). In particular, in this way the coefficients
$\displaystyle\lambda_{1}$, $\displaystyle\lambda_{2}$ and
$\displaystyle\lambda_{3}$ were obtained in flat space-time for massless spin
1/2 field (9). The key is the second of the relations (29). The factor
$\displaystyle\lambda_{\Lambda}$ cannot be obtained in the same way from the
density operator in flat space-time, but we can obtain it simply using the
duality (29) from the flat space transport coefficient
$\displaystyle\lambda_{2}$. In particular, using (9) and (29), we obtain the
following expression for the current of massless Dirac fields

$\displaystyle\displaystyle j_{A}^{\mu(3)}=\left(\
-\frac{\omega^{2}}{24\pi^{2}}-\frac{a^{2}}{8\pi^{2}}+\frac{R}{96\pi^{2}}\right)\omega^{\mu}\,,$
(30)

where we used $\displaystyle\Lambda=\frac{R}{4}$. Thus, we obtained a new term
in the axial current proportional to the scalar curvature in the hydrodynamic
gradient expansion. This current has the same conductivity as the acceleration
term, according to (29), which adds an additional element to the
hydrodynamic/gravitational duality. We will also return to this issue in the
Discussion section. The same expression for the axial current in AdS space-
time was obtained by another method in [36].

##  3 Anomalous transport: gravitational anomaly and cosmological constant

###  3.1 Riemann tensor and gradient expansion

In this section we consider the more general case of space-time with a non-
zero Weyl tensor and generalize the derivation of [9] to the case with a non-
zero Ricci tensor proportional to the cosmological constant. Our starting
point will be a general decomposition for the Riemann tensor [9]

$\displaystyle\displaystyle R_{\mu\nu\alpha\beta}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
u_{\mu}u_{\alpha}A_{\nu\beta}+u_{\nu}u_{\beta}A_{\mu\alpha}-u_{\nu}u_{\alpha}A_{\mu\beta}-u_{\mu}u_{\beta}A_{\nu\alpha}+\epsilon_{\mu\nu\lambda\rho}u^{\rho}\left(u_{\alpha}B^{\lambda}_{\>\beta}-u_{\beta}B^{\lambda}_{\>\alpha}\right)+$
(31)
$\displaystyle\displaystyle+\epsilon_{\alpha\beta\lambda\rho}u^{\rho}\left(u_{\mu}B^{\lambda}_{\>\nu}-u_{\nu}B^{\lambda}_{\>\mu}\right)+\epsilon_{\mu\nu\lambda\rho}\epsilon_{\alpha\beta\eta\sigma}u^{\rho}u^{\sigma}C^{\lambda\eta}\,.$

The tensors $\displaystyle A_{\mu\nu}$, $\displaystyle B_{\mu\nu}$ and
$\displaystyle C_{\mu\nu}$ are covariant generalizations of similar three-
dimensional tensors from [37]

$\begin{cases}A_{\mu\nu}=u^{\alpha}u^{\beta}R_{\alpha\mu\beta\nu}\,,\vspace{0.2
cm}\\\
B_{\mu\nu}=\frac{1}{2}\epsilon_{\alpha\mu\eta\rho}u^{\alpha}u^{\beta}R_{\quad\beta\nu}^{\eta\rho}\,,\vspace{0.2
cm}\\\
C_{\mu\nu}=\frac{1}{4}\epsilon_{\alpha\mu\eta\rho}\epsilon_{\beta\nu\lambda\gamma}u^{\alpha}u^{\beta}R^{\eta\rho\lambda\gamma}\,.\end{cases}$
(32)

These tensors have properties

$\displaystyle\displaystyle A_{\mu\nu}=A_{\nu\mu},\quad
C_{\mu\nu}=C_{\nu\mu}\,,\quad B^{\mu}_{\mu}=0\,,\quad
A_{\mu\nu}u^{\nu}=C_{\mu\nu}u^{\nu}=B_{\mu\nu}u^{\nu}=B_{\nu\mu}u^{\nu}=0\,.$
(33)

The tensors $\displaystyle A_{\mu\nu}$ and $\displaystyle C_{\mu\nu}$ can be
decomposed into a traceless and a trace parts, and the pseudotensor
$\displaystyle B_{\mu\nu}$ can be decomposed into symmetric and anti-symmetric
parts

$\displaystyle\displaystyle A_{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{A}_{\mu\nu}+\frac{A}{3}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)\,,$
$\displaystyle\displaystyle C_{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{C}_{\mu\nu}+\frac{C}{3}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)\,,$
$\displaystyle\displaystyle B_{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle B^{s}_{\mu\nu}+B^{a}_{\mu\nu}\,,$ (34)

accordingly, we have

$\displaystyle\displaystyle\widetilde{A}_{\mu}^{\mu}=\widetilde{C}_{\mu}^{\mu}=0\,,\quad\widetilde{A}_{\mu\nu}u^{\nu}=\widetilde{C}_{\mu\nu}u^{\nu}=B^{s}_{\mu\nu}u^{\nu}=B^{a}_{\mu\nu}u^{\nu}=0\,,$
$\displaystyle\displaystyle A_{\mu}^{\mu}=A\,,\quad C_{\mu}^{\mu}=C\,,\quad
B^{s}_{\mu\nu}=B^{s}_{\nu\mu}\,,\quad B^{a}_{\mu\nu}=-B^{a}_{\nu\mu}\,.$ (35)

Ricci tensor has the form

$\displaystyle\displaystyle
R_{\mu\nu}=g^{\alpha\beta}R_{\alpha\mu\beta\nu}=u_{\mu}u_{\nu}\left(A+C\right)+\left(A_{\mu\nu}+C_{\mu\nu}\right)-Cg_{\mu\nu}+B^{\lambda\alpha}u^{\rho}\left(u_{\nu}\epsilon_{\alpha\lambda\mu\rho}+u_{\mu}\epsilon_{\alpha\lambda\nu\rho}\right)\,.$
(36)

To illustrate the new elements, we now consider two cases: the Ricci tensor is
equal to zero (the case considered in [9]), and the Ricci tensor is equal to
the cosmological constant (present case).

1) Ricci tensor is zero, Weyl tensor is not zero: $\displaystyle
R_{\mu\nu}=0,\,C_{\mu\nu\alpha\beta}\neq 0$.

The condition $\displaystyle R_{\mu\nu}=0$ leads to additional properties for
the tensors $\displaystyle A_{\mu\nu},B_{\mu\nu}$ and $\displaystyle
C_{\mu\nu}$ such as (compare with [37])

$\displaystyle\displaystyle A_{\mu\nu}=-C_{\mu\nu}\,,\quad
A^{\mu}_{\mu}=0\,,\quad B_{\mu\nu}=B_{\nu\mu}\,.$ (37)

This means that

$\displaystyle\displaystyle A=C=0\,,\quad
B^{a}_{\mu\nu}=0\,,\quad\widetilde{A}_{\mu\nu}=-\widetilde{C}_{\mu\nu}\,.$
(38)

Then, writing the axial current in terms of all possible pseudo-vectors
arising in the third order of gradients, we obtain

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\xi_{1}(T)w^{2}w_{\mu}+\xi_{2}(T)\alpha^{2}w_{\mu}+\xi_{3}(T)(\alpha
w)\alpha_{\mu}+\xi_{4}(T)\widetilde{A}_{\mu\nu}w^{\nu}+\xi_{5}(T)B^{s}_{\mu\nu}\alpha^{\nu}\,.$
(39)

By considering covariant derivative of the (39) and considering that it should
lead to anomaly (6), a key relation (8), fixing duality with gravitational
anomaly was obtained [9].

We note that there could be some extra terms in the third order, including,
for example

$\displaystyle\displaystyle\nabla_{\lambda}\widetilde{A}_{\mu\nu}\,,\quad\nabla_{\lambda}B^{s}_{\mu\nu}\,,\quad\nabla_{\lambda}R_{\mu\nu\rho\sigma}\epsilon^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}\,,\quad\nabla_{\lambda}R_{\mu\nu\rho\sigma}\epsilon^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}u^{\alpha}u^{\beta}\,,\quad\alpha_{\lambda}R_{\mu\nu\rho\sigma}\epsilon^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}\,.$
(40)

But all the index contractions which makes these terms pseudovectors will be
equal zero because of Bianchi identities

$\displaystyle\displaystyle
R_{\mu\nu\rho\sigma}+R_{\mu\sigma\nu\rho}+R_{\mu\rho\sigma\nu}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0\,,$
$\displaystyle\displaystyle\nabla_{\lambda}R_{\mu\nu\rho\sigma}+\nabla_{\sigma}R_{\mu\nu\lambda\rho}+\nabla_{\rho}R_{\mu\nu\sigma\lambda}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 0\,,$ (41)

and the condition $\displaystyle\nabla_{\lambda}R_{\mu\nu}=0$. In the case
$\displaystyle R_{\mu\nu}=0$, considered in [9], this condition is satisfied.
However, it is also valid in the more general case $\displaystyle
R_{\mu\nu}=\Lambda g_{\mu\nu}$ due to the metricity condition
$\displaystyle\nabla_{\lambda}g_{\mu\nu}=0$. Also there is possibility to add
another type of terms like

$\displaystyle\displaystyle\left(\nabla_{\mu}\alpha_{\nu}\right)\omega^{\nu}\,,\quad\left(\nabla_{\mu}\omega_{\nu}\right)\alpha^{\nu}\,,\quad\left(\nabla\alpha\right)\omega^{\mu}\,,\quad\left(\nabla\omega\right)\alpha^{\mu}\,,$
(42)

but they can be transformed into terms already contained in the expression
(39) for the axial current.

2) Ricci tensor is given by the cosmological constant, Weyl tensor is not
zero: $\displaystyle R_{\mu\nu}=\Lambda g_{\mu\nu},C_{\mu\nu\alpha\beta}\neq
0$.

Now fixing $\displaystyle R_{\mu\nu}=\Lambda g_{\mu\nu}$ doesn’t affect the
condition $\displaystyle\nabla_{\lambda}R_{\mu\nu}=0$, and the conditions
mentioned above don’t change either and there will be no extra terms such as
(40) and (42) in the current. But the equations (37) and (38) are no longer
valid and the tensors have the form (34). Therefore, using (36), (35) and
$\displaystyle R_{\mu\nu}=\Lambda g_{\mu\nu}$ we can write the following
equations and define the constrains and symmetries of tensors $\displaystyle
A_{\mu\nu}$, $\displaystyle B_{\mu\nu}$ and $\displaystyle C_{\mu\nu}$

$\begin{cases}u^{\mu}u^{\nu}R_{\mu\nu}=\Lambda=\frac{R}{4}=A\,,\\\
g^{\mu\nu}R_{\mu\nu}=4\Lambda=4A=2(A-C)\Rightarrow A=-C\,,\\\
u^{\mu}R_{\mu\nu}=\Lambda
u_{\nu}=-Cu_{\nu}+B^{\lambda\alpha}\varepsilon_{\alpha\lambda\nu\rho}u^{\rho}\Rightarrow
B^{\lambda\alpha}=B^{\alpha\lambda}\,.\end{cases}$ (43)

Substituting these expressions into the Ricci tensor (36), we obtain the final
conditions for the tensors $\displaystyle A_{\mu\nu}$, $\displaystyle
B_{\mu\nu}$ and $\displaystyle C_{\mu\nu}$

$\begin{cases}A_{\mu\nu}=-C_{\mu\nu}\,,\\\
B_{\alpha\lambda}=B_{\lambda\alpha}=B^{s}_{\lambda\alpha}\quad(B^{a}_{\lambda\alpha}=0)\,,\\\
\Lambda=\frac{R}{4}=A=-C\,.\end{cases}$ (44)

As can be seen, the tensors $\displaystyle A_{\mu\nu}$ and $\displaystyle
C_{\mu\nu}$ are related, so we don’t need to add a term with tensor
$\displaystyle C_{\mu\nu}$ to the expression for the current. Since we have
separated the traceless part from the tensor $\displaystyle A_{\mu\nu}$, a new
term proportional to the cosmological constant should appear in the current,
since $\displaystyle
A_{\mu\nu}\omega^{\nu}=\widetilde{A}_{\mu\nu}\omega^{\nu}+\Lambda\omega_{\mu}$.
Therefore we have

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\xi_{1}(T)w^{2}w_{\mu}+\xi_{2}(T)\alpha^{2}w_{\mu}+\xi_{3}(T)(\alpha
w)\alpha_{\mu}+\xi_{4}(T)\widetilde{A}_{\mu\nu}w^{\nu}+\xi_{5}(T)B^{s}_{\mu\nu}\alpha^{\nu}+\xi_{\Lambda}(T)\Lambda
w_{\mu}\,.$ (45)

###  3.2 Axial current divergence

Due to the quantum anomaly, the axial current isn’t conserved in the presence
of external gravitational fields. The gravitational chiral anomaly (6) can be
expressed through the tensors $\displaystyle\widetilde{A}_{\mu\nu}$ and
$\displaystyle B^{s}_{\mu\nu}$

$\displaystyle\displaystyle\nabla_{\mu}j_{A}^{\mu}=\mathcal{N}\epsilon^{\mu\nu\alpha\beta}R_{\mu\nu\lambda\rho}R_{\alpha\beta}^{\quad\lambda\rho}=16\mathcal{N}\left(A^{\mu\nu}-C^{\mu\nu}\right)B^{s}_{\mu\nu}=32\mathcal{N}\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\,.$
(46)

In the case under consideration with a nonzero Weyl tensor, instead of (25),
we obtain

$\begin{cases}\nabla_{\mu}T=T^{2}\alpha_{\mu}\,,\\\
\nabla_{\mu}u_{\nu}=T\left(\epsilon_{\mu\nu\alpha\beta}u^{\alpha}w^{\beta}+u_{\mu}\alpha_{\nu}\right)\,,\\\
\nabla_{\mu}w_{\nu}=T\left(-g_{\mu\nu}(\alpha
w)+\alpha_{\mu}w_{\nu}\right)+\frac{1}{T}B^{s}_{\mu\nu}\,,\\\
\nabla_{\mu}\alpha_{\nu}=T\left(w^{2}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)-\alpha^{2}u_{\mu}u_{\nu}-w_{\mu}w_{\nu}-u_{\mu}\eta_{\nu}-u_{\nu}\eta_{\mu}\right)-\frac{1}{T}\left(\widetilde{A}_{\mu\nu}+\frac{\Lambda}{3}\left(g_{\mu\nu}-u_{\mu}u_{\nu}\right)\right)\,,\\\
\nabla^{\mu}\left(\widetilde{A}_{\mu\nu}w^{\nu}\right)=-3TB^{s}_{\mu\nu}w^{\mu}w^{\nu}+\frac{1}{T}\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\,,\\\
\nabla^{\mu}\left(B^{s}_{\mu\nu}\alpha^{\nu}\right)=3T\widetilde{A}_{\mu\nu}w^{\mu}\alpha^{\nu}-\frac{1}{T}\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}-TB^{s}_{\mu\nu}w^{\mu}w^{\nu}-TB^{s}_{\mu\nu}\alpha^{\mu}\alpha^{\nu}\,.\end{cases}$
(47)

Now for clarity, let us write out the derivatives of each term of the axial
current (45)

$\begin{cases}\nabla_{\mu}\left(\xi_{1}w^{2}w^{\mu}\right)=(\alpha
w)w^{2}\left(T^{2}\xi_{1}^{\prime}-3T\xi_{1}\right)+B^{s}_{\mu\nu}w^{\mu}w^{\nu}\left(\frac{2}{T}\xi_{1}\right)\,,\\\
\nabla_{\mu}\left(\xi_{2}\alpha^{2}w^{\mu}\right)=(\alpha
w)\alpha^{2}\left(T^{2}\xi_{2}^{\prime}-3T\xi_{2}\right)+\widetilde{A}_{\mu\nu}\alpha^{\mu}w^{\nu}\left(-\frac{2}{T}\xi_{2}\right)+\Lambda(\alpha
w)\left(-\frac{2}{3T}\xi_{2}\right)\,,\\\ \nabla_{\mu}\left(\xi_{3}(\alpha
w)\alpha^{\mu}\right)=(\alpha
w)\Bigg{[}\alpha^{2}\left(T^{2}\xi_{3}^{\prime}-T\xi_{3}\right)+w^{2}\left(2T\xi_{3}\right)+\Lambda\left(-\frac{4}{3T}\xi_{3}\right)\Bigg{]}+\widetilde{A}_{\mu\nu}\alpha^{\mu}w^{\nu}\left(-\frac{1}{T}\xi_{3}\right)+B^{s}_{\mu\nu}\alpha^{\mu}\alpha^{\nu}\left(\frac{1}{T}\xi_{3}\right)\,,\\\
\nabla^{\mu}\left(\xi_{4}\widetilde{A}_{\mu\nu}w^{\nu}\right)=\widetilde{A}_{\mu\nu}\alpha^{\mu}w^{\nu}\left(\xi_{4}^{\prime}T^{2}\right)+B^{s}_{\mu\nu}w^{\mu}w^{\nu}\left(-3T\xi_{4}\right)+\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\left(\frac{1}{T}\xi_{4}\right)\,,\\\
\nabla^{\mu}\left(\xi_{5}B^{s}_{\mu\nu}\alpha^{\nu}\right)=B^{s}_{\mu\nu}\alpha^{\mu}\alpha^{\nu}\left(T^{2}\xi_{5}^{\prime}-T\xi_{5}\right)+B^{s}_{\mu\nu}w^{\mu}w^{\nu}\left(-T\xi_{5}\right)+\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\left(-\frac{1}{T}\xi_{5}\right)+\widetilde{A}_{\mu\nu}\alpha^{\mu}w^{\nu}\left(3T\xi_{5}\right)\,,\\\
\nabla^{\mu}\left(\xi_{\Lambda}\Lambda w_{\mu}\right)=\Lambda(\alpha
w)\left(T^{2}\xi_{\Lambda}^{\prime}-3T\xi_{\Lambda}\right)\,.\end{cases}$ (48)

Combining (46) and (48) we obtain

$\displaystyle\displaystyle\nabla_{\mu}j^{\mu(3)}_{A}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle(\alpha
w)w^{2}\left(T^{2}\xi_{1}^{\prime}-3T\xi_{1}+2T\xi_{3}\right)+(\alpha
w)\alpha^{2}\left(T^{2}\xi_{2}^{\prime}-3T\xi_{2}+T^{2}\xi_{3}^{\prime}-T\xi_{3}\right)+\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\left(\frac{1}{T}\xi_{4}-\frac{1}{T}\xi_{5}\right)+$
(49)
$\displaystyle\displaystyle+\widetilde{A}_{\mu\nu}\alpha^{\mu}w^{\nu}\left(-\frac{2}{T}\xi_{2}-\frac{1}{T}\xi_{3}+3T\xi_{5}+T^{2}\xi_{4}^{\prime}\right)+B^{s}_{\mu\nu}w^{\mu}w^{\nu}\left(\frac{2}{T}\xi_{1}-3T\xi_{4}-T\xi_{5}\right)+$
$\displaystyle\displaystyle+B^{s}_{\mu\nu}\alpha^{\mu}\alpha^{\nu}\left(T^{2}\xi_{5}^{\prime}-T\xi_{5}+\frac{1}{T}\xi_{3}\right)+\Lambda(\alpha
w)\left(T^{2}\xi_{\Lambda}^{\prime}-3T\xi_{\Lambda}-\frac{4}{3T}\xi_{3}-\frac{2}{3T}\xi_{2}\right)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
32\mathcal{N}\widetilde{A}^{\mu\nu}B^{s}_{\mu\nu}\,.$

As in (26) all pseudo-scalars are independent, so we can write a system of
equations

$\begin{cases}T^{2}\xi_{1}^{\prime}-3T\xi_{1}+2T\xi_{3}=0\,,\\\
T^{2}\xi_{2}^{\prime}-3T\xi_{2}+T^{2}\xi_{3}^{\prime}-T\xi_{3}=0\,,\\\
-\frac{2}{T}\xi_{2}-\frac{1}{T}\xi_{3}+3T\xi_{5}+T^{2}\xi_{4}^{\prime}=0\,,\\\
\frac{2}{T}\xi_{1}-3T\xi_{4}-T\xi_{5}=0\,,\\\
T^{2}\xi_{5}^{\prime}-T\xi_{5}+\frac{1}{T}\xi_{3}=0\,,\\\
\frac{1}{T}\xi_{4}-\frac{1}{T}\xi_{5}=32\mathcal{N}\,,\\\
T^{2}\xi_{\Lambda}^{\prime}-3T\xi_{\Lambda}-\frac{4}{3T}\xi_{3}-\frac{2}{3T}\xi_{2}=0\,.\par\end{cases}$
(50)

By redefining dimensionless constants as we did it in (28), we obtain

$\displaystyle\displaystyle\xi_{1}=T^{3}\lambda_{1}\,,\quad\xi_{2}=T^{3}\lambda_{2}\,,\quad\xi_{3}=T^{3}\lambda_{3}\,,\quad\xi_{4}=T\lambda_{4}\,,\quad\xi_{5}=T\lambda_{5}\,,\quad\xi_{\Lambda}=T\lambda_{\Lambda}\,.$
(51)

Then

$\begin{cases}\lambda_{3}=0\,,\\\
-2\lambda_{2}+3\lambda_{5}+\lambda_{4}=0\,,\\\
2\lambda_{1}-3\lambda_{4}-\lambda_{5}=0\,,\\\
\lambda_{4}-\lambda_{5}=32\mathcal{N}\,,\\\
-2\lambda_{\Lambda}-\frac{2}{3}\lambda_{2}=0\,.\\\ \end{cases}$ (52)

From here we can finally write the solution

$\displaystyle\displaystyle\lambda_{1}-\lambda_{2}=32\mathcal{N},\quad\lambda_{4}=\frac{\lambda_{1}}{2}+8\mathcal{N},\quad\lambda_{5}=\frac{\lambda_{1}}{2}-24\mathcal{N},\quad\lambda_{\Lambda}=-\frac{\lambda_{2}}{3}\,.$
(53)

The first three equalities in (53) are consistent with the work [9], while the
last one tells us that the coefficients between the cosmological constant term
the squared acceleration term square are related as in the simpler case with
the zero Weyl tensor (29). Thus, if we know the gravitational chiral anomaly,
then the current in third order in gradients is determined to within one
arbitrary coefficient. It is convenient to choose $\displaystyle\lambda_{1}$
as this arbitrary coefficient, since it can be calculated in flat space-time.
As a result, knowing only one transport coefficient in flat space-time, we
complete the full expression for the current in curved space in the third
order. As a result, we have for the general case

$\displaystyle\displaystyle j^{A(3)}_{\mu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(\lambda_{1}\omega^{2}+(32\mathcal{N}-\lambda_{1})\left[\frac{R}{12}-a^{2}\right]\right)\omega_{\mu}+\left(8\mathcal{N}+\frac{\lambda_{1}}{2}\right)R_{\alpha\mu\beta\nu}u^{\alpha}u^{\beta}\omega^{\nu}+$
(54)
$\displaystyle\displaystyle+\left(\frac{\lambda_{1}}{4}-12\mathcal{N}\right)R^{\eta\rho}_{\quad\beta\nu}\epsilon_{\alpha\mu\eta\rho}u^{\alpha}u^{\beta}a^{\nu}\,,$

where we moved from the tensors $\displaystyle A_{\mu\nu}$ and $\displaystyle
B_{\mu\nu}$ to the Riemann tensor by means of (32).

Formula (54) is our main result. It describes the axial current of massless
particles in a vortical and accelerated medium in global thermodynamic
equilibrium in curved space with a nonzero Weyl tensor, and the Ricci tensor
of the form (14). One can also assume that by adding also the linear order
(2), the formula will be accurate (all higher orders will be zero) at
sufficiently high temperatures111This polynomiality is a fairly rare and
interesting phenomenon for quantum field theory. The absence of higher order
terms has been directly shown in a number of approaches for specific cases
[38, 27, 39, 23] and is characteristic only for the massless theory. Formally,
the absence of higher order terms corresponds to the absence of negative
powers of temperature. At low temperatures, however, the phase transitions
associated with the singularity of modes at the horizon should be expected
[40] at least for half integer spins, which will modify the obtained
formulas..

In the particular case of massless Dirac fields, we know the anomaly
$\displaystyle\mathcal{N}=1/(384\pi^{2})$ (10) and the coefficient
$\displaystyle\lambda_{1}=-1/(24\pi^{2})$ (9). As a result, the current will
have the form

$\displaystyle\displaystyle
j_{\mu}^{A(3)}=\left(-\frac{\omega^{2}}{24\pi^{2}}+\frac{1}{8\pi^{2}}\left[\frac{R}{12}-a^{2}\right]\right)\omega_{\mu}-\frac{1}{24\pi^{2}}R^{\eta\rho}_{\quad\beta\nu}\epsilon_{\alpha\mu\eta\rho}u^{\alpha}u^{\beta}a^{\nu}\,,$
(55)

which generalizes the formulas (9) and (30).

##  4 Discussion

###  4.1 Thermodynamic equilibrium

From the very beginning we used the assumption that the system is in the so-
called global thermodynamic equilibrium, which corresponds to the entropy
maximum (as well as invariance of the density operator with respect to the
choice of hypersurface) [21]. Beyond this equilibrium our results may be
unfair (or give only zero order contribution). We will discuss the global
thermodynamic equilibrium in more detail.

A consequence of this equilibrium is that the inverse temperature vector
should be the Killing vector (15). Since its square
$\displaystyle\beta_{\mu}\beta^{\mu}=T^{-2}>0$ is related to the temperature,
this vector should be time-like and also future-directed, since
$\displaystyle\beta_{0}=u_{0}/T>0$ [41]. The spaces containing timelike
Killing vector are well known stationary spaces. In particular, any static
space-time whose components do not depend on time is stationary. Thus, our
derivation, is valid only for stationary spaces. In Section 2 we considered
the case of (A)dS spaces. For (A)dS space we can choose a static metric

$\displaystyle\displaystyle
ds^{2}=\left(1-\frac{\Lambda}{3}r^{2}\right)dt^{2}-\frac{1}{1-\frac{\Lambda}{3}r^{2}}dr^{2}-r^{2}\left(d\Theta^{2}+\sin^{2}\Theta\,d\phi^{2}\right)\,.$
(56)

where the positive sign $\displaystyle\Lambda>0$ corresponds to the metrics of
a static patch in de Sitter space-time, and $\displaystyle\Lambda<0$
corresponds to static coordinates of the anti-de Sitter space-time.

Not all types of space-times are stationary and, accordingly, global
equilibrium is not always possible. In particular, it is not obvious whether
global equilibrium is possible in the expanding Universe, since, for example
the Friedman metrics clearly depends on time (but local thermodynamic
equilibrium should be possible [42]).

The conditions of global equilibrium (15) are consistent with other known
criteria of thermodynamic equilibrium. In particular, there is a well-known
Tolman-Ehrenfest thermodynamic equilibrium criterion [43, 44], according to
which the temperature in space with the metric $\displaystyle
g_{\mu\nu}(\mathbf{x})$ is distributed according to the law

$\displaystyle\displaystyle
T(\mathbf{x})=\frac{T_{0}}{\sqrt{g_{00}(\mathbf{x})}}\,,$ (57)

which for stationary space-times can be written in covariant form [45, 44]

$\displaystyle\displaystyle
T(\mathbf{x})=\frac{T_{0}}{\sqrt{g_{\mu\nu}(\mathbf{x})\xi^{\mu}(\mathbf{\mathbf{x}})\xi^{\nu}(\mathbf{x})}}\,,$
(58)

where $\displaystyle\xi^{\mu}$ is a dimensionless Killing vector. It is clear,
that (58) is in full agreement with the global equilibrium, according to which
the temperature is also defined in exactly the same way by the Killing vector
as $\displaystyle\beta_{\mu}\beta^{\mu}\equiv
g_{\mu\nu}\beta^{\mu}\beta^{\nu}=T^{-2}$.

Also, it is easy to see that in global equilibrium (15) the dissipative
contribution to the stress-energy tensor vanishes, which also corresponds to
equilibrium [46]

$\displaystyle\displaystyle T_{\mu\nu}^{\text{d}iss}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta(\nabla_{\mu}u_{\nu}+\nabla_{\nu}u_{\mu}-u_{\mu}u^{\alpha}\nabla_{\alpha}u_{\nu}-u_{\nu}u^{\alpha}\nabla_{\alpha}u_{\mu})-\left(\zeta-\frac{2}{3}\eta\right)\nabla^{\alpha}u_{\alpha}(g_{\mu\nu}-u_{\mu}u_{\nu})$
(59) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta\left[T(\nabla_{\mu}\beta_{\nu}+\nabla_{\nu}\beta_{\mu})+u_{\mu}\left(\frac{\nabla_{\nu}T}{T}-a_{\nu}\right)+u_{\nu}\left(\frac{\nabla_{\mu}T}{T}-a_{\mu}\right)\right]+$
$\displaystyle\displaystyle+\left(\zeta-\frac{2}{3}\eta\right)\left(T\nabla_{\alpha}\beta^{\alpha}+\frac{\nabla_{\alpha}T}{T}u^{\alpha}\right)(g_{\mu\nu}-u_{\mu}u_{\nu})=0\,.$

The same, of course, applies to dissipative effects in a current.

Thus, the conditions of global equilibrium look like a fairly general and
natural assumption about the properties of the system. Note also that,
generally speaking, equilibrium can lead to additional restrictions on the
external field, as was in the case with the electromagnetic field [19, 20].
Such limitations are not significant in this paper, but we hope to explore
them in the future.

###  4.2 Qualitative description

Current (54) (as well as (55)) does not explicitly depend on the properties of
the medium, such as temperature or chemical potential, being poorly kinematic.
This distinguishes it from CVE (2) or CME [16]. Therefore, they are associated
with quantum vacuum fluctuations. Note that transport effects of vacuum kind
are known (for example in external electromagnetic fields [47] or there is
well-known vacuum contribution to the stress-energy tensor in accelerated
frame [48]).

This approach allows us to obtain an approximate qualitative picture for these
effects. Consider for example the term with scalar curvature and acceleration
in the current (55)

$\displaystyle\displaystyle
j_{\mu}^{A}=\frac{1}{2}\frac{|a|^{2}+R/12}{(2\pi)^{2}}\omega_{\mu}\,,$ (60)

To understand at a qualitative level the meaning of (60), it is enough to take
into account two phenomena. Firstly, according to the famous Unruh effect
[49], thermal radiation occurs in an accelerated medium with the Unruh
temperature

$\displaystyle\displaystyle T_{U}=\frac{|a|}{2\pi}\,.$ (61)

Similarly, in de Sitter space the vacuum state is also characterized by a
certain temperature [50]

$\displaystyle\displaystyle T_{R}=\frac{\sqrt{R/12}}{2\pi}\,.$ (62)

Temperatures (61) and (62) can be combined [33, 32] if we take into account,
that the de Sitter space is embedded in a five-dimensional flat space. In this
case, the radiation temperature is determined by the five-dimensional
acceleration

$\displaystyle\displaystyle
T_{aR}=\frac{|a_{5}|}{2\pi}=\frac{\sqrt{|a|^{2}+R/12}}{2\pi}\,,$ (63)

which is also true for anti-de Sitter space.

On the other hand, we need to recall the well-known CVE (9), according to
which an axial current

$\displaystyle\displaystyle j^{A}_{\mu}=\frac{T^{2}}{6}\omega_{\mu}\,,$ (64)

arises in a rotating (in flat space) medium at finite temperature. CVE can be
associated with the alignment of spins in a magnetic field, taking into
account that rotation is similar to a magnetic field. Combining (64) and (63),
that is, considering radiation in a vacuum as a medium with temperature (63),
we automatically obtain that a current should arise, which, up to the overall
coefficient, coincides with (60).

Of course, the picture described is rough and is more intended to give an
intuitive qualitative understanding of the effects occurring. And to be more
precise, it is necessary to take into account all the peculiarities of the
space with a horizon (which is not quite equivalent to a thermal medium [51]),
which, in particular, is the reason for the difference in the overall
coefficient.

###  4.3 Duality between hydrodynamics and gravity

The result we obtained continues to develop the entropy approach to gravity,
which explains the effects of general relativity using the statistical
properties [52]. Actually, the possibility of such a dual description suggests
that the effects in gravitational fields should be accompanied by similar
partner-phenomena within statistical approaches.

There are a number of examples where such a relationship has been revealed,
and the effects of gravity appear essentially in the limit of zero Newton’s
constant or zero curvature. This includes, for example, the well-known Unruh
effect [49] \- despite the fact that the accelerated system has zero
curvature, it, like a black hole, is filled with thermal radiation bath. In
this regard, we note a number of works [53, 54, 41], where the Unruh effect
was analyzed from the point of view of statistical approaches. Another example
is the statistical derivation of the Einstein equation [55], in which, for
example, Newton’s constant itself arises as the inverse entropy.

Our result adds more elements to this duality. Indeed, on the one hand, there
is an essentially gravitational effect - a gravitational chiral anomaly (6).
Let us emphasize that this anomaly occurs only when space has non-zero
curvature $\displaystyle R_{\mu\nu\alpha\beta}\neq 0$, while upon transition
to a non-inertial system, the curvature remains equal to zero $\displaystyle
R_{\mu\nu\alpha\beta}=0$. However, we see, according to (53), that this
anomaly is ‘‘hidden’’ in the properties of the kinematic current (7) in an
accelerated and vortical medium.

We also note the relationship (29) between the current $\displaystyle
j^{A}_{\mu}\sim a^{2}\omega_{\mu}$ in an accelerated vortical medium, and
$\displaystyle j^{A}_{\mu}\sim R\omega_{\mu}$, in a vortical medium in the
presence of a gravitational field, which indicates the similarity between
scalar curvature $\displaystyle R$ and acceleration $\displaystyle a^{2}$.

The statements made above can be made more concrete, if we remember both
theories, within the framework of which the gravitational anomaly (6) and
current (55) can be obtained. The anomaly is associated with fundamental
gravitational interaction. In the lowest order, by the definition, the
corresponding vertex contains the stress-energy tensor

$\displaystyle\displaystyle\delta S=-\frac{1}{2}\int d^{4}x\,\delta
g_{\mu\nu}T^{\mu\nu}\,.$ (65)

On the other hand, current (7) can be obtained using the statistical density
operator [34, 26]

$\displaystyle\displaystyle\hat{\rho}=\frac{1}{Z}\exp\Big{\\{}-\beta_{\mu}\hat{P}^{\mu}+\zeta\hat{Q}-\alpha_{\mu}\hat{K}^{\mu}_{x}-w_{\mu}\hat{J}^{\mu}_{x}\Big{\\}}\,,$
(66)

where $\displaystyle\zeta=\mu/T$, $\displaystyle\hat{P}^{\mu}$ is the four-
momentum operator and $\displaystyle\hat{Q}$ is the charge operator. The first
two terms correspond to the usual grand canonical distribution with , while
the last two terms contain effective vertices, which describe statistically
induced interaction, with the angular momentum operator
$\displaystyle\hat{J}^{\mu}_{x}$, associated with vorticity $\displaystyle
w_{\mu}=\omega_{\mu}/T$ and the boost operator
$\displaystyle\hat{K}^{\mu}_{x}$, associated with acceleration
$\displaystyle\alpha_{\mu}=a_{\mu}/T$ (the operators are shifted by the vector
$\displaystyle x_{\mu}$). Moreover, (66) is exact for global thermodynamic
equilibrium (15), which further illustrates the statistical nature of (66).
Current (7) can be obtained within the framework of the equilibrium
perturbation theory [34]. For example, the term
$\displaystyle\lambda_{1}\omega^{2}\omega_{\mu}$ in (7) is given by a
correlator of the form [26, 29]

$\displaystyle\displaystyle\lambda_{1}=-\frac{1}{6}\int_{0}^{|\beta|}d\tau_{x}d\tau_{y}d\tau_{z}\langle
T_{\tau}\hat{J}^{3}_{-i\tau_{x}}\hat{J}^{3}_{-i\tau_{y}}\hat{J}^{3}_{-i\tau_{z}}\hat{j}_{A}^{3}(0)\rangle_{\beta(x),c}\,,$
(67)

where the operators are shifted along the axis of the imaginary time along
which integration is carried out from 0 to the inverse temperature
$\displaystyle|\beta|=1/T$, and the index
$\displaystyle\langle...\rangle_{\beta(x),c}$ means that the averaging is
taken using grand canonical distribution (i.e. using (66) with
$\displaystyle\alpha_{\mu}=w_{\mu}=0$). Note that (66) and (67) are written in
flat space-time. Moreover, they are written in ordinary Minkowski metrics.
However, as we have shown in the main part of our work, the effects of the
types (66) and (65) are related by the equations (53).

At the same time, our result can be interpreted as a confirmation of the
equivalence principle, according to which, acceleration effects are equivalent
to the effects in gravitational field. In most cases only the ‘‘linear order’’
is considered, that is, the equivalence at the level of acceleration and
Christoffel symbols. Our derivation demonstrates the relationship between
inertial effects and gravity at higher orders of metric and velocity
derivatives.

Thus, apparently, it can be argued that there is a class of theories that
reproduce gravitational properties, despite the fact that they themselves do
not contain gravity. Such theories can be considered as a limiting case when
Newton’s constant tends to zero. The role of such a limit is also evidenced by
recent results on the algebra of observables [56].

###  4.4 Quantum or classical?

The conclusion we presented is completely unrelated to quantum theory
calculations, which could create the illusion of the classical nature of the
current (54). Evidently, this is not the case, since hydrodynamics, like
thermodynamics, is a universal approach for describing both classical and
quantum phenomena (just remember black hole radiation or superfluidity).

The quantumness of (55) becomes evident, if we analyse the way it can be
directly obtained using the perturbation theory. In particular,
$\displaystyle\omega^{2}\omega_{\mu}$ and $\displaystyle a^{2}\omega_{\mu}$
terms are expressed through loop diagrams of the form (67), which, of course,
are elements of quantum theory.

##  5 Conclusion

In this paper we generalize the previously obtained kinematic vortical effect
[9] to the case of a nonzero Ricci tensor, proportional to the cosmological
constant. We obtained not only the conservation of a previous result for the
KVE, but also an entirely new term in the axial current, generated by the
scalar curvature in a vortical fluid. Analyzing the conservation of axial
current, we relate this new term with the transport coefficient of the purely
kinematic quadratic acceleration term.

We discover relationship between kinematic and gravitational effects,
expressed in a system of equations connecting transport coefficients of two
types, which indicates the gravitational-hydrodynamic duality and confirms the
equivalence principle in the higher orders of gradient expansion. This
relationship allows one by considering kinematic effects in a relativistic
fluid (for example, in a quark-gluon plasma, in which sufficiently large
vorticity and acceleration can be generated), to simulate quantum field
effects in the gravitational field in the absence of real strong gravity.

Acknowledgements
The authors are thankful to Jian-Hua Gao for stimulating discussions. G.P.
thanks Zuo-Tang Liang and Shandong University for the hospitality during the
visit to Qingdao. The work of G.P. and V.Z. was supported by Russian Science
Foundation Grant No 24-22-00124. The work of V.Z. is partially supported by
grant No. 0657-2020-0015 of the Ministry of Science and Higher Education of
Russia.

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