ЖЭТФ, 2022, том 162, вып. 5 (11), стр. 680-685

DIMENSIONLESS PHYSICS: CONTINUATION

G. E. Volovika,b*

^a Low Temperature Laboratory, Aalto University,

FI-00076 Aalto, Finland

^b Landau Institute for Theoretical Physics,

142432, Chernogolovka, Moscow region, Russia

Received August 21, 2022,

revised version August 21, 2022

Accepted for publication September 09, 2022

DOI: 10.31857/S0044451022110086

The metric g_μν has dimension n = 2, while the con-

EDN: KZDZIL

travariant metric g^μν has dimension n = -2:

Several approaches to quantum gravity (including

[g_μν ] =

, [gμν ] = [L]2 .

(4)

[L]²

the model of superplastic vacuum; Diakonov tetrads

emerging as the bilinear combinations of the fermionis

The tetrad determinant has dimension n = 4 in the

fields [1-4]; BF -theories of gravity; and effective acous-

4-dimensional spacetime and dimension n = N in the

tic metric [5, 6] ) suggest that in general relativity the

N-dimensional spacetime, where the dimensions of the

metric must have dimension 2, i.e. [gμν ] = 1/[L]2, ir-

metric elements are the same as in Eq.(4):

respective of the dimension of spacetime. This leads

to the "dimensionless physics" discussed in the review

[e] = [√-g ] =

(5)

[L]^N

paper [7]. We continue to exploit this issue.

Elasticity tetrads. The 3 + 1-dimensional vacuum

Eq.(5) makes the spacetime integration dimensionless:

crystal is the plastic (malleable) medium [8], described

∫

[

in terms of the elasticity tetrads [9-12]:

d^N x√-g] = [1] = 0 ,

(6)

∂X

E^aμ =

(1)

which leads to the dimensionless Lagrangian L:

∂x^μ

∫

[

]

[

where equations X^a(x) = 2πn_a are equations of the

d^N x√-g L] =

d^N x√-g]·[L] = [1]·[1] = [1] .

(deformed) crystal planes. The functions X^a play the

(7)

role of the geometric U(1) phases and are dimension-

Classical dynamics of particle is described by action:

less. The elasticity tetrads play the role of the gauge

∫

fields (translation gauge fields) and have the same di-

S = M ds,

(8)

mension 1 as the dimension of gauge fields:

[E^aμ] =

(2)

where with Eq.(3) the interval is dimensionless:

[L]

The dimension n of quantity A means [A] = [L]^-n,

ds² = g_μν dx^μdx^ν , [s²] =

· [L]² = [1] = 0 .

(9)

[L]²

where [L] is dimension of length. The matrix E^aμ is not

necessarily quadratic. The extension of tetrads to the

The variation of action gives the Hamilton-Jacobi

rectangular vilebein is considered in Ref. [13].

equation in terms of the contravariant metric:

Elasticity tetrads in Eq.(1) give rise to the metric,

g^μν∂_μS∂_νS + M² = 0 .

(10)

which is the bilinear combination of tetrads:

Since the action and the interval are dimensionless, the

g_μν = η_abE^aμE^bν .

(3)

mass M in Eq.(8) is also dimensionless, [M] = [1] = 0,

^* E-mail: grigori.volovik@aalto.fi

for any dimension N of spacetime.

680

ЖЭТФ, том 162, вып. 5 (11), 2022

Landau Institute

In the spacetime crystal, the interval between the

one obtains Eq.(15) for normalization:

events is counted in terms of the lattice points, and this

∫

is the geometric reason why the interval is dimension-

1 =< ψ | ψ >= dN-1r

√γ |ψ|² .

(18)

less. One may say that dynamics comes from geom-

etry. In the Diakonov theory, the interval determines

From Eq.(18) it follows that the wave function is di-

the dynamics of the particle, rather than the geometric

mensionless, which is the consequence of the presence

distance, i.e. the geometry follows from dynamics.

of the metric field. This demonstrates the connection

Scalar fields. The quadratic terms in the action for

between quantum mechanics and general relativity.

the scalar field Φ in the N-dimensional spacetime are:

The action (13) and Lagrangian (14) do not contain

∫

ℏ. The role of ℏ in the conventional relation between

(

)

S = d^Nx

√-g

g^μν∇_μΦ^∗∇_νΦ + M²|Φ|²

(11)

the energy levels and frequency, E_m - E_n = ℏω_mn,

√

is now played by

g00 in the red shift equation

√

From Eqs. (4) and (6) it follows that the scalar field is

M_m - M_n =

g⁰⁰ ω_mn [15]. The dimensional met-

dimensionless, [Φ] = [1] = 0, for arbitrary spacetime di-

ric leads to the difference between the dimensional fre-

mension N. This universal zero dimension differs from

quency, [ω_mn] = 1/[L], and the dimensionless mass:

the N-dependent dimension n = (N - 2)/2 of scalar

√

fields in the conventional approach.

[M] = [

g⁰⁰][ω] = [L] ·

= [1] = 0 .

(19)

[L]

Wave function. Expanding the Klein-Gordon

equation over 1/M one obtains the non-relativistic

Weyl and Dirac fermions. The dimensional tetrads

Schrödinger action for the wave function ψ:

[E^aμ] = 1/[L] are obtained directly from the zero dimen-

(

√

)

sion of wave functions, which gives rise to the dimen-

Φ(r, t) =

√ exp iMt/

-g⁰⁰

ψ(r, t) ,

(12)

sionless Weyl and Dirac fields, [Ψ] = 0, in the action:

∫

S = d⁴xeeμa Ψγ^a∇_μΨ,

(20)

S_Schr = d³xdt√-g L, (13)

√

where e is the tetrad determinant. Since the ac-

2L = i

-g⁰⁰ (ψ∂_tψ^∗ - ψ^∗∂_tψ) +

∇_iψ^∗∇_kψ . (14)

tion is dimensionless, then assuming that the quan-

tum field operators Ψ are dimensionless, one obtains

The normalization condition for the wave function is:

∫

[e e^μa] = 1/[L]³, which gives the dimensional tetrads:

d³r

√γ |ψ|² = 1 ,

(15)

[e^μa] = [L] , [E^aμ] =

, [e] =

(21)

[L]

[L]⁴

where

√γ =

√-g^√-g00 is the determinant of the

This is in agreement with the Diakonov theory [1-4],

space part of the metric. This corresponds to the parti-

where tetrads emerge as the bilinear combinations:

cle number conservation in the nonrelativistic quantum

mechanics, see e.g. Eq.(13) in Ref. [14].

E^aμ ∝ <

^Ψγ^a∇_μΨ > , [E^aμ] =

(22)

Since the dimension of this determinant is

[L]

[√γ ]

, the wave function is dimensionless.

[L]³

and metric g_μν is the quadrilinear combination of the

This is distinct from the conventional Schrödinger

fermionic fields, <

ΨΨΨΨ >. This approach also al-

equation without gravity, where the dimension of ψ

lows the rectangular vilebein [13], where spin a and

is [ψ] = [L]-(N-1)/2 for the N dimensional space-

coordinate μ spaces have different dimensions.

time. Inclusion of gravity provides the natural zero

The Hamiltonian for massless Dirac fermions has di-

dimension for the probability amplitude in quantum

mension 1, i.e. the same as the dimension of frequency:

mechanics, [ψ] = 0, for any spacetime dimension.

∫

The same result can be obtained from overlap of the

H =

d³r e e^ia Ψγ^a∇_iΨ , [H] = [ω] =

quantum states, which is naturally dimensionless:

x0=const

[L]

(23)

< r|r^′ >=

δ(r - r^′) .

(16)

The dimension of the Hamiltonian does not coincide

√γ

with the dimension of mass M, which is dimensionless.

Then for the wave function

Gauge fields. The action for the U(1) gauge field in

∫

the N-dimensional spacetime is:

ψ(r) =< r | ψ > , | ψ >= dN-1r

√γ ψ(r)| r > ,

∫

S ∼ d^Nx

√-g g^μν g^αβ F_μαF_νβ .

(24)

(17)

681

G. E. Volovik

ЖЭТФ, том 162, вып. 5 (11), 2022

In case of the conventional dimensionless tetrads, the

q is the vacuum variable, and μ plays the role of the

action in Eq.(24) is dimensionless only for N = 4.

chemical potential in the vacuum thermodynamics. In

With dimensionful tetrads the action (24) is dimen-

the expanding Friedmann-Robertson-Walker universe:

sionless for arbitrary N, since

da/dτ

ds² = g_μν dx^μdx^ν = -dτ² + a²(τ)dr² , H(τ) =

, (31)

[g^μν ] = [L]2 , [Fμν ] =

, [√-g] =

(25)

a(τ)

[L]²

[L]^N

where τ is the conformal time; a(τ) is the scale factor;

Acoustic metric also has dimension 2. The effective

and H(τ) is time-dependent Hubble parameter. The

acoustic metric [5,6] describes propagation of sound in

scale factor a(τ) has dimensions 1, [a(τ)] =

, while

a non-homogeneous flowing fluid and also phonons in

[L]

the following quantities are dimensionless:

moving superfluids and other Goldstone modes, such as

magnons and collective modes of magnon Bose conden-

[q] = [μ] = [ϵ] = [R] = [G_N ] = [Λ₀] =

sate [16]. The action for Goldstone mode (the phase φ

= [H] = [τ] = [ψ] = [M] = [1] = 0 .

(32)

of the Bose condensate) is similar to the action (11):

∫

√

Some of the dimensionless quantities can be fundamen-

S = d⁴x

-g g^μν∇_μφ∇_νφ.

(26)

tal, or correspond to some integer valued topological in-

variants. For example, the "chemical potential" μ may

From the action (26) it follows that the effective con-

correspond to the topological invariant, μ = ±1, which

travariant metric g^μν has the conventional dimension

changes sign at the Big-Bang quantum phase transition

-2, i.e. [g^μν] = [l]². This is also seen form the effective

[20]. Since masses of particles are dimensionless, and

interval in terms of hydrodynamic variables [6, 17]:

there is no fundamental mass scale, one can choose any

ds² = g_μνdx^μdx^ν =

[-s²dt²+(dxⁱ-vⁱdt)(dxⁱ-vⁱdt)] .

convenient mass as a unit mass.

Note also that the dimensionless interval in Eq.(9)

(27)

does not mean the existence of the fundamental length,

Here n is the density of atoms in the liquid; m is the

such as Planck length. First, because the gravitational

mass of the atom; s is the speed of sound; and vⁱ is the

coupling 1/G_N is not necessarily fundamental. Sec-

velocity of the liquid, which coincides with the shift vec-

ond, in the model of the superplastic vacuum there is

tor Nⁱ in the Arnowitt-Deser-Misner formalism. Using

no equilibrium value of the distance between the neigh-

the conventional dimensions of hydrodynamic variables

bouring lattice points. As distinct from the solid state

one obtains the dimension 2 for the covariant metric:

crystals, arbitrary deformations of the vacuum crys-

[g_μν] = [n] ·

· [l] =

(28)

tal are possible. In Diakonov model [1] the metric is

[m]

[l]³

[l]²

emergent, and on the fundamental level the distance

and the dimensionless interval. The dimension of

between the spacetime points is not determined.

avoustic metric follows from the dynamics of the su-

Unruh and Hawking. In terms of the dimensionful

perfluid: geometry comes from dynamics.

metric, the acceleration is dimensionless [7]:

General relativity. Let us consider the GR action on

example of q-theory - the class of theories which avoid

d²x^μ d²x^ν

a²

=g_μν

(33)

the cosmological constant problem. The huge contribu-

ds²

tions of zero point energy to the cosmological constant

is cancelled in the equilibrium state of the vacuum due

[a²] = [g_μν ][x^μ][x^ν ] =

· [l]² = [1] = 0 .

(34)

[l]²

to thermodynamics [18-20]. For the particular q-theory

This leads to the dimensionless Unruh temperature:

on “brane” the action is [21,22]:

∫

[

]

√

T_U =

, [TU] = [1] = 0 .

(35)

S=- d⁴x

-g ϵ(q) +

+ Λ₀ + L^M[ψ,q]

2π

16πG_N(q)

∫

The Gibbons-Hawking temperature of the cosmological

+μ d⁴x n , q =

(29)

horizon is also dimensionless, as follows from Eq.(32):

√-g

Here n is the 4D analog of the particle density in the

T_H =

, [TH ] = [H] = [1] = 0 .

(36)

2π

quantum vacuum (density of the "spacetime atoms"),

which has the same dimension 4 as tetrad determinant

Eqs.

(35) and (36) look fundamental: they do not

√

contain parameters. However, for the temperature of

[n] = [

-g] =

(30)

[L]⁴

the Hawking radiation from the black hole horizon,

682

ЖЭТФ, том 162, вып. 5 (11), 2022

Landau Institute

T_BH = 1/8πG_NM, situation is different. Although

Einstein-Cartan, Barbero-Immirzi, Nieh-Yan and

the Hawking temperature is dimensionless ([T_BH] = [1],

topology. Topological invariants relevant for the quan-

since [G_N ] = [M] = [1]), it does not look fundamental,

tum vacuum are known in the crystalline matter

since it depends on the dimensionless parameter G_N .

[10, 11, 32] and can be extended to the superplastic

The same concerns the Bekenstein-Hawking entropy:

vacuum. The topology in the crystalline quantum

vacua is enriched due to the dimensional elasticity

S_BH =

(37)

tetrads in Eq.(1), which come from the geometric U(1)

4G_N

phases. This topological approach may take place in

It is dimensionless due to dimensionless horizon area:

the Einstein-Cartan-Sciama-Kibble theory, which is ex-

√

pressed in terms of tetrads, and thus is more funda-

dA =

dS^ikdS_ik , [A] = [1] = 0 ,

(38)

mental than the conventional Einstein gravity based on

∫2π

∫ π

metric. Such type of gravity emerging in superplastic

A = dθ dφ

√g_φφg_θθ , [g_φφ] = [g_θθ] = [A] = 0 .

crystals has been discussed in Ref. [33]. The action in

(39)

the Einstein-Cartan gravity can be expressed in terms

The Bekenstein-Hawking entropy (37) determines the

of the differential forms, which contain the elasticity

black hole thermodynamics, but similar to the Hawking

tetrads as the translational gauge fields:

temperature it does not look as fundamental, since it

∫

contains the gravitational coupling 1/G_N . Also it is not

S_EC ∼ ϵ_abcd d⁴xE^a ∧ E^b ∧ R^cd .

(41)

clear why the microscopic degrees of freedom responsi-

ble for the black hole entropy should be characterized

This action is dimensionless because the one-form

by the Planck length [23]. In the superplastic vacuum

tetrad has dimension 1, [E^aμ] =

, while the curva-

[L]

[8] the Planck length scale is absent, since there is no

ture two-form R^ab has dimension 2:

equilibrium value of the distance between the lattice

points: this vacuum can be arbitrarily deformed.

[R^abμν ] =

(42)

[L]²

On the other hand, since the area is dimensionless,

one may suggest that the entropy of the black hole hori-

With the dimensional elasticity tetrads the topology

zon can be expressed in terms of the area only:

of the 3 + 1 crystalline phases [10,11,32] may provide

the fundamental topological prefactor in Eq.(41), with

S_BH = ηA , [η] = [S_BH] = [A] = [1] = 0 .

(40)

1/G_N as integer or fractional topological number.

Here η is some fundamental dimensionless parameter,

The same can be valid for the dimensionless param-

like the topological invariant. In this case one may take

eter in the Barbero-Immirzi action:

∫

the point of view that Einstein’s gravity equations can

be derived solely from thermodynamics [24]. The con-

S_BI ∼ d⁴xE^a ∧ E^b ∧ R_ab .

(43)

stant of proportionality η between the entropy and the

area determines gravitational coupling 1/G_N = 4η. In

Eq.(43) looks similar to the Nieh-Yan term in the ac-

this thermodynamic approach, 1/G_N becomes funda-

tion, see e.g. Ref. [34]. Due to dimensional tetrads the

mental due to the fundamentality of the parameter η.

prefactors in the Nieh-Yan and in the Barbero-Immirzi

However, in the thermodynamic approach to grav-

actions are dimensionless, and thus can be fundamen-

ity there is the "species problem" [25]: the gravita-

tal [7]. It is not excluded that these parameters are

tional coupling G_N may depend on the number of

the topological invariants similar to that in topological

fermionic and bosonic quantum fields [26-28]. This

insulators, semimetals and superconductors [10].

destroys many conjectures, which are based on posi-

The dimensional metric and tetrads appear also in

tivity of the gravitational coupling [29], and prevents

such topological field theories as the BF -theoryl. For

1/G_N to be the fundamental parameter. But this "no-

example, the composite metric (Schönberg-Urbantke

go theorem" can be avoided, if 1/G_N is the quantum

metric [35-40]) is formed by triplet of the 2-form fields:

number related to symmetry and/or topology. Then

the parameter 1/G_N does not depend on interaction

√-gg_μν =

e_abce^αβγδB^aμαB^bβγB^cδν .

(44)

between gravity and quantum field, though it may ex-

∫

perience jumps during the topological quantum phase

The 2-forms in the BF action

B ∧F have dimension

transitions. This takes place in topological materials

2, [B] = [F] = 1/[L]². Then the composite metric in

when one varies the parameters of interaction [30, 31]

Eq.(44) has also dimension 2, [g_μν ] = 1/[L]². In the

and may take place when the Big Bang is crossed [20].

same way the two-form field B can be represented as

683

G. E. Volovik

ЖЭТФ, том 162, вып. 5 (11), 2022

the bilinear combination of the tetrads [37]: B = E∧E.

This follows from Eqs.(45), (48) and (51) for dimen-

These one-form tetrads have dimension 1, [E^aμ] = 1/[L].

sions of the ADM metric elements in 3 + 1 spacetime.

Arnowitt-Deser-Misner (ADM) formalism [41] is

The corresponding quadratic Hamiltonian for the

used for the Hamiltonian formulation of general relativ-

electromagnetic field is:

ity. Let us consider this formalism and its application

∫

(

)

d³r

using the dimensional metric. One has the following

H =

BⁱB^k

(58)

√γγik αDiD^k +

metric elements and their dimensions:

The Hamiltonian has dimension 1, i.e.

[H] = 1/[L].

g_ik = γ_ik , [γ_ik] =

(45)

[L]²

Both the Hamiltonian in Eq.(58) and the Poisson

bracket in Eq.

(54) do not contain the gauge poten-

g0i = N_i = γ_ikN^k , [N_i] =

, [Ni] = 0 , (46)

tials. The gauge potentials also do not enter the Pois-

[L]²

son brackets for charged particle, {p_i, p_j} = qF_ij and

g₀₀ = γ_ikNⁱN^k - N² = NⁱN_i - N² , [N] =

(47)

{p_i(r), D^k(r^′)} = -qδ^kiδ(r - r^′), where q is the dimen-

[L]

sionless electric charge of the particle in terms of the

g⁰⁰ = -

[g⁰⁰] = [L]² , (48)

electric charge of the electron.

N²

The quantization of electromagnetic field is ob-

g0i =

[g0i] = [L]² , (49)

tained by the substitution of the Poisson brackets (54)

N²

by commutation relations between D and B. The Pois-

NⁱN

g^ik = γ^ik -

, [γik] = [L]2 , (50)

son brackets in Eqs. (53) and (54) look as fundamental.

N²

They do not depend on the metric and do not contain

√-g = N√γ , [√γ ] =

(51)

physical parameters of the quantum vacuum. However,

[L]³

the function D in Eq.(55) breaks this fundamentality.

γ^ikγ_kl = δ^il . (52)

It is the phenomenological variable, which describes the

response of the vacuum to electric field. This response

Here N and Nⁱ are lapse and shift functions corre-

contains the electromagnetic coupling 1/α, which is

spondingly, and γ_ik are space components of metric.

not fundamental because of the corresponding "species

The ADM formalism allows to consider dynamics

problem": it depends on the fluctuating bosonic and

in curved space in terms of the Poisson brackets. Let

fermionic fields in the quantum vacuum, and is space-

us consider this on example of Poisson brackets for the

dependent. While the gravitational coupling 1/G_N can

classical 3 + 1 electrodynamics in curved space:

be fundamental due to topology, there are no topologi-

{A_i(r), D^k(r^′)} = δ^kiδ(r - r^′),

(53)

cal invariants which could support the fundamentality

of the electromagnetic coupling 1/α. This is in favour

which in terms of the gauge invariant fields is:

of the scenario in which the quantum electrodynamics

{Bⁱ(r), D^k(r^′)} = e^ikl∇_lδ(r - r^′).

(54)

is the effective low-energy theory, where for example

the gauge fields emerge as the bilinear combinations of

Here B is magnetic field, and the vector D is the electric

the fermionic fields, or/and the gauge fields emerge in

induction of the quantum vacuum (electric displace-

the vicinity of the topologically stable Weyl points in

ment field). The electric induction D is expressed in

the fermionic spectrum [17,42-44]. This, however, does

terms of the electric field E_i = F0i:

not exclude the other possible pre-quantum and pre-

√γ

spacetime theories, see Ref. [45] and references therein.

D^k =

γ^ikE_i .

(55)

α N

Conclusion. Several approaches to quantum gravity

(including the model of superplastic vacuum; Diakonov

Here α is the dimensionless fine structure constant,

which determines the dielectric constant - the electric

tetrads emerging as the bilinear combinations of the

fermionis fields; BF -theories of gravity; and effective

permittivity of the relativistic quantum vacuum, ϵ_vac,

and the magnetic permeability of the vacuum, μ_vac:

acoustic metric) suggest that in general relativity the

metric has dimension 2, i.e. [g_μν ] = 1/[L]², irrespective

ϵ_vac =

(56)

of the dimension of spacetime. One consequence of such

μ_vac

dimension of the metric is that the wave function in

In spite of the dimensional metric, electric induc-

quantum mechanics is dimensionless, [ψ(x)] = [1] = 0.

tance D has the same dimension 2 as electric field E:

This also leads to the dimensionless quantum fields.

On the other hand, if one starts with the conjecture

[Dⁱ] = [E_i] =

(57)

[L]²

that in quantum mechanics the wave function is natu-

684

ЖЭТФ, том 162, вып. 5 (11), 2022

Landau Institute

rally dimensionless, one obtains dimension 2 for metric.

19.

F.R. Klinkhamer and G.E. Volovik, Phys. Rev. D 78,

This suggests the close connection between quantum

063528 (2008).

mechanics and general relativity.

20.

F.R. Klinkhamer and G.E. Volovik, Phys. Rev. D

Acknowledgements. This work has been sup-

105, 084066 (2022).

ported by the European Research Council (ERC) under

the European Union’s Horizon 2020 research and inno-

21.

F.R. Klinkhamer and G.E. Volovik, JETP Lett. 103,

627 (2016).

vation programme (Grant Agreement No. 694248).

22.

F.R. Klinkhamer, arXiv:2207.03453.

The full text of this paper is published in the English

version of JETP.

23.

E.P. Verlinde and M.R. Visser, arXiv:2206.03161

24.

T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995).

REFERENCES

25.

T. Jacobson, arXiv:gr-qc/9404039.

D. Diakonov, arXiv:1109.0091.

26.

A.D. Sakharov, Sov. Phys. Dokl. 12, 1040 (1968).

A.A. Vladimirov and D. Diakonov, Phys. Rev. D 86,

27.

V.P. Frolov, D.V. Fursaev and A.I. Zelnikov, Nuclear

104019 (2012).

Physics B 486, 339 (1997).

28.

M. Visser, Mod. Phys. Lett. A 17, 977 (2002).

A.A. Vladimirov and D. Diakonov, Physics of Parti-

cles and Nuclei 45, 800 (2014).

29.

G.E. Volovik, Mod. Phys. Lett. A 37, 2250034 (2022).

Y.N. Obukhov and F.W. Hehl, Phys. Lett. B 713,

30.

G.E. Volovik, Springer Lecture Notes in Physics 718,

321 (2012).

31 (2007).

W.G. Unruh, Phys. Rev. Lett. 46, 1351 (1981).

31.

G.E. Volovik, Physics-Uspekhi 61, 89 (2018).

M. Visser, Class. Quantum Grav. 15, 1767 (1998).

32.

J. Nissinen, T.T. Heikkilä and G.E. Volovik, Phys.

Rev. B 103, 245115 (2021).

G.E. Volovik, JETP 132, 727 (2021).

33.

M.A. Zubkov, arXiv:1909.08412.

F.R. Klinkhamer and G.E. Volovik, JETP Lett. 109,

34.

G. Calcagni and S. Mercuri, Phys. Rev. D 79, 084004

364 (2019).

(2009).

I.E. Dzyaloshinskii and G.E. Volovick, Ann. Phys.

35.

M. Schönberg, Rivista Brasileira de Fisica 1,

125, 67 (1980).

(1971).

10.

J. Nissinen and G.E. Volovik, Phys. Rev. Research 1,

36.

H. Urbantke, J. Math. Phys. 25, 2321 (1984).

023007 (2019).

37.

R. Capovilla, J. Dell, T. Jacobson and L. Mason,

11.

J. Nissinen, arXiv:2009.14184.

Class. Quantum Grav. 8, 41 (1991).

12.

L. Gioia, Chong Wang, and A.A. Burkov, Phys. Rev.

38.

Yu.N. Obukhov, S.I. Tertychniy, Class. Quantum

Research 3, 043067 (2021).

Grav. 13, 1623 (1996).

13.

G.E. Volovik, arXiv:2205.15222.

39.

F.W. Hehl and Yu.N. Obukhov, Foundations of clas-

sical electrodynamics, Birkhauser, Boston, 2003.

14.

V. Kabel, A.-C. de la Hamette, E. Castro-Ruiz and

C. Brukner, arXiv:2207.00021.

40.

L. Friedel and S. Speziale, SIGMA 8, 032 (2012).

15.

G.E. Volovik, JETP Lett. 90, 697 (2009).

41.

R. Arnowitt, S. Deser and C.W. Misner, Gen. Rel.

Grav. 40, 1997-2027 (2008).

16.

J. Nissinen and G.E. Volovik, JETP Lett. 106, 234

(2017).

42.

G.E. Volovik, JETP Lett. 43, 551 (1986).

43.

G.E. Volovik, JETP Lett. 43, 693 (1986).

17.

G.E. Volovik, The Universe in a Helium Droplet,

Clarendon Press, Oxford (2003).

44.

P. Horava, Phys. Rev. Lett. 95, 016405 (2005).

18.

F.R. Klinkhamer and G.E. Volovik, Phys. Rev. D 77,

45.

T.P. Singh, Eur. Phys. J. Plus 137, 664 (2022).

085015 (2008).

685