Axion electrodynamics on the Bianchi spacetime platform: fingerprints of shear of the aether velocity

The Bianchi models, which describe evolution of the homogeneous anisotropic Universe, are well known and very fruitful in the context of modeling of the physical system dynamics. We study the properties of the axion-photon coupling mediated by the dynamic aether [1-4]. When we deal with the anisotropic Universe, the velocity of the aether U ^J" is characterized not only by the expansion scalar О = V _k U^k (e.g., as in the case of Friedmann Universe), but also by the nonvanishing shear tensor o _pq. Thus, the Bianchi spacetimes become the unique models for the extended analysis of the aether dynamics and its impact on the electrodynamic systems. However, the constraint for the ratio of the velocities of the gravitational and electromagnetic waves, which has been obtained due to the observation of the binary neutron star merger (see [5]), crossed out the shear tensor from the basic scalar K^li ^mHV i U _m V j U _H introduced into the Lagrangian of the dynamic aether theory [6]. This means that in the canonic theory of the dynamic aether the shear tensor is a. hidden quantity, and we have to search for another way to

“The work was supported by Russian Foundation for Basic Research (Grant No. 20-52-05009).

reveal the shear fingerprints. In this short note we suggest to consider a. specific model of interaction between the axion field ф and aether velocity, namely, we introduce the square of the shear tensor into the axion field potential V. Since the axion field is coupled to the electromagnetic field, the shear of the aether velocity affects the structure of magnetic and electric fields in the Universe. In other words, we suggest the mechanism of the degeneration removal with respect to the shear of the aether velocity, based on the axion-aether interaction. Keeping in mind this idea, below we derive the extended master equations for the axion, vector, electromagnetic and gravitational fields.

• 1.    The formalism

We start with the action functional of the Einstein-Maxwell-axion-aether theory

^ (eaa) — J Fz _x' g (^ \ R +^+ X ( g _mn U^mU " —1)+K ^abmnV _Q U _m V _b U _n^ +

(1.1)

+ 4 ( Fmn + фгт„ ) F^m + |ф0 [V(ф, О, ст²)—g^mnVmфV„ф] I .

As usual, the term with the Lagrange multiplier A guaranties that the aether velocity four-vector is timelike and unit, g _mn U^mU ⁿ—1. The standard decomposition of the covariant derivative of the aether velocity four-vector

ViUk — UiDUk + а _гк + _{Ш гк} + _3i0^_lk ,                         (1.2)

whore D — U^kV _k is the com-eel ive derivative. A_ik — g _ik —UiU_k is the prejector. О — V _k U^k is the expansion scalar, w _ik — 2ApAk(V_pU_q—V_qU_p) is the skew symmetric vorticity tensor, involves into consideration the symmetric traceless shear tensor a _ik—2 A^Ak ( V _p U _q+ V _q U _p)) — 3ОA_гk (g^mnCT_mn—0). Using the Jacobson’s constitutive tensor

_K a^bmn — _Ciga^bg^mn + _C 2 g^ _m g^bn + C ^^^ g^ + C^a Щ ^Ь ^^^^ ,

(1.3)

one can rewrite the kinetic term V = K^abmn ( V _a U _m)( V _b U _n) in the following form

• £ — ( C i+ C 4) DUkDU^k +( C i+ C 3) CTik ff^{ti k} + ( C i —C 3) ^ik^' +| (С1+ЗС2+С3) О² .      (1.4)

Based on the results presented in [5] and on the theoretical predictions about the gravitational wave velocity in the aether [6], one has to put C₃ — —C 1. This means that the shear tensor o _ik disappears from the decomposition (1.4). The term in (1.1), which contains the Maxwell tensor F_mn and its dual F^^^ produces the equations of axion electrodynamics, when one uses the variation procedure with respect to the potential of the electromagnetic field Aj. The potential of the axion field V is assumed to depend on the axion field ф itself, on I lie expansion scalar О and I lie square of the shear tensor ст ² — a_pqa^pq-. the parameter Ф0 is reciprocal to the coupling constant of the axion-photon interaction. Due to the definition F _ik — V _i A _k —V _k A_i, we have to keep in mind the equation V _k F * ^ik — 0. The standard procedure of variation of (1.1) with respect to А _г gives the main equation of axion electrodynamics

Vk [Fik + фF*ik] — 0 ^ VkFik — —F*ikVkф .(ho)

Clearly, the properties of the magneto-electric configurations depend on the state of the axion field.

Variation of (1.1) with respect to ф gives the equation

1 d1

дmnVmV„ф +2— V(ф, О, ст2) — — 4-^2F   F .(l.G)

The term in the right-hand side of this equation can be interpreted as the electromagnetic pseudoscalar source of the axion field; the second term in the left-hand side shows that the aether regulates the behavior of the axion field via the scalars О and ст2.

Variation with respect to U ³ gives us the equations of the aether dynamics in the canonic form

Va Ja3 = AU3 +13, A = U3 [Va Ja3 - 13 ] ,(1.7)

but now the four-vector quantity 1³ and the tensor J^a3 are, respectively, of the form

13 = C4(DUm)(V3Um) - КФ2^T^DUm ,(1.8)

d(cr2)

J4 = J(0).3 + кф2 Ц^ ^ + 1 |v д^] , J(^ = Kab3n(vbUn) .(1.9)

Equations for the gravitational field as the result of variation with respect to metric is of the form p 1                  (EM)   (A)^) । тУ) । тУ)

(1.10)

(1.11)

^Кгк ^- 2 К ^дгк = ^{Л д}гк ⁺ У4   ⁺ У4 ⁺ -Чк ⁺ Чк •

The stress-energy tensors of the electromagnetic and pure axion field have the standard form

t ( E^M ) = 4 дгкFmnF^mn-FimFk^m ,  T^ = Ф0 V_гфVkФ + 2 дгк ( V-VnфVⁿф )

Other contributions contain the derivatives of the axion field potential with respect to О and cr2:

• T% ) = 2 дгк K ^abmnVaUmVbUn + UгUk U3 V aJ^ + C4 ( DUгDUk-UгUk DUmDU^m ) +

+V^m [u₍ г J^-Jm⁰₍⁾_гUk)-J ^Um ] +C1 [(VmUг )( V^mUk ) - ( VгUm )(Vk U^m)] ,      (1.12)

• ^Т г ^у ) = ^кФ0 ।² 5(^2) ^^Crm(г^crm)^-crm(г^Vk)^{Um +} 3^Оcrгkj ^{-( D +O)} ^гк _g (^) + 2 ^дгк уд] } "    (^1ЛЗ)

• 2.    Bianchi-I spacetime platform

When one considers the anisotropic spatially homogeneous Universe with the metric ds2 = dt2 - a2(t)dx2 - b2(t)dg2 - c2(t)dz2 ,(2.1)

and with the aether velocity four-vector U³ = §0, one obtains immediately that

VгUk = Vk Ui = 2 дгк ^ DUг = 0 ,  LJmn = 0 ,  О= | + Ь + С ,  ^к =0 ,(2.2)

• 1 a 1^     2 b 1^     3 С           2     nA2 b\    (c\21 ^

• C1 = q0 , C2 = T- q0 , C3 = “- q0 , C          + I +(" ) - Q0 .t^'^

n 3           b 3           c 3           \a Ф bM    Cc J3

Next, we will have a. thorough analysis of the complete system of model equations taking into account I lie term c2 (2.3).

Conclusion

The full-format analysis of the system of equations for gravitational, electromagnetic, vector and pseudoscalar (axion) fields is, unfortunately, out of the frames of this short note. However, based on the presented results, we can see that the incorporation of the square of the shear tensor cr² into the axion field potential V ( ф, О, cr²) removes the degeneration with respect to the shear tensor, attributed to the aether velocity.