Axionic extension of the Einstein-Dirac-aether theory: degeneracy removal with respect to shear of the aether flow

We study the guiding role of dynamic aether in the processes of interaction between spinor and pseudoscalar fields. This sector of cosmic interactions attracts attention, when one considers the coupling between the axionic dark matter and fermions (massive Dirac particles), as well as, the coupling between axions and massless neutrinos [1]. Since the dynamic aether is mathematically associated with the global timelike unit vector field U ^j, the effect of this cosmic substratum on the axion and spinor field could be associated with the acceleration four-vector o_t = U ^j V j U _t, with the expansion scalar О = V_kU ^k,

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

with the skew-symmetric vorticity tensor w_mn and the symmetric traceless shear tensor cr_mn. When we work with the isotropic Friedmann type cosmology, the only О turns out to be not equal to zero. The nonvanishing vorticity tensor appears in the Godel’s rotating Universe. When we deal with the Bianchi type anisotropic homogeneous cosmological models, the nonvanishing shear tensor inevitably appears as an important characteristic of the dynamic aether flow. Five years ago we were faced with specific degeneracy with respect to the shear of the aether velocity. In 2017, the observation of the binary neutron star merger [2] has shown that the ratio of the velocities of the gravitational and electromagnetic waves is highly likely equal to one, so that two Jacobson’s constants C 1 and C3 [3] are linked by the relationship C1+C₃=0. This means that the shear tensor happens to be crossed out from the aether kinetic term, thus disappearing from the Lagrangian of the dynamic aether and becoming the hidden quantity.

• 1.    The formalism

The action functional of the Einstein-Dirac-aether-axion theory contains three groups of terms:

S =

^ ^d4 x^—g ^^^ [Л+2^Л+А(зтп

U ^m U ⁿ - 1)+ K ^abmn ^_aUm ^bUn ] +

(1.1)

+

[ 2 W

D_k^-D_k^7 ^k ф ) mr^

+ |ф0 U—gmnVm^Vn<^]| .

The first group located in square brackets with the multiplier 2^ in front, relates to the version of the vector-tensor theory of gravity known as the Einstein-aether theory [3]. In this theory the variation with respect to the Lagrange multiplier A provides the aether velocity four-vector U^k to be unit, g_mnU ^m U ⁿ=1. The kinetic term K = K ^abmn V _aU_mV _bU_n with the constitutive tensor

Kabmn=Cigabgmn+C2g^mgbn+C3g^ngbm+C4UaUbgmn,(1.2)

in which we have to put C₃ = -C 1 due to results of the observations [2], can be rewritten as

K = (Ci+C4)DUkDUk +2C1ШгkШгk +C2О2 .(1.3)

Here we use the decomposition of the covariant derivative of the aether velocity four-vector

• V,Uk = U,DUk+c,k +шгк + |0A,k , D = U3V3 , Ak = g^-U.Uk ,(1-4)

where the symmetric traceless shear tensor and the vorticity tensor have, respectively, the form

Ck = ^AfAk (VpUq +VqUPY) -|0A,k , шгk = jA^Ak(VpUq-VqUp).(1.5)

The second group of terms relates to the contribution of the Dirac field. It includes the spinor field ф and its Dirac conjugated ф, the Dirac matrices 7” and 75. The covariant derivatives of the spinor field are taken in the Fock-Ivanenko form based on the tetrad four-vectors X ³ 7 ^(a)

Dkф = 8kф - ^ф,   Dkф = 8kФ + фTk , rk = 4 gmnX^7 ^s 7 ⁿ VkXm ) .        (1-0

The third group of terms with the square of the coupling constant Ф0 in front, describes the contribution of the pseudoscalar (axion) field ф. The potential of the axion field is considered to be the function of seven arguments (see their definitions above) V = V фф, S, P, ш, fi, О,ст2). The arguments S and P describe the specific coupling of the axion and spinor fields; the arguments , and fi describe trilateral interactions between axion, spinor fields and aether; the arguments О and ст2 describe the direct control over the axion field, which is carried out by the aether.

Master equation for the axion field, obtained by the variation of the action functional (1.1) with respect to ф, keeps the standard form:

9^mnV_mV„, = -¹   V (ф, S, P,, fi, О, ст²).                         (1.7)

2 оф

Dirac equations acquire new terms due to variation of the potential V with respect to fi and fi:

~ dV dV 5  /dV

M= dsE+ dP7 + (aJE+ ty nDnfi — ^mE — 2ф0

■ Mf^ fi = 0 , iDnifyn	+ fi	2 mE — 2ф 0 М) =0 ,	(1-8)
-1|75) Uk7k+ ^E + У fi            oS	d^7b+ f d^E + |V75)jk7k . dP    \ouj    dfi		(1-9)

The matrix term M = (mE — 2Ф0М) plays the role of the effective mass of the spinor field interacting with the axion field; E is the unit matrix (see [1] for details).

Master equations for the unit rector field U³ undergo the following extensions:

V„ J^aj = AU³ + 1³, A = U \ VaJ^a3 — 1³ ] ,

(1.10)

,,        ,        „г dv,- ■ , dv,- ■ c , dv 1

1³ =C4(DU_m)(V³U^m) + _KФ2        ■     ■ — . 75fi) —       ■         ,      1.11

ow             ofi               о (ст2)

_J4 = J^(0)a3 + _кф2 d-V^ar³¹³ + 2Юg^a/| , J^(2)a3 = к^ab3n(V_bU_n) .         (1.12)

Equations for the gravity field as the result of variation with respect to metric is of the form

Rik — 2 R д_гк = Лд^ + ЕГ^ + ET^ +T^ ) + T^ ) .              (1.13)

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

^Tik) = 2 \ d7(₃^Dk)^{fi— D}(i^fi7k)^fi]   '/a

12 VhⁿDnfi—(Dnfihⁿfi ] —mfifi^ ,

(1.14)

T⁽_k^ = Ф0 [ViCVkC + 2 gik (V —Vn,Vⁿc)j .

The term T^ ) contains the derivatives of the ax ion field potential with respect to О arid ст²:

(1.15)

m⁽U ) _ ^Tik

2 д_гк К^abmnVaU_mVbUn+U_iU_kU3 V .J ^ +C4 (DU_tDU_k—UfiJ_kDU_mDU^m) +

+V^m Jjj<-JmmliJk)-J^)Jm] +C1 [(VmUi)(V^mUk) — (ViUm)(VkU^m)] ,

The term T^). in addition, contains the derivatives of V with respect to , at id fi:

₍w)       ₂ 9V              dV           5

^Tik = ^2кфо у, \^fi;U(i7k)^fi] ⁺ dfi \^fi;U(»7k)7^5fi] ⁺

(1.16)

(1.17)

9V       _{m         m} 1       1            9V 1 9V

⁺ d^ ^CTm№)^—CTm(i^{Vk)U +}з^Ост^^{к —} 2 ^{( d +0) ст}^^кдст)⁺2^дгкэе   .

The complete system of Master equations for gravitational, spinor, vector and axion fields and is ready for analysis; unfortunately, it is out of the frames of this short note.

is

derived

Conclusion

We plan to apply the formulated theory to the anisotropic homogeneous cosmological model of the Bianchi-I type, for which cr^ = 0. For this case the structure of the potential of the axion field V(ф, S, Р,ш, fi, O, c²) guarantees the degeneracy removal with respect to the shear tensor, attributed to the aether velocity.