Aetheric control over axionically active quasi-electrostatic system and latent structure of the rotating Godel universe

In 1949 Kurt Godel has published the paper [1], in which a discussion about the cosmological solutions of a new type has been opened. Over the next seventy years various problems of the rotating Universe was studied (see, e.g., [2] for review and references). As a prologue for this short note we would like to recover some classical results.

The metric of the Godel spacetime includes one parameter q:

曲² = q ² }

dt² - dx² + I e^2xdy² - dz² + 2e^xdtdy^ .

⑴

* Работа поддержана РФФИ (грант № 20-52-05009)

The Ricci scalar is constant R=厶，and five nontrivial Einstein's equations can be written as follows: —     1                                  1    . о _3

rToo = 2 — л。=呂loze , 凡丁11 = 2 + А。=呂爲3 ,  2 — л。= 2呂Т22£   ■(2)

Л is the cosmological constant, 呂 =8t tG, c =1. In the work [1] Godel assumed that the stress-energy tensor is of the form T^dust) =pU_pU_q and describes the dust with the velocity four-vector U ' and the mass density p ： the mentioned quantities are found to be

и)=-用,Uk = a (彼 + S2 e”), p =— , Л =-—.

。                                   kq2

The covariant derivative of the velocity four-vector in the spacetime with the metric (1) is the antisymmetric traceless tensor orthogonal to the velocity four-vector Uk

This means that the flow of the dust in the Godel Universe is characterized by vanishing acceleration, shear and expansion. Only the antisymmetric vorticity tensor ®k ( = VkU is nonvanishing; it has only one independent component ^12= 2e” ， and its square ®² 三 ®k ( ®^k' = 表 is constant. The angular velocity ^ ^m 三 ^*^mnU_n. where ^ *^mn is t he pseudot ensor dual t o t he ^ki- happens t o be of t he form 口 但 = — 6 争 , 。 2. In other words, the Universe rotates around the z-axis, and the angular velocity of this rotation is proportional to 1. Also, it is clear that Vk (pU ^k ) = 0 and V^q T (du^St ) = 0. i.e…the laws of conserA ； al.ion of the particle number and of the energy density hold.

In this work we consider the spacetime platform of the Godel type and add three new players to the known model. First, we introduce the unit timelike vector field, which describes the velocity four-vector of the dynamic aether [3]; we use the same symbol U' for this quantity. Second, we consider the pseudoscalar field, associated with the axionic dark matter (see, e.g., the review [4] for basic ideas and references). Third, we work with the electromagnetic field coupled both to the aether and to the axions. In other words, we add elements of the aetheric extension of the axion electrodynamics. But the principally new feature of our work is that the dynamic aether carries out the control over the evolution of the axion electrodynamic system. What does this mean? As it was established in the works [5], the dynamic aether guides the evolution of the axionically active electrodynamic system, first, via the aetheric effective metric Q^mn=g^mn+^U^mUⁿ with the guiding function of the first type W, incorporated into the kinetic terms of the axions and photons; second, via the guiding function of the second type Ф* inserted into the potential of the axion field [6]. Generally, the guiding functions 我 and Ф* depend on four differential invariants of the aether velocity four-vector. When we deal with the Godel spacetime platform, only the square of the vorticity tensor is nonvanishing, J= 表. In this context, we take the general model elaborated in [5], reduce it to the Godel case and solve the obtained master equations.

• 1.    The formalism and its application to the Godel model

As it was advocated in [5] the aetherically extended action functional of the Einstein-Maxwell-axion theory contains two guiding functions 我 and Ф*. Both guiding functions enter the master equations for the electromagnetic field

V_k [ғ2^k + 我 (pc^qU^kU_q - F^kqU⁴U_q⁾ +F *泳 Ф * sin (T)] = 0 .

⑸

The second subset of the electrodynamic equations remains standard, VkF*^2k=0. We assume that there exists the electromagnetic field with the potential Aj =60/o(n)+62&(n), for which there are only two nonvanishing components of the Maxwell tensor, F10 and F12, and thus F*^m1 = 2^-| 石 但 ¹P^qF^q=0. We obtain two nontrivial master equations

£ [2F21 + (1 -〃 ) F10e ” ] = 0 ,  £ [F10 + e^- ” F21] =0 ,                   (G)

ax                          ax the solutions to which depend on the assumption about the value of the guiding function of the first type H. The functions 我(®2) and Ф*(®2) happen to be constant since ®2=表=const. When 我 = —1, the appropriate solution to the second equation (6) is Fio=£-"Fi2； the solution to the first equation is 用2 =，where £ is constant. When H = —1. the solulions for Ғю and F12 have the same form, but now £ is not constant obligatory. The four-vector of the electric field Ep 三 FpkU“ is equal to Ep = bp I£-"; the four-vector of the magnetic field is vanishing, Вp 三 Ғ*pqUq=0. The first invariant of the electromagnetic field is F^nFmn=2EpEp = — 2^2e-2^, and the second (pseudo)invariant vanishes, Ғ^*ьпҒтп=0. The stress-energy tensor of the electromagnetic field coupled to the dynamic aether (see [5]) has now five non-zero components, which happen to be proportional to the multiplier (1+H):

T (EM) — T ( EM) — T ( EM) — ^{(1+H) £ 2} -2^    t ( EM) _ ^{3 (1+H) £} 2 f(EM) _ ^{(1+H) £} 2 -%

.

⑺

丄 ⁰⁰   =   ¹¹   = 丄 ³³   =    2 q 2   ^e , 丄 ²²   = —而 ² — ,  丄 ⁰²   =    2 q 2   ^e

In the context of the Godel model the master equation for the axion field transforms into

!(。〃 + 。 ' ) + * sin (督) =0 .

⑻

For our example we choose the exact solution to this equation in the form 0=^@*=const, which corresponds to the localization of the axion field in the minimum of the potential with the serial number n. For this solution all the components of the stress-energy tensor of the axion field take zero values ：

T(A) = T(^A) = — T3A ) =n ,   4(A) = ； Пе²? ,北 (^А ) =Пе? , П 三 ； 田 0п²Ф*2=О .       ⑼

Master equations for the unit vector field in the context of the Godel model with 0=пФ*, U ' = | 品， and the electric field presented above, can be written as follows (see [5]):

—= 人 uf^HU ‘小% +m (工小3).

These equations are compatible with the assumption that U 3 = | 矶 if ^2 =0, and we obtain that

J3 = ( Ci — Сз ) 3 。 ³ ,   ▽ a ， ¹⁷ =—席 G—5 3 ) ,

入 =Uj ▽ a ， ¹⁷—'HEmE^ .

The stress-energy tensor, associated with the aether velocity field (see [5]) has now the form

4(U) = 2 (Ci — Сз) ,  7*) = 2 (Ci — Сз)= — T3U ) FU ) = ： (Ci — Сз) e²% , 喘) = ； (Ci — Сз) e% . (12)

Conclusions

Master equations for the gravitational field (2) contain now the total stress-energy tensor, which has the form

必* = 7(U) + '7((A) + '7(EM 」 7(TT) + '7(dust).                  (ІЗ)

L                     L                     L                     L                          L

Clearly, the contribution of the dynamic aether (12) into the total stress-energy tensor vanishes, i.e_ 7^)=0噜 when two Jacobson constants coincide. Ci=C3- Keeping in mind that the sum of these parameters has been estimated after the event GRB 170817A, one could fix the constraint |Ci| = IC3I < 10-i5. Since the square of the vorticity tensor is constant and thus the second guiding function also is constant, we obtain that the contribution of the axion field, which is in one of minima of the axion potential, 0=пФ*, also vanishes (see (9)). When the guiding function of the first type takes the critical value H=—1 the contribution of the electromagnetic field (7) also disappears. '7(EM) = 0. Finally, the part of the total stress-energy tensor indicated in [5] as 7(INT) also vanishes, when ^2=0 al. H=—1. Formally speaking, the condition H=—1 means that the dielectric pe门nit tivity of the aether is equal to zero (see [5]). In other words, if we accept all the mentioned conditions, we obtain that 7^=7^dust); this means that the solution obtained by Godel in [1] describes not only the simple dust, but also the specific hidden configuration of the aetheric, axion and electric fields presented above as an example of exact solutions to the master equations of the Einstein-Maxwell-aether-axion model.