arXiv:1308.0896v3 [gr-qc] 1 Aug 2014 pc eeto operator reflection Space the bibliography). for complete [3] and (see retrospective [2] the Pomeranchuk historical and for mir- Okun developed the Kobzarev, was Later by in concept reflection. processes matter mirror the ror the of under equivalence universe of the sector”) the “hidden restores the theory also field (called sector formulated matter the they mirror where of [1], idea Yang and the Lee of article mous xsec fthe of existence C ini h e-agform Lee-Yang the in tion omttvt ihatm-rnlto [ time-translation a with commutativity operator version eto irrmte,weeoperator where matter, mirror of cept ilsit h irroe.I result, In ones. mirror the into ticles le h osre unu ubr orsodn to corresponding number, quantum I conserved the plies I endb h orsodn hrlt operators chirality corresponding the are by components These defined functions. wave the of components and ticles, and adMdlcnan nytelf-addcomponents left-handed Ψ the spinors only of contains Model dard n rmtehlct prtr endas defined operator, helicity the from ent P ilswr rpsdadeaoae.Nurn mixing Neutrino elaborated. par- and mirror proposed Different and were wormholes ticles physics. of signatures theoretical observational in problems challenging limit, the high-energy to the tends in operator only helicity operator The chirality under changed reflection. are mirror particles the of helicity and chirality Both ~p ˆ ∗ † r r R = -al [email protected] e-mail: [email protected] e-mail: h adufr fteoperator the of form Landau The . Ψ rey h ocp fmro atri sfollows. as is matter mirror of concept the Briefly, fa- the in proposed was matter mirror of concept The oaas omoe n irrprilsaethe are particles mirror and wormholes Nowadays, stecag-ojgto,i o aifcoydeto due satisfactory not is charge-conjugation, the is o example, For . R ~p/p γ i Ψ = r ia arcs h arnino h Stan- the of Lagrangian The matrices. Dirac are , ~ σ L ′ R ( = hr h rm eoe h irrpar- mirror the denotes prime the where , L omoe nteUies r icse.Sm o-rva a non-trivial signa the Some of astrophysical considered. discussed. are throat The are electrodynamics the Universe fluid. at critica the the exactly in The for wormholes located traversable is versa. are fluid vice holes perfect and the of particle flow mirror corresponding a faypril o lsia bet staesn through traversing is object) classical (or particle any If and eeaietecmaiiiyo h irrmte ocp wi concept matter mirror the of compatibility the examine We σ = 1 .INTRODUCTION I. P σ , I CP P P r voaini ekitrcin.The interactions. weak in -violation L L L 2 o-retbewrhlsa otl otemro world mirror the to portals as wormholes Non-orientable nteqatmpril pc.The space. particle quantum the in σ , .Teciaiyoeao sdiffer- is operator chirality The Ψ. voain lentv representa- Alternative -violation. r,rsetvl,tergtadleft and right the respectively, are, (1 = 3 ,and ), ~ r − 1 nttt o ula eerho h usa cdm fSci of Academy Russian the of Research Nuclear for Institute − → γ 0hOtbrAnvrayPopc a 132Mso,Russi Moscow, 117312 7a, Prospect Anniversary October 60th I 5 r ) σ / = i ,where 2, ~ r r h al matrices. Pauli the are orsod otein- the to corresponds R P ..Dokuchaev V.I. ed otecon- the to leads R I I r r γ ,I H, Ψ rnltspar- translates 5 ~ σ = L CP ≡ · r CP ~p/ Ψ = ˆ Dtd coe 0 2018) 10, October (Dated: im- 0 = ] iγ -violation ,where 2, 0 E where , P γ R ′ L 1 ≫ γ 1, and and 2 m ∗ γ 3 . n uN Eroshenko Yu.N. and h nenlsmer operator symmetry internal the o-retblt,wihi eae ihthe with related space is the which only non-orientability, here consider We [18]. space- time-non-orientable even or time-non-orientable space-non-orientable, the around versa transformation 20]. [19, vice winding string” and the “Alice one to mirror analogous the of is into transformation particle that ordinary corre- Note, the the into particles. but mirror the Model, sponding into Standard the not the through of transformed out passing anti-particles are particles, pointed wormhole, if was absent, non-orientable it is Meainwhile, problem where particles. with this elementary 285), compatible of Model not Standard are the (p. wormholes [18] non-orientable discussed the of that are book view wormholes the in Non-orientable justify in to an- difficult violation. its is CP into process was but transformed It but worm- is hole, non-orientable [16]. the particle through in moving that while later tiparticle [17] discussed was in topology. idea stated non-orientable similar general [15] a a Novikov Quite I. with and universe Ze’ldovich the Ya. for by idea proposed This first wormhole. while was versa non-orientable vice the and through particle moving mirror corresponding a [14]. into in studied were bottle with Klein solutions the in non-orientable of the discussed topology of is a Properties The wormholes 13]. through matter. [12, mirror particles the of of passage aspects cosmologi- astrophysical quantitative also and for See cal therein the [2]. references in between and proposed interaction was [8–11] sectors gravitational Lagrangian). mirror and the the ordinary of in terms type mir- interaction is This and (the oscillations our particles these between ror for interaction gravitational reason were local world The the our [4–7]. and in mirror considered the between oscillations and atcefo h riayt irroe n ieversa. vice and one, reason mirror this to For ordinary the from particle irrmte oesteoperator the models mirror-matter n osntvoaethe violate not does and h irrmte etr fi xss hne nythe only changes exists, it if some of sector, sense matter mirror The rmthe from npicpe h o-retbewrhlsmyb the be may wormholes non-orientable the principle, In transformed is particle any that argue we article this In h o-retbewrhl,i un into turns it wormhole, non-orientable the ue ftepeec fnon-orientable of presence the of tures pcso h o-retbewormholes non-orientable the of spects C oi on ntehydrodynamical the in point sonic l omoe hsmasta worm- that means This wormhole. eas the because htennoinal wormholes. non-orientable the th PT CP R P 1, prtosudrtemro reflection mirror the under operations † scnevd The conserved. is CP PT CP ences a snta nrknsymmetry, unbroken an not is term aey nthe in Namely, -theorem. R hc rnfrsany transforms which , P sdtrie pto up determined is R prto differs operation P operation. 2 and the nature is not CP -invariant. By this reason, in Reissner-Nordstr¨om black hole the mirror matter models any particle, while travelling 2 e2 − around the non-orientable space, transforms not to its f1(x)= f2(x)=1 + 2 , (2) antiparticle but to the mirror particle. x x The most direct astrophysical observational signature where e = Q/M is an electric charge of the black hole of the existence of non-orientable wormholes, besides the in the dimensionless units. The wormhole metric is often gravitational effects, is a possible excess of matter in the written in the form [24]: vicinity of the wormhole throats, ejected from the other 2Φ(x) side of wormholes with the mirror–ordinary transforma- f1(x)= e , (3) tions. In particular, it is interesting to explore the idea of the mirror matter as a candidate for the dark mat- K(r) S(x) f2(x)=1 − =1 − , (4) ter of the universe. In fact, the dark matter in the form r x of the mirror matter particles was widely discussed and where the dimensionless function S(x) is introduced. The elaborated in details in many papers (see, e. g., the dis- S(x) is related to the (Misner-Sharp or Hawking) mass cussion of different aspects of the mirror dark matter in function. Some specific substance, violating the weak the recent review paper [21]). energy conditions, is required for the construction of the The mirror particles have their own mirror electric wormhole throat to support a wormhole in the steady charge and photons with a corresponding mirror elec- state. For example, the phantom energy is used for this tromagnetic interactions. Therefore, during traversing construction [25]. the non-orientable wormhole throat, the ordinary charge In the oriented spaces, a smooth gluing of the angular disappear, but, instead of, the wormhole throat becomes coordinates at the edges of metric map produces a total charged. The winding up of the electric field lines around geometry with the globally fixed orientation. In the case the wormhole throats were considered by J. A. Wheeler of wormholes, the gluing of separate maps produced at with relation to the “charge without charge” concept [22], the throats r = K(r): [23]. In the case of the non-orientable wormhole, the ′ ′ ordinary-to-mirror transformation will also alternate the θ = θ, φ = φ. (5) field lines from the ordinary to the mirror ones, as we This is illustrated in Fig. 1 (see the left panel), where discuss in this paper. only a spatial part of the global wormhole geometry is The paper is organized as following. In the Section II qualitatively presented. However, in principle the inver- we present the mathematical description of the non- sion of orientation is possible in the form orientable wormholes. In the auxiliary Section III we demonstrate the hydrodynamical traversability through θ′ = −θ, φ′ = φ. (6) some definite types of wormholes. This traversability is needed for the justification of the baryons flows with This inverse gluing of the corresponding maps is similar mirror–ordinary transformations, considered as a par- to the construction of the M¨obius strip. In the wormhole ticular example in the Section IV. When in the Sec- case, when there are two throats in the same universe, tion V we discuss non-trivial aspects of the non-orientable the inverse gluing produces a non-orientable wormhole wormholes electrodynamics in the presence of the mirror– (see the right panel in Fig. 1). Therefore, any wormhole ordinary charged particles transformations. Finally, in solution with the both throats in our universe can be the Section VI we write some conclusions. transformed into the non-orientable one by the inverse gluing of space coordinates at the throat. While con- sidering the ordinary and mirror particles in the same II. NON-ORIENTABLE WORMHOLES space-time, we may describe them for simplicity and ob- viousness on the opposite sides of the sheets in Fig. 1, which are, respectively, the ordinary an mirror worlds. It was indicated in [15] that any particle, after travers- Note that a similar procedure of the inverse gluing pro- ing the universe with a non-orientable topology on a cer- duces the Reissner-Nordstr¨om or Kerr black hole with tain spatially closed trajectory, can turn into the mirror an inverse orientation of the internal universes inside the one. The very specific representation of a non-orientable corresponding inner horizons. space is the non-orientable wormhole [18], which can be The calculations in the non-orientable spaces require constructed as a simple generalization of the ordinary some caution. For example, in the curved space-time wormhole discussed in details by [24]. Let us write the case the γ5 operator is changed as (see, e. g., [18],p. 286) most general form of the spherically symmetric static ε metric: γ5 ≡ i [µνλρ] eµ eν eλ eρ γ(a)γ(b)γ(c)γ(d), (7) 4 (a) (b) (c) (d) 2 2 −1 2 2 2 2 2 ds = f1(x)dt − f2(x) dr − r (dθ + sin θdφ ), (1) The chirality operator is changed in a similar way. In ad- dition, the direct using of the Gauss theorem reveals the where x = r/M, M is a mass parameter, and G = c =1 non-trivial electrodynamic properties of non-orientable units are used. For example, in the case of a charged wormholes, which will be discussed in the Section V. 3

mation of the test fluid, following the general method of Michel [27]. We define an auxiliary function dn/n = dρ/(ρ + p), which corresponds to the number density of particles in the case of the atomic gas. We solve equa- µν tions of motion T ;ν = 0 in the given background metric. From a zero µ = 0 component of these equations, and µν from the projection equation uµT ;ν = 0, it follows the two integrals of motion:

2 1/2 2 2 (ρ + p) f2 + u γ x u = C1, (9)
2 nuγx = −An∞, (10)

1 1/2 where u ≡ u , γ(x) ≡ (f1/f2) , and n∞ is the value of n at x → ∞. The rate of mass flow toward the coordinate ˙ 2 r center, M = −4πr T0 , is expressed in the form [29]:

2 M˙ =4πAM [ρ∞ + p(ρ∞)]. (11)

The numerical constant A can be found by the fixing of the mass flow rate at the critical sound point, and usually A ∼ O(1). To calculate the fluid parameters at the critical point, it is convenient to divide (9) into (10). This procedure gives: Figure 1. Ordinary and non-orientable wormholes. 2 1/2 (ρ + p) f2 + u γn/n∞ = C2, (12)
III. HYDRODYNAMICAL TRAVERSABILITY 2 where C2 ≡ −C1/A. We denote by V = dp/dρ the OF WORMHOLES square of the sound speed of the accreted fluid. The critical point is calculated by the standard method from We describe here the hydrodynamical flows of matter [27]. We take the logarithmic differential of (10) and (12), between the mirror and our worlds through a wormhole exclude dn/n and put the multipliers of dx/x and du/u throat. We will use the approximation of the perfect fluid to zero. In this way, we find the system of equations for moving in the background metric. The corresponding the critical point: energy-momentum tensor of the perfect fluid 2 2 u V = 2 (13) Tµν = (ρ + p)uµuν − pgµν , (8) f2 + u where ρ and p are, respectively, the energy density and µ µ ′ ′ ′ pressure of the fluid in the rest frame, u = dx /ds is 2 γ x γ x f2x 2V 2+ =2 + 2 , (14) the 4-velocity. The adiabatic equation of state p = p(ρ)  γ  γ f2 + u is assumed. For the wormhole metric we use an exact solution [25] with the phantom energy supporting the where ‘primes’ denote the derivatives with respect to x. wormhole throat. The phantom energy in this model is It is easy to check that in the case, when f1 and f2 corre- concentrated in the finite region r0 r1 the system of equations yields the known parameters of the wormhole metric coincides with the Schwarzschild black critical point [28]. hole metric. The influence of the wormhole on the matter Let us now consider the critical point for the worm- at r > r1 can be considered similar to the black hole case. hole. Recall, that we consider the test fluid without a We are interesting in the possibility of the stationary ac- back-reaction on the metric. Gravity of the accreted liq- cretion flow through the wormhole throat. A stationary uid would lead to some shift of both the metric param- accretion is possible, if there is a critical sonic point in eters and the critical point position. From the system the flow. The accretion of perfect fluid on the wormhole of equations (13), (14), we find the square of the fluid was discussed in [26] without the reference to the critical velocity at the critical point: point. Note, that a direction of the accretion flow (inflow 1 or outflow) for individual wormhole will depend on the u2 = Φ′(x − S). (15) 2 initial conditions in the ambient plasma in the vicinity of the wormhole. It is clear a stationary accretion takes place in the case We consider the stationary accretion in the approxi- of finite Φ′ and x>S (outside the throat). However, 4 the fluid velocity at x = S should not be zero, while the −1, the roots are < 1 or even complex. This means fluid passes through the throat. The stationary accre- that solution with y < 1 does not correspond to any tion condition is possible only if Φ′ ∝ (x − S)−1, and critical (sonic) point and, therefore, is non-physical. In Φ′ should remain positive. This condition is satisfied for result, we demonstrate that there is a critical point in the wormhole supported by the phantom energy [25]. In- the fluid flow, which is located exactly at the throat of deed, the corresponding Einstein equations were written the wormhole. This means that the described wormhole in [25] for the wormhole metric. In our notations, the is traversable for the perfect fluid. There are may be corresponding equations have the form: another sonic points outside the phantom energy region, where the Schwarzschild solution is valid, see [29]. In ′ 2 2 S =8πx ρM˜ , (16) this case the fluid must flow subsequently through all the sonic points. This is the condition of the stationary 8πx3pM˜ 2 + S self-consistent solution. Φ′ = , (17) At the second throat of the wormhole, a fluid flow 2x(x − S) would have the same stationary density distribution as whereρ ˜ andp ˜ are, respectively, the energy density and in the accretion flow toward the first throat, but with pressure of the supporting phantom energy. It is clear the reversed direction of flow, i. e., with the outward di- from (17), that condition Φ′ ∝ (x − S)−1 is valid for rected fluid velocity u> 0. x → S. Note, that the a similar traversable wormholes, supported by the phantom energy, were considered also IV. ACCRETION OF BARYONS BY THE in [36]. NON-ORIENTABLE WORMHOLE A radial position x of the critical point obeys the fol- lowing equation Astrophysical signatures of the mirror matter in the V 2(2+Φ′x)=Φ′x. (18) Galaxy were considered in [30], where the gravitational influence of the mirror matter on the clustering of or- ′ By substituting Φ from (17), we obtain the following dinary baryons were investigated. We consider now the equation, which is non-linear in general, physical effect, which may be the observational signature of the existence of non-orientable wormhole. The dif- 2 − 3 2 3 2 V [4(x S)+8πx pM˜ + S]=8πx pM˜ + S. (19) ference between the ordinary matter and the mirror one is related with the asymmetry of the inflation potential, It’s immediately seen that x = S = x0 obeys this Equa- 2 −1/2 see, e. g., [31], [6]. As an example, we will assume that tion, where x0 = (−8πp˜0M ) is a throat radius, and mirror matter is the dark matter in the our universe [21]. p0 is a pressure of the phantom energy at the throat. Therefore, one of the hydrodynamical sonic point lies ex- The dark matter density in the Galactic halo is a rather actly on the throat of the wormhole. well described by the Navarro-Frenk-White profile [32] If a near uniform distribution of the supporting phan- ρ0 ρH(r)= , (24) tom energy is realized in the vicinity of the throat [25], (r/R )(1+ r/R )2 the solution of equation (16) in the vicinity of the throat s s is where Rs = 20 kpc is a scale radius, Rh = 200 kpc is −3 2 3 3 a virial radius of the halo, ρh(r⊙)=0.3 GeV cm is S = x0 +8πρM˜ (x − x )/3. (20) 0 an average density at the distance of the Sun, R = r⊙ = Respectively, equation (19) in this case takes the form 8.5 kpc, from the Galactic center. The baryonic matter dominates at the distances r ≤ r⊙. In this model the 1 − V 2 1 4V 2 1+3w Navarro-Frenk-White profile gives the density of mirror y3 + + y − = 0 (21) 1+3V 2 3w  1+3V 2 3w matter. If the density and pressure of matter in the one cho- where y ≡ x/x0, w ≡ p/˜ ρ˜ < −1. It is easy to see that sen world, our or mirror one, is much greater than the y1 = 1 is the root of equation (21) for any V and w. The corresponding values in the other world, the collision of other two roots can be found from the Vieta relations: the counter-propagating streams in the wormhole throats can be ignored. Under this condition, there will be a net 1+3w 1+ y2 + y3 =0, y2y3 = − (22) flow from the world with a greater density into the world 3w with a lower density through the wormhole throats. For in the form example, let us consider the case, when the first throat is located in the halo of our Galaxy far beyond the stellar 1 1 4+15w 1 1 4+15w disk at some distance r = r from the Galactic center, y = − ± , y = − ∓ . w 2 2 2r 3w 3 2 2r 3w and the second throat of the same whormhole is located (23) at the place, where the baryon density is ≪ ρH(rw). In One of the roots, y2,3, could be ≥ 1 only for 0 < w ≤ this case, the mirror matter will flow through the worm- 1/3. However, in the considered phantom case with w< hole and will turn into the ordinary one at the second 5 throat. In result, the region of the moving matter with electric field could be very strong near the wormhole an unusual chemical composition will be observed near throat without existence of any real charge, mimicking the second throat. the violation of the Gauss theorem. In fact, the standard We suppose that mirror dark matter forms the major formulation of the Gauss theorem is applicable only to part of the extended Galactic halo in the our world. For the closed or topologically simply connected space-time simplicity, we suppose also that a mean density of the backgrounds. Meantime, the winding problem for the mirror gas in the Galactic halo is of the same order as electric (or magnetic) field becomes nontrivial in the case a mean density of the mirror stars. The heavy elements of the non-orientable wormholes and with the presence define the size of the forming baryonic halo of Galaxy, be- of mirror matter. In the ordinary case, i. e., without the cause the epoch of the halo stabilization depends on the mirror sector, any field line, that goes out of the throat, star formation history, which, in turn, strongly depends corresponds to the ordinary electric field line. Respec- on ambient chemical composition. This means that the tively, the net “charge without charge” of the wormhole mirror matter is stabilized at the Galactic halo outskirts, throat is proportional to the number of winding loops of avoiding contraction into the disk. As a consequence, these lines. the heavy elements would be more abundant in the mir- Quite the contrary, this correspondence changes rad- ror universe, because the mirror stars are formed more ically, if one takes into account the mirror-to-ordinary efficiently in the halo. A similar idea, underlying the transformation in the presence of the mirror sector with specific properties of the mirror world, is elaborated in mirror electric charges and mirror photons and if the [33]. wormhole is non-orientable. To elucidate this problem As an illustrative example, we consider the Bondi- one needs to consider the “charge without charge” con- Hoyle accretion onto the wormhole with a mass M lo- cept with the electric charge q moving repeatedly along cated at the distance l = 30 kpc from the Galactic center, the winding up trajectory. Let us suppose that we see −26 −3 where the local density (24) is ρH ≃ 5.7 × 10 g cm . the ordinary electric charge, which several times enters The mass rate of the Bondi-Hoyle accretion, i. e., the rate the left throat and exits from the right throat. After the of transformation of the mirror matter into the our bary- first passage through the non-orientable wormhole, the onic matter is charge becomes the mirror one. After the second pas- 2 2 sage the charge becomes the ordinary, and so on. The ˙ 4πG M ρ M = 3 (25) effective electric charge (“charge without charge”) of the v throat can be expressed through the electric flow as usual M 2 v −3 ≃ 1.6 × 1010 g s−1, 2 −1 1 10 M⊙  200 km s  Q = EdS. (26) 4π I where v is a sound speed or the wormhole throat velocity through the ambient gas. We choose the virial velocity As a result, in the case of the non-orientable wormhole, as a distinct and representative value. The accretion en- we will see only half of the “charge without charge” in ergy is radiated outward if the throat radius is smaller comparison with the net charge of the corresponding ori- than the accretion disk radius. The corresponding lumi- entable wormhole. More exactly, after the n passages the ˙ 2 nosity of this object is LX = ηMc , where η ≃ 0.1 is left and the right throats will have the charges a typical efficiency of the accretion mass-energy conver- sion. For the normalization parameters used in (25), the n +1 n − ∼ × 30 −1 Ql = q ,Qr = q , (27) corresponding luminosity is LX 1.4 10 erg , i. e.,  2  h 2 i it is at the level of faint stars luminosity. This radiation is emitted in the mirror world, passed partially through the where [...] is the integer part of a number. This is the wormhole and is emitted finally in the our world. The specific feature of the non-orientable wormhole electro- passage of light through a wormhole was examined in dynamics. Note that similar conclusion is true for the [34, 35]. Therefore, some non-identified X-ray objects in lepton, baryon and other conserved quantum charges. the Galaxy can be related, in principle, with the mirror Now we consider the following thought experiment. matter outflows from the non-orientable wormholes. Let there are two close Coulomb interacting charges q1 and q2. Lets the q2 went through the non-orientable wormhole and approached the q1 again. The q2 is now V. ELECTRODYNAMICS OF THE the mirror charge, so the direct interaction through the NON-ORIENTABLE WORMHOLES short line is impossible. But there is still the long way through the non-orientable wormhole. The charges can The mirror-to-ordinary transformations of the charged exchange by virtual photons (they transform into the particles and fields would modify also the “charge with- mirror photons on the way) through the non-orientable out charge” concept, proposed by J. A. Wheeler [22, 23]. wormhole and can experience some interaction through Let us consider the “charge without charge” construc- the long way. By other words the wormhole’s throat be- tion, where electric field lines are winded up through two comes electrically charged q = q2, and the q1 interacts wormhole throats. The strength of the resulting global with the charged wormhole’s throat. If the charges are 6 not far from the wormhole’s throat, the throat may not particles. This nonlocal or topological transformation is be the point charge, so the charge q1 can feel the small- additional to the local gravitational interaction between scale structure of the field, while it “sees” the charge q2 the ordinary and mirror matter. through the wormhole’s throat. In another thought ex- We described the hydrodynamical traversability of the periment both the charges q1 and q2 fly together through wormholes by calculating the stationary accretion flow of the non-orientable wormhole. The fundamental con- perfect fluid through the wormhole throat. The worm- stants values are the same in our and mirror worlds, hole, supported by the phantom energy, appears to be the therefore the charges will continue to interact one with traversable for this hydrodynamics flows. As the possible another with the permanent strength all the time. But astrophysical signatures of the non-orientable wormhole before the fly by the interaction is mediated by the ordi- hypothesis we suggest the observation of mass fluxes and nary electric field, and after the fly by the interaction is the non-identified X-ray sources, assuming that the dark mediated by the mirror field. Note also that there is no matter in the universe is composed of the mirror matter. any “switching” point along the way, because the ordi- As an alternative to the non-orientable wormhole, one nary or mirror nature of the mediated forces is the only can consider the Reissner-Nordstr¨om or Kerr black hole question of the side of view. global geometry with a modified connection between the external and internal universes similar to the M¨obius strip (6). The resulting non-orientable black hole would connect our universe with the internal mirror universe. VI. CONCLUSION Matter inflowing through this black hole would be con- verted into the mirror matter traversing to the internal In this paper we consider the relations between the universe. However, this is only the one way flow without non-orientable wormholes and mirror particles. The the possibility to return back. In principle, we may see main idea of our paper is in the fusion of the non- these outflows of mirrir matter, if there are white holes orientable wormholes and the mirror matter concept. in the our universe. ACKNOWLEDGMENTS In particular, it is stated that the mirror particles are transformed into the ordinary ones and vice versa while traversing through the non-orientable wormhole. This Authors acknowledge Z. Berezhiani and A. Gazizov for statement was previously applied only to the universe the enlightening discussions of the mirror matter prob- with a non-orientable topology [15]. Here this concept is lems. This research was supported in part by the grants applied to a particular solution of the Einstein equations NSh-3110.2014.2 and RFBR 13-02-00257-a. in the form of the non-orientable wormhole. In the frame- work of this hypothetical solution we expect the existence of matter flows through the wormhole throats, accompa- REFERENCES nied by the transformations between ordinary and mirror

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