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Non-Orientable Wormholes As Portals to the Mirror World

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Non-Orientable Wormholes As Portals to the Mirror World Non-orientable wormholes as portals to the mirror world V.I. Dokuchaev1, ∗ and Yu.N. Eroshenko1, † 1Institute for Nuclear Research of the Russian Academy of Sciences 60th October Anniversary Prospect 7a, 117312 Moscow, Russia (Dated: October 10, 2018) We examine the compatibility of the mirror matter concept with the non-orientable wormholes. If any particle (or classical object) is traversing through the non-orientable wormhole, it turns into a corresponding mirror particle and vice versa. The critical sonic point in the hydrodynamical flow of the perfect fluid is located exactly at the throat of the wormhole. This means that worm- holes are traversable for the fluid. The astrophysical signatures of the presence of non-orientable wormholes in the Universe are discussed. Some non-trivial aspects of the non-orientable wormholes electrodynamics are considered. I. INTRODUCTION and oscillations between the mirror and our world were considered in [4–7]. The reason for these oscillations is The concept of mirror matter was proposed in the fa- the local gravitational interaction between our and mir- mous article of Lee and Yang [1], where they formulated ror particles (the interaction terms in the Lagrangian). the idea of the P -violation in weak interactions. The This type of the gravitational interaction between the mirror matter sector (called also the “hidden sector”) of ordinary and mirror sectors was proposed in [2]. See also field theory restores the equivalence of the processes in [8–11] and references therein for quantitative cosmologi- the universe under the mirror reflection. Later the mir- cal and astrophysical aspects of the mirror matter. The ror matter concept was developed for the CP -violation passage of particles through wormholes is discussed in by Kobzarev, Okun and Pomeranchuk [2] (see [3] for the [12, 13]. Properties of the non-orientable solutions with historical retrospective and complete bibliography). a topology of the Klein bottle were studied in [14]. Briefly, the concept of mirror matter is as follows. In this article we argue that any particle is transformed Space reflection operator ~r →−~r corresponds to the in- into a corresponding mirror particle and vice versa while version operator Ir in the quantum particle space. The moving through the non-orientable wormhole. This idea commutativity with a time-translation [H, Ir] = 0 im- was first proposed by Ya. Ze’ldovich and I. Novikov [15] plies the conserved quantum number, corresponding to for the universe with a general non-orientable topology. Ir. The Landau form of the operator Ir = CP , where Quite a similar idea was discussed later in [16]. It was C is the charge-conjugation, is not satisfactory due to stated in [17] that particle is transformed into its an- existence of the CP -violation. Alternative representa- tiparticle while moving through the non-orientable worm- tion in the Lee-Yang form Ir = P R leads to the con- hole, but but process is difficult to justify in view of the cept of mirror matter, where operator R translates par- CP violation. Non-orientable wormholes are discussed ′ ticles into the mirror ones. In result, IrΨL = ΨR and in the book [18] (p. 285), where it was pointed out ′ IrΨR = ΨL, where the prime denotes the mirror par- that non-orientable wormholes are not compatible with ticles, and R and L are, respectively, the right and left the Standard Model of elementary particles. Meainwhile, components of the wave functions. These components are this problem is absent, if particles, passing through the defined by the corresponding chirality operators PL and non-orientable wormhole, are transformed not into the 5 5 0 1 2 3 arXiv:1308.0896v3 [gr-qc] 1 Aug 2014 PR. For example, PL = (1−γ )/2, where γ ≡ iγ γ γ γ anti-particles of the Standard Model, but into the corre- i and γ are Dirac matrices. The Lagrangian of the Stan- sponding mirror particles. Note, that transformation of dard Model contains only the left-handed components the ordinary particle into the mirror one and vice versa of spinors ΨL = PLΨ. The chirality operator is differ- is analogous to the winding transformation around the ent from the helicity operator, defined as ~σ · ~p/ˆ 2, where “Alice string” [19, 20]. ˆ ~p = ~p/p, ~σ = (σ1, σ2, σ3), and σi are the Pauli matrices. In principle, the non-orientable wormholes may be the Both chirality and helicity of particles are changed under space-non-orientable, time-non-orientable or even space- the mirror reflection. The helicity operator tends to the time-non-orientable [18]. We consider here only the space ≫ chirality operator only in the high-energy limit, E m. non-orientability, which is related with the P operation. Nowadays, wormholes and mirror particles are the The mirror matter sector, if it exists, changes only the challenging problems in theoretical physics. Different sense of some CPT operations under the mirror reflection observational signatures of wormholes and mirror par- and does not violate the CPT -theorem. Namely, in the ticles were proposed and elaborated. Neutrino mixing mirror-matter models the operator P is determined up to the internal symmetry operator R, which transforms any particle from the ordinary to mirror one, and vice versa. ∗ e-mail: [email protected] For this reason P R is conserved. The R operation differs † e-mail: [email protected] from the C because the CP is not an unbroken symmetry, 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.
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