In the Quest for Cosmic Rotation

Abstract

This paper analyzes the problem of global rotation in general relativity (GR) theory. Simple cosmological models with rotation and expansion are presented, which give a natural explanation of the modern values of the acceleration parameter at different red shifts without involving the concepts of “dark energy” and “dark matter”. It is shown that due to the smallness of the cosmological rotation, for its detection one should use observations that do not depend on the magnitude of the angular velocity of the Universe. Such tests include the effects of the cosmic mirror and the cosmic lens. For the first time on the basis of modern electronic catalogs the search on the celestial sphere of images of our Galaxy and other galaxies is made. Viable candidates for both effects have been found.

1. Introduction

Rotational, vortex, and chiral phenomena are widely observed in physics on all scales including astrophysics, high-energy physics, and the heavy-ion collisions. Quoting E.T. Whittaker [1]: “Rotation is a universal phenomenon; the earth and all the other members of the solar system rotate on their axes, the satellites revolve round the planets, the planets revolve round the Sun, and the Sun himself is a member of the galaxy or Milky Way system which revolves in a very remarkable way. How did all these rotary motions come into being? What secures their permanence or brings about their modifications? And what part do they play in the system of the world?” Currently, the search for the similarities of vortex structures encountered in the heavy-ion physics and in astrophysical conditions attracts a lot of attention in the literature.

The hypothesis of a universe’s rotation, in our opinion, belongs to the most intriguing issues of the modern cosmology. This idea is quite natural from the physical viewpoint and it does not contradict any of the astrophysical observations and still remains a challenging unsolved mystery.

In observational cosmology, the main difficulty for detecting a global rotation is its smallness—less than 10⁻¹³ yr⁻¹ according to the generally accepted assessment. It is impossible in the Universe to distinguish the direction corresponding to the axis of rotation, with respect to which one could notice deviations (in the standard tests) from the Friedman standard cosmology.

In theoretical cosmology, the main difficulties are related, on the one hand, to the lack of simple models of an expanding and rotating Universe in general relativity (GR) similar to Friedman–Robertson–Walker models. On the other hand, there are no convincing predictive effects of cosmic rotation that are consistent with the capabilities of the equipment of modern astronomical observatories.

Observing rotating planets, stars, galaxies, and clusters of galaxies, we naturally come to two hypotheses. The first, the rotation was originally inherent in the content of “embryonic singularity” (“primeval atom” [2]). Following Einstein’s assumption which linked properties of matter and properties of a space-time, it is natural to assume that physical rotation of matter (orbital and intrinsic) generates geometrical rotation of space-time (universal and local). Eventually, the rotation slows down and global orbital rotation becomes barely noticeable. Rotating cosmic objects are traces of that initial global rotation.

It is difficult to build a model of matter with orbital and inner angular momentum. The best candidate is the model of the perfect Weyssenhoff–Raabe fluid [3], whose elements have a “classical” spin. It remains to add the global rotation of the fluid as a whole (as a solid body for example) to the energy-momentum tensor of the Weyssenhoff fluid to obtain a realistic source for cosmology with rotation. One could attempt to do it directly, in the form of global rotation energy but this is hindered by a determination of the (local) orbital moment of inertia. Then one should look for an indirect introduction of orbital rotation.

This can be done in several ways. One of them is to consider a fluid with additional properties beyond a perfect fluid (for example, with viscosity, heat flow, or electric charge), taking into account the spin-spin and spin-electromagnetic interaction [4]. The next step is the consistent introduction of known physical fields into the energy-momentum tensor of the Universe matter. A promising generalization of the Weissenhoff fluid model from the point of view of adequacy to cosmological models with rotation, in our opinion, is the antisymmetric third rank tensor field, the sources of which are extended objects (cosmic strings, shells, or bags) in a perfect fluid (see [5] and the literature cited there) [4].

According to the second hypothesis, after the explosion of the “embryonic singularity”, the matter particles run in different directions and gravitational interaction (within the framework of the Newtonian theory) inevitably leads to rotation relative to center of mass (and relative to each other (orbital rotation)) and to their own intrinsic rotation (in GR in harmonic coordinate system [6,7]). As a result, there are no non-rotating objects in the Universe. The latter is an instructive proof of the validity of GR and the Big Bang theory. The angular velocity of global and local rotation in this case increases from zero to modern values.

Let us start with the requirement of spatial homogeneity, since the fact of large-scale homogeneity of the Universe is considered to be proven in observational astronomy. We restrict ourselves to metrics of the form (t—cosmological time; xⁱ, i = 1,2,3—three spatial coordinates)

$$d{s}^2=d{t}^2-2R{n}_{i}d{x}^{i}dt-R^2{\mathsf{\gamma }}_{ij}d{x}^{i}d{x}^j,i,j=1,2,3.$$

(1)

We assume the same law of changing of scale factor R = R(t) in all directions, therefore, all metrics (1) are shear-free. In all other aspects, (1) is space-time interval of the most general form. Here

$$n_i={\mathsf{\nu }}_a{e}_i^{\left(a\right)},{\mathsf{\gamma }}_{ij}={\mathsf{\beta }}_{ab}{e}_i^{\left(a\right)}{e}_j^{\left(b\right)},$$

(2)

where ${\mathsf{\nu }}_a,$ ${\mathsf{\beta }}_{ab}\left(\det {\mathsf{\beta }}_{ab}\ne 0\right),$ a, b = 1,2,3 are constant coefficients, $n_i=n_i\left(x^k\right),{\mathsf{\gamma }}_{ij}={\mathsf{\gamma }}_{ij}\left(x^k\right)$ are functions of spatial coordinates xⁱ on the t = const hypersurfaces, and

$$e^{\left(a\right)}=e_i^{\left(a\right)}\left(x\right)d{x}^i$$

(3)

are the invariant 1-forms with respect to the action of a three-parameter group of motion which is admitted by the space-time (1). One can prove that the isotropic expansion in (1) guarantees the isotropy of microwave background radiation (MBR) in such models [8]. Accordingly, the angular velocity of rotation should not be estimated on the basis of models with shear effects, contrary to the statements of [9].

The spatial homogeneity condition imposes a restriction on the group of motions which should act simplyransitively on the spatial (t = const) hypersurfaces. Such metrics are called Bianchi metrics, there are nine types of them. These manifolds are classified according to the Killing vectors ${\mathsf{\xi }}_{\left(a\right)}$ and their commutators $\left[{\mathsf{\xi }}_{\left(a\right)},{\mathsf{\xi }}_{\left(b\right)}\right]=C_{ab}^c{\mathsf{\xi }}_{(c)}.$ The invariant forms (3) solve the Lie equations $L_{{\mathsf{\xi }}_{\left(b\right)}}{e}^{\left(a\right)}=0$ for each Bianchi type, so models n (1) are spatially homogeneous.

Besides the three Killing vector fields, space-times (1) admit a nontrivial conformal Killing vector ${\mathsf{\xi }}_{conf}=R{\partial }_t.$The explicit form for all Bianchi metrics are given in [10]. Standard cosmologies (with ${\mathsf{\nu }}_a=0$ in (2)) are known to belong to types I, V, and IX.

The rotation tensor ${\mathsf{\omega }}_{\mathsf{\mu }\mathsf{\nu }}$ and the volume expansion scalar θ for (1) are determined in the co-moving matter characterized by (average) 4-velocity vector $u^{\mathsf{\mu }}={\mathsf{\delta }}_0^{\mathsf{\mu }}$. They read as follows:

$${\mathsf{\omega }}_{\mathsf{\mu }\mathsf{\nu }}=\{\begin{array}{l}{\mathsf{\omega }}_{0j}=0,\\ {\mathsf{\omega }}_{ij}=-\frac{R}{2}{\stackrel{⌢}{C}}_{{}_{ij}}^k{n}_k,\end{array}\mathsf{\theta }=\frac{3\dot{R}}{R},\mathsf{\mu },\mathsf{\nu }=0,1,2,3,$$

here, ${\stackrel{⌢}{C}}_{ij}^k=e_{\left(a\right)}^k\left({\partial }_i{e}_j^a-{\partial }_j{e}_i^a\right).$ The explicit form of the structure constants $C_{bc}^a,$ of the anholonomity objects ${\stackrel{⌢}{C}}_{ij}^k$ and also of ${\mathsf{\xi }}_{\left(a\right)}$ and $e^{\left(а\right)}$ are given in [10]. It is easy to notice that ${\mathsf{\omega }}_{\mathsf{\mu }\mathsf{\nu }}=0$ for Bianchi-I models.

The acceleration vector a_μ for (1) is nontrivial

$$a_{\mathsf{\mu }}=\{\begin{array}{l}{a}_0=0,\\ a_i=n_i\dot{R},\end{array}$$

for non-zero $\dot{R}$. It is easy to see that u^μ is not orthogonal to the hypersurface of homogeneity (“tilted” models).

Cosmological models (1) are parallax-free [10] (in terminology of [11]), and hence velocity of rotation cannot be estimated from parallax effects, contrary to the statements of [12,13].

It is worthwhile to mention an unusual physical property of metrics (1): there is no shear despite the non-zero rotation. In other words, despite the rotation, the expansion has the same magnitude in all directions. One usually thinks of a two-dimensional expanding and rotating elastic sphere and expects that at the poles the sphere should expand more slowly than at the equator. However imagine a two-dimensional expanding and rotating cylindrical surface: it is obvious that relative to any point on the axis of rotation, the expansion velocity is the same. The expansion along the axis of rotation can be arbitrary in magnitude; the observed MBR isotropy requires that the expansion along the axis of rotation is the same as in directions perpendicular to the axis of rotation. Note that this line of reasoning gives preference to models with non-zero initial spin in “embryonic singularity”.

The correct causal structure of space-time, that is, the absence of closed time-like curves is considered as a serious restriction on the properties of a space-time manifold with rotation. K. Gödel drew attention to the “non-physicality” of such models with rotation in his classical work [14]. S. Maitra found a simple and elegant way to describe such curves [15]. For (1) the absence of closed time-like geodesics is guaranteed when the matrix β_ab is positive-definite. Indeed, let a closed curve $x^{\mathsf{\mu }}\left(s\right),0\le s\le 1,x^{\mathsf{\mu }}\left(0\right)=x^{\mathsf{\mu }}\left(1\right)$, be everywhere time-like, i.e., for arbitrary $s,$

$$g_{\mathsf{\mu }\mathsf{\nu }}\frac{d{x}^{\mathsf{\mu }}}{ds}\frac{d{x}^{\mathsf{\nu }}}{ds}>0.$$

(4)

Choose $s_0$ as the value of the parameter s for which $\frac{dt}{ds}=0$. Such a point necessarily exists by assumption that the curve is closed: with the growth of s, the coordinate t at first increases (decreases), and then decreases (increases). We compute the square of the modulus of the 4-velocity vector tangent to $x^{\mathsf{\mu }}\left(s\right)$ at point $s_0$ for (1):

$${g_{\mathsf{\mu }\mathsf{\nu }}\frac{d{x}^{\mathsf{\mu }}}{ds}\frac{d{x}^{\mathsf{\nu }}}{ds}|}_{s=s_0}=-R^2{\mathsf{\beta }}_{ab}{e}_i^{\left(a\right)}{e}_j^{\left(b\right)}\frac{d{x}^i}{ds}\frac{d{x}^j}{ds}.$$

(5)

The right-hand side expression in (5) is always negative for a positive-definite ${\mathsf{\beta }}_{ab},$ thus contradicting the initial condition (4) that $x^{\mathsf{\mu }}\left(s\right)$ is time-like. It is important to emphasize that the presence or absence of causality is not connected in any way (except for the historic aspect [14]) with the presence or absence of cosmic rotation.

There are no other restrictions for (1).

2. Metric and Matter vs. Dark Energy and Oscillating Physics

Different cosmological scenarios can be described by the following special cases of metrics (1)

$$d{s}^2=d{t}^2-2R\left(t\right)\sqrt{B}\left(dx-zdy\right)dt-R^2\left(t\right)\left(A{\left(dx-zdy\right)}^2+d{y}^2+d{x}^2\right),$$

(6)

$$d{s}^2=d{t}^2-2R\left(t\right)\sqrt{C}{e}^{mx}dydt-R^2\left(t\right)\left(d{x}^2+D{e}^{2mx}d{y}^2+d{z}^2\right),$$

(7)

$$d{s}^2=d{t}^2-2R\left(t\right)\sqrt{K}{e}^{\left(1\right)}dt-R^2\left(t\right)\left(L{\left(e^{\left(1\right)}\right)}^2+{\left(e^{\left(2\right)}\right)}^2+{\left(e^{\left(3\right)}\right)}^2\right).$$

(8)

In (6)–(8) the following notation is introduced: A, B, C, D, m, K, L – const; $e^{\left(1\right)}=\cos y\cos zdx-\sin zdy;e^{\left(2\right)}=\cos y\sin zdx+\cos zdy;e^{\left(3\right)}=-\sin ydx+dz$. These metrics belong to Bianchi-II, Bianchi-III; Bianchi-IX classes respectively. In all metrics the rotation slows down with the growth of the scale factor R(t). The modified spinning fluid [4] describes the material source in these cosmological models. This is a continuum whose elements are charged particles with spin: in the modern epoch these are galaxies or clusters of galaxies [16,17], whereas in the early cosmological stages these are fundamental particles with spin [18]. In general, the spin-electromagnetic and spin-spin coupling between fluid’s elements is mediated by the scalar fields and antisymmetric tensor fields. As a result, the energy-momentum tensor T_μν encompasses the usual contributions from the electromagnetic and scalar fields and different types of interactions between the matter elements are described by antisymmetric tensors: the electromagnetic field F_μν; the third-rank tensor field F_μνα whose sources are extended objects (strings, shells, and bags) [5]; and F_μναβ that is dynamically equivalent to the scalar field. The coupled system of the gravitational field equations and the equations of motion of physical fields and their sources [19,20,21,22] can be consistently solved for the class of cosmological metrics with expansion and rotation. These models take into account the cosmological constant (Λ-term) of either Einstein type (Λ > 0, which corresponds to negative pressure and repulsion) or Gödel type (Λ < 0, which corresponds to positive pressure and attraction).

Let us consider the nearest non-stationary cosmological generalization of Gödel’s solution [19,21], when the gravitational field is described by the metrics (6) and (8).

In particular, for (6) we have [19]: for scale factor evolution (0 < A << 1, A/B << 1–constants)

$${\dot{R}}^2=c{R}^2-\frac{B}{4},$$

(9)

(where c is an integration constant) and for matter state equation

$$p=-\mathsf{\epsilon }=\frac{1}{\mathsf{\kappa }}\mathsf{\Lambda },$$

(10)

where $\mathsf{\kappa }=8\mathsf{\pi }G-$ Einstein’s gravitational constant of GR and rotation is

$${\mathsf{\omega }}^2={\mathsf{\omega }}_{\widehat{2}\widehat{3}}{\mathsf{\omega }}^{\widehat{2}\widehat{3}}=\frac{B}{4R^2}.$$

(11)

Analyzing (9), we see that c cannot have a negative value since B > 0. Differentiating (9) over t, we obtain

$$\frac{\ddot{R}}{R}=c>0,$$

(12)

consequently, this solution describes the accelerated expansion of Metagalaxy, which is apparently currently observed. The change of acceleration is of the Hubble type: the farther is the galaxy, the greater is its acceleration. It is assumed that scale factor is proportional to the distance between galaxies. The presence of acceleration and repulsion forces in a rotating Universe are absolutely natural physical factors to explain the accelerated scattering of galaxies. (Although it is more common to introduce dark matter and dark energy paradigm.) Rewrite (9) with (11) as

$${\dot{R}}^2+{\mathsf{\omega }}^2{R}^2=c{R}^2.$$

(13)

Equation (13) is very similar to the equality of kinetic energy (the sum of translational expansion and rotation) and the potential energy of the Universe (as in a tensile and rotating spring). Relation (10) reproduces the equation of state of matter in the Gödel model: Λ < 0; ε and p remain constant (as time goes on) despite the expansion and rotation. The relations (10) and (13) agree perfectly with each other in physical sense if we interpret the left-hand side of (13) as ε and the right-hand side as minus p. That is, at any time at small R and at large R the sum of kinetic and potential energy is zero. In this sense, this solution can be called stationary. It is also clear why the potential energy of gravity is proportional to R²: this is because the potential energy is computed inside a homogeneous cylinder of a perfect fluid.

A completely similar solution exists for the metric (8) [21], in this sense, such a solution can be considered as typical one for models of the Universe with rotation and expansion.

Let us analyze the cosmological model (6) in which the source of the gravitational field is a neutral spinning Weyssenhoff-Raabe fluid and a scalar field φ = φ(t) which determines the spin-spin coupling between elements of the fluid [4,19]. This model can be considered as the nearest generalization of the model (9) and (10). Indeed, in order to make the model (9) and (10) look more realistic, it is necessary to take into account that all the elements of the cosmological fluid (galaxies) are rotating. Therefore, Weyssenhoff-Raabe fluid provides an appropriate description for the cosmological matter. However, since one should also take into account the spin of galaxies, we cannot ignore their spin-spin interaction. It is known [6] that in GR rotating spherically symmetric gravitational bodies affect the rotation of each other. Such an interaction can be effectively modeled by means of a scalar field. Now instead of (9) and (10) we find (0 < А << 1, А/B << 1)

$${\dot{R}}^2=c{R}^2-\frac{B}{4}+\frac{2\mathsf{\alpha }}{R},$$

(14)

$$p=\frac{1}{\mathsf{\kappa }}\mathsf{\Lambda },\mathsf{\epsilon }=2{\mathsf{\chi }}_{2}S{\mathsf{\phi }}^2-\frac{1}{\mathsf{\kappa }}\mathsf{\Lambda },$$

(15)

where ${\dot{\mathsf{\phi }}}^2=12\mathsf{\alpha }/R^3,\mathsf{\alpha }=const>0,$ $S=\frac{S_0}{R^3}-$ spin density of matter, $S_0$,$-const$, ${\mathsf{\chi }}_2-$ constant of spin-spin interaction (for more details see [4,23]); it is clear that (11) is the same; c and B are positive similarly to (9).

Differentiating (14) by time, we find instead of (12)

$$\frac{\ddot{R}}{R}=c-\frac{\mathsf{\alpha }}{R^3}>0,$$

(16)

that is, if $R<{\left(\mathsf{\alpha }/c\right)}^{1/3}$ then the acceleration is negative (deceleration); if $R={\left(\mathsf{\alpha }/c\right)}^{1/3}$then the acceleration is zero, finally, if $R>{\left(\mathsf{\alpha }/c\right)}^{1/3}$ the acceleration is positive; so (16) is in a full agreement with modern observations on the dependence of the acceleration parameter on the red shift (in 1998–1999 [24,25] the conclusion about the accelerated expansion of the universe was made; over the past decade, the results [24,25] have been repeatedly tested with ever-improving statistics but the main result remained unchanged: relatively recently at z < 0.5, the Universe had the transition from decelerated expansion to accelerated one).

Let us rewrite (14) with (11) as

$${\dot{R}}^2+{\mathsf{\omega }}^2{R}^2=c{R}^2+\frac{2\mathsf{\alpha }}{R}.$$

(17)

We see that the equation again has the form of the law of conservation of energy. As compared to (13), a term corresponding to the potential energy of the gravitational field (there is no electromagnetic field) appeared in the ratio (17). However, this is exactly what was expected: to simulate (“effectively”) the gravitational interaction of spins of galaxies. The structure of (17) fully confirms the validity of this approach.

A remark is in order about the equation of state of matter. (i) For Gödel-type Λ-term we have: p < 0; ε > 0; p + ε > 0 which is to a large extent similar (10). (ii) For Λ = 0 we find: p = 0; ε > 0 – dust. (iii) For Einstein-type Λ-term: p > 0; ε ≥ 0. As one can see, there are several variants of possible states of matter, and different signs of the Λ-term are allowed. However, following the logic of gradual complication of the model the first variant looks more preferable.

In 1990 [26] an apparently periodic structure of the number of sources was demonstrated as a function of red shift in the large-scale distribution of galaxies. A possible explanation of this fact was discussed in [27]. In particular, it was noticed that the periodic structure of the universe may arise from the “oscillating physics”, e.g., oscillations of the gravitational or fine structure constants, or of a dark matter (scalar) fields. In [28] we pointed on global cosmological rotation as a possible reason of this observational effect.

Along with many known exotic and beautiful hypotheses in the history of cosmology, the cosmological rotation on the one hand does not contradict any of the known observations, on the other hand, it gives natural explanations to new discoveries in astrophysics and therefore it deserves closer attention from observational astronomy.

3. In the Theoretical Quest for Cosmic Rotation

The idea of cosmic rotation is revisited (perhaps less often than it deserves) when a certain anisotropy is detected in the Universe. The most famous attempt to explain such observations by a global rotation was due to P. Birch [29] in 1982. As it turned out later after a long discussion [30,31,32] the discovered effect, if it really exists, is not related to Universe rotation but this interpretation had nevertheless caused an explosion of interest in the theoretical development of this subject (particularly in the group of D. Ivanenko in Moscow).

Classical cosmological tests, such as apparent magnitude-red shift (m-z), number counts-red shift (N-z), angular size-red shift relations, and some other, reveal specific dependence of astrophysical observables on the angular coordinates (θ, φ) in a rotating world. Thus, a careful analysis of the angular variations of empirical data over the whole celestial sphere is necessary. The knowledge of null geodesics makes it possible to obtain the explicit form of the area distance r between an observer at a point P and a distant star S, which is a crucial step in deriving formulas for classical cosmological tests [33]. For metric (7) we derive an apparent magnitude-red shift relation (m-z).

$$\begin{array}{c}m=M-5{\log }_{10}{H}_0+5{\log }_{10}z+\frac{5}{2}\left({\log }_{10}e\right)\left(1-q_0\right)z-5{\log }_{10}\left(1+\sqrt{\frac{C}{C+D}}\sin \mathsf{\theta }\sin \mathsf{\phi }\right)\\ -\frac{5}{2}\left({\log }_{10}e\right)\frac{{\mathsf{\omega }}_0}{H_0}\frac{\sin \mathsf{\theta }\cos \mathsf{\phi }\left(\sqrt{\frac{C}{C+D}}+\sin \mathsf{\theta }\sin \mathsf{\phi }\right)}{{\left(1+\sqrt{\frac{C}{C+D}}\sin \mathsf{\theta }\sin \mathsf{\phi }\right)}^2}z+o\left(z^2\right),\end{array}$$

(18)

and number of sources-red shift relation (N-z)

$$\begin{array}{c}\frac{dN}{d\mathsf{\Omega }}=\frac{n_0{z}^3}{3H_0^3{\left(1+\sqrt{\frac{C}{C+D}}\sin \mathsf{\theta }\sin \mathsf{\phi }\right)}^3}(1-\frac{3}{2}\left(1+q_0\right)z\\ -3\frac{{\mathsf{\omega }}_0}{H_0}\frac{\sin \mathsf{\theta }\cos \mathsf{\phi }\left(\sqrt{\frac{C}{C+D}}+\sin \mathsf{\theta }\sin \mathsf{\phi }\right)}{{\left(1+\sqrt{\frac{C}{C+D}}\sin \mathsf{\theta }\sin \mathsf{\phi }\right)}^2}z+o\left(z^2\right)),\end{array}$$

(19)

here $M=-\frac{5}{2}{\log }_{10}{L}_s$ is the absolute magnitude of a light source with an intrinsic luminosity $L_s$ and $n_0$ is the modern value of number density of $n=n(t)$ (as usual, (19) is derived under the assumption of the absence of source evolution). The (N-z) relation describes the number of sources observed in a solid angle dΩ up to the value z of red shift. One can estimate the global difference of the number of sources visible in two hemispheres of the sky, $N^{+},N^{-}$ by integrating (19):

$$\frac{N^{+}-N^{-}}{N^{+}+N^{-}}=\frac{1}{2}\sqrt{\frac{C}{C+D}}\left(3-\frac{C}{C+D}\right)+o\left(z^2\right).$$

(20)

Note that (20) is independent of z and (18)–(20) obtained for nearby galaxies with z < 1. Similar formulas are not difficult to obtain for (6) and (8).

The main difficulty of using classical cosmological tests for the discovery of space rotation is that the angular velocity of the Universe is apparently very small and it is not easy to separate the effects of the evolution of the physical properties of the sources from the effects of rotation. The only possibility for observational astronomy is to detect some angular dependencies like in (18)–(20) or in similar metrics of Bianchi-type with rotation.

An experiment similar to Foucault’s pendulum, which proves the rotation of the Earth, carried out in space rocket would be also inconclusive because of the very slow rotation.

Therefore, it is necessary to look for such effects of cosmic rotation, which would not depend on how great the velocity of rotation is, but would be possible only in the rotating Universe. Such effects were predicted in 1999 [34] for Bianchi-II (6) cosmological model.

In the geometric optics approximation light rays are null geodesics, i.e., curves x^µ(λ) with an affine parameter λ and tangent vector k^µ:

$$k^{\mathsf{\mu }}=d{x}^{\mathsf{\mu }}/d\mathsf{\lambda },k^{\mathsf{\mu }}{k}_{\mathsf{\nu };\mathsf{\mu }}=0,k^{\mathsf{\mu }}{k}_{\mathsf{\mu }}=0.$$

(21)

The metric (6) has three Killing and one conformal Killing vectors:

$${\mathsf{\xi }}_{\left(1\right)}^{\mathsf{\mu }}=\left(0,1,0,0\right),{\mathsf{\xi }}_{\left(2\right)}^{\mathsf{\mu }}=\left(0,0,1,0\right),{\mathsf{\xi }}_{\left(3\right)}^{\mathsf{\mu }}=\left(0,y,0,1\right),{\mathsf{\xi }}_{\left(conf\right)}^{\mathsf{\mu }}\equiv {\mathsf{\xi }}_{\left(4\right)}^{\mathsf{\mu }}=\left(R,0,0,0\right).$$

(22)

Therefore, we easily get four first integrals of null geodesics (21):

$$k_{\mathsf{\mu }}{\mathsf{\xi }}_{\left(J\right)}^{\mathsf{\mu }}=-q_J,$$

(23)

where $q_J$ are constants along the null geodesics (J = 1, 2, 3, 4). From (22) and (23) we obtain

$$k^{\widehat{0}}=-\frac{q_4}{R},k^{\widehat{1}}=\frac{q_1+\sqrt{B}{q}_4}{\sqrt{A+B}R},k^{\widehat{2}}=\frac{q_2+q_{1}z}{R},k^{\widehat{3}}=\frac{q_3-q_{1}y}{R}.$$

(24)

It is convenient to use the local orthonormal tetrad $h_{\mathsf{\mu }}^a$ (Greek indices µ, ν, ... = 0, 1, 2, 3 refer to the local coordinates; a, b, ... = $\widehat{0},\widehat{1},\widehat{2},\widehat{3}$ are tetrad indices):

$$h_0^{\widehat{0}}=1,h_1^{\widehat{0}}=-\sqrt{B}R,h_2^{\widehat{0}}=\sqrt{B}Rz,h_1^{\widehat{1}}=\sqrt{A+B}R,h_2^{\widehat{1}}=-\sqrt{A+B}Rz,h_2^{\widehat{2}}=R,h_3^{\widehat{3}}=R.$$

(25)

We assume that an observer has coordinates P(t = t₀, x = 0, y = 0, z = 0). The null geodesics through P are labeled with the help of spherical angles θ and φ which define direction of a light ray in the local Lorentz basis of the observer at P:

$$k_P^{\widehat{0}}=1,k_P^{\widehat{1}}=\sin \mathsf{\theta }\cos \mathsf{\phi },k_P^{\widehat{2}}=\sin \mathsf{\theta }\sin \mathsf{\phi },k_P^{\widehat{3}}=\cos \mathsf{\theta }.$$

(26)

From (25) and (26) we find the constants q_J:

$$q_4=-R_0,q_1=R_0\left(\sqrt{B}+\sqrt{A+B}\sin \mathsf{\theta }\cos \mathsf{\phi }\right),q_2=R_0\sin \mathsf{\theta }\sin \mathsf{\phi },q_3=R_0\cos \mathsf{\theta },$$

(27)

where $R_0\equiv R\left(t_0\right).$ Using (21), $k^0=dt/d\mathsf{\lambda },$ we can eliminate $\mathsf{\lambda }$ for $t$ and convert (24) to the following system

$$k^0=\frac{dt}{d\mathsf{\lambda }}=\frac{R_0}{R}\left(1+\sqrt{\frac{B}{A+B}}\sin \mathsf{\theta }\cos \mathsf{\phi }\right)\equiv \frac{q_0}{R},$$

(28)

$$\frac{dx}{d\mathsf{\tau }}=\frac{q_1-R_0\sqrt{B}}{q_0\left(A+B\right)}+\frac{z\left(q_2+q_{1}z\right)}{q_0},$$

(29)

$$\frac{dy}{d\mathsf{\tau }}=\frac{q_2+q_{1}z}{q_0},$$

(30)

$$\frac{dz}{d\mathsf{\tau }}=\frac{q_3-q_{1}y}{q_0},$$

(31)

where $d\mathsf{\tau }\equiv \frac{dt}{R}.$ Solving (29)–(31), we get the exact form of null geodesics equations for q₁ ≠ 0:

$$\begin{array}{c}x=-\frac{q_2{q}_3}{2q_1^2}+\mathsf{\tau }\left(\frac{q_1-R_0\sqrt{B}}{q_0\left(A+B\right)}+\frac{q_2^2+q_3^2}{2q_1{q}_0}\right)+\frac{q_2{q}_3}{q_1^2}\cos \left(\frac{q_1}{q_0}\mathsf{\tau }\right)-\\ -\frac{q_2^2}{q_1^2}\sin \left(\frac{q_1}{q_0}\mathsf{\tau }\right)-\frac{q_2{q}_3}{2q_1^2}\cos \left(\frac{2q_1\mathsf{\tau }}{q_0}\right)+\frac{q_2^2-q_3^2}{4q_1^2}\sin \left(\frac{2q_1\mathsf{\tau }}{q_0}\right),\end{array}$$

(32)

$$y=-\frac{q_3}{q_1}\cos \left(\frac{q_1}{q_0}\mathsf{\tau }\right)+\frac{q_2}{q_1}\sin \left(\frac{q_1}{q_0}\mathsf{\tau }\right)+\frac{q_3}{q_1},$$

(33)

$$z=\frac{q_2}{q_1}\cos \left(\frac{q_1}{q_0}\mathsf{\tau }\right)+\frac{q_3}{q_1}\sin \left(\frac{q_1}{q_0}\mathsf{\tau }\right)-\frac{q_2}{q_1}.$$

(34)

From (33) and (34) we get

$${\left(y-\frac{q_3}{q_1}\right)}^2+{\left(z+\frac{q_2}{q_1}\right)}^2=\frac{q_3^2+q_2^2}{q_1^2},$$

(35)

i.e., null geodesics in Bianchi-II metric (6) are circular helix and pitch of these helices that may be found from (32) (Figure 1). It is easy to see that all the loops of (35) touch the x-axis.

Figure 1. One of the helix-geodesics (32)–(35).

Time-like geodesics are circular helices too and all massive test particles in the Bianchi-II space-time (6) move along spirals [34].

Here we come to the particular case of the null-geodesics set in Bianchi-II rotating world: the existence of closed 3-curves. The necessary condition of their existence is

$$\frac{q_1-R_0\sqrt{B}}{q_0\left(A+B\right)}+\frac{q_2^2+q_3^2}{2q_1{q}_0}=0.$$

(36)

Then (32)–(34) describe periodic functions of τ. The condition (36) gives the directions (θ, φ) on the celestial sphere, along which the light rays are closed (compare (36), (27)–(28)):

$$\sin \mathsf{\theta }\cos \mathsf{\phi }=\frac{\sqrt{-A}-\sqrt{B}}{\sqrt{A+B}}<0,$$

(37)

(negative values of A are acceptable for correct causality space-time structure [32]). From (32)–(34) it is clear that at the point of observation we have: τ = τ₀ = 0 and t = t₀, x = 0, y = 0, z = 0. Light returns to the same spatial point x = 0, y = 0, z = 0 at the moments of time

$${\mathsf{\tau }}_n=\frac{2\pi q_0}{q_1}n;n=1,2,\mathrm{...}$$

(38)

The existence of the closed null 3-curves gives rise to a number of new observational effects in rotating and expanding models of the Universe independent on the magnitude of the angular velocity of the Universe.

In what can be considered as a cosmological lens effect, let us assume that some galaxy is located at a point on the closed light ray (null geodesic). Then an observer can see the same galaxy in the two opposite directions (Figure 2b). In a generic case the observed galaxy is asymmetrically located relative to the opposite directions of observation. As a result, a detection of the two images of identical galaxies at different distances (with different redshifts) visible at the opposite semi-spheres of the sky would likely mean that this is the same galaxy, and we therefore observe the lens effect due to rotation of the Metagalaxy. Thus, the Universe as a whole becomes a lens.

Figure 2. Possible observational effects in the rotating and expanding models independent of the magnitude of the angular velocity of the Universe: (a) cosmological mirror effect; (b) cosmological lens effect; (c) cosmological shadow effect.

In a different situation which could be described as a cosmological shadow effect, let us assume that two galaxies (or more) are located on the closed light ray (Figure 2c). Then the observer can see only the two closest galaxies, which screen one another and the other galaxies on this ray.

The lens (which was described above) is “broken” and the image of a galaxy is not duplicated.

This would also mean (though unlikely) that we may not be able to observe many galaxies that are in shadow of other galaxies. This effect can qualitatively (at least partially) contribute to the problem of the hidden mass.

Finally, one can think of cosmological mirror effect as follows. Suppose, that there are no galaxies on a closed light ray (Figure 2a). Then an observer would be able to see his own galaxy from the different sides. Thus, provided we are living in a rotating world, we would have a chance to look at ourselves from aside. The Universe as a whole becomes a mirror (similar local mirror effect can take place in the Kerr gravitational field of a rotating compact source). An astronomer on the Earth, by discovering absolutely identical images of galaxies in the pairwise opposite directions on the sky sphere, may in fact happen to observe just one and the same native Galaxy. Additionally, there could be many such mirror reflections! This would be an observational evidence of the existence of the universal rotation, even for a small value of the cosmic vorticity. In practice of course all the closed light paths may be blocked by other galaxies, which breaks the cosmic mirror.

These effects are due to the geometry and topology of our physical space and are inherent only to the rotating (no matter how fast) world. In contrast, the standard Friedman type universe with expansion does not have closed light rays, and the same is true for general expanding non-rotating models. It is worthwhile to clarify the understanding of “the galaxy is on the closed null-geodesic”. Here we consider a null-geodesic, satisfying the condition (37) and labeled by angles θ and φ on the celestial sphere. The set of such curves forms a cone with a vertex at the observation point and intersecting with the celestial sphere along a circumference with the radius $\sqrt{2/\left(1-\sqrt{-B/A}\right)}$ and the center on x-axis: $x=\left(\sqrt{-A}-\sqrt{B}\right)/\sqrt{A+B}$ the points of which satisfy the condition n (37).

The real galaxy is an extended object, its image is a spot—not a point. The observer receives a bundle of light rays from the object of observation. In geometric optics approximation, rays of light are null-geodesics: the telescope receives a truncated light cone with one base at the galaxy and another base at the telescope lens (Figure 3).

Figure 3. Formation of an image of the galaxy on a photographic plate of the telescope located in an arbitrary place on the celestial sphere. G—Galaxy; T—Telescope of an observer.

Let us call the middle ray GT in Figure 3 central, let it correspond to the central null-geodesic. The path of the rays in the formation of the image in the telescope lens with a closed central null-geodesic is shown on Figure 4a. It is clear that only a central geodesic “returns” exactly to the point from which it “started”.

Figure 4. Loopback of the light cone: (a) the path of the rays which form the image of the Galaxy (cosmic mirror effect) in the telescope lens with a closed central null-geodesic; a thin dotted line shows the light cone between the Galaxy and the observer, the thick dotted line is the central null-geodesic; (b) an illustration of the fact that only the central ray is closed; (c) the Ouroboros (figure copied from the site “Ya-webdesign.com”—images for free download).

Other geodesics, taking part in the formation of the image in the telescope lens, have labels which do not satisfy (37), so they are not closed. Indeed, consider the “inverse” problem: imagine that an observer on Earth does not have a telescope, but a flashlight to detect the Galaxy in the darkness of the Universe (Figure 4b). Taking into account a continuity property, null-geodesics continuously fill the light cone around the central null-geodesic.

Some of them fall on the Galaxy and they form an image on the photo-plate of the telescope (light rays coming back); the rest is scattered in the Universe; and only a bunch of null-geodesics passing through the points (37) of the segment crossing the Galaxy will return exactly to the original places.

Figure 4a,b looks like the Ouroboros—a snake eating its own tail (Figure 4c).

It is thus reasonable to search for the effects of the cosmic lens and cosmic mirror on our sky. Clearly there are many galaxies and many other opaque objects in the Universe that can block the Milky Way image. However, the snapshots of the Galaxy are sent out along the (formally infinite) set of null-geodesics (37), so there is a non-zero possibility that at least one of them can return to us.

Analyzing the catalogs of galaxies and quasars in the quest for cosmic rotation, it is necessary to take into account that the equations of null-geodesics in the real world may differ from the equations of ideal homogeneous models. Local gravitational inhomogeneities can disrupt the smoothness of curves (the observer on the Earth is located at the edge (at the tail of spiral arm) of a spiral Galaxy, in the center of which there is the strongest center of gravity, possibly a black hole).

Important, though indirect, evidence of the rotation of the Universe could be the detection of anisotropy in: (i) the distribution of the directions of the axes of galaxies, i.e., the existence of some predominant direction of orientation of the axes; and (ii) the directions of rotation of galaxies: clockwise or counterclockwise. Theoretically, the existence of such anisotropy follows from the use of the Weyssenhoff–Raabe fluid model [3] as a cosmological material medium [17,35].

4. In the Experimental Quest for Cosmic Rotation

Let us now turn to the analysis of the experimental data accumulated in several online electronic catalogs of galaxies and quasars. We focus on looking for two effects, namely the cosmic mirror effect and the cosmic lens effect, described above. These effects will show up themselves if the Earth crosses a closed null geodesic and if we manage to find two similar object images, located on two opposite sides of this null geodesic. Every pair of candidate objects must fulfill the condition of being mutually opposite on the celestial sphere.

For the cosmic lens effect, both objects in the pair must be of the same type (an elliptical galaxy or a spiral galaxy or a quasar). Their redshift may be different as those objects can actually be a double image of the same object located at an arbitrary point of the closed null geodesic.

However when looking for the cosmic mirror effect, we require both objects in the pair to have also the same redshift values (ideally, all other astrometric parameters should be the same as well). According to modern ideas, most of elliptical galaxies are created as a result of spiral galaxies collisions. On the celestial sphere we observe galaxies younger than their present state. For that reason, when searching for the effect of cosmic mirror, we limit ourselves to spiral galaxies and quasars.

For every catalog we obtain several series of pair sets characterized by their positional accuracy with which the locations of the two objects in the pair are opposite on the celestial sphere. The largest set in every series is always in the table for the cosmic lens effect. Tables for the cosmic mirror effect are subsets of tables for the lens effect.

4.1. RCSED (Reference Catalog of Spectral Energy Distributions of Galaxies) Catalog

We begin with the data of the electronic catalogue RCSED (Reference Catalog of Spectral Energy Distributions of galaxies) [36], which became public in 2016. This catalog contains spectroscopic and photometric data for 800,299 galaxies with low and intermediate redshift (0.007 < z < 0.6) selected from the Sloan Digital Sky Survey DR7 spectroscopic sample. A special program written to work with this catalog compares declination (dec), right ascension (ra), and redshift of galaxies (z). It turned out that the information about galaxies in the opposite parts of the sky with respect to celestial equator is extremely uneven. Only 65,610 galaxies (8.2%) out of 800,299 have a negative declination. That is, despite the impressive size of the catalog, we are able to use for our purpose only a small part of its volume which, of course, dramatically reduces the chances of detecting the effects of the cosmic mirror and lens. A second restriction comes from uneven angular distribution of observations done in northern and southern celestial hemispheres. Minimal value of the declination angle in this catalog is only −11.25° what means that all possible candidates for pairs are distributed close to the celestial equator. The last restriction is that for the effect of the cosmic mirror, only spiral galaxies may be taken into account.

First we look for the cosmic mirror effect. With an accuracy of 0.01° for ra and dec and 0.01 for z, we have found 187 pairs of galaxies, composed of 371 galaxies, where three galaxies are found in more than one pair (Appendix A, Table A1). To reduce the number for pair candidates we can require a better accuracy. Within an accuracy of 0.001° for ra and dec and 0.01 for z, we have three pairs of galaxies (Table 1).

Table 1. Pairs of opposite galaxies for the effect of cosmic mirror [36]. Id SDSS DR7—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and 0.01 for z.

	Id SDSS DR7	Ra	Dec	z
(1)	587730818432041321	328.93761990	−7.06320875	0.0592138
	587732579378724931	148.93835864	7.06251179	0.0602015
(2)	587726879409635950	311.33617957	−5.13025247	0.100322
	587732703390204231	131.33587944	5.12942269	0.094418
(3)	588848899931963795	224.12922043	−0.10150706	0.376950
	588015509288648969	44.128540880	0.10090797	0.371090

All three pairs of galaxies in Table 1 may be interpreted not as real galaxies, but only as images of the Galaxy on the celestial sphere. All three pairs are not too far from us (z < 0.4), so they can be quite “fresh” pictures of the Galaxy (from the recent past), provided they are of spiral type.

Note that the accuracy in z in all three pairs in Table 1 is actually better than 0.006. The first pair has an even better match for z with accuracy value 0.001 (Table 2).

Table 2. The pair with the best positional and redshift accuracy for the effect of cosmic mirror in catalog [36]. Id SDSS DR7—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and 0.001 for z.

	Id SDSS DR7	ra	dec	z
(1)	587730818432041321	328.93761990	−7.06320875	0.0592138
	587732579378724931	148.93835864	7.06251179	0.0602015

RCSED catalog [36] contains also redshift error values zerr. For example, the first and second galaxy listed in Table 2 have zerr values of 0.000106854 and 0.000167721, respectively. Other objects in this catalog have similar zerr values so our search for galaxy pairs requiring they match in redshift z up to 0.001 is reasonable. These redshift values are from SDSS DR7 and in newer data releases they may change. For example, in SDSS DR15 the first galaxy in Table 2 has a listed redshift value z of 0.0591773 with zerr value of 0.0000133428. We see that these data are mutually consistent and that newer data are more precise.

However, the spectroscopic observations are not always available to determine an object’s redshift with good precision. For faint objects it is very difficult to measure good spectra. In those cases the redshift may be estimated from colors, it is then called the photometric redshift. To illustrate possible differences in precision between outcomes of these two methods, let us take as an example the first galaxy from Table 2 (named SDSS J215545.02-070347.5). Its photometric redshift in DR7 is 0.074914 with error value zerr of 0.019358. In DR15 we find photomeric redshift of 0.076662 with zerr value of 0.017785. zerr/z ratio for photometric redshift is 0.26 in DR7 and 0.23 in DR15. This is 100–1000 times greater than the same ratio computed for spectroscopic redshift values. Our next example is the galaxy with maximum listed redshift value of z = 0.599891 (zerr = 0.0002402) in RCSED catalog. Its name is SDSS J125400.43+523835.4 and the listed photometric redshifts in DR7 and DR15 are z = 0.554839 with zerr = 0.018274 and z = 0.559419 with zerr = 0.026632, respectively. Although zerr/z ratio here is slightly better, the spectroscopic redshift value (0.6) is not even in the interval given by z and zerr values.

Next we will work also with objects having only photometric redshifts. To be safe, when filtering photometric redshifts, we should not consider the listed redshift values to be “good data” in the sense that at least several most significant digits in z values are valid. Instead, the listed photometric z values should be considered as only estimates of unknown true redshift values. We will work also with some other catalogs, which are collections of objects from various sources. In these catalogs, the situation is even worse, because the order of zerr estimations may vary considerably from one source to another and at the same time, we are not sure how good these estimates of z and zerr are.

This is why we use also an alternative approach for filtering objects when looking for the effect of cosmic mirror. In this approach for all objects with the listed redshift values we use the same estimate for zerr value equal to z/2. Redshift z then represents an interval of values from 0.5z to 1.5z, and we consider two objects to be possible pair candidates if their redshift intervals overlap. There are cases when this more conservative approach may actually prove to be more realistic compared to our first approach in which z values are considered to be accurate numbers.

Filtering RCSED catalog [36] in this way with ra and dec accuracy of 0.001° gives 18 candidate pairs for the effect of cosmic mirror (Appendix A, Table A3). Among them, there are also all three galaxy pairs from Table 1 (pairs 7,9,18).

Requiring a relaxed z accuracy, one can then require better positional accuracy for the effect of cosmic mirror. In RCSED, one has three galaxy pairs with ra and dec accuracy of 0.0005° (Table 3).

Table 3. Pairs of opposite galaxies for the effect of cosmic mirror [36]—alternative approach. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.0005° for ra and dec and z/2 for z.

	Id SDSS DR7	Ra	Dec	z
(1)	587726879416648036	327.43326873	−7.24482558	0.120794
	587732770503458957	147.43340352	7.24510724	0.172101
(2)	587722981749620931	201.81109033	−1.15290891	0.0672574
	587731514217726115	21.810947880	1.15332655	0.0825304
(3)	588848899373138039	173.86518822	−0.55499635	0.105208
	588015509803565082	353.86519822	0.55470851	0.241575

Let us note that when filtering galaxies which are not too distant (like those in the RCSED catalog) in the search for opposite pairs there is no sense trying to get better positional (angular) accuracy than 0.0001° for ra and dec parameters. For example, the galaxy SDSS J215545.02-070347.5 (first galaxy in Table 2) has ra, dec coordinates of 328.93761990, −7.06320875 in DR7 and of 328.937618974, −7.063200482 in DR15. These are actually the same coordinates because their decimal parts have only five significant digits. Although in newer catalogs decimal parts of ra, dec may have more significant digits, for extended objects like galaxies it has no meaning. For example, when looking at the image of SDSS J215545.02-070347.5 we see its angular size is about 20 arcseconds = 0.0056°. Even the most distant galaxy in the RCSED catalog (SDSS J125400.43+523835.4 mentioned above) has the angular size of several arcseconds and 1 arcsecond = 0.00028°.

When searching for the effect of cosmic lens, it is natural to remove the restriction (filter) on the equality of galaxies redshifts z. With an accuracy of 0.01° for ra and dec, we get 2127 pairs of galaxies, composed of 4174 galaxies (76 galaxies are found in more than one pair). If we restrict ourselves to the accuracy of 0.001° for ra and dec, we have 22 pairs of galaxies (Appendix A, Table A2). These are all pairs from Table A3 plus four new pairs (1,14,19,22). Within accuracy of 0.0005° for ra and dec, we get four pairs of galaxies (Table 4).

Table 4. Pairs of opposite galaxies for the effect of cosmic lens [36]. Id SDSS DR7 is objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.0005° for ra and dec.

	Id SDSS DR7	Ra	Dec	z
(1)	587726879416648036	327.43326873	−7.24482558	0.120794
	587732770503458957	147.43340352	7.24510724	0.172101
(2)	587722981749620931	201.81109033	−1.15290891	0.0672574
	587731514217726115	21.810947880	1.15332655	0.0825304
(3)	588848899373138039	173.86518822	−0.55499635	0.105208
	588015509803565082	353.86519822	0.55470851	0.241575
(4)	587731511542349980	42.407177420	−0.84245813	0.023485
	587722984443019391	222.40757112	0.84279922	0.211561

Of course, Table 4 must list all pairs from Table 3 plus those pairs which do not fulfill the condition for the redshift values from Table 3. There is only one such pair—number 4.

Note that (1) all pairs in Table 4 are relatively close to us (z < 0.3), so each pair can be actually composed of doubled pictures of one nearby galaxy (different in each case); (2) any type of galaxies may be taken into account, the only limitation is that in every pair, both galaxies should be of the same type; so all the four pairs are still possible candidates because their type is still unknown, for now we only know that these eight objects are galaxies.

4.2. Kuminski and Shamir Catalogs

Next we turn to catalogs created by Kuminski and Shamir [37,38,39,40] in 2016. These catalogs contain about 3,000,000 galaxies. Catalogs [38,40] are almost of the same size (2,911,899 vs. 2,912,341 objects) and contain mostly the same objects, differing only by morphological type parameters, calculated by computer algorithm. For our purpose, there is an angular restriction for pair candidates as the minimum declination angle is only −24.9° with only 22.03% of objects having negative declination angles. In the catalog of [39], the possible angular distribution of our pair candidates is even narrower as the minimum declination angle is only −19.7° with only 11.36% from 2,638,883 objects having negative declination angles. Moreover, a lot of objects in the catalog of [39] are classified as stars. The next complication is that many celestial objects (defined by their angular position ra and dec on the sky) are contained in these catalogs several times with different IDs. As for our purpose the position of an object is a key feature, we need to remove those redundant objects. For the catalogs of [38,39,40] we found 222,647, 18,739, and 222,655 redundant objects, respectively.

Our search in the catalogs of [38,40] gave the same resulting pair objects. Accordingly, we report only results of searching in the catalog of [40]. Without the redshift parameter in this catalog, we were looking for pair candidates with possible lens effect. With an accuracy of 0.001° for both ra and dec parameters, we find 1222 pairs of galaxies, composed of 725 galaxies (385 galaxies are found in more than one pair)1. If we restrict ourselves to the accuracy of 0.0005° for ra and dec, we have 605 pairs of galaxies, composed of 265 galaxies (176 galaxies are found in more than one pair). This is still too much uncertainty, so we require even better positional accuracy. Within an accuracy of 0.0002° for ra and dec, we get 32 pairs of galaxies, composed of 30 galaxies, where 12 galaxies are found in more than one pair (Appendix B, Table A4). For the lens effect both galaxies in a pair should be of the same type. Let us define, that a galaxy is elliptical when parameter “elliptical” > 0.5 and spiral when parameter “spiral” > 0.5. Then from 32 pairs in Table A4, only pairs 3, 6, and 8 are not composed from galaxies of the same type. We are then left with 29 valid candidates for the cosmic lens effect in Table A4.

Finally, searching in [40] for pairs with a positional accuracy of 0.00015° for ra and dec, we get five pairs of galaxies, composed of eight different galaxies and only two galaxies are found in more than one pair (Table 5).

Table 5. Pairs of galaxies for the effect of cosmic lens [40]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.00015° for ra and dec.

	Id SDSS DR8	Ra	Dec	Elliptical	Spiral
(1)	1237678888521237047	26.5696157538807	−3.036690882053760	0.009671	0.990329
	1237674469000413487	206.5697430121670	3.036761704248740	0.147295	0.852705
(2)	1237646748740026643	62.1747511145106	−1.016781419414090	0.097682	0.902318
	1237648705676575427	242.1748881053750	1.016716451442130	0.092952	0.907048
(3)	1237646748740026643	62.1747511145106	−1.016781419414090	0.097682	0.902318
	1237648705676575428	242.1748844511030	1.016720360108510	0.078252	0.921748
(4)	1237655177615638991	128.6682349968410	−0.926547264053684	0.883225	0.116775
	1237649942587703356	308.6681411440870	0.926418898103171	0.861631	0.138369
(5)	1237648720134537650	128.6682518312900	−0.926538367783351	0.902699	0.097301
	1237649942587703356	308.6681411440870	0.926418898103171	0.861631	0.138369

It is interesting that every pair in Table 5 has a high probability of both galaxies being of the same type. These five galaxy pairs are thus quite realistic candidates for discovering the cosmic lens effect.

The catalog of [41] adds a photometric redshift parameter zphot for objects in [40] and this is what we need to search for possible pair candidates for cosmic mirror effect. With an accuracy of 0.001° for ra and dec and 0.01 for zphot we get 115 pairs of galaxies with 52 galaxies in more than one pair. However, for cosmic mirror effect, both galaxies in pair should be spiral. Again, let us consider galaxy to be spiral if parameter “spiral” > 0.5. Then, keeping only spiral galaxies, we are left with 77 candidate pairs for cosmic mirror effect, with 36 objects in more than one pair (Appendix B, Table A5).

If we search the catalog [41] with a better accuracy for zphot, namely 0.001, we find seven pairs of galaxies with two galaxies in more than one pair (Table 6).

Table 6. Pairs of galaxies for the effect of cosmic mirror from catalog [41]. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.001° for ra and dec and 0.001 for zphot.

	Id SDSS DR8	Ra	Dec	Elliptical	Spiral	Zphot
(1)	1237666409525739588	21.8109112443	1.15334613372	0.398	0.602	0.0798
	1237651709428367652	201.811088818	−1.15291424464	0.185	0.815	0.0789
(2)	1237678617427378399	21.8109254471	1.15335605428	0.343	0.657	0.0794
	1237651709428367652	201.811088818	−1.15291424464	0.185	0.815	0.0789
(3)	1237660237652623499	21.8109336601	1.15333316620	0.356	0.644	0.0710
	1237648702974525748	201.811090479	−1.15290686150	0.179	0.821	0.0713
(4)	1237666340801871988	21.8109380881	1.15333261091	0.413	0.587	0.0717
	1237648702974525748	201.811090479	−1.15290686150	0.179	0.821	0.0713
(5)	1237646588221128874	21.8109525056	1.15334337704	0.465	0.535	0.0707
	1237648702974525748	201.811090479	−1.15290686150	0.179	0.821	0.0713
(6)	1237646748740026643	62.1747511145	−1.01678141941	0.098	0.902	0.0756
	1237648705676575428	242.174884451	1.01672036011	0.078	0.922	0.0757
(7)	1237660776104657068	171.086130380	−0.99648723534	0.212	0.788	0.0294
	1237650011315241011	351.087087401	0.99670903212	0.738	0.262	0.0287

Here we searched [41] without filtering out elliptical galaxies. For the cosmic mirror effect we need both galaxies in the pair to be spiral. As we can see, only the seventh pair does not fulfil this condition. However the best probability for both galaxies be of spiral type has the sixth pair. Actually this is the third pair from Table 5 and at the same time it is the only pair from [41] with opposite position accuracy of 0.00015° for ra and dec and zphot accuracy of 0.001 (Table 7).

Table 7. The pair with the best positional accuracy for the effect of cosmic mirror in catalog [41]. At Table 6. this pair has the best probability for both galaxies to be of spiral type. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.00015° for ra and dec and 0.001 for zphot.

	Id SDSS DR8	Ra	Dec	Elliptical	Spiral	Zphot
(1)	1237646748740026643	62.1747511145	−1.01678141941	0.098	0.902	0.0756
	1237648705676575428	242.174884451	1.01672036011	0.078	0.922	0.0757

An alternative approach (described in Section 4.1) for searching candidate pairs for the effect of cosmic mirror, using only spiral galaxies in the catalog of [41] with positional accuracy of 0.001° for both ra and dec, gives 171 pairs, with 58 galaxies in more than one pair (Appendix B, Table A6). Three pairs with the best positional accuracy of 0.00015° for both ra and dec within this approach are collected in Table 8.

Table 8. Pairs with the best positional accuracy for the effect of cosmic mirror in catalog [41]—alternative approach. Id SDSS DR8—objID identification number in the Sloan Digital Sky Survey. Accuracy is of 0.00015° for ra and dec and zphot/2 for zphot.

	Id SDSS DR8	Ra	Dec	Elliptical	Spiral	Zphot
(1)	1237678888521237046	26.5696157538807	−3.036690882053760	0.009671	0.99033	0.0757
	1237674469000413486	206.5697430121670	3.036761704248740	0.147295	0.85271	0.1149
(2)	1237646748740026643	62.1747511145106	−1.016781419414090	0.097682	0.90232	0.0756
	1237648705676575428	242.1748844511030	1.016720360108510	0.078252	0.92175	0.0757
(3)	1237646748740026643	62.1747511145106	−1.016781419414090	0.097682	0.90232	0.0756
	1237648705676575427	242.1748881053750	1.016716451442130	0.092952	0.90705	0.0839

This table should be the subset of Table 5 and indeed it is2.

Finally we searched the catalog of [39]. In this catalog, parameters whose values are expressed by floating-point numbers have only six significant digits. Therefore, the best accuracy we can reasonably require in right ascension angle is 0.001°. Requiring accuracy 0.001° in ra and dec with z arbitrary we obtain 513 galaxy pairs (25 galaxies in more than one pair) as candidates for the cosmic lens effect, providing both galaxy types in the pair are the same. This is too many candidates to be useful, so we have to make some refinements in our search. We begin by selecting only elliptical galaxies first which we define as objects with parameter “elliptical” > 0.5. Searching for pairs of opposite galaxies in this selection with 1,185,705 objects requiring accuracy 0.001° for both ra and dec gives 106 pairs with two galaxies in more than one pair (Appendix B, Table A7). Then we select only spiral galaxies, defined as objects with parameter “spiral” > 0.5. Searching in this selection with 274,416 objects with accuracy 0.001° for ra and dec gives zero number of pairs. Searching with accuracy of 0.005° for ra and dec gives 58 pairs with two galaxies in more than one pair (Appendix B, Table A8). One can do the same also for stars (we define them as objects with parameter “star” > 0.5). There are 569,643 stars in the catalog. Although stars are too close to us and their angular velocities may be too large to be seriously considered as members of pair candidates, possibly lying on one null-geodesic, it is interesting to do such search just to compare the number of star pairs in catalog with the number of galaxy pairs. With an accuracy of 0.001° for ra and dec one finds 30 star pairs with two stars in more than one pair (Appendix B, Table A9). One can ask why searching among spiral galaxies with their half count compared to number of stars in this catalog returned no pairs. This is explained by the difference in distribution of stars and galaxies on the celestial sphere. While galaxies are distributed evenly across the whole celestial sphere, most of the stars are located near the celestial equator. With higher density near the celestial equator, stars have a better chance to participate in pairs of opposite objects.

When searching for the cosmic mirror effect, the two opposite galaxies in our pair should be spiral. We have already seen that positional accuracy of opposite spiral pairs in the catalog of [39] is within the limit of 0.005° in ra and dec. If we add requirement for equal redshift z with accuracy of 0.01, we get six pairs (Table 9).

Table 9. Pairs of spiral galaxies for the effect of cosmic mirror [39]. Accuracy is of 0.005° for ra and dec and 0.01 for z. SpecObjID is ID of optical spectroscopic object in SDSS.

	SpecObjID	Ra	Dec	z	Elliptical	Spiral
(1)	774688716978415616	5.28659	−1.03586	0.1077	0.29	0.71
	324403517626279936	185.29	1.03245	0.0993	0.29	0.71
(2)	743129709168060416	19.6432	−10.9021	0.1409	0.17	0.83
	1911855335291774976	199.642	10.8989	0.1491	0.16	0.84
(3)	1399629972804495360	57.6003	0.0114088	0.0839	0.37	0.63
	385078137475590144	237.596	−0.00662002	0.0830	0.46	0.54
(4)	1393917732184942592	155.36	8.6444	0.0450	0.13	0.87
	809694968183547904	335.363	−8.64717	0.0373	0.24	0.75
(5)	306369326917642240	156.167	0.893717	0.0937	0.41	0.58
	422226519742507008	336.167	−0.897112	0.0990	0.42	0.51
(6)	309696723461105664	161.174	−0.503556	0.1161	0.13	0.87
	425721592429963264	341.171	0.498878	0.1068	0.22	0.78

Some pairs in Table 9 have quite a high probability ratio spiral/elliptical to be considered as serious candidates for the effect of cosmic mirror, even if positional accuracy is not very good. If one requires a better accuracy in z, eventually one obtains the best pair in parameters ra, dec, and z (Table 10).

Table 10. Spiral galaxy pair with the best accuracy in ra, dec, and z in catalog [39] for the effect of cosmic mirror. Accuracy is of 0.005° for ra and dec and 0.001 for z. SpecObjID is ID of optical spectroscopic object in SDSS.

	SpecObjID	Ra	Dec	z	Elliptical	Spiral
(1)	1399629972804495360	57.6003	0.0114088	0.0839	0.37	0.63
	385078137475590144	237.596	−0.00662002	0.0830	0.46	0.54

When using an alternative approach for filtering redshifts of spiral galaxies in catalog [39] for the effect of cosmic mirror, with the same accuracy of 0.005° for ra and dec one gets 36 candidate pairs (Appendix B, Table A10). This is of course a subset of pair candidates for the effect of cosmic lens in Table A8.

4.3. GAIA Data Release 2 Quasar Catalog

GAIA Data Release 2 [42,43,44] is based on 22 months of space observations (25 July 2014–23 May 2016). This catalog is known for its most precise astrometric parameters to date. Errors of right ascension, declination, and parallax are given in miliarcsecond units and typical error values are only fractions of mas. Specific to GAIA Data Release 2 celestial object database (compared to other catalogs we searched) is the use of the ICRS reference system, implemented by the International Celestial Reference Frame through the coordinates of a defining set of quasars. Resulting objects coordinates are given in reference epoch J2015.5 unlike old reference standard J2000.0. However, the difference between ICRS coordinates and mean J2000.0 equatorial coordinates is only ≈25 mas = 0.0000069°. This is sufficient for our purpose in the sense that it allows us to identify the same celestial objects across different catalogs, using their position coordinates. Then we can directly compare pairs found in GAIA to pairs from other catalogs.

Although the main objective of GAIA space observatory is to create detailed 3D map of objects in the Milky Way, its database contains many extragalactic objects as well. We are interested in GAIA DR2 quasar catalog database, to which we will refer as GAIADR2Q catalog. A list of 555,934 GAIA DR2 objects (their source_id parameters), matched to the AllWISE AGN catalog can be downloaded from http://cdn.gea.esac.esa.int/Gaia/gdr2/, together with the list of 2880 objects matched to the ICRF3-prototype. However, in the second list, only 935 records are unique, the rest are already contained in the first list. This gives together 556,869 unique GAIA DR2 sources, matched to quasars or AGNs. The corresponding quasar database (GAIADR2Q) with astrometric and photometric parameters (97 parameters in total) can be downloaded from GAIA archive https://gea.esac.esa.int/archive/.

Being located in space, GAIA observatory produces almost symmetrical data with respect to declination sign. This is in big contrast to the asymmetry of data that is present in all catalogs of the ground-based observatories. Therefore, with a seemingly small volume of 556,869 quasars, we have almost the same number of quasars with negative (271,063 or 48.68%) and positive (285,806 or 51.32%) declinations. Angular distribution of quasars in declination angle is from −89.77° to 89.96°. All this significantly increases the chance of detecting the effects of the cosmic mirror and lens. Unfortunately, the database does not contain redshifts yet.

According to modern ideas, quasar is a pre-galaxy, i.e., a galaxy at an early stage of evolution. Therefore, when searching for opposite pairs of quasars assuming the cosmic mirror effect is real, then the young image of our Galaxy may well be found among these pairs. With the improvement of the quality of observation equipment in the future we would see our Earth and everything that happened on it in the distant past, recorded on null-geodesic as on a film.

Searching in GAIADR2Q catalog for pairs of opposite quasars with positional accuracy of 0.01° in ra and dec coordinate gives 1945 pairs, with 22 quasars in more than one pair. Searching with accuracy 0.005° in ra and dec gives 485 pairs, with two quasars in more than one pair. To reduce the count of candidate pairs to a reasonable number, we require a still better positional accuracy. In Table A11 (Appendix C), 17 pairs of opposite quasars with accuracy of opposite positions equal to 0.001° for ra and dec are given. Unfortunately, the data do not contain the information on the redshifts of quasars, so one cannot separate the quasars involved in the effect of the cosmic mirror from the quasars in the lens effect.

Finally, with an accuracy of 0.0005° in ra and dec, we get four pairs of quasars for mirror and lens effect (Table 11). The absence of redshift slightly reduces the effect of the result, but we can say that all of these four pairs can participate in the effect of the cosmic lens.

Table 11. Pairs of quasars from GAIADR2Q catalog for mirror and lens effects. Source_id—the unique identification number in GAIA DR2 database. Accuracy is of 0.0005° for ra and dec. Redshift z is unknown.

	Allwise_Name	Source_Id	Ra	Dec
(1)	J210304.14-762224.9	6368528062846000640	315.7673682	−76.37360527
	J090304.41+762224.0	1125452302531483520	135.7684166	76.37343393
(2)	J120429.60-215735.5	3493448195802870400	181.1233688	−21.95989953
	J000429.68+215736.0	2847161965440082176	1.123715043	21.9600063
(3)	J124510.71-180851.9	3522237773904469248	191.294554	−18.1478664
	J004510.62+180851.9	2782801120299995520	11.29451879	18.14802663
(4)	J035733.19-083041.3	3194958963846806144	59.38814002	−8.511587953
	J155733.13+083042.9	4454299557503538688	239.3880677	8.511930041

As quasars are point-like objects on the celestial sphere, it is important that their positions in the GAIADR2Q catalog are determined with an error (of order of 1 mas = 2.78 × 10⁻⁷ degrees), which is much smaller compared to the accuracy (0.0005°) with which objects in pairs are opposite.

4.4. Milliquas 6.3 Catalog

The Million Quasars (MILLIQUAS) catalog is maintained by Eric W. Flesch [45]. We have used the version 6.3 (16 June 2019) [46,47]. This version already uses GAIA-DR2 astrometry where available (approx. 63% of all objects in database). The number of used decimals in right ascension and declination coordinates is seven, which can then (for those 63% of all objects) be considered as a number of significant digits in the fraction part of coordinates.

Milliquas 6.3 database contains 1,986,800 objects, 525,349 (26.44%) with negative declination and 1,461,451 (73.56%) with positive declination. Positional distribution is over the whole sky, with declination values from −89.77° to 89.96°, the same as in GAIADR2Q.

We first search for pair candidates for the effect of cosmic lens. It means that we are interested only in positional accuracy with which two objects in pair are opposite and ignore their redshifts. Search with accuracy 0.01° for ra and dec gives 23,024 pairs with 1037 objects in more than one pair. This is nearly 12× more compared to the same search in GAIADR2Q catalog. Next we require the accuracy of 0.001° for ra and dec. This gives 219 pairs with every object in only one pair (Appendix D, Table A12). One can now compare Table A12 with Table A11 where corresponding 17 pairs from GAIA DR2 are given. We see that pairs 1, 2 in Table A11 are the same as pairs 2, 3 in Table A12. Pair 3 from Table A11 is missing in Table A12. Then pairs 4, 5 in Table A11 are the same as pairs 8, 9 in Table A12. Pairs 6, 7 in Table A11 looks almost like pairs 27, 29 in Table A12, but they are not the same. Thus, we see that not all quasars from GAIADR2Q catalog are contained in Milliquas 6.3 catalog as well.

Search with an accuracy 0.0005° for ra and dec gives 50 candidate pairs for the cosmic lens effect (Appendix D, Table A13). Comparing Table A13 with Table 11 we see that number of corresponding pairs in GAIADR2Q is only four. Pairs 1, 2, 4 in Table 11 are the same as pairs 1, 21, 29 in Table A13. Pair 3 from Table 11 is similar to pair 23 in Table A13, but they are not the same.

Eventually we searched Milliquas 6.3 with an accuracy 0.0001° for ra and dec which gives two pairs (Table 12).

Table 12. Pairs of quasars from Milliquas 6.3 catalog with the best positional accuracy for lens effect. Accuracy is of 0.0001° for ra and dec. Descrip column is the classification of object, see [47]. Redshift z is given, although not essential here.

	Ra	Dec	Name	Descrip	z
(1)	75.7744245	−40.7936211	WISEA J050305.91-404736.8	q	2.3
	255.7743683	40.7937126	SDSS J170305.84+404737.3	q	0.3
(2)	43.6995104	−17.1177057	WISEA J025447.88-170703.8	q	1.9
	223.699538	17.1176163	SDSS J145447.88+170703.4	q	1.1

These two pairs are not found in the GAIADR2Q catalog, because it contains only the first and the third quasar from Table 12. However, the number of significant digits in coordinates of second and fourth quasar in Table 12 may be questioned. Hopefully, the GAIA DR3 database will have also these two quasars so their coordinates will be known with better accuracy.

We now turn to the search for pair candidates for the effect of cosmic mirror. Objects in Milliquas 6.3 catalog are collected from various sources so redshift values are computed by various methods and with different relative errors. Thus, some redshifts in catalog are rounded to 0.1z, while others may have error up to 0.5z. Many redshift values in Milliquas 6.3 are estimated by Eric W. Flesch using the four-color based method described in his original HMQ article [45] Appendix 2. Therefore, although Milliquas 6.3 gives redshift values for 1,906,535 objects, it is hard to compare them using some kind of fixed filter value. We used 0.2 as the best redshift accuracy we can reasonably have and even this value may be too optimistic.

We begin with an accuracy of 0.001° for ra and dec and 0.2 for z. Search in Milliquas 6.3 then gives 27 pairs (Appendix D, Table A14). Search with an accuracy of 0.0005° for ra and dec and 0.2 for z gives four pairs (Table 13).

Table 13. Pairs of opposite quasars with the best accuracy in position and redshift found in Milliquas 6.3 catalog as candidate pairs for cosmic mirror effect. Accuracy is of 0.0005° for ra and dec and 0.2 for z. Descrip column is the classification of object, see [47].

	Ra	Dec	Name	Descrip	z	Rmag	Bmag
(1)	5.8514071	−7.0936418	SDSS J002324.33-070537.1	q	0.6	21.75	21.73
	185.8513794	7.093749	SDSS J122324.33+070537.4	qX	0.6	21.82	21.61
(2)	356.0054037	−4.4881222	SDSS J234401.30-042917.1	q	2.1	19.79	20.52
	176.0057901	4.4880115	SDSS J114401.38+042916.8	q	2.2	21.32	21.32
(3)	215.9122427	−3.1485981	SDSS J142338.93-030854.9	q	2	21.54	21.34
	35.911743	3.1485686	SDSS J022338.81+030854.7	q	2.173	20.03	20.59
(4)	148.4235067	−0.1395639	2QZ J095341.5-000822	q	0.284	19.06	19.96
	328.4236129	0.1398213	SDSS J215341.66+000823.3	q	0.1	22.55	23.38

Here we added two columns Rmag, Bmag to show also red and blue colors magnitudes to see striking similarity in these parameters in the first pair, which has also the best match in redshift.

We now repeat our search in Milliquas 6.3 database for the effect of cosmic mirror with an alternative filtering of redshift, described in Section 4.1. With an accuracy of 0.001° for ra and dec we found 135 pairs (Appendix D, Table A15). Requiring the accuracy of 0.0005° for ra and dec we found 30 pairs (Appendix D, Table A16). The pair with the best positional accuracy of 0.0001° for ra and dec is listed in Table 14. As always, every table created by alternative filtering of redshift is a subset of corresponding set of pairs for the lens effect with the same positional accuracy.

Table 14. Pair of quasars from Milliquas 6.3 catalog with the best positional accuracy for mirror effect—alternative approach. Accuracy is of 0.0001° for ra and dec and z/2 for z. Descrip column is the classification of object, see [47]. Redshift z is given, although not essential here.

	Ra	Dec	Name	Descrip	z
(1)	43.6995104	−17.1177057	WISEA J025447.88-170703.8	q	1.9
	223.699538	17.1176163	SDSS J145447.88+170703.4	q	1.1

4.5. KQCG (Known Quasars Catalog for GAIA Mission)

Authors of catalog [48] compiled QSOs and AGNs from several sources, resulting in 1,842,076 objects in total. According to the authors, the purpose of this compilation is to provide positions of known QSOs, which can be used for cross matching with the GAIA observations.

After removing two objects with duplicate positions, the KQCG catalog has actually 1,842,074 unique objects, 595,535 of them with redshift value. There are 721,960 objects (39.19%) with negative declination and 1,120,114 objects (60.81%) with positive declinaton. The catalog has good declination angle distribution (from −89.82° to 89.97°).

When searching for pair candidates for the effect of cosmic lens, we first search KQCG catalog with the accuracy of 0.01° for ra and dec which gives 19,250 pairs with 849 objects in more than one pair. Search with an accuracy of 0.001° for ra and dec gives 167 pairs with three objects in more than one pair (Appendix E, Table A17). When searching with accuracy of 0.0005° for ra and dec one gets 36 pairs (Appendix E, Table A18). The best positional accuracy we can achieve here is 0.0001° for ra and dec which gives two pairs (Table 15).

Table 15. Pairs of opposite quasars from the KQCG catalog [48] with the best positional accuracy for the lens effect. Accuracy is of 0.0001° for ra and dec. Redshift z is given when available, although not essential here.

	Name	Ra	Dec	z
(1)	J013702.04-513252.3	24.2585004	−51.5478865
	133702.07+513252.1	204.2586596	51.54780646	3.123
(2)	J035233.36-211221.7	58.1390267	−21.2060385
	J155233.38+211221.7	238.1391178	21.2060426

Next we are looking for opposite quasar pair candidates for the effect of cosmic mirror. With only one-third of objects in KQCG database having nonempty redshift value, we can not expect big numbers of pairs. Search with an accuracy of 0.01° for ra and dec and 0.1 for z gives 109 pairs (Appendix E, Table A19). Search with an accuracy of 0.001° for ra and dec and 0.1 for z gives only three pairs (Table 16).

Table 16. Pairs of opposite quasars from the KQCG catalog [48] with the best accuracy in position and redshift for the cosmic mirror effect. Accuracy is of 0.001° for ra and dec and 0.1 for z.

	Name	Ra	Dec	z
(1)	LQAC_179-001_037	179.9377083	−1.372166667	1.2752
	235945.22+012216.3	359.9384346	1.371206026	1.2224771
(2)	004034.86-003821.0	10.14525273	−0.63918039	1.8359182
	LQAC_190+000_011	190.1443333	0.638194444	1.8557
(3)	LQAC_352-000_049	352.8172404	−0.117229211	1.69638
	LQAC_172+000_041	172.8182083	0.118	1.7844

The last quasar in Table 16 may not have a required precision in the dec coordinate so the third pair in that table should be taken with care.

An alternative filtering of redshift for quasar pairs with a positional accuracy of 0.001° for ra and dec gives 16 pairs (Appendix E, Table A20). Three pairs with the best possible positional accuracy of 0.0005° in ra and dec are listed in Table 17.

Table 17. Pairs of opposite quasars from the KQCG catalog [48] as candidates for the cosmic mirror effect with the best positional accuracy—alternative approach. Accuracy is of 0.0005° for ra and dec and z/2 for z.

	Name	Ra	Dec	z
(1)	LQAC_038-009_025	38.5649606	−9.392240877	3.15853
	LQAC_218+009_031	218.5647428	9.392719278	1.79609
(2)	022538.91-012519.8	36.41214596	−1.422178345	0.7273704
	LQAC_216+001_018	216.4125417	1.4225	1.1363
(3)	LQAC_322-001_003	322.04071	−1.110389	2.0165
	LQAC_142+001_019	142.0406947	1.109906236	0.69137

5. Numerical Experiments

We searched several databases of galaxies and quasars for the presence of pairs of objects, with opposite location on the sky sphere and with certain angular accuracy. This is main feature of the null geodesics which reveals their existence in our part of the Universe. However, in a large sample of celestial objects, pairs of opposite objects may exist by pure chance also in a non-rotating Universe. With rotation, their number just should be greater and the difference should be due to the presence of closed null geodesics.

Therefore it would be useful if we could estimate number of pair objects in our part of the Universe considering the standard three-dimensional space without rotation using randomly generated objects positions on the celestial sphere. For each real global catalog, we can create a set of randomized counterparts with the same number of objects. Then, their filtering with a certain angular precision in dec and ra coordinates, will give us a set of counts of opposite pairs, from which we can estimate the corresponding values for the real world catalogs in the case of a non-rotating Universe.

Let us begin with the RCSED catalog. Although it contains 800,299 galaxies, they are located mostly on the northern part of the celestial sphere and only 221,502 of them, located within the interval from −11.25° to 11.25° in dec angle have a chance to participate in opposite pairs. Therefore it makes sense to create random catalogs of only 221,502 objects, located randomly in dec interval from −11.25° to 11.25°. Results of this simulation with four random catalogs are collected in Table 18.

Table 18. Number of opposite pairs for selected angular accuracy values eps in the Reference Catalog of Spectral Energy Distributions of galaxies (RCSED) catalog and in four random catalogs with objects evenly distributed on the celestial sphere within interval of dec values from −11.25° to 11.25°.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001	№ of Objects	Dec Interval
RCSED	214,050	2127	542	22	4	0	800,299	−11.25 … 70.29
Random 1	121,672	1204	288	14	2	0	221,502	−11.25 ... 11.25
Random 2	121,918	1261	332	14	4	0	221,502	−11.25 ... 11.25
Random 3	121,391	1269	323	14	4	0	221,502	−11.25 ... 11.25
Random 4	121,263	1250	284	11	0	0	221,502	−11.25 ... 11.25
Random 5	121,192	1163	288	6	0	0	221,502	−11.25 ... 11.26

Apparently the RCSED catalog contains a suspiciously large number of opposite pairs. However this has nothing to do with closed null geodesics. It only means that objects in the RCSED catalog within dec interval from −11.25° to 11.25° are not evenly randomly distributed in the dec coordinate. Of course this is due to the fact that data in the RCSED catalog were obtained in observatories located mostly on the northern hemisphere. The value of eps = 0.1 is not suitable for the search of closed null geodesics, because it corresponds to cone with too wide solid angle. However, it is useful for testing if our random catalogs correspond to the real world catalog3. Objects located on the null geodesic should be opposite with good angular accuracy so for our purpose values of eps from 0.001 to 0.0001 are more useful. For such eps values, our random catalogs and RCSED catalog give similar numbers of pairs. However, we should consider this with caution, because we know from results for eps = 0.1 that our random catalogs do not emulate real world data in the RCSED catalog.

Our simple experiment with evenly distributed random catalogs does not help to decide if the data in the RCSED catalog could indicate the possible presence of null geodesics. The situation is similar for other catalogs with the data from the Earth-based observatories. Fortunately, the GAIA observatory is located in space so we can expect better angular distribution of quasars in the GAIA DR2 catalog.

Let us then repeat our numerical experiment with evenly distributed random catalogs for the GAIA DR2 quasar catalog. Objects in this catalog have dec positions from −89.77° to 89.96° so our random catalogs can use the full range of dec coordinates and they should have the same number of objects as the GAIA DR2 catalog (Table 19).

Table 19. Number of opposite pairs for selected angular accuracy values eps in GAIA DR2 quasar catalog and five random catalogs with the same number of objects, evenly randomly distributed on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
GAIADR2Q	198,621	1945	485	17	4	0
Random 1	150,489	1506	401	15	5	0
Random 2	150,358	1512	381	15	3	0
Random 3	150,550	1495	382	13	5	0
Random 4	151,013	1520	372	14	4	1
Random 5	150,306	1503	357	16	6	0

Again we see that, although for small eps values, opposite pairs counts in the GAIA DR2 catalog are similar to those from random catalogs, results for higher eps values are different. This is due to the fact that objects positions in the GAIA DR2 catalog are not quite evenly distributed across the celestial sphere. Indeed, the histogram of GAIA DR2 dec angular position values shows the difference from an ideal angular distribution in dec angle, represented by the cosine function (Figure 5). We should expect also some deviation from an ideal (linear) distribution of the ra coordinate since GAIA cannot see quasars hidden behind the Milky way.

Figure 5. Histograms of dec coordinate distribution in RCSED and GAIA DR2 quasar catalogs. While Table 2. catalog shows some difference from an ideal dec angular distribution represented by the cosine function, the histogram of RCSED catalog clearly demonstrates its big angular nonuniformity.

This means that our test catalogs with evenly distributed random positions on the celestial sphere are not good emulations of GAIA DR2 positions data.

Since in all real world catalogs which we processed in Section 4, the number of opposite pairs for small eps values is very small, to find some hints for the presence of closed null geodesics, we should use as good random catalogs as possible. On a large angular scale, positions in a reasonably good test catalog should copy deviations of the real world catalog from ideal angular distribution. On a small scale, its angular positions should be distributed randomly.

The following two tables summarize the results of such a simulation for two different angular distributions. The first (Table 20) is simulating an uneven angular distribution in the RCSED catalog. The second (Table 21) is a simulation of an almost evenly distributed quasars in the GAIA DR2 catalog. Every random catalog named RandomN_x where x = 1, …, 5 contains the same number of positions as its real world counterpart, distributed to an Nx2N angular mesh in dec and ra coordinates with number of objects in every cell which copies the corresponding real world catalog. In order to provide a sufficient space for local randomness, we limited the range of N to a maximum of 180 mesh cells in dec coordinate (and 360 cells in ra coordinate).

Table 20. Number of opposite pairs for selected angular accuracy values eps in the RCSED catalog and in several random catalogs with the same number of objects and the same angular distribution on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
RCSED	214,050	2127	542	22	4	0
Random18_1	176,055	1676	417	21	4	2
Random18_2	176,584	1723	440	20	6	1
Random18_3	175,916	1728	428	22	9	0
Random18_4	176,006	1826	429	15	6	0
Random18_5	175,810	1789	425	13	4	0
Random30_1	185,460	1794	423	14	5	1
Random30_2	185,681	1881	467	26	7	0
Random30_3	187,713	1827	428	20	8	1
Random30_4	185,976	1862	466	18	5	0
Random30_5	184,818	1883	462	17	4	0
Random90_1	208,266	2062	496	27	11	0
Random90_2	208,645	2092	562	23	6	0
Random90_3	207,692	2051	491	22	4	0
Random90_4	208,046	2029	513	15	6	0
Random90_5	208,289	2077	517	25	10	1
Random180_1	212,404	2107	591	32	11	1
Random180_2	213,252	2201	559	22	4	0
Random180_3	212,848	2080	529	29	7	1
Random180_4	212,768	2077	482	16	5	0
Random180_5	213,511	2077	529	19	8	1
Average of 20 runs with N = 180				21.95	5.3	0.15
Standard deviation				3.43	1.95	0.36

Table 21. Number of opposite pairs for selected angular accuracy values eps in the GAIA DR2 quasar catalog and in several random catalogs with the same number of objects and the same angular distribution on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
GAIADR2	198,621	1945	485	17	4	0
Random18_1	196,114	1893	483	17	3	0
Random18_2	197,388	1972	479	16	5	0
Random18_3	196,415	1972	502	18	4	1
Random18_4	197,010	1984	475	22	3	0
Random18_5	198,107	1961	511	12	2	0
Random30_1	196,396	1979	470	16	2	0
Random30_2	198,441	2039	491	17	4	0
Random30_3	198,034	1926	476	19	5	0
Random30_4	198,060	2009	490	21	5	0
Random30_5	198,763	1984	510	23	3	0
Random90_1	198,481	2020	491	11	2	0
Random90_2	198,438	2044	500	21	7	0
Random90_3	197,632	2020	517	21	2	0
Random90_4	198,300	1958	472	20	2	0
Random90_5	199,184	1967	501	23	4	0
Random180_1	197,900	1888	489	14	4	1
Random180_2	198,556	2002	511	21	2	0
Random180_3	198,563	2007	487	18	5	0
Random180_4	199,210	1980	492	21	7	0
Random180_5	199,532	1962	499	19	4	0
Average of 20 runs with N = 180				19.2	4.65	0.5
Standard deviation				3.72	1.49	0.81

As expected, with higher number of mesh cells, the number of opposite pairs for larger eps values in random catalogs is closer to the corresponding values of the real world catalog. Thus random catalog with a more detailed angular mesh is indeed a better simulation of the real world catalog. Next, we see that the better the uniform angular distribution of the real world catalog, the less detailed mesh is needed for a good reproduction of its opposite pairs number for given eps value.

Tables with filtering results for real world catalogs together with data from of our random catalogs have a twofold purpose. Firstly, they show that our simulation catalogs can be useful as imitations of real world catalogs. Secondly, for large N values they show how the numbers of opposite pairs for the given real world catalog should look like in the case of a non-rotating Universe. To give a better prediction for the case of a non-rotating Universe, we attached two lines with results of 20 simulations using random catalogs with N = 180 for small eps values, together with corresponding standard deviation values.

In Table 17 and Table 18 we see that, in RCSED and GAIA DR2 catalogs, the observed number of pairs for small eps value does not exceed predicted values for a non-rotating Universe. However, it is not clear, if those eps values are small enough for our task, because observing null geodesics may require a very narrow observation cone. To register the presence of closed null geodesics, we need to estimate number of pairs in a non-rotating Universe for as small eps values as possible. From our random catalogs we see the feature shared with the real world catalogs, namely that number of opposite pairs for small eps values can be predicted from the number of pairs for greater eps values. If, for example, number of pairs for eps = 0.1 is 200,000, then we can expect about 20 pairs for eps = 0.001. This feature is expected for evenly distributed objects and from our simulations we see it remains valid for non-even angular distributions as well.

To observe several pairs for eps = 0.0001, we need hundreds of pairs for eps = 0.001 and millions for eps = 0.1. Therefore, let us proceed to emulate larger databases.

The biggest database we are dealing here with is Kuminski and Shamir [38,39,40]. We chose the catalog of [40], cleaned from repeated objects, containing only galaxies (and not stars like in [39]). The number of objects is 26,899,686 and dec values start at −24.93. Here we are filtering it without distinguishing galaxy type (Table 22).

Table 22. Number of opposite pairs for selected angular accuracy values eps in Kuminski and Shamir catalog [40] and in several random catalogs with the same number of objects and the same angular distribution on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
K&S [40]	7,612,266	76,867	18,024	1222	605	0
Random18_1	4,475,466	45,104	11,333	441	126	7
Random18_2	4,468,199	44,838	11,320	456	114	6
Random18_3	4,526,555	44,884	11,162	448	118	2
Random18_4	4,475,244	44,714	11,163	462	113	4
Random18_5	4,478,613	44,762	11,166	473	137	6
Random90_1	6,533,538	65,955	16,568	720	185	6
Random90_2	6,534,673	65,300	16,417	616	161	6
Random90_3	6,538,237	65,311	16,266	674	187	5
Random90_4	6,538,688	65,529	16,250	663	155	5
Random90_5	6,536,033	65,938	16,531	643	152	8
Random180_1	7,005,872	70,357	17,657	702	178	3
Random180_2	7,010,050	70,260	17,538	663	172	6
Random180_3	7,006,156	70,465	17,578	729	174	6
Random180_4	7,008,239	70,197	17,455	678	182	3
Random180_5	7,006,665	70,257	17,669	708	178	11
Average of 20 runs with N = 180				697.3	174.5	7.35
Standard deviation				26.7	13.8	2.52

From Table 22 we see that according to our random catalogs, number of pairs for eps = 0.0001 should be about seven. However, we found zero pairs in [40], although for close eps value of 0.0002 (not in the table) we found 32 pairs. On the other hand, the number of pairs for eps = 0.0005, 0.001 is too large compared to our random catalogs. Finally, while data in the first row for eps = 0.1–0.005 follow the approximate square rule dependence of pairs numbers on eps (like 7,612,266/76,867 ≈ (0.1/0.01)²), data for eps = 0.001–0.0001 do not follow this rule.

We have no explanation for these strange results of catalog [40] filtering. Our random catalogs have the same number of objects and the same angular distribution. Nevertheless, the number of opposite objects for eps values 0.001, 0.0005, 0.0001 is quite different even for detailed angular mesh. While it may be just large statistical deviations, we did not see such behavior in other catalogs.

Next we will emulate the quasar catalog Milliquas 6.3. Its angular distribution is not far from uniform so it should be easy to reproduce outcomes of its filtration with our random catalogs. The results are in Table 23.

Table 23. Number of opposite pairs for selected angular accuracy values eps in the Milliquas 6.3 catalog and in several random catalogs with the same number of objects and the same angular distribution on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
Milliquas 6.3	2,272,887	23,024	5795	219	50	2
Random90_1	2,261,589	22,473	5564	214	60	1
Random90_2	2,259,751	22,559	5581	215	66	3
Random90_3	2,261,151	22,391	5496	245	70	2
Random90_4	2,262,077	22,672	5702	264	55	2
Random90_5	2,262,742	22,643	5683	239	59	2
Random180_1	2,270,577	22,426	5527	219	59	2
Random180_2	2,270,319	22,829	5722	219	59	2
Random180_3	2,269,632	22,637	5630	224	53	3
Random180_4	2,270,186	23,136	5814	233	48	1
Random180_5	2,270,844	22,680	5738	237	59	1
Average of 20 runs with N = 180				225.7	57.25	2.3
Standard deviation				13.28	8.12	1.7

According to our simulations, the number of opposite pairs in Milliquas 6.3 catalog for small eps values is standard for a non-rotating Universe so there is no evidence for the presence of closed null geodesics.

Our last studied database is the the KQCG quasar catalog (Table 24).

Table 24. Number of opposite pairs for selected angular accuracy values eps in the KQCG catalog and in several random catalogs with the same number of objects and the same angular distribution on the celestial sphere.

Eps	0.1	0.01	0.005	0.001	0.0005	0.0001
KQCG	1,933,692	19,250	4735	167	36	2
Random90_1	1,930,091	19,298	4838	217	49	3
Random90_2	1,928,432	19,588	4806	201	50	5
Random90_3	1,930,965	19,463	4774	191	46	1
Random90_4	1,927,537	19,340	4821	196	48	0
Random90_5	1,930,231	19,183	4787	189	50	2
Random180_1	1,929,688	19,326	4745	182	53	2
Random180_2	1,933,921	19,436	4810	175	49	4
Random180_3	1,931,871	19,442	4892	213	43	3
Random180_4	1,930,751	19,212	4800	206	53	1
Random180_5	1,934,750	19,483	4785	161	36	4
Average of 20 runs with N = 180				190.9	47.8	1.5
Standard deviation				16.14	6	1.2

Again numbers of observed opposite pairs in KQCG catalog for small eps values are not greater than corresponding estimations from the analysis of similar random databases. We should therefore conclude that observations cannot be definitely interpreted as a consequence of the presence of closed null geodesics in our part of Universe.

We conclude this section with two remarks.

Firstly, in order to get better statistics for small eps values, we need bigger catalogs than are available today. Only then there is a hope to see them on the background of random coincidence where two objects are opposite just by chance.

Secondly, perhaps one does not need statistical methods at all. When the number of closed null geodesics is very small, say one or two, what we actually need is to observe the opposite pair in which both objects are astrophysically identified as being the images of the same cosmic object, possibly from different time of its evolution. Astrophysical analysis is even more important than the angular accuracy with which the two objects are opposite, as a light path across the Universe may be diverted by gravitational lensing of other cosmic objects. For example, let us consider the first quasar pair in Table 13. These two quasars are not only opposite with a good accuracy, but also have close values of Rmag, Bmag and redshift z parameters as well. This may well be a pure coincidence and a further astrophysical analysis may eventually prove that they are different objects. However, if one finds a similar pair and the two images are indeed from the same object, then one should seriously consider the existence of a closed null geodesic even for a single pair. If a small number of closed null geodesics is a reality, then we need to catalogize a very large number of distant cosmic objects and be lucky to find two of them located precisely on such geodesic.

6. Conclusions

The study of the rotational and chiral phenomena is of fundamental importance in astrophysics, high-energy physics, and the heavy-ion collisions [49,50,51]. In particular, it is worthwhile to mention the recent discovery of observational evidence for the coherence between galaxy rotation and neighbor rotational motions on the scale of several megaparsecs [16,17]. Moreover, the search for the similarities of vortex structures in the heavy-ion physics and in astrophysical conditions attracts considerable attention in the current research [52,53,54,55].

Here we considered possible observational manifestations of the global cosmic rotation. The goal of our analysis of modern electronic catalogues of galaxies and quasars was to find, among the observed galaxies and quasars, pairs of images that might not be real different objects, but instead that could be several images of a single object. This might have been a demonstration that the Universe is not only expanding, but also rotating. Our analysis of the number of opposite pairs did not find a clear statistical evidence of the global rotation. However, currently available catalogs are far from being perfect. They collect only a small number of very distant celestial objects with an uneven angular distribution (except for the GAIA DR2 quasar catalog). Therefore further work is needed in order to verify and improve our current results for bigger catalogs with better distribution of objects. One cannot exclude a possibility that an appearance of closed null geodesics, as a manifestation of the cosmic rotation, is rare and is not statistically significant. In that case, one needs more thorough astrophysical observations which could reveal that two opposite objects are actually images of the same object, possibly in different epoch of its evolution.