Let us assume an expanding Universe described by the CC metric in the following form (Grøn and Johannesen, 2011; Vavryčuk, 2022a): d s 2 = a 2 t − c 2 d t 2 + d r 2 1 − k r 2 + r 2 d Ω 2 , d Ω 2 = d θ 2 + sin 2 ⁡ θ d ϕ 2 , (5) where a(t) is the scale factor defining the cosmic expansion, t is the comoving (contravariant) time, c is the speed of light, k is the Gaussian curvature of the space, r is the comoving (contravariant) distance, and θ and ϕ are the spherical angles.

To avoid confusions, we have to discuss the CC metric described by Eq. 5 in more detail. This form of the metric is often used in cosmology but in a modified notation and in a different physical context. Time t is called the “conformal time” and it is usually denoted as η. However, the physical meanings of the comoving time t and the conformal time η are essentially different. Here, we consider time t in Eq. 5 as the physical comoving cosmic time. Consequently, the time-time component of the metric tensor g ₀₀ is time dependent. The rate of time varies and Eq. 5 encompasses the space expansion as well as the time dilation during the evolution of the Universe. By contrast, the conformal time η is commonly assumed to be a rescaled proper time with no physical meaning and no effects on the metric. The g ₀₀ component is still time independent as for the FLRW metric. In this way, the metric with the conformal time η describes the space expansion with a uniform rate of time and no time dilation during the evolution of the Universe.

When introducing the FLRW metric, it is often argued that g ₀₀ can be assumed to be time invariant, because we have a freedom to rescale time to keep g ₀₀ constant (Weinberg, 1972). From this point of view, the FLRW and CC metrics would be equivalent (Visser, 2015). However, this is not correct, because we cannot rescale arbitrarily a cosmological coordinate system without physical consequences. A trivial example is a relation between the space metrics for a static and expanding Universe. Both metrics can be transformed each into the other. However, this does not mean that the model of the static and expanding Universe are physically equivalent. Similarly, the CC metric and the FLRW metric are not physically equivalent, because the FLRW metric is characterized by a uniform rate of time, but the CC metric is characterized by a varying rate of time during the evolution of the Universe.

Using Eq. 5, the equation of the null geodesics, which describes propagation of photons, ds ² = 0, reads a 2 t − c 2 d t 2 + d l 2 = 0 , d l 2 = d r 2 1 − k r 2 + r 2 d Ω 2 . (6)

Consequently, we get for the comoving velocity v and the proper velocity V of photons (Vavryčuk, 2022a, his Eq. A6) v = d l d t = c , V = a c . (7) The propagation velocity of massive particles is described by the geodesic equation d 2 x μ d s 2 + Γ α β μ d x α d s d x β d s = 0 . (8) Substituting the distance element ds by the time element dt, we obtain d 2 x μ d t 2 = − Γ α β μ d x α d t d x β d t + Γ α β 0 d x α d t d x β d t d x μ d t . (9) Considering the metric tensor g _μν needed for calculating the Christoffel symbols Γ α β μ in Eq. 9 defined by Eq. 5, we get a c 2 v ̇ + a ̇ c 2 v − a ̇ v 3 = 0 , (10) hence v ̇ v 1 − v 2 / c 2 = − a ̇ a . (11) Consequently, for a massive non-relativistic particle (v ≪ c) we write v ̇ v = − a ̇ a (12) and the comoving velocity v and the proper velocity V read v = v 0 a − 1 , V = a v = V 0 , (13) where subscript “0” refers to quantities at present.

Hence, the proper velocity of photons is not constant as in the FLRW metric, but it increases in the CC metric with the expansion as a(t)c, where c is the speed of light for a = 1. By contrast, the proper velocity of massive non-relativistic particles is not affected by the Universe expansion. This is in contradiction with behaviour of non-relativistic massive particles in the FLRW metric, where the comoving velocity v depends on a as a ⁻² and the proper velocity V as a ⁻¹.

Note that the varying speed of light in Eq. 7 for the CC metric is an inevitable consequence of the time dependence of g ₀₀. The varying speed of light seems apparently against the basic principles of theory of the Special and General Relativity, but according to Einstein (1920): “The law of the constancy of the velocity of light in vacuo, which constitutes one of the fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity”...“its results hold only so long as we are able to disregards the influences of gravitational fields on the phenomena (e.g., of light)”. Since we do not study the speed of light in free-falling inertial systems but in non-inertial systems, the acceleration due to gravity is not cancelled with the gravitational field. Hence, the varying (coordinate-dependent) speed of light in the CC metric is fully consistent with GR being analogous, for example, to the varying (coordinate-dependent) speed of light known in the famous Schwarzschild solution (Weinberg, 1972).

Assuming the FLRW metric described by Eq. A-1, the Friedmann equation for the perfect isotropic fluid reads (Peacock, 1999; Ryden, 2016) a ′ a 2 = 8 π G 3 ρ − k c 2 a 2 , (14) where a′ = da/dT is the derivative of the scale factor a(t) with respect to the proper time T, G is the gravitational constant, ρ is the mean mass density, and k is the spatial curvature of the Universe at present.

In order to express Eq. 14 in the CC metric, we have to substitute the proper time T by the comoving time t and time derivative a′ = da/dT by a ̇ = d a / d t = a a ′ . Hence, the conformal Friedmann equation reads a ̇ a 2 = 8 π G 3 ρ a 2 − k c 2 , (15) where a ̇ denotes the derivative with respect to the comoving time t. Considering the matter-dominated Universe, we get 8 π G 3 ρ = H 0 2 Ω m a − 3 . (16) Eq. 15 is rewritten as (Vavryčuk, 2022a) H 2 a = H 0 2 Ω m a − 1 + Ω k (17) with the condition Ω m + Ω k = 1 , (18) where H ( a ) = a ̇ / a is the Hubble parameter, H ₀ is the Hubble constant, Ω _m is the normalized matter density, and Ω _k is the normalized space curvature. As shown in Vavryčuk (2022a), Eq. 17 describes the Type Ia supernova (SNe Ia) dimming well with no need to introduce dark energy, which is necessary in the standard ΛCDM model in order to fit the SNe Ia data.

Considering a = 1/(1 + z), the comoving time t is expressed from Eq. 17 as a function of redshift as follows d t = 1 H 0 1 + z Ω m 1 + z + Ω k − 1 / 2 d z , (19) and the proper time T related to comoving t as dT = a(t) dt reads d T = 1 H 0 1 + z 2 Ω m 1 + z + Ω k − 1 / 2 d z . (20) These relations are needed for relating observations of redshift to cosmic time.

Next, we study the influence of the expanding Universe on the local gravity field produced by a point mass. The gravity field is assumed to produce a small perturbation of the metric tensor g _μν describing the expanding space. So far, this problem has been studied under the assumption that the space expansion is defined by the standard FLRW metric (Carrera and Giulini, 2010), see Appendix A. Here, we derive equations for the gravitational orbits for the space expansion defined by the CC metric.

The homogeneous and isotropic expanding space characterized by the CC metric will be disturbed by a spherically symmetric gravitational field produced by a point mass M situated in the origin of coordinates. Since we limit ourselves to the weak gravitational effects of the point mass only, Eq. 5 is modified as follows d s 2 = a 2 t − c 2 1 + 2 α d t 2 + 1 − 2 α d r 2 1 − k r 2 + r 2 d Ω 2 , (21) where α = − G M r c 2 , | α | ≪ 1 (22) is the Newtonian gravitational potential normalized to c ², r is the comoving distance of an orbiting particle from the point mass M, and G is the gravitational constant.

Let us assume a massive non-relativistic particle (v ≪ c) orbiting in the gravitational field in the plane defined by θ = 0. The metric tensor g _μν is defined by Eq. 21. Calculating the Christoffel symbols Γ α β μ in Eq. 9, we get r ̈ − r ϕ ̇ 2 − α c 2 + a ̇ a r ̇ − α 2 ϕ ̇ 2 r − 3 r ̇ 2 r − 1 c 2 a ̇ a r ̇ r ̇ 2 + ϕ ̇ 2 r 2 = 0 , (23) r ϕ ̈ + 2 r ̇ ϕ ̇ + a ̇ a r ϕ ̇ + 2 α r ̇ ϕ ̇ − 1 c 2 a ̇ a ϕ ̇ r ̇ 2 + ϕ ̇ 2 r 2 = 0 , (24) where dots over quantities mean derivatives with respect to the comoving time t. Since |α|≪ 1 and c ² ≫ 1, terms multiplied by α or by 1/c ² in Eqs. 23, 24 can be neglected and we get the following approximate equations 1 a d d t a v r = − G M r 2 + v ϕ 2 r = f g + f c , (25) 1 a d d t a r v ϕ = 0 , (26) where v r = r ̇ is the radial comoving velocity, v ϕ = r ϕ ̇ is the tangential comoving velocity, and f _g and f _c are the radial gravitational and centrifugal forces in the comoving coordinate system. If we assume that the orbit of the particle is stationary, the radial and centrifugal forces are balanced (f _g = −f _c ) and the RHS of Eq. 25 equals zero. Consequently, we get a v r = V r = c o n s t , (27) a r v ϕ = r V ϕ = c o n s t , (28) where V ^r and V ^ϕ are the radial and tangential components of the proper velocity V. For a circular orbit in the comoving coordinates, Eqs. 27, 28 are further simplified as V r = 0 , V ϕ = c o n s t , (29) r = c o n s t , R = a R 0 , (30) where subscript “0” refers to the quantity at present.

Eq. 30 is surprising and in contradiction to the common opinion that the expansion of the Universe is without any appreciable effect on local gravitational systems (Carrera and Giulini, 2010). This opinion is based on equations for gravitational orbits in an expanding space described by the FLRW metric (see Appendix A). The equations were derived by many authors (Dicke and Peebles, 1964; Pachner, 1964; Callan et al., 1965; Cooperstock et al., 1998; Faraoni and Jacques, 2007; Sereno and Jetzer, 2007; Carrera and Giulini, 2010) and they differ from the standard Newtonian equations for orbits in the static Universe only slightly. The only difference is that the equation for the radial acceleration of an orbiting body in the gravitational field imbedded in the space with the FLRW metric contains a term a ̈ a R , which is related to the space expansion (see Appendix A, Eq. A-7). It can be shown that this term might be appreciable in dynamics of large-scale structures as galaxy clusters, but the orbits within galaxies are affected negligibly. Consequently, all gravitationally bound systems with size of galaxies or smaller should be unaffected by the space expansion.

By contrast, considering the CC metric for the expanding Universe, the evolution of the local gravitational systems is essentially different from that obtained for the FLRW metric. In particular, the CC metric predicts:

1 An increase of the proper orbital radius R _orb with the expansion irrespective of the size of the local system and an increase of the proper orbital period T _orb with the expansion. Consequently, the size of galaxies (and all other local gravitational systems) must growth in the expanding Universe. The rate of growth is (1 + z)⁻¹. The orbit is stationary in comoving coordinates but non-stationary in proper coordinates.

For modelling, we need to specify the expanding history of the Universe described by the Hubble parameter H(z). Since we do not use the FLRW metric, we cannot adopt the ΛCDM model. Instead, the CC metric described by Eq. 17 must be used. According to Vavryčuk (2022a), we use parameters Ω _m = 1.2, Ω_Λ = 0 (no dark energy), and Ω _k = −0.2 (closed Universe). The Hubble constant is H ₀ = 69.8 km s⁻¹ Mpc⁻¹, obtained by Freedman et al. (2019) from observations of the SNe Ia data with a red giant calibration. Obviously, Eqs. 17, 19 predict a quite different evolution of the Hubble parameter and a time-redshift relation than the ΛCDM model. For example, redshift z = 4 corresponds to the cosmic time of 12 Gyr for the ΛCDM model but 8.2 Gyr for the CC model (see Figure 1).

The galaxy is assumed to be formed by a bulge and disk with the following exponential density profiles (McGaugh, 2016) Σ b R = Σ b R b 0 e − r − R b 0 / R b , (31) Σ d R = Σ d R d 0 e − r − R d 0 / R d , (32) where Σ _b (R) and Σ _d (R) are the surface densities of the bulge and disk, respectively. The required parameters are for the bulge Σ _b (R _b0) = 3.6 M_⊙ pc⁻², R _b0 = 2.5 kpc, R _b = 0.3 kpc, and for the disk Σ _d (R _d0) = 99.2 M_⊙ pc⁻², R _d0 = 8.0 kpc, R _d = 2.8 kpc. The total masses of the bulge and disk are M _b = 8.5 × 10⁹ M_⊙ and M _d = 8.5 × 10¹⁰ M_⊙, respectively. The bulge is usually modelled as a prolate, triaxial bar (Binney et al., 1997; Bissantz and Gerhard, 2002), but we do not focus on modelling a 3D geometry of the bulge and its evolution, so the simplified approximation described by Eq. 31 is satisfactory.

The density profiles and the Newtonian rotation curves predicted for the used parameters are shown in Figure 2. The total rotation curve (Figure 2B, black line) is separated into two domains. The separation (critical) distance R _c is about 4 kpc (Figure 2B, dashed vertical line) and corresponds to the maximum rotation velocity associated with the disk density profile (Figure 2B, red line). For shorter distances (bulge-bar domain), the behaviour of the rotation curve is complex being affected by both the bulge and disk. For larger distances (spiral domain), the initial rotation curve is monotonously decreasing affected mostly by the total mass in the bulge-bar area. Obviously, the critical distance R _c may vary for different galaxies being dependent on their mass and density profiles. As discussed below, the rotation curve evolves in time and the final rotation curve becomes flat in the spiral domain (Figure 2B, green line).

Figure 3 shows the bulge-bar and spiral regions for the spiral galaxy UGC 6093 together with a scheme suggesting a possible origin of spirals and their temporal evolution. The bulge-bar region is characterized by a high concentration of stars and gas. Consequently, gravitational forces maintain the shape of the galaxy inside this region, irrespective of its rotation. Since the bar is continuously increasing due to the space expansion, its ends cross the region boundary and outflow from the bulge-bar region into the spiral region. In this region, the radial gravitational forces become dominant and stars and gas move as bodies in a spherically symmetric gravitational field. Their motion will be described by Eqs. 29, 30: the rotation velocity will be constant with time but the orbital radius will increase. Consequently, the rotating bar will outstrip the material in the spiral region, which will form a pattern of trailing spirals. The material will display flat rotation curves in all spirals, and the size of galaxies will increase with time.

The form of spirals depends on several factors: 1) The radius of the bulge-bar area, 2) the age of the galaxy, 3) the mass of the galaxy, density profile and its rotation velocity, and 4) the expansion history of the Universe. Therefore, we assume several alternative scenarios in the numerical modelling, where we vary some of these parameters. The radius of the bulge-bar area R _c will be considered 3, 4, and 5 kpc. The age of a galaxy will be 8.0, 8.5, and 8.8 Gyr. This age will correspond to the redshift z of 3.2, 5.6, and 11.2. The masses of the bulge and disk of a galaxy are M _b = 8.5 × 10⁹ M_⊙ and M _d = 8.5 × 10¹⁰ M_⊙, but we will also model a galaxy with a half of this mass and with a twice higher mass.

In order to simulate a more realistic galaxy evolution, we assume random conditions for the bulge-bar outflow. The critical distance R _c is not defined by a single value, but it varies according to the Gaussian distribution with the standard deviation of 0.05 kpc. Similarly, the bulge-bar outflow does not cross the domain boundary at two single points defined at the two ends of the bar by angles ϕ = 0° and 180°. Instead, angle ϕ obeys the Gaussian distribution centred at 0° and 180° with the standard deviation of 20°.

In modelling, we calculate orbits of stars and gas flowed out from the bulge-bar domain into the spiral domain and the evolution of the orbits in time. Geometry of orbits is defined by Eqs. 29, 30, in which we specify the rotation velocity at the boundary between the bulge-bar and spiral regions and the expansion history defined by the Hubble parameter H(z). Figure 4 shows the evolution of a galaxy with age of 8.8 Gyr. The galaxy started to evolve at redshift z of 11.2. The galaxy was formed just by the bulge and bar with no spirals at the beginning of the simulation. At this time, the radius of the galaxy was 4 kpc being the same as the radius of the bulge-bar area. The material, which outflowed from the bulge-bar domain due to the space expansion, formed typical trailed spirals during the galaxy evolution (Figure 4B). Hence, the radius of the galaxy increased from 4 kpc to 49 kpc during its life (Figure 4A). The rotation velocity of the material in the spirals is uniform and attains a value of 217.7 km/s. The classical problem in the galaxy dynamics called the “winding paradox” of spirals (Ferreras, 2019) cannot appear, because the radius of spirals is continuously increased and spiral arms are moving away from the bulge-bar region.

Figure 5 demonstrates a dependence of the spiral pattern on the size of the bulge-bar area. The age of the galaxy is again 8.8 Gyr and the evolution started at redshift z of 11.2. For a smaller radius of the bulge-bar area (Figure 5A, R _c = 3 kpc), the rotation velocity is slightly higher being 218.3 km/s. As a result, the orbital period is smaller and the recession of the spirals from the central part of the galaxy is not so pronounced. By contrast, for a larger radius of the bulge-bar area (Figure 5C, R _c = 5 kpc), the rotation velocity is slightly lower being 215.2 km/s. The orbital period is higher and the recession of the spirals from the central part of the galaxy is more distinct.

Figure 6 shows how is the spiral pattern affected by the age of the galaxy. The figure shows three galaxies with the age of 8.0, 8.5, and 8.8 Gyr. The corresponding redshifts are 3.2, 5.6, and 11.2. The radius of the bulge-bar area is R _c = 4 kpc and the rotation velocity is 217.7 km/s for all three galaxies. As expected, the younger the galaxy, the less evolved the spiral pattern. Consequently, the final radius of the galaxy increased from 4 kpc to 17, 26, and 49 kpc, respectively.

Finally, Figure 7 presents how is the spiral pattern affected by the mass of a galaxy. Figure 7B shows a galaxy with masses of the bulge and disk of a galaxy are M _b = 8.5 × 10⁹ M_⊙ and M _d = 8.5 × 10¹⁰ M_⊙ (rotation velocity at R _c = 4 kpc is 217.7 km/s). These values were used in all previous simulations (Figures 4–6). Figure 7A shows a galaxy, which has a twice higher mass. The corresponding rotation velocity at R _c = 4 kpc is 307.9 km/s. By contrast, Figure 7C shows a galaxy, which has a twice lower mass. The corresponding rotation velocity at R _c = 4 kpc is 153.9 km/s only. The age of the three galaxies is 8.8 Gyr and their evolution started at redshift z = 11.2. Since the galaxies have the same age, their size increases in the same way: from 4 kpc to 49 kpc. Still the spiral pattern is remarkably different for all three galaxies. The massive galaxy rotates fast and the spirals are receding slowly from the galaxy centre (Figure 7A). The galaxy with the smallest mass rotates slowly and the recession of spirals from the galaxy centre is high (Figure 7C).

As shown in the previous sections, the size of galaxies should increase with redshift as (1 + z)⁻¹ with no change of the galaxy mass. Based on observations, it is accepted that the size of galaxies evolves rapidly during the cosmic time (van Dokkum et al., 2008; van Dokkum et al., 2010; Williams et al., 2010), see Figure 8. Using observations from the Hubble Space Telescope (HST), galaxy sizes defined by the effective radius, R _e , have been extensively measured with the Advanced Camera for Surveys (ACS) and the Wide Field Camera 3/IR channel on board HST for massive galaxies at z < 3 (van der Wel et al., 2014) and z ≥ 3–4 Lyman break galaxies (LBGs) selected in the dropout technique (Trujillo et al., 2006; Dahlen et al., 2007; McLure et al., 2013). The average size is reported to evolve according to R _e ∼ (1 + z)^−B , with B ranging most frequently between 0.8 and 1.2 (Bouwens et al., 2004; Oesch et al., 2010; Holwerda et al., 2015). For example, Shibuya et al. (2015) studied the redshift evolution of the galaxy effective radius R _e obtained from the HST samples of ∼190,000 galaxies at z = 0–10, consisted of 176,152 photo-z galaxies at z = 0–6 from the 3D-HST + CANDELS catalogue and 10,454 Lyman break galaxies (LBGs) at z = 4–10 identified in the CANDELS, HUDF 09/12, and HFF parallel fields. They found that R _e values at a given luminosity decrease toward high z, as R _e ∼ (1 + z)^−B , with B = 1.10 ± 0.06 for median, see Figure 8A.

Since it is believed that the size of galaxies cannot be affected by the expansion of the Universe, the observed expansion of galaxies is explained by other mechanisms. The most popular theory suggests the growth of galaxies being produced by galaxy mergers (Naab et al., 2009; Kormendy and Ho, 2013; McLure et al., 2013; Conselice, 2014). An important role in merging of galaxies play dark matter haloes (Kauffmann et al., 1993; Mo et al., 1998). Two types of mergers are distinguished: a major merger where the stellar masses of the galaxies are comparable, and a minor merger where the stellar mass of one galaxy is much lower.

However, the idea of the galaxy expansion due to galaxy mergers is controversial for several reasons (Lerner, 2018). First, observations indicate that the major and minor merger rates are much lower to explain the galaxy expansion (Taylor et al., 2010; Man et al., 2012; Man et al., 2016). For example, Mundy et al. (2017) report approximately 0.5 major mergers at z < 3.5 representing an increase in stellar mass of 20%–30% only when considering constant stellar mass samples. As regards minor mergers, Newman et al. (2012) studied 935 galaxies selected with 0.4 < z < 2.5 and concluded that minor merging cannot account for a rapid growth of the size seen at higher redshifts. Man et al. (2016) studied massive galaxies using the UltraVISTA/COSMOS catalogue, complemented with the deeper, higher resolution 3DHST + CANDELS catalogue and estimated ∼1 major merger and ∼0.7 minor merger on average for a massive (M _* ≥ 10^10.8 M_⊙) galaxy during z = 0.1–2.5. The observed number of major and minor mergers can increase the size of a massive quiescent galaxy by a factor of two at most. Hence, additional mechanisms are needed to fully explain the galaxy evolution. Second, mergers cannot explain the growth of spiral galaxies, because mergers destroy disks as shown by Bournaud et al. (2007). Third, the idea of mergers implies an increase of stellar mass in galaxies over cosmic time. However, observations show no or slight mass evolution in time (Bundy et al., 2017; Kawinwanichakij et al., 2020).

Another basic characteristics predicted by the presented theory are flat rotation curves of spiral galaxies. In Newton theory, a velocity of stars in a rotating galaxy is controlled by gravitational and centrifugal forces only. Assuming the Newton’s gravitation law, a balance between the forces implies a decay of the orbital speed V(R) of a star with its distance R from the galaxy centre V 2 R = G M R R , (33) where M(R) is the mass of the galaxy as a function of R. Hence, the rotation curve V(R) decays as R ^−1/2 provided the most of mass is concentrated in the galaxy centre. The same applies to the improved Newtonian equations in an expanding Universe described by the standard FLRW metric. Obviously, the flat rotation curves predicted by the improved Newtonian equations in the Universe with the CC metric point to fundamental differences between both the metrics. Importantly, the flat rotation curves of spiral galaxies are observationally confirmed.

Several theories have been proposed to explain the origin and evolution of structure of spiral galaxies and to predict basic properties of spiral arms (Toomre, 1977; Dobbs and Baba, 2014; Shu, 2016). Lindblad (1962) was the first, who assumed that the spiral structure arose from interaction between the orbits and gravitational forces of the stars of the disk, and suggested to explain the spiral arms as density waves. This idea was further elaborated by Lin and Shu (1964) and Lin and Shu (1966) in their hypothesis of the quasi-stationary density waves, in which the spirals are formed by standing waves in the disk. They assume that the spiral pattern rotates in a particular angular frequency different from the rotation velocity of stars, which depends on the star distance from the galaxy centre. The formation of the global spiral pattern is considered as an instability of the stellar disk caused by the self-gravity. The density-wave theory was further developed and extended (Shu, 1970; Roberts et al., 1975; Sellwood and Carlberg, 1984; Elmegreen et al., 1999; Sellwood, 2011) and it is now the main tool for studying the gravitational stability of disk galaxies. The results are not decisive, but N-body simulations suggest that the spiral arms are transient and recurrent rather than quasi-stationary (Baba et al., 2009; Baba et al., 2013).

The density-wave theory faces, however, with several open questions and limitations. First, the theory is based on the classical Newtonian gravity, which neglects the space expansion. Second, predictions of the theory are uncertain. Still it is not clearly resolved, whether the spiral arms must be dynamic or whether the quasi-static arms are a feasible solution. Third, the observationally documented growth of the spiral galaxies with cosmic time is completely ignored and unexplained in this theory.

All the mentioned difficulties with modelling of spiral arms arise from the fact that the theory starts with rejecting an idea of spirals formed by stars and gas that remain fixed in the spirals. The reason is the so-called winding problem, when objects moving with the same orbital speed in the disk cause a differential rotation of material in galaxies (Ferreras, 2019). Since the length of orbits is shorter near the galaxy centre, the inner part of spirals winds up tighter than its outer part. Hence, a typical spiral pattern disappears after a few rotations. This idea is, however, simplistic and incorrect, because the GR effects in the galaxy evolution are ignored. If the space expansion and time dilation are considered, the galaxy size is growing and galaxy rotation speed varies with time. The spiral pattern is not destroyed, because it is continuously expanding. Consequently, the winding problem does not occur as demonstrated by numerical modelling in Section 3.

Observations confirm that the global expansion affects also the solar system. Here we mention some of prominent examples (Anderson et al., 1998; Krizek, 2012; Iorio, 2015; Krizek et al., 2015; Krizek and Somer, 2015).