In the $\mathsf{\Lambda }$CDM (Lambda Cold Dark Matter) cosmological model, based on the Friedmann-Lemaître-Robertson-Walker metric, matter moves synchronously with the expanding space. Perturbed metrics, [1,4] are a generalization of the FLRW metric. Deviations from standard cosmology are investigated by considering flat FLRW space-time in Painlev$\acute{e}$–Gullstrand coordinates [5]. With this coordinate choice the space is not expanding, although the distance between galaxies is certainly increasing. In Ref. refcite5,6 is used a distorted stereographic projection of hyperconical universes, which are 4D hypersurfaces embedded into 5D Minkowski spacetime. This model is a perturbed flat FRW metric with linear expansion, that takes the perspective of explaining Hubble tension [8]. In our model Universe is a radially inhomogeneous and described by the example of perturbed FLRW metric.

Determination of the Hubble constant using the cosmic distance ladder [9,10] and gravitational lensing [11,12] yields a difference of about 1/10 of its value. We consider the dependence of the change rate of photon energy on the direction of its motion. It is found applying the principle of the extremum of the photon’s energy integral [13,14,15,16,17] and the relationship between it and the energy of a material particle obtained using Lagrange mechanics [13].

A generalization of the flat FLRW metric is obtained by transforming the time interval where $f\left(t\right)$ is an additional metric function. The linear element takes the following form: For small r, this metric is approximated by the metric of flat expanding spacetime. The components of the Einstein tensor are given by where ($\dot{}$) denotes differentiation with respect to $x^1=ct$.

The components of the contravariant energy-momentum tensor correspond to the energy density, momentum density, energy flux through unit surfaces per unit time and stresses [18]. The energy-momentum tensor of matter in contravariant form is given by The non-zero contravariant coefficients for the metric (3) will be as follows: In the absence of angular momentum of matter ($u^3{=u}^4=0$), the non-zero components of the energy-momentum tensor (8) are given by Let’s consider a model with a cosmological constant $\mathsf{\Lambda }=0$. We will seek a solution for a region where the radial velocity $u^2$ is small, and the term of component $T^{22}$ containing its square is negligible and can be disregarded. Next, we will justify this assumption. Also, we assume that the terms of the Einstein tensor components (4)-(7) containing $r^2$ in the numerator are small and can be neglected. Given the satisfaction of this condition and for $g^{11}$, Einstein equations $G^{ij}=\chi T^{ij}+g^{ij}\mathsf{\Lambda }$ with the energy-momentum tensor components (11)-(14), yield The absence of additional energy and momentum influx implies the equality of the corresponding densities $T^{12},T^{21}$ to zero, i.e., the right-hand side of equation (16). From its left-hand side we obtain Solving this equation for f under the condition $a\left(t_0\right)=1$ we find with $f\left(t_0\right)=f_0$. Substituting this into equation (15) under the condition with average density of the Universe at the present time ${\mathsf{\rho }}_0=\mathsf{\rho }\left(t_0\right)$ yields Expressing the integral from here and differentiating it with respect to $x^1$, from the obtained equality we find This equation has a solution where $A_1$ is a constant.

From equations (17) and (18), we obtain the pressure which, due to (23), turns out to be $p=0$. In this case, from (16) under the condition of the absence of energy and momentum influx from the outside (18), we find the radial component of the matter velocity in the comoving reference frame. Thus, it turns out to be a valid assumption that the radial velocity in the region under consideration is small and may not be taken into account in the component of the energy-momentum tensor (13).

The function (19), with the scale factor of space (23), takes the form and equation (21) yields

Let’s find how the energy of a photon changes in the considered space-time applying the principle of the extremum of the photon’s energy integral [13,14,15,16,17]. The Euler-Lagrange equations for the covariant components of the energy-momentum vector of a light-like particle of unit energy with corresponding to the energy, take the form under the designation From the equality of interval (3) to zero, we obtain where $\sigma$ takes on values of ±1 depending on the direction of the light. For small deviations from radial motion: we find The expression for the time component of the covariant velocity vector is as follows: Further we will consider $fr,\dot{a}/a,\dot{f}/f$ to be small quantities. After substituting the components $u_{ph}^2$ and $u_{1ph}$ into equation (30) without accounting for higher order small quantities, it is rewritten in the form Since the covariant and contravariant temporal components of the generalized momenta of a light-like particle are related by the rate of energy change can be written in the form Substituting $u_{ph}^2$ from equation (34) and $d{p}_1/d\mu$ from equation (36) into this expression, we find Assuming that the terms in this equation containing r are small in the region under consideration, we can write In the case of the motion of light emitted towards the observer, we choose $\sigma =-1$. Then, upon substitution of (23) and (26), this expression is transformed to become In the region under consideration, the energy of a radially moving photon is given by where ${\mathsf{\rho }}_0$ - is the value of the photon frequency at the observation point. Since in the cosmological model described by an orthogonal metric, the Hubble constant is expressed as from equation (27), when comparing it with the Friedman equation it follows that $A_1$ can be a quantity of the same order as the Hubble parameter. After substituting the function f (26) without small quantities, equation (43) takes the form Since in the region under consideration the spacetime metric (3) approximates the Minkowski metric, the affine parameter will be close to $\mu =ct$.

The Lagrangian of a material particle [13] in space-time (3) is given by The motion of a material particle in a gravitational field, like that of a photon, in accordance with equation (28) at $\mu =s$ can be represented [10,14,16] as the result of the action of generalized forces which, however, do not generally form a first-order tensor. The components of the vector of generalized forces associated with $F_{\lambda }$, are related to gravitational forces ${\mathrm{F}}^k=c{F}^k$. The equations of motion (28) for the contravariant momentum are transformed into the form where represents the energy imparted to the gravitational field.

Let’s consider a particle of matter in a comoving frame, in which it is motionless (25). The component of the force vector (50) corresponding to energy, $F_m^1$, becomes zero. The rate of change of energy transferred to the gravitational field (52) will be Due to (51), the energy of the material particle changes at a rate of The derivative of the function $f\left(t\right)$ has the form Now we can express the change in energy of the material particle at the present time as In the considered region, the energy of the material particle will be

The equation (27) is rewritten in the form We assume that $f_0$ is of the same order of magnitude as $A_1$ and, consequently, with the Hubble constant. Since in the considered region the quantity $fr$ is small, taking into account the expressions for the rate of change of particle energies (47) and (56), the change in the energy of a material particle will be small in comparison with the change in the equivalent energy of a photon (44) because of where the differentiation parameters are The observed redshift of photon energy is considered as a result of the change in the ratio of their energy (44), (47) to the energy of atoms (57). In a small neighborhood without small higher-order quantities due to (59), this will be from which the value of the Hubble constant at present time can be deduced, obtained as a result of registering radially moving photons.

In this case, equation (58) takes the form

The length element with $p,q=1,2,3$ in the space-time (3) is given by that is, the space in its proper frame is homogeneous. However, the observed space in the coordinate frame appears radially inhomogeneous. The radial elements of the length of the metric (3) at constant time is less than obtained from (65) with $d\theta =d\phi =0$ and approach it with decreasing r.

Let’s consider how the energy of a photon will change in the presence of gravitational lensing. We denote the change in the energy of the photon when moving along the path $L_l$, which includes gravitational lensing, as and along the radial path as We assume that the coordinate distance $\mathsf{\Delta }r$ from the beginning of the paths to the observer is the same. Under the gravitational lensing, neglecting small quantities, the Hubble constant is determined as follows: where $\delta {\mu }_l$ and $\delta {\mu }_r$ are the change in the affine parameter along each of the paths.

The coordinate system is chosen such that when the photon moves along the path $L_l$ the condition $\theta ={\displaystyle \frac{\pi }{2}}$ is satisfied. If we take into account only the influence of the deviation from radial motion, then the difference in the time taken for the photon to travel along both paths will be where in the first approximation we consider Due to condition (33), the integral (70) transforms into the form In the considered region, we have By substituting the corresponding value from (47) into (67) and (68), we find the additional difference in energy change caused by the discrepancy in the rate of its during radial and angular motion The change in the value of the Hubble constant when light moves along a curved path will be Since for the considered motion of the photon, the relations hold, by substituting the values (73) and (9) into (75), we obtain Let’s estimate the values of $A_1$ and $f_0$ using the results of obtaining the Hubble constant with and without gravitational lensing. The Hubble constant was found through gravitational lensing of the cosmic microwave background: 68.3±1.5 $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ [11] and of radiation from the supernovae: ${66.6}_{-3.3}^{+4.1}\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$[12]. The values obtained through the extragalactic distance ladder to supernovae are 73.5±1.1 $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$[9] and ${75.4}_{-3.7}^{+3.8}$$\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$[10]. These results yield the value $\mathsf{\Delta }{H}_0={cf}_0={-7}_{-2.6}^{+2.4}$$\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, and due to relation (62) corresponding to the radial motion of the photon we obtain $c{A}_1=81.5\pm 2.4$$\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$.

When the value of $f_0$ is negative, the photon moving towards observer from the star additionally loses energy compared to the motion tangential to the circle from the center of observation or in the opposite direction. The dependence of the spacetime properties on the observer’s position is similar to the manifestation of the spacetime curvature in the non-flat FLRW model. In view of (63), the negative value of the coefficient $f_0$ excludes the possibility of explaining the presence of dark energy using this perturbed metric.

In the considered model in region with small perturbations of the FLRW metric, while conserving energy and momentum, this part of the Universe expands exponentially with zero pressure. Solution is obtained that relates the constant of relative expansion of space for a model with a zero cosmological constant. According to this model, in addition to the actual expansion of space, its additional movement in the radial direction is also established. The considered metric can also be applied to a cosmological model with a nonzero $\mathsf{\Lambda }$ -term.

The redshift in the spectra of galaxies is found using Lagrange mechanics in its application to the principle of the extremal integral of photon energy. The light used to obtain redshift through the ladder distance travels along a radial path, unlike gravitational lensing, where its pass has a tangential component. In the considered spacetime, from the observer’s standpoint, a photon loses energy faster when moving radially towards them compared to its motion with a tangential component. Based on the difference in value of the Hubble constant obtained using gravitational lensing and distance ladder the metric coefficients of considered space-time were found in a local region in which their perturbations are small. The value of these coefficients corresponds to the additional movement of space away from the observer.

This perturbation model explains the Hubble tension but does not come close to explaining the existence of dark energy. Certain parameters correspond to a region with low redshift.