SOLUTIONS OF THE STANDARD DIFFERENTIAL EQUATION OF THE DYNAMIC GRAVITATIONAL FIELD OF A FLAT GALAXY

The solution to the gravitational field equations of a flat galaxy has been found. It is shown that at the edge of the galaxy the excessively strong ordinary (unreduced) centrifugal pseudo-forces of inertia are compensated mainly by centripetal pseudo-forces of evolutionary self-contraction of matter in the background Euclidean space of expanding Universe, and not by the weak gravitational pseudo-forces at the edge of the galaxy. The strength of the dynamic gravitational field of spiral and other flat (or superthin) galaxies, according to their two-dimensional topology, is inversely proportional to the radial distance, not to its square. And this is the case, despite the inverse proportionality of the strength of individual gravitational fields of all spherically symmetric astronomical objects of the galaxy exactly to the square of radial distance. The general solution of the equations of the gravitational field of the galaxy with an additional certain parameter n is found. At possible values of n < 1, the velocity of the orbital motion of stars is slightly less than the highest possible velocity even at the edge of the galaxy. According to the General Relativity (GR) equations and the Relativistic Gravithermodynamics (RGTD) equations, the configuration of the dynamic gravitational field of a flat galaxy in a quasi-equilibrium state is standard (canonical in RGTD). That is because it is not determined at all by the spatial distribution of the average mass density of its non-continuous matter. After all, this spatial distribution of the average mass density of the galaxy's matter is itself determined by the standard configuration of its dynamic gravitational field. The standard value of the average density of mass of matter at the edge of a galaxy is determined by the cosmological constant Λ and by the difference between unity and the maximum value of the parameter b_c. And it is a non-zero standard value, despite the gravitational radius at the edge of a galaxy takes the zero value. Therefore, in the RGTD, in contrast to the GR, there can be no shortage of baryonic mass.

Keywords: galactic dynamics; spiral structure; General Relativity; flat (or superthin) galaxy; dark non-baryonic matter; Relativistic Gravithermodynamics.

1. Introduction

In 1934, Swiss astronomer Fritz Zwicky discovered that the total gravitational mass of a galaxy – calculated using the gravitational virial theorem – was significantly greater than the luminous mass observed through optical methods. This suggested that the majority of a galaxy's gravitational mass is invisible [1, 2]. According to Newton's law of universal gravitation, the rotational speed of stars within a galaxy should decrease as their distance from the center increases. However, in the 1970s, American astronomer Vera Rubin and her colleagues observed that the rotational velocity of gas in spiral galaxies remains nearly constant at varying radii. This phenomenon, known as the asymptotically flat rotation curve, remains one of the key mysteries in astrophysics [3 – 5].

The problem of the lack of enough amount of baryonic matter in galaxies has been the subject of many research works [6 – 18]. But it has not yet been solved. This research considers the possibility of a radical solution to the problem without the use of ghostly dark non-baryonic matter based on the fact that the decrease in gravitational field strength along the planes of flat galaxies is inversely proportional to the radial distance, and not to its square, and based on the spatiotemporal non-invariance of the gravitational constant [19, 20]. And the possibility of this is confirmed by the corresponding solution of the gravitational field equations of a flat galaxy [19 – 22].

Due to the fact that evolutionary decrease of the radius r of the star´s orbit is fundamentally unobservable in the intrinsic frames of references of coordinates and time (FR) of matter, this radius is the same (r=invar) in all FRs. The orbital velocities of galaxies and their stars that are observed on an exponential physically homogeneous scale of intrinsic time t [19, 23] of any observer should also be considered real in the observer´s FR. Taking this into account, a dynamic gravitational field [22] of galaxies is examined in this paper: the field in which the velocities v of the hypothetical equilibrium circular motion (r=const) of astronomical objects in equilibrium state do not depend directly on the radial coordinate r, but depend only on the value of the coordinate vacuum velocity of light  of general relativity (GR) or on the equivalent limit velocity of matter motion v_l (or v_lс) of the relativistic gravithermodynamics (RGTD) [19, 20, 23]. Thus, unlike the modified Newtonian dynamics proposed by Mordechai Milgrom [24, 25], both in the orthodox GR and in its modification by the RGTD, the speed of orbital motion of astronomical objects in a flat galaxy, albeit indirectly, still depends on their radial distance to the center of the galaxy.

Because of this, the Λ-reduced (evolutionarily weakened) centrifugal pseudo-force of inertia [19, 20]:

which “balances” (compensates) the gravitational pseudo-force in a rigid FR of matter, depends in GR and RGTD on the cosmological fundamental constant  and, therefore, on the Hubble fundamental constant . The fundamental invariability of these constants in the intrinsic time t of matter ensures the continuity of the intrinsic space of a rigid FR [19, 20].

Here:  is ordinary (unreduced) centrifugal pseudo-force of inertia;  is centripetal evolutionary pseudo-force, which pushes matter towards the center of the galaxy, thereby compensating within the galaxy (when r<Λ^-1/2) the centrifugal gravitational pseudo-force, which is responsible for the evolutionary distancing of other galaxies from it according to Hubble's law;  is the radius of the event pseudo-horizon, which covers the entire infinite fundamental space of the Universe [19] in the FR of any matter due to the fundamentally unobservable in FR of people´s world evolutionary self-contraction (in fundamental space) of matter spiral-wave microobjects, which are the so-called elementary particles [19, 26 – 28].

Therefore, astronomical objects in distant galaxies move in stationary, rather than divergent spiral orbits precisely due to the presence (in the observer's FR) of the action on them not only of gravitational, but also of evolutionary centripetal pseudo-force. And it is precisely this evolutionary centripetal pseudo-force that causes these same astronomical objects to move in convergent spiral orbits in the comoving with expanding Universe FR (CFREU) [19] (due to fundamental unobservability (in the observer's FR) of evolutionary self-contraction of matter in fundamental space of the CFREU).

The dependence of Λ-reduced centrifugal pseudo-force of inertia exactly on the intrinsic value of the object's orbital velocity  actually compensates for the non-identity of its inertial mass  to the much larger gravitational mass m_gr and thereby provides the possibility of using a single galactic value  of the gravitational constant in the FR_g of the galaxy. But in the FR_si of each of the stars of this galaxy there may be their own values  of the gravitational constant [19 – 22], according to which the planets and satellites rotate relative to them. Similarly, in the FR_E of the Earth, each of the distant galaxies may also have its own gravitational constant . The failure to take this into account, together with the failure to take into account the two-dimensional topology of flat galaxies, are the main reasons for the imaginary need for dark non-baryonic matter in the Universe. After all, compensation for the mutual non-identity of the inertial and gravitational masses of only the most distant galaxies does not provide compensation for the mutual non-identity of the inertial and gravitational masses of their stars.

The second reason for the imaginary need for dark non-baryonic matter may be the erroneous conclusion about the presence of relativistic dilation of intrinsic time for objects of distant galaxies. Because of this, objects of distant galaxies should move in their intrinsic time at much higher speeds than according to the clock of an observer from the Earth. In fact, galaxies “fall” onto the event pseudo-horizon by inertia, and therefore their objects, which also rotate relative to centers of galaxies by inertia, do not have dilation of intrinsic time at all [19, 20]. The galaxies that cooled down, and therefore were previously much larger, always had (and still have) non-rigid FRs. The variable function u(v) [22], which corresponds to a non-rigid FR and at which there will be no need in dark non-baryonic matter in a flat galaxy, can be determined using the parameter “ε” for any such galaxy.

It turned out that this dynamic gravitational field corresponds well to flat (or superthin) galaxies in which, for not very large values of the parameter n <1 [19 – 22], the velocity v of the orbital motion of stars is slightly less than the highest possible velocity even at the edge of the galaxy [29 – 31].

Based on the hypothesis of the spiral-wave nature of both microobjects of matter [26, 27] and the entire Universe as a whole [28], the author came to the conclusion that the coordinate vacuum velocity of light in GR and the equivalent limit velocity of motion of matter in RGTD are hidden intranuclear parameters of matter. It is precisely with this hypothesis that the another hypothesis is connected: the hypothesis of a quantum change in the location (discrete microscopic shift of the coordinates of the spiral waves nuclei) of matter in space, which occurs with the de Broglie frequency in the process of orbital motion of matter in a quasi-equilibrium state (its motion by inertia in a quantum-variable gravitational field). The quantum change in the parameters of the gravitational field of the galaxy occurs together with the quantum change in the quasi-equilibrium thermodynamic state of matter, which quantumly cools with the de Broglie frequency [19, 32].

Richard Tolman [33] showed that in the quasi-equilibrium state of continuum matter, the values of its extranuclear thermodynamic parameters and, above all, the absolute temperature T are correlated (coordinated) with the intranuclear parameter v_cv (Tv_cv=const(r)). Of course, such correlation is absent in the entire non-continuous matter of the galaxy. That is why Albert Einstein [34] was right when he stated that, unlike the metric tensor, the energy-momentum tensor he proposed is similar to low-quality wood. This is what prompted the author to endow matter with other hidden intranuclear parameters and potentials [19]. Analysis of the relationships between these intranuclear parameters and potentials led to the conclusion that the gravitational and inertial masses of matter are equivalent to each other only according to the intrinsic clock of this matter [19, 35]. It turned out that in the general case, the gravitational mass is equivalent to the Lagrangian of the ordinary total energy of matter, and the inertial mass is equivalent to the Hamiltonian of only the matter inert free energy, only which is similar to the Gibbs free energy of thermodynamics [19, 35]. This is what allowed author to confirm Dirac's hypothesis about the existence of an evolutionary change in the gravitational constant [36]. The justification of the spatiotemporal non-invariance of the gravitational constant also contributed to this [19, 20]. If its value  is conditionally invariant according to the intrinsic clock of matter, its effective value  depends on the value of the limit velocity of motion v_lc, and therefore, in distant galaxies it increases with the approach to the event pseudo-horizon. But in the observer's FR this is revealed not only in a way of increase in the gravitational constant, but also in a way of increase in the value of the gravitational mass of matter relatively to the value of its inertial mass m_gr=m_inc²v_lc^-2. It is precisely ignoring of this that leads to the imaginary need for dark non-baryonic matter in a flat galaxies.

The use of the difference between the values of the densities of gravitational and inertial masses (instead of the negligible value of pressure p in cosmic space) in the differential equations of the dynamic gravitational field (spatial distribution of potentials of this field is formed by moving matter) made it possible to obtain a standard (canonical in RGTD) differential equation, in which the spatial distributions of the gravitational and inertial masses of galactic matter are absent. Thus, the gravitational field equations themselves began to determine standard distributions of the mass of matter in a flat galaxy, depending on the variable function u(v), on the certain value of the parameter n and on the value of mass of the loose nucleus of the galaxy.

In these equations, as in the equations of Lorentz-invariant thermodynamics [19, 37 – 39], the kinematic parameters v and Г(v, v_cv) and the parameters a and b of the hypothetical static gravitational field are hidden internal parameters of matter. Not only all fundamentally measurable thermodynamic parameters, but also the parameters а_c and b_c (or analogous to them parameters а_s and b_s) of the dynamic gravitational field depend on those hidden internal parameters. Therefore, the direct use of v and Г(v, v_cv) of galaxy objects and а and b of the hypothetical static gravitational field in these equations is unnecessary. And this is an important feature of both the GR gravitational field equations and the RGTD gravitational field equations derived from them. In the very solution of the canonical differential equation of the gravitational field, both the logarithmic potential of the gravitational field [19, 20], and the identity (for all observers) of the spatial distribution of the relative value of limit velocity of motion precisely according to the gravity-quantum clock of a separate point i in its pseudo-centric ^icFR [19, 20] are used:

where:  and  are the intrinsic and observed external values of the gravitational “constant” of a galaxy, respectively;  is a parameter of the dynamic gravitational field in the intrinsic FR₀ of a galaxy.

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