Black Holes or objects without event horizon?

In this paper, we investigate the question of whether the generally accepted interpretation of the supermassive object in M87, investigated by EHT collaboration, is the only possible one. There are grounds for this, in particular, due to the detection of a magnetic (cid:28)eld in its vicinity. For this purpose, the stability of self-gravitating objects is investigated based on the correct de(cid:28)nition of the energy of gravitation in the framework of the bi-metric approach to the theory.


Introduction
In work [1], bi-metric equations of gravitation were considered, according to which the gravitation of a point mass does not have a singularity. The force of gravity acting on a test particle at rest is where r g = 2GM/c 2 is the Schwarzsschild radius of mass M, r is distance from the center, and f = (r 3 g + r 3 ) Singularity is missing in this model. The gravitational force decreases within the Schwarzschild radius, and eventually tends to zero along with the distance from the center.
Typically, the Schwarzschild radius is small compared with the dimensions of celestial objects. That is, almost always the condition r r g is executed and therefore F coincides with Newton's law.
However, this solution leads to the possibility of the existence of supermassive objects without an event horizon. The possibility of this was considered even before such an object was discovered in the center of our galaxy [2]. In   [3] it was proposed as an alternative to black holes. Later, this possibility was studied in more detail in [13, 2, ?, 14]. However, the nature of such objects remained unclear.
Here new result is presented in viewed on EHT [4,5] collaboration results.
2 Energy of gravitation A correct expression for the density of the gravitational eld in the bi-metric model can be obtained directly from the expression for the force (1) in Minkowski space.
We dene the potential of the force F as U (r) =´r 0 As a result, we obtain that function U (r) satises the dierential equation where the mass density is ρ = t 00 /c 2 .
This equation has a clear physical meaning. This is the Poisson equation for a spherically symmetric eld, in which ρ is the mass density of the gravitational eld created by the mass M , since in the absence of another matter only the density of the gravitational eld can lead to the fact that the potential will dier from zero.
We can also write where κ = 4πG/c 4 which gives, when integrating over entire spacê where E k is the internal inergy of the M 0 formed by N protons in the volume V = 4 /3πR 3 . The second term V t 00 is the gravitational binding energy in volume V . Here and r g = 2GM 0 /c 2 . These graph are shown in g. 2 for several values of mass M 0 . In equation 4, E k is equal [10]: m e and m n are mass of electron and neutron, respectively.
These graphs show that all congurations of self-gravitating objects are stable. The graphs also show that the mass of the completely degenerate Fermi gas not exceeding 10 39 g. Apparently, objects with large masses have a more complex structure, their outer layers have a low density.
It is also impossible to exclude gross errors in determining the mass of such objects [17]. However, in any case, one might think that objects with masses less than 10 39 g are a viable alternative to black holes.

Light deection
Let's nd the radius of the ring of light from the object in M87. This can be done in two ways. First, the deection angle, dened as the angle between the asymptotic incoming and outgoing trajectories, obtained in [6] for an arbitrary Lagrangian like used in [15] based on some Weinberg method.
, r is the distance from the object center, B = f 2 , r g is the closest approach distance of the light, u is the impact parameter, dened as the distance between the black hole and each of the asymptotic photon trajectories, related to r m by u = B(rm) Second, we can use Nemirov's simple and elegant method [7], suitable for at space. This gives In this equation R is the radius of the object,r g is its Schwarzschild radius, D is the distance from an observer.
The both method gives the same result -0.22 µuas that is consistent with observation [4]. However, a quantitative comparison is not possible for the result [5] due to the strong blurring of the inner boundary of the photon ring.
In addition, it should be noted that the existence of a magnetic eld in the vicinity of the object is dicult to reconcile with the generally accepted interpretation of the M87 object as a Black Hole.
Thus, in fact, we do not have a quantitative test for a condent conclusion about the nature of M87.
There is, however, another method for this, which is discussed in the following sections.
5 Schwarzschild radius of the visible Universe

Motion of a uid
The expanding Universe can be viewed as an isentropic uid. It is shown in [15] that the macoscopically small elements (particles) of such a spherically symmetric uid are described in exact accordance with the relativistic Euler equations by the following Lagrangian where G αβ = χ 2 g αβ , η αβ is a solution of our vacuum equations of gravitation in Minkowski space, χ = w/ρc 2 where w is the volume density of the enthalpy where w is the enthalpy per unit volume, ρ is the density, t 00 is the density of gravitational binding energy for mass M , p is the pressure.
The integral of motion is This equation is the relativistic Bernoulli equation in Minkowski space.
The constant has a meaning of the energy of a particle and , therefore, this equation can be read as where the constant E = E/mc 2 is the dimensionless energy E of the particles of the uid and v =ẋ. At present, the pressure is negligible and therefore χ = 1 + /ρc 2 , and for small distance from an observer and atĒ = 1, the velocity v(r) is proportional to the distance r from the observer:

Acceleration of Universe as a test gravity properties
The Lagrange equation obtained from (7)  This graph shows that the acceleration of distant particles rapidly tends to zero at distances to observed objects in excess of the value R g = c 3 8πGρ 1/2 were ρ is the density of the observed Universe because R g = 1.6 · 10 28 cm at ρ = 6 · 10 −30 g/cm 3 .  The same should be said about the force of gravity, which is the acceleration of a unit mass.
Therefore, this value R g can be called the gravitational radius of the visible Universe.
At any point in the innite universe, only gravity within the radius R g aects the observed matter.
The graph shows that the acceleration of the expansion of the universe, which cannot explain general relativity, is a consequence of the behavior of gravitation near the Schwarzschild radius R g of the Universe.
The reason for the acceleration of the expansion of the Universe is that its observable size approaches the Schwarzschild radius R g of the Universe. This is a measurable quantity. And it is this that makes it possible to test the conclusion about the existence of black holes and the existence of a singularity.
For example, you can nd the deceleration parameter: gives some condence in the above result.
The observation gives the present-day value of this parameter q 0 as approximately −0.55. Graph 4 shows q as a function of distance from the observer in the theory used here.
Graph 4 shows that the observed value of pre sent-day q 0 ≈ −0.55 corresponds to the geometrical distance of the remote objects of about 10 28 cm. It is consistent with g. 3 that shows the acceleration of distance object.

Appendix
Physical meaning of the bimetricity In paper [1], the opinion was substantiated, which goes back to Rosen [8], that the theory of gravitation should be bi-metric. However, this assumption needs serious substantiation. The book [11] is devoted to this, and below is a brief substantiation of the description of gravitation in the Mikowski space.
In Einstein's classical theory, coordinates play a twofold and dicultly compatible role. On the one hand, coordinates are only a way to parameterize events, that is, points in space-time. From this point of view, they are completely arbitrary. On the other hand, they play the role of gauge transformations.
Rosen [8] was the rst to recognize the need for introducing Minkowski space into theory. The possibility of considering Einstein's equations in at space was also considered by some authors after the paper of Tirring [9].
The physical meaning of bimetrism used in the present paper is based on ideas going back to Poincaré, who realized that there is a strange situation: In order to characterize the properties of the geometry of space, we must know the properties of measuring instruments, and in order to characterize the properties of instruments, we must know the geometric space properties. In the modern interpretation, this can be summarized as follows: The physical, operational sense has only the aggregate "space-time geometry + properties of measuring instruments" [11]. In this form, this fact has never been realized in physics.
However, a step in its implementation can be made if we notice that it is the reference frame used by the observer is the measuring tool that is necessary to establish the geometric properties of space-time. Therefore, it should be assumed that the following statement is true: Only the combination space-time geometry + properties of the reference frame used has physical meaning.
Of course, by reference frame we mean here not a coordinate system, but a physical device consisting of a reference body and a clock attached to it.
Remembering all the above, we now consider a classical eld F in an inertial reference system (I RF ), where space-time according to experience is Minkowski space. The world lines of the particles of mass m moving under the action of the eld F form the reference body of a non-inertial reference which can be named the proper reference frame (P RF ) of the eld F .
If an observer in a P RF of the eld F is at rest, his world line coincides with Du α /dσ = 0.
(We mean that an arbitrary coordinate system is used.) The same should occur in the P RF used. That is, if the line element of space-time in the P F R is denoted by ds, the 4-velocity vector ζ α = dx α /ds of the point-masses forming the reference body of the P RF should satisfy the equation Dζ α /ds = 0 (10) The equation (10) For example, if the eld F is electromagnetic, then the space-time in such reference frames is Finslerian [1,11]. And the space-time in the reference frames comoving to an isentropic ideal uid is conformal to the Minkowski space. 1 We use notations and definitions, following the Landau and Lifshitz book [16].
In the case of gravity, we proceed from Thirring's assumption that gravity is described by a tesor eld ψ αβ (x) and the Lagrangian describing the motion of test particles has the form L = −mc[g αβ (ψ)ẋ αẋβ ] 1/2 .
Then it is obvious that space-time in the PRF is Riemannian with a linear element of the form ds = g αβ (x)dx α d β 1/2