PROBLEM OF MATHEMATICAL DEDUCTION OF THE EXISTENCE OF BLACK HOLES

The mathematical proof of existence of Black Hole is based on the assumption of mass being independent of speed. Considering the effect of special relativity of the dependence of mass with speed there is no Black hole.

A particle with mass m falls from R by the gravitational force of Newton toward the star M, with the initial velocity v 0. The equation of motion is then d dR GMm --(m-)

R2
(2.1) In Newtonlan mechanics, mass does not change with speed, m const.
(2.2) so m can be cancelled out to get d2R GM dt 2 R 2 The integration of (3) with the initial conditions: dR t 0, R , -0, (2.3) (2.4) gives the result: (2.7) R Now, if the radius r of the star is so small such that fi-> c, (velocity of light) then the light particles (considered as materal mass points In Newtonlan sence) emitted from the surface of the star cannot reach point at infinity and will be attracted back eventually to the star again.
or The critical radius RLN of a star wth mass M then satlsfes (2.8) 2GM RLN ----. (2.9) So far all we have done is wthln Newtonlan physics.

SPECIAL RELATIVITY DENIES THE EXISTENCE OF LAPLACE-NEWTON BLACK HOLE.
In the two centuries that have past, no Laplace-Newton black holes have been detected.
Theoretically the reasoning for existence of Laplace-Newton black hole can be discarded as follows: In speclal relatfvlty mass is not a constant as Newton assumed in (2.2), but changes with speed in the following way: m 0 so that m cannot be cancelled out in (2.1), and one gets the equation: Hence, based on special relativity, the Laplace-Newton Black Hole does not exist.
The crucial point is that the existence of critical radius 2GM RLN -of the Laplace-Newton black hole i deduced mathematical without considering the effect of mass changing with speed.
This is quite understandable, because special relativity appears long after Laplace.

4.
THE PROBLEM OF MATHEMATICAL DEDUCTION OF CRITICAL RADIUS ROS OF OPPENHEIMER- SCHAWARZCHILD BLACK HOLE.
In 1939 Oppenheimer deduced the theoretical existence of a black hole based on Schwarzschild Line Element: ds 2 In the standard textbook of relativity by P.G.Bergman [I] with forward by A. Einstein there is a remark: "In nature, mass is never sufficiently concentrated to permit a Schwarzschild Singularity to occur in empty space." This remark has not obtained sufficient attention from scientists working on black holes.
Many experiments are conducted to find Oppenheimer-Schwarzschild black holes in the universe, but few people considered the assumptions leading to the theoretical prediction of the existence of Oppenhelmer-Schwarzschild black holes.
Although the theory of relativity is fundamentally different from the theory of space-time of Newton-Galileo, a crucial question is why the critical radius RLN o Laplace-Newton black hole is identically equal to the critical radius ROS of Oppenhelmer-Schwarzschild black hole?Is this a relation by chance or a certainty by same assumption?The answer is that they are the same mathematical deductions based on the same assumption that mass does not change with speed.
Let us look at how the constant of Gravitation G gets into the theory of General Relativity.
In mathematical deduction of $chwarzschild llne element (4.1) one arrives at the metric tensor wtth a component: g44 eB(l ) (4.4) with two integration constants a and B to be determined.lim g44 I, so B 0.
To determine the integration constant a, the Newtonlan potentlal of a point of mass M which creates the filed with (4.4) is G M -.
(4. s) On the other hand, the Newtonlan potential of a point with mass M is given by CM R" (2.5) Hence one gets the Integration constant a as It is clear that the gravitational constant G gets into the theory of general relatlvlty through the Newtonlan potentlal G M of (4.5), a result of (2.5) based on the assumption (2.2).
Mass does not change with speed.
Hence the identity (.3) becomes reasonably expectable, because the two critical radius RLN and ROS are deduced mathematically on the same assumption (2.2).
Since speclal relatively replaces (2.2) by (3.1) to discard the existence of the Laplace-Newton black hole, it is naturally to expect that replacing (2.2) by (3.1) will also solve the difference of the mathematlcal deduction of existence of the Oppenhelmer-Schwarzschild black hole based on the assumption of (2.2) and the reallty that in nature one has not found any real Oppenhelmer-Schwarzschild black hole, because mass does change with speed according to (3.1).

MATHEMATICAL DEDUCTION OF SCHWARZSCHILD LINE ELEMENT FROM SPECIAL RELATIVITY AND NEWTONIAN MECHANICS.
We use three simple steps to deduce it.
(i) Let a heavy mass M be situated at the origin (0,0,0).

From Special
Relativity one has: ds 2 dT (dXL 2 + dy L + dz L) for the local tlme-space coordinates (xL,YL,ZL), or in polar coordinates (R L,0,): (il) Let an Einstein elevator falling from point at infinity with zero initial speed toward this mass M under the Newtonlan gravitation law.Then the speed is: {ere R means measured by a rod at infinity, i.e. no contraction in Newtonian mechanlcs, or V v /ZGM (5.3) 2" (ill)From the Einstein elevator, by special relativity one has dT L / V2dT=, and dR These are standard realtions of special relativity for local time-space. (5.4) (5.5) There is no relative velocity in (,0) directions, so there is no Lorentz factor for (,O).
Putting (5.3), (5.4) and (5.5) into (5.2),one gets at once the Schwarzschild line element: From this type of deduction we can see that the relative velocity V comes from Newtonian mechanics.
Now if we replace (2.3) by (3.1) as we have done in section llI, we shall have Hence there is no singularity in the llne element (5.7) except the origin, which will be discussed In the last section.
AgaLn we get line element without a black hole by considering the effect that mass does change with speed.
6. THE EXPERIMENTAL TESTS OF LINE ELEMENT (5.7).exp 2__) 2GM + Rc 2 Rc 2 So (5.7) is the same as Schwarzschild llne element in first approximation, the gravitational red shift and the deflection of light are just the same.The crucial test is the precession ofthe perlherlon of Mercury, which is an effect of second order approximation.
The discrepancy is 24 3a2 2 ( e )(or) 2 (Angle in radiants per period) 2 The llne element (5.7) glves of this value.The problem is now: Where is the-of this value?
The answer comes from the special relativity again.
From the Lorentz transformation: X-VT T -VX X* T* so in one turn of the oriblt there is a time delay of the magnitude: not change wlth speed so it can be cancelled from (3 mass M and with radius smaller than R is an Oppenhelmer-Schwarzschild os Black Hole.

(
T. Change this angle into a radian system, i.e. mulltlply it by 27