When one introduces these values of the functions f in the expression (9) of the line element
and goes back to the usual polar co-ordinates one gets the line element that forms the exact solution of Einstein’s problem:
What incomprehensible blather are you going on about?
Expression (9) in Einstein’s paper is not a line element but a component of the Einstein tensor Guv.
The polar coordinate equations which are defined below equation (2) define r as the radius.
The problem with the paper is that in equation (1)
r is not a physical radius but in isotropic coordinates defined by the transformation r → r(1+m/2r)².
For space-time that is for intents and purposes flat such as in our solar system beyond Mercury’s orbit, or when it becomes asymptotically flat at large distances from the gravitational source, the isotopic version of r is the same the radius r but not so in the case of black holes.
Which as his original paper shows is not even remotely the same metric you have come to know as the Schwarzschild metric. What is claimed today as his metric is not even his metric, but is the corrupted version by others.
https://arxiv.org/pdf/physics/9905030.pdf
That's all any of them are. Mathematical models..... None of them are physical models.....
http://www.ptep-online.com/2006/PP-05-10.PDF
"What the theoreticians routinely fail to state clearly is that the black hole comes from a solution to Einstein’s field equations when
treating of the problem of the motion of a test particle of negligible mass in the vicinity of a single gravitating body. The gravitational field of the test particle is considered too small to affect the overall field and is therefore neglected. Therefore, Hilbert’s solution is a solution for one gravitating body interacting with a test particle. It is not a solution for the interaction of two or more comparable masses. Indeed, there is no known solution to Einstein’s field equations for more than one gravitating body. In fact, it is not even known if Einstein’s field equations actually admit of solutions for multi-body configurations. Therefore, there can be no meaningful by theoretical discussion of black hole binaries or colliding black holes, unless it can be shown that Einstein’s field equations contain, hidden within them, solutions for such configurations of matter. Without at least an existence theorem for multi-body configurations, all talk of black hole binaries and black hole collisions is twaddle"
You are not fooling anyone in providing links that are clearly beyond your level of understanding.
Unbeknown to you have contradicted Crothers’ paper in the second link by claiming General Relativity works in our solar system.
The “corrupt” or Hilbert version of the Schwarzschild metric goes beyond being a model for a static black hole but also explains the perihelion advance of Mercury’s orbit, predicted the gravitational bending of light and gravitational redshift of emission lines in the laboratory both of which have been confirmed by observation and experiment respectively.
The reason why it works in our solar system is that most of the mass resides with the Sun and any two body problem involving the Sun and a planet, the planet can be approximated as a point mass of a test particle.
General relativity being a non linear theory will only work for a two body problem if the mass difference is considerable.
Since the 1960’s much work has been done on linear approximations to General Relativity which have been successful in describing the interaction of bodies of similar masses such as black holes and neutron stars which produce gravitational waves.
Not to mention all black hole solutions set the energy tensor to zero. This means the black hole is alone in a universe devoid of all other matter. Einstein’s equations require an energy tensor describing the gravitational field for all bodies of mass.
This is what made it even a theoretical possibility in the first place. It was alone in a universe devoid of all other matter. Hence the term coined “singularity”. Single, one, etc.
They also require an asomptotically flat universe. The second you add another black hole it is no longer asomptotically anything. And also is not compatible with elucidian space.....
Do you actually understand what you write or simply paraphrase Crothers’ nonsense because of some vague ideological/religious connection?
Crothers' mistake is using Newtonian gravity in his line of argument. The notion of gravity in General Relativity is very different. Newtonian gravity has only one source, mass. General relativity has two sources mass and the gravitational field itself. We can illustrate the differences using small masses as there is an overlap between Newtonian physics and General Relativity for weak gravitational fields.
The equations that describe Newtonian gravity are linear. The theory would predict that if we measure the gravitational force between two small masses, we would find that if one of the masses was broken up into pieces, the sum of the forces between each piece and the unbroken mass would equal the total force between the two unbroken masses.
In General Relativity the equations are non linear. General Relativity would predict the sum of the forces to be greater than the total force between the masses.
In fact the sum of the masses of the pieces would be greater than the combined mass. The "missing mass" m is taken up by the binding energy E required to break up the mass into smaller pieces. The relationship between the missing mass and binding energy is the well known equation E=mc².
This has been confirmed experimentally. We find the atomic mass of atoms to be less than the sum of the masses of the individual protons, neutrons and electrons.
We can conclude the energy E is a source of gravity.
We can extend this idea to the field equations Guv=0.
These equations are non linear and tells us the gravitational field is also a source of gravity. Hence we don't need to include a mass term to have a gravitational field as claimed by Crothers. In fact to do so contradicts the Schwarzschild metric which excludes the mass source as it is defined as an exterior solution.
The second problem is that if we put a mass term into the right hand side of the equation as Crothers suggests, the resulting field is acting externally on the intrinsic field. This mass term does not generate the intrinsic field. Since the mass occupies space time, the geometry of space time is no longer flat as is indicated by the non zero term.
Since you naively believe that this can be all explained by an Electric Universe model try answering a question that every other EU enthusiast has run away from over the years; how can a three body problem involving only electromagnetic forces be stable?