The mathematician Riemann wrote the main and important elements of the geometry used by Einstein in General Relativity long before Einstein wrote his theory of relativity; that in and of itself does not diminish Einstein's work .  The issue is that Riemann's geometry of curves of multi-dimensions is purely speculative theory.  In the real universe there are only three detectable dimensions, therefore it is impossible to know what a 4th dimension would be like and no way to know what the math for our real universe would be if it had a 4th dimension.  Mathematicians have speculated on what four and higher dimensional math would be like for the real world, but that is all that is possible, speculation.  There is no way to test a 4th dimension in a three dimensional world.  On a two dimensional piece of paper I can draw what looks like three dimensions, but it's not three dimensions, every point on that piece of paper can be described by only two numbers, an X axis number and a Y axis number; the point is that in a two dimensional world (eg. the surface of a piece of paper) you cannot create anything that has a third dimension.  Similarly, you could give billions of dollars to the world's most acclaimed engineers and physicists and they could NOT build anything that is four dimensional; for whatever they would create, each point could be described by only three numbers, an X axis distance, a Y axis distance and a Z axis distance.

There is no way to know what a fourth dimension would look like or act like, as it is impossible to test a 4th dimension in a three dimensional world.  Consequently, all of Riemann's four and more dimensional geometry and all of the four and more dimensional geometry created after him and used in Einstein's work is 100% speculation, as it is not testable, and therefore, it is unverifiable.

Just as Riemann did not discover a 4th dimension, he postulated it, Einstein did not discover a space-time continuum, he postulated it; Einstein having postulated a space-time continuum did not cause it to come into existence.  Before Einstein, Riemann (whose geometry is the foundation for Einstein's space-time continuum postulation) postulated the geometry of four and more dimensional curved spaces.  Riemann's postulations and the math he developed based on his postulations, did not cause a fourth dimension (or higher dimensions) to come into existence.  The point is that Riemann’s and Einstein’s postulations (ie. their imagined creations) are neither correct nor incorrect, they are unverifiable speculations which could be correct or incorrect.

Euclidean geometry is verifiable in our universe.  The Earth is not flat, the Earth does not go around the Sun in a rectangular orbit, but Euclidean geometry applies perfectly to the Earth and its orbit because every point on and inside of the Earth and between the Earth and the Sun can be described perfectly using only three coordinates, each of which is at a right angle to each of the other of the coordinates.  Those coordinates are usually called the X axis, Y axis and Z axis.

You have all seen the drawings of the sheet of rubber with the ball placed on it, and the curve which the ball makes in the sheet of rubber, and you have been told that the sheet of rubber is the space-time continuum of General Relativity and that the curve in the rubber is what we call gravity.  If there was a space-time continuum and there was only one object in the entire universe, that ball on the sheet of rubber diagram would be reasonably accurate.  But the reality is that there are at least trillions of balls (stars, planets, moons, comets, asteroids), and the sheet of rubber is (according to Einstein) a sphere or an ellipse, therefore, it is not just a surface area but it also has an interior, with trillions of interior curved deformations many of which overlap.  My imagination isn't sufficient to do the following, but if you can, Imagine a golf ball that is 1,000 meters in diameter, with 1,000,000 irregular shaped and regular shaped different sized dimples along its entire surface and in all of its insides, some of which overlap and some of which do not overlap, that would be a very simplified example of the space-time continuum universe as envisioned by Einstein.  It is mathematically provable that such a universe could not exist, but that is an effort I would not go to for free.

As I am not willing to provide such a time consuming mathematical proof here, if what I have written above has not already convinced you that Einstein’s geometry for the universe is very unlikely to be correct, consider the following:

In Einstein's Book, at page 104, to calculate the mass of a distant dwarf star Einstein used Euclidean geometry and Newton's gravity equations, which are for a Euclidean Universe; Einstein did NOT use his equations or a non-Euclidean Universe model.

When dealing with objects that are large enough to be seen without the aid of magnification (ie. a magnifying glass or a microscope) the universe as we experience it is Euclidean, meaning that where point "B" is in relation to point "A" can be stated by giving three coordinates, the x coordinate, the y coordinate and the z coordinate.  It does not matter if points “B” and “A” are on a flat plain, in a cube, on the surface of a sphere or cylinder or even inside a cylinder or sphere, or on or inside of an object of any shape. Where point “B” is in relation to point “A” can always be given by three numbers each of which represents the length of a straight line; that is Euclidean geometry.

For many astrophysicists, cosmologists and theoretical physicists, to continue receiving their salaries and research grants they use Einstein's "Theory of Relativity" to do very complex calculations to come up with speculative "answers" which can never be proven to be correct or proven to be wrong because they are dealing with things that are not able to be checked or verified, as they are either far beyond our solar system or they are too small to be physically verified.  However, when they want an answer that is as close as people can come to being correct, just as Einstein did at page 104 of his book, they use a Euclidean Universe model and Newton's gravity equations.

Riemann’s math and the geometric math of the mathematicians that came after him, as well as Einstein’s math, is not the issue. Regarding their equations, the issue is, was their vision of what a fourth dimension would look like and how it would function, correct ?  Because if their visions of how dimensions beyond three dimensions will exist and will function is wrong, then the math based on those visions is only valid in the multi-dimensional worlds of their imagination, not in the real universe.

It is correct that on the surface of a sphere, cylinder or other curved 3d object the shortest distance between points “A” and “B” is a geodesic (a curved line on the surface of the 3d object which connects points A and B in the shortest path travelable on the surface), but that is not relevant to the gravity of the 3d object or of the universe, as all direct physical gravity experiments have proven that gravity acts linearly, in a Euclidean geometric fashion.  The curving of light, including gravitational lensing, can be accounted for by linearly acting gravity of the correct amount.  The fact that not all of the mass needed to account for the gravity needed for Mercury’s orbit, using Newton’s linear gravity equation, is physically detectable, is much less of an issue than the fact that 95% of the mass is missing that is needed to account for the results of Einstein’s gravity equations.

Despite his "Theory of Relativity" Einstein knew that the universe was Euclidean and that gravity acted linearly, that is why, as shown above, on page 104 of Einstein's Book, when he wanted to get an as accurate as possible calculation of the mass of a distant dwarf star Einstein used Euclidean geometry and Newton's gravity equations, which are based on gravity acting linear, in a Euclidean Universe.

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Einstein Theory of Relativity Misconceptions and Errors paper.