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“Some scientists and physicists” belies the fact that virtually all of them believe in Special Relativity. Not I. Technically, I am not a scientist, I am an engineer. However, that is not relevant to my refutation of Special Relativity. First, let’s dispense with the most common rationalization: “It has passed every test … blah, blah, blah” Every experiment ever performed to “test Special Relativity” actually confirmed the Lorentz Transform. This transform was predicting correct results of measurements while it was part of Lorentz’s failed Aether Theory. So it is no surprise that it continues to predict correct results as part of Einstein’s failed theory. Name one single person who has observed a rigid measuring rod contract. Just one. … I thought not.We are told to forego common sense and logic because “That’s Relativity for you!” This conveniently rules out logical arguments against Special Relativity. Then there are the propagandists who claim that Einstein didn’t mean what he wrote, although his writing was meticulous and deliberate. According to them, physics has reinterpreted Einstein’s work so much it isn’t even his. Crafting an argument against Special Relativity is like trying to shoot a moving target. They keep moving the goal posts. It’s bad enough that we cannot trust common sense. But common sense is not a proof. A proof requires logic, and even logic sometimes defies common sense. For example, if you start a conditional statement with a false premise, then whether the following argument is true or not, the conditional statement is automatically true. But the die-hard defenders of the dogma of relativity have taken our most powerful weapon away, and logic itself can’t be trusted.Before a fair evaluation of Special Relativity is possible, this prohibition must be reversed. As long as it stands, however, it prevents logical arguments from gaining any traction. However, these ad hoc rationalizations of a faulty theory have not always existed. They are just a little over a century old. Long time, huh? Compared to logic, which was invented by ancient Greeks, these prohibitions are fly-by-night extremisms. So, instead of trying to prove that the prohibitions are baseless, I went backwards to a time before Relativity, before Einstein, before Lorentz, before their century, before the last millenium, even before 1 AD. That’s right. All the way back to classical Greek Geometry, and its primary tool, straightedge and compass construction. Needless to say, logic (something else those Greeks invented) also applies. Even Einstein had positive things to say about Greek geometry,“…Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the ‘truth’ of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the “truth” of a geometrical proposition in this sense we understand its validity for a construction with ruler and compasses. “Before you stop reading, google ruler and compass construction. You will find that the Greeks had procedures (or algorithms) for all the basic calculator functions, including square root and conjugate. There is no math in Einstein’s book any more complicated than these functions. After all, in Special Relativity, with its inertial frames, velocity is constant and derivatives are simply ratios. In theory, it is possible to translate Special Relativity into ruler and compass construction. But this doesn’t reveal how simple and common sensical this technique actually is.To describe all the properties of the derivation would require a book, but I present here an outline which covers the major points. So grab a ruler, compass, pencil and piece of paper (preferably graph paper) and follow along. For starters, we select an origin, preferably near the bottom left of the page. Construct a horizontal axis that goes through the origin. Then construct a perpendicular axis that also goes through the origin. Select a radius for the compass that is roughly half the width of the page, and scribe an arc from the vertical axis through the horizontal axis. Now construct another perpendicular that goes through the intersection point of the arc and the horizontal axis. Now bisect the angle between the horizontal axis and the perpendicular axis at the origin. This is the upper asymptote of the hyperbola. Technically, a hyperbola cannot be drawn with just ruler and compass. And if we do use only a ruler and compass, the curve cannot be drawn in a finite number of operations. However, the end result of the construction is a point which is on the unit hyperbola. The process can be repeated an arbitrary (finite) number of times, creating a set of points, which, if packed closely enough, fills the curve to the point where it is a hyperbola for all practical purposes.The unit from the origin to the tangent line between the two curves is an invariant. Depending on which curve, the definition of the invariant changes. For the circular arc, the invariant is the radius. The coordinates of a point on the circle are of the form (r, rφ). For the unit circle, this is (1, φ). This is an invariant base unit and a circular arc of length, φ. These two vectors add, and the point at the free end of the arc has Cartesian projections (cos(φ), sin(φ)). Draw a line from the origin through some point on the circle at angle φ to the horizontal. We’ll call this angle, tilt. Extend the radius to this point until it intersects the tangent line. The triangle thus formed has the same invariant unit base, which does not change as the tilt angle varies. The coordinates of the apex, given the included angle φ, are (1, tan(φ)). Next, construct a line parallel to the base through the intersection point on the tangent line, extending it to the right of the tangent line. Set the compass to the length of the hypotenuse of the right triangle, and swing an arc down to the horizontal axis using the origin as a pivot. Construct a vertical line that passes through this point on the horizontal axis, and intersects the horizontal line from the apex of the right triangle. This point is on the unit hyperbola, and it has Cartesian projections, (sec(φ), tan(φ)). Finally, construct a line parallel to the base that passes through the point on the circle, (cos(φ), sin(φ)), and intersects the tangent line. The coordinates of this new point are (1, sin(φ)). If your drawing is not sloppy, the origin, the last point, and the point on the hyperbola, are collinear. Draw a vector from the origin through all three points. For every non-zero angle φ, there is a unique set of points described by this construction. (When φ=0, all 4 points degenerate into the initial unit point on the horizontal axis. In a nutshell, this is the error of Special Relativity. While these 4 points represent 4 different vectors, Special Relativity assumes that they are still one and the same vector when the angle is non-zero, too. This is illogical.What is the significance of the angle φ as it regards Special Relativity, you may ask. Well, it is defined by the relative velocity, v = c sin(φ). Then in familiar symbols, the point on the circle is (1/γ, β), the apex of the triangle is (1, βγ), the point on the hyperbola is (γ, βγ), and the last point is (1, β). Now it is clear that the apex of the triangle is the extension of the unit radius γ(1/γ, β) = (1, βγ). Similarly, the lower point on the tangent line is a projection of the vector to the hyperbola (γ, βγ)/γ = (1, β). At this point, the construction is finished. The angle φ is defined by the relative velocity, and the skeleton graph is constructed as above. But this is a pure mathematical construction, and is dimensionless. Each tilt angle has a unique graph mapped to it, but one exists for every possible tilt angle.The relevant projection ratios for a relativistic effect all appear in the same phase drawing, regardless of relative velocity. All the ratios so defined are actually in the phase drawing, as trig functions of the triangles in the drawing. Relativity is Geometry. A rule of Geometry is that two triangles are similar if they have the same 3 angles. They may not have the same length sides but they have all the same trig ratios. What this means is we can scale a triangle in the graph with any multiplier, either a numerical unit or a physical unit, such as c.It is informative to introduce Complex geometry to the ruler and compass drawing. This typically makes the math seem more complicated. But is it illogical? Of course not. By the same token, if the structure is Complex to begin with, reducing it to only Real components makes it simpler. But is it logical? No, it’s Special Relativity.Once we scale the triangle by c, its legs are c, c tan(φ), and c sec(φ). The base is c, the velocity in time for any and all observers, according to the principle of relativity. For the time being, we can assume c is invariant (since we do know that), but we will see that this stems from the geometry, too. In any case, the Pythagorean Theorem says c²sec²(φ)-c²tan²(φ)=c². The same formula as the relativistic invariant for what physics calls the 4-velocity {c sec(φ), c tan(φ)}, and its invariant magnitude is c. But what it really is is the semi-major axis of the light speed hyperbola. All the points on this hyperbola represent a 4-velocity. All points on a given hyperbola share the same semi-major axis, because it is defined as the x coordinate when the y coordinate is 0, at the vertex of the hyperbola. Every 4-velocity represents an inertial frame of reference. Each of these is a point on the same hyperbola, so shifting from one frame to another involves sliding a point from one place on the hyperbola to another, analogous to a circular rotation sliding a point on the circumference to another point on the circumference. If we didn’t already know that such a hyperbolic rotation is a Lorentz Transform, we could take advantage of another hyperbolic identity.Of all the variables that relate to velocity, the only one that is linear is the hyperbolic angle, called rapidity, w, when used in reference to velocity. So, if we are going to move a point on the hyperbola, we want to express coordinates in hyperbolic functions. As it happens, there is a function which maps every hyperbolic angle to a tilt angle (and vice versa) called the gudermannian (and its inverse). It is not so special that a function connects these two parameters. What’s unique is that sec(tilt)=cosh(w) for every possible pair of angles. It is trivial to demonstrate that the other trig functions also agree: tan(tilt)=sinh(w), sin(tilt)=tanh(w), cos(tilt)=sech(w), cot(tilt)=csch(w), and csc(tilt)=coth(w). The relationship between tilt and w is such that as w approaches infinity, tilt angle asymptotically approaches Pi/2. Then as Proper velocity = c sinh(w) approaches infinity with w, Newtonian velocity, c sin(tilt), asymptotically approaches c. As Proper velocity, which is Complex, routinely exceeds c, the idea that it is a universal speed limit is false. By the same token, the cosine projection of Proper velocity is Newtonian velocity. We have already established that Newtonian velocity is limited to c, now we see why. A Newtonian velocity greater than c would require a Proper velocity greater than infinity. That’s a contradiction.Anyway, two distinct points on the hyperbola represent two distinct Proper velocities. Thanks to the gudermannian, the coordinates of points on the light speed hyperbola can also be expressed as functions of w: (c cosh(w), c sinh(w)). Since both points are on the same hyperbola, they have the same magnitude, c. But they are distinguished by the rapidity, w. So, let’s say we start with a point, (c cosh(w1), c sinh(w1)). We want to change to point, (c cosh(w2), c sinh(w2)). Again, assuming we don’t know that it’s the Lorentz Transform, we exploit simple hyperbolic identities. Start by realizing that w2 = w1 - w3, where w3 is the difference between the two rapidities, assuming w1 does not equal w2. Then (c cosh(w2), c sinh(w2)) = (c cosh(w1 - w3), c sinh(w1 - w3)).c cosh(w1 - w3) = c (cosh(w3)cosh(w1) - sinh(w3)sinh(w1))c sinh(w1 - w3) = c (-sinh(w3)cosh(w1) + cosh(w3)sinh(w1))If we examine these carefully, it is just the expansion of the matrix product│c cosh(w2)│ │cosh(w3) -sinh(w3) ││c cosh(w1)││ c sinh(w2)│=│-sinh(w3) cosh(w3) ││c sinh(w1)│The 2x2 matrix is a typical Lorentz Transform. The two column vectors are 4-velocities, and the square matrix is a Lorentz matrix. Notice that we just derived the Lorentz Transform from the property of linearity of the hyperbolic angle. There was no reference to light signals or simultaneity. It’s a property of the geometry.You might ask, “What about the velocity addition rule?” In the first place, it’s already answered. In hyperbolic angle w, Newtonian velocity is c tanh(w) (= c sin(tilt)). Again, exploiting the property of linearity, c tanh(w3) = c tanh(w1+w2) =c sinh(w1+w2)/cosh(w1+w2) =c (sinh(w2)cosh(w1) + cosh(w2)sinh(w1))/(cosh(w2)cosh(w1) + sinh(w2)sinh(w1)) =c (sinh(w2)/cosh(w2) + sinh(w1)/cosh(w1))/(1 + sinh(w2)sinh(w1)/cosh(w2)cosh(w1)) =c (tanh(w2) + tanh(w1))/(1 + tanh(w2)tanh(w1)) =c (sin(tilt2) + sin(tilt1))/(1 + sin(tilt2)sin(tilt1)) =c (v2/c + v1/c)/(1 + v2/c v1/c) =(v2 + v1)/(1 + v2v1/c²). QEDAnother property of the geometry.So far, I haven’t highlighted the use of Complex dimensions. The significance is more apparent when discussing momentum. When Einstein published in 1905, he described mass with a longitudinal and a transverse component. So when relativistic mass was proposed, about 4 years later, it was an improvement. Even so, Einstein preferred the use of relativistic momentum and energy instead of relativistic mass. So when we scale the phase triangle by Newtonian momentum, the three sides of the triangle are mv, mv tan(tilt), and mv sec(tilt). If Newtonian momentum is the base, the hypotenuse, which is Complex, is relativistic momentum. From tons of experimental evidence, whenever we observe a Newtonian velocity v, and expect Newtonian momentum, mv, we also measure a relativistic momentum, γmv. This is where the idea of relativistic mass originated. Einstein called it. He speculated that momentum is not linearly proportional to velocity. In fact, it is directly proportional to γv, which is mass x Proper velocity. The mathematical difference between these two momenta is literally a pure Imaginary momentum vector. So, although we can’t measure the Imaginary component of the velocity, we can’t ignore the relativistic momentum of the Proper velocity.It would take a book to completely describe all the implications of 8 dimensional hypercomplex geometry. What they won’t admit is that there is no Special Relativity in 8 dimensions. Everything we identify with SR is an intrinsic feature of the geometry. The choice of coordinate system is not supposed to affect the laws of physics. Similarly, the various observers may disagree about measurements of an event relative to another one, but they will agree that there was an event. The choice of an 8 dimensional hypercomplex geometry eliminates Special Relativity completely. This logically implies that Special Relativity is merely an artifact of a coordinate system that doesn’t have enough dimensions. More to the point, the derivation presented here was logical, made sense, and was purely mathematical, yet it leads to every feature of Special Relativity as we know it. This refutes the propaganda of the physics establishment that relativity clashes with common sense.It also restores all the rules of logic, since none were required to be broken in this derivation. Under the full set of rules, what physicists glibly refer to as “That’s Relativity for you!”, are actually CONTRADICTIONS that invalidate the Einstein Interpretation.