The first draft of "21st Century Relativity: a Primer" is now available as a Kindle eBook. Since the last update of the site, a much more sophisticated view of relativity has emerged. If you are a Physics professional, you may be wondering, "Why?" In simple terms, relativity is illogical and it is based on false assumptions. Assumptions that go back to Einstein's first paper, "On the Electrodynamics...". Before he even talks about relativity, he goes into great detail about measurement. In his arrogance, he spells out what he considers to be an ideal measuring system. He describes a grid of rigid rods with an identical clock at every corner. His assumption is that with a perfect measuring system, any interval of time or space can be measured to an arbitrary degree of accuracy. He allows that such a grid may be impractical for moving reference frames or great distances, and accepts alternative means, like trigonometry, as long as they get the same results as would his grid if it were possible. The details of his protocol are simple. Place the rigid rods end to end along the axis of the measurement. The problem starts right here. The technique he describes is a mathematical procedure called the dot product. Except that as he describes it, the procedure is only valid in Newtonian physics. And here he is, trying to develop special relativity, where velocities are not only relative, but also relativistic. And his first gedanken experiment proves it. After applying a Lorentz transformation to arbitrary measurements of time and space, he finds that according to a fast-moving observer, his rigid rods turn to silly putty and his identical clocks drift apart. He made a reasonable logical assumption. A logical argument resulted in a contradiction. His rigid rods did not stay rigid and his ideal clocks did not keep accurate time. By the rules of logic, his premise, that it was possible to accurately measure moving intervals of time and space with an ideal measuring system, must be false. If you start a logical argument with a false premise, anything is possible. You might even stumble upon truth, but you can't prove anything that way.

So, if you can't get an accurate measurement with an ideal grid, what can you do? Aside from thinking he could use a Newtonian measurement protocol in the first place, the bigger truth is that the universe simply does not allow any relatively moving observer to measure 100% of any interval, perfect measuring system or not. But the protocol is not entirely wrong. In the context of the dot product, it is a perfectly accurate description of measurement of the real, cosine projection of a rotated vector. In short, the universe only allows any observer to measure what is real to that observer, regardless of what is real to any other observer. And given any two vectors, what is real is the dot product of the two. Geometrically, the dot product is defined as the simple product of the magnitude of the two vectors with the cosine of the included angle. In Euclidean geometry, the included angle is obvious. But in relativity, the included angle is 4 dimensional. This would pose a dilemma if there were not a trivial way to determine it. The details are in the book, but the short story is that the Lorentz transformation, which maps coordinates from one frame of reference to another one moving at some relative velocity, v, is characterized by a hyperbolic rotation angle. When this angle is referenced to a stationary origin, it is called rapidity, but an increment of rapidity associated with a Lorentz transformation is called a boost. This is still 4 dimensional and hard to visualize, but every hyperbolic angle has a "twin". An ordinary circular angle which I call tilt. Physics recognizes this fact, but does not teach it. They typically call it θ, an arbitrary parameter of convenience. The fact is, given a hyperbolic angle, it is NOT arbitrary. Tilt is the gudermannian of boost. Each angle is at the center of a 6-group of trig functions. That would seem to imply that there are 12 trig projections between them. However, there are 6 identities that map each hyperbolic function to a circular function. The one that applies here is sin(tilt) = tanh(boost). Then, v = c sin(tilt) = c tanh(boost). The other 5 identities can be derived from this one. The point is, this tilt angle IS the included angle of the dot product measurement protocol. All we need to know is the relative velocity, and that is what relativity is all about, anyway. So, the result of any measurement is the dot product of the unknown moving interval magnitude with a real unit in the stationary observer frame, with the Arcsin(v/c) as the included angle.

It is no coincidence that when v = c sin(tilt), the sec(tilt) = γ, the Lorentz factor, and the cosine of the included angle is just 1/γ. So if two frames are synchronized, and two vectors are measured as ct and r, both observers will get the same measurements for both vectors. But if one frame starts moving, it introduces a tilt angle between the two frames. When we represent velocity this way, the origins remain coincident, and the included angle becomes a simple rotation between the two frames. When the stationary observer attempts to measure the two vectors in the moving frame, he gets ct' = ct cos(tilt) and r' = r cos(tilt). These are equivalent to ct = ct' sec(tilt and r = r' sec(tilt). But these are also equivalent to ct = γct' and r = γr', time dilation and length contraction as defined by Einstein in his failed gedanken experiment, but without the bullshit. The dot product protocol supersedes the relativity of simultaneity and the coincidence of clock location. All of these are window dressing that was required to fix the error in Einstein's assumption that he could measure relativistic intervals with a Newtonian protocol. "21st Century Relativity" starts with a clean sheet of paper and the dot product measurement protocol. Ultimately, the gudermannian function, which is a hyperbolic identity, predicts the existence of a maximum observable speed, and an associated momentum that approaches infinity, because the dot product protocol applies to force, which is applied in the frame of the moving object, and distance which is measured in the stationary frame of the observer. The same tilt angle that defines time dilation and length contraction also exists between force and distance. Even in 3D, work is the dot product of these two vectors. It is no different in 4D, and the tilt angle is determined by the instantaneous velocity. Nothing gets heavier as it speeds up. Less and less of the assumed force can be applied to linear momentum as the tilt angle grows. This is what makes it harder to accelerate any mass.

But if Einstein was the White Rabbit, Minkowski led us all down the rabbit-hole to Wonderland, where it is normal to "believe 6 impossible things before breakfast". He said that relativity required us to abandon Euclidean geometry and the sum of squares, that the real axes of his light-cone weren't actually perpendicular, that time and space changed their angle with relative velocity, that the Pythagorean Identity was not valid, just to mention a few perversions. The justification was that his spacetime and its rules predicted the correct measurements. What a load of horseshit! In the first place, all isomorphisms of any given theory predict the same results. As someone once told me, "If it ain't isomorphic to special relativity, it ain't special relativity." Getting the predictions right proves nothing. It is a minimum condition that qualifies a theory for consideration, that's all. The things that make isomorphisms different have no effect on the outcome of any experiment. Those are the rules, and the rules of one isomorphism have nothing to do with the rules of any other one. It is more fitting to use Occam's Razor to choose between isomorphisms. Let's see what we can do by ignoring Minkowski's rules.

We start with the unit circle, and a real axis. It has a constant radius that is 1 unit. The point on the circumference of the circle has two perpendicular projections, and the sum of their squares is 1 for any angle of rotation. A simple-minded assumption is that the hypotenuse is invariant with respect to the rotation. Through the point on the circumference, we construct a tangent line. This line intersects the real axis, forming another, similar right triangle. The naive assumption that the hypotenuse is the invariant is not true. In this triangle, the invariant unit is not the hypotenuse, it is the base. The tangent line segment is the altitude and the hypotenuse is the sum of the squares of the base and the altitude, like any other right triangle. This is straight-up Euclidean geometry. If we multiply every edge by the same invariant scalar, c, it becomes a velocity vector diagram. The radius of the circle is c, and the sine projection of an arbitrary radius in the right half-plane is relative velocity. The cosine projection represents the velocity of propagation of light in a medium where the index of refraction is the cosecant of the tilt angle of the radius. They don't necessarily apply at the same time, because the tilt angle of the medium has nothing to do with the tilt angle of a boost. However, we aren't interested in them right now. The altitude of the bigger triangle is now the celerity, u, associated with some relative velocity, v = c sin(tilt). So u = c tan(tilt). The hypotenuse represents total velocity in spacetime. As it has no common name, let's just call it, w. By the Pythagorean identity, w² = c²+u², and c, velocity in time, is w cos(tilt), while u, celerity in space is w sin(tilt). But u = c tan(tilt), so w² = c²+(c tan(tilt))² = c²(1+tan²(tilt)) = c²sec²(tilt), and total velocity in spacetime is w = γc. If we want to measure w, we can only measure its cosine projection, and w cos(tilt) = c. This is the reason for the myth that the faster you go in space, the slower you travel in time. In point of fact, the speed of time is invariant just as the speed of light is in a vacuum. But the universe won't allow us to measure all of w. And the cosine projection is invariant with respect to velocity in space. Now, if we multiply by another relativistic invariant, mass, the graph becomes a vector momentum graph. The base of the large triangle is now mc, momentum in time, and the invariant of 4-momentum. The altitude is mu = mc tan(tilt) = mc sin(tilt) sec(tilt) = γmv, relativistic momentum in space. Interestingly, the cosine projection of this is just mv, Newtonian momentum. When v is small, γ ≈ 1, and v ≈ u, mv ≈ mu = p. But as v increases, so does γ and the two definitions of momentum diverge. Newton's formula was technically wrong, although it is a very good first-order approximation. But with the correct formula, there is no reason to invent relativistic mass. So, (mw)² = (mc)²+(mu)² = m²(c²+u²) = m²(c²+(c tan(tilt))²) = m²c²(1+tan²(tilt)) = γ²m²c² = (γmc²)²/c² = E²/c² = (E/c)² = (mc)²+p². While this is still a Pythagorean Identity for momentum, it is more familiar after clearing the fractions, E² = (mc²)²+c²p² or E²-c²p² = (mc²)², the so-called dispersion relationship for energy and momentum. And we arrived at it logically while breaking all the rules of Minkowski spacetime.

There are many more details in the eBook, but to give you an idea, here is a Sampler. A free page from every chapter. If you want more, you'll have to buy the book. But, you don't have to take my word for it. The sampler has links to verbatim transcripts of chat sessions with 4 different AIs. I know they make mistakes, and sometimes even lie, but 4 different AIs were trained on 4 different sets of data by 4 different sets of people. The odds that they would all get the same thing wrong are slim, just as the odds that if they all agree on something that it would be wrong. I'm happy to report that all 4 agree that my math is rigorous and logical. All 4 describe the work as "elegant", and all 4 offered to assist in preparing a version suitable for formal peer review. The transcripts are not fabricated. They are the result of hundreds of screenshots spliced together. Although it is called "21st Century Relativity", I believe it will still be taught in the 22nd Century and beyond. After all, it is based on 3000 year old geometry and logic. That's a solid foundation.