From time to time, I will be adding copies of posts made on Social Media. It will almost always be the same information, but it will often be focused on a specific issue, instead of the whole nine yards. Since nearly all my writing is extemporaneous, it also reflects what most recently has come to my attention. The variation in focus helps to flesh out the picture, and one post may appeal more to some readers than others. As they say, variety is the spice of life. Today’s post is about the weirdness of Einstein’s Relativity. The relativists won’t call a contradiction a contradiction, but they use other euphemisms, like weirdness. I have omitted some of the details which are included in the Hyperbolic, Hypercomplex tab. Eventually, there will be pop-ups linked to the less familiar keywords. Until then, there is a list of links under the Invocation tab.

About the weirdness of Einsteinian Relativity

The “weirdness” of Einsteinian Relativity stems from the physicists’ dislike of infinity and insistence on explaining everything in terms of real numbers and real dimensions, a distaste inherited from Rene Descartes, who did not like the idea that negative numbers could have square roots. Physics employs a double standard with regard to complex numbers. In quantum mechanics, they describe numerous properties with complex numbers, but are quick to point out that they do not refer to anything physical. It is just coincidence that the mathematical structure is consistent with the observed physical interactions. With regard to spin, for example, the complex math correctly describes the results of interactions, but nothing is supposedly actually spinning. They like to have their cake and eat it, too.

You can write me off as a crank, but I have a mathematical explanation for why Einstein’s 2nd Postulate is valid. It is, after all, just an educated guess that turned out to be consistent with every attempt to verify it. For physics, if the numbers agree, that’s good enough. More than one has told me that “Why?” is for philosophers, not physicists. I have no objection to being called a philosopher, however, and that’s who were the earliest physicists. If they want to deny their roots, that’s their business. But when I present the mathematical argument, it is not philosophy, it is logic, it is mathematics.

They like to say that mathematics is not physics, and ignore the math. Perhaps, in general, that is true. But when the mathematics is in 100% agreement with the physical observations, that’s a special case, like spin in quantum mechanics. The math is the physics. In fact, while it is true that there is much mathematics that is not physics, the converse is not true. All physics IS mathematics. In some cases, we haven’t yet identified which math. This is in part due to the phobia of physicists with regard to complex numbers.

I promised a mathematical basis for a limiting velocity. I was not fantasizing. It is remotely possible that I am wrong, but to prove that, you’ll have to follow the argument. It is based on an explicit, closed-form solution to a simple differential equation. The solution exists in mathematics. The only question is, does it apply to physics? It is based on the isomorphism between circular rotation angles and hyperbolic angles, both of which have an unquestioned existence in mathematics. In physics they are called a parametric angle and rapidity, which masks the fact that they are inescapably entwined by the diffeq. The parametric angle is the gudermannian of the hyperbolic angle. Both are used to refer to velocity, without acknowledging that they are functionally related.

If we let the parametric angle be θ, and the hyperbolic angle be w, then the mathematical relationship between them is θ = gd(w), a transcendental function. In physics, the usage is slightly different, because it is based on a closed form solution of the diffeq. In physics, relative velocity, v, is equal to c sin(θ) = c tanh(w), for all pairs (w,θ) where θ = gd(w). Then, dθ = sech(w) dw, implying that dθ/dw = sech(w) or dw/dθ = sec(θ). Notice that the differential equation is invariant to a swap of circular and hyperbolic arguments and functions.

Both forms are easily solved by a lookup in standard tables of integrals, but I like to demonstrate that they are also solved by geometric means that are more intuitive. This approach uses analytic geometry and the unit circle and the unit hyperbola, fundamental figures of geometry that were known for centuries before physics and calculus were invented. Of course, this was before analytic geometry was invented, too, but at least that came long before relativity.

We start with the most general case, an arbitrary point on the unit circle, and an arbitrary point on the unit hyperbola. It takes 4 coordinates to define two arbitrary points, and the fact that they are elements of specific figures means we can specify two equations that define them. For the circle, x²+y² = 1, and for the hyperbola, t²-z² = 1. Since the hyperbola is a curve with two disjoint branches that are mirror symmetric, we can restrict our attention to the right half plane with no loss of generality. In other words, x and t are greater than 0. In fact, both curves are also mirror symmetric in the vertical direction. Since, strictly speaking, neither one is a function, we can consider only the 1st quadrant, in which both curves are single-valued functions, and all coordinates are positive, with no loss of generality.

The minimum of z² is actually 0, so t can never be 0 and still represent a point on the hyperbola. This means that for every point on the hyperbola, we can rearrange the equation by adding z² to both sides and dividing both sides by t². The rearranged formula for the hyperbola is then (1/t)²+(z/t)² = 1. From the symmetry, we can argue that for every point, (t,z), on the hyperbola, there is some point on the semi-circle, (x,y), for which x = 1/t and simultaneously, y = z/t. Because, if x = 1/t, then x²+y² = 1 = (x)²+(z/t)², and by the transitive property of equality, y² = (z/t)², or y = z/t. Then, excluding division by zero, t = 1/x, t/z = 1/y, x/y = 1/z and y/x = z.

We have imposed 3 conditions on the 4 variables. This leaves a 4th condition. If we now express these variables in polar coordinates, we have 6 equations in hyperbolic and circular functions. This 4th condition is the independent value of w. Specifically, x = cos(θ), y = sin(θ), t = cosh(w) and z = sinh(w). The 6 identities become:

cosh(w) = sec(θ) = γ, the Lorentz factor
coth(w) = csc(θ) = 1/β
csch(w) = cot(θ) = 1/βγ
sech(w) = cos(θ) = 1/γ
tanh(w) = sin(θ) = β = v/c
sinh(w) = tan(θ) = βγ = Proper velocity/c

If we implicitly differentiate any one of these 6 identities with respect to either angle, the result is either the diffeq or its reciprocal. For example, d/dw tanh(w) = d/dw sin(θ). This transforms to 1/cosh²(w) = cos(θ) dθ/dw. From the identity table, sech(w) = cos(θ), or dθ/dw = sech(w). And this is just the reciprocal of dw/dθ = 1/sech(w) = cosh(w) = sec(θ). I have appended the appropriate physical quantities to trig functions in the table.

Since the 6 circular functions and the 6 hyperbolic functions are all members of 6-groups, any one of them can be uniquely determined by any other, and knowledge of only 1 value uniquely determines all of the others. In mathematics, it is illogical to accept velocity, but to deny Proper velocity, because they are isomorphisms of each other. But physics does it all the time, because they are obligated to prop up the ultimate speed limit.

So, whether you agree that Proper velocity is physical or not, there is no disagreement about its magnitude. Getting back to the diffeq, dθ/dw = sech(w) = cos(θ). Or dθ = cos(θ) dw. At very small velocities, θ ≈ 0, cos(θ) ≈ 1 and sin(θ) ≈ θ. Then dθ ≈ dw. A small increment in the hyperbolic angle results in an equal increment of the circular angle. It is easy enough to show from the geometric definition of rapidity, that the composition of two rapidities is by linear addition.

In other words, the composite of two rapidities is w3 = w1+w2, for all values of rapidity, which is unbounded. This applies to small increments as well, dw3 = dw1+dw2. Since dw ≈ dθ, small increments in the gudermannian also add linearly. Since sin(θ) ≈ θ, the sine projections also add linearly. Since velocity is just the sine projection scaled by c, small velocities also add linearly, v3 = v1+v2, the Newtonian velocity addition law. As θ increases, all of the approximations break down.

However, the original formula, w3 = w1+w2, was not an approximation, so it remains valid. The gudermannian identity allows velocity to be represented as v = c tanh(w). Inserting the composition rule, v3 = c tanh(w3)
= c tanh(w1+w2) = ((c tanh(w1)+(c tanh(w2))/(1+(c tanh(w1)(c tanh(w2)/c²)), which can be found in any table of trig identities. The result is just v3 = (v1+v2)/(1+v1*v2/c²), the relativistic velocity addition law. It is non-linear in precisely the way it is because it is a transformation of rapidity addition, which is exactly linear.

The same differential equation says that as w approaches infinity, θ approaches Pi/2. This implies that as Proper velocity (c sinh(w)) approaches infinity , measured velocity (c sin(θ)) approaches c, because sin(Pi/2) = 1. And there it is. Lightspeed is the mathematical limit of the cosine projections of Proper velocity as rapidity approaches infinity. It is a constant, because a limit is unique, and infinity is the same everywhere, for all observers. This justifies Einstein’s postulate.

In more general terms, the linear addition of rapidity explains the other weirdness about the invariance of c with respect to relative velocity. To compose velocities we can transform them to their equivalent rapidities, add them and transform back to a velocity. Since the lightspeed limit is uniquely mapped to infinite Proper velocity, all finite Proper velocities, which result from finite rapidities, must transform to sublight velocities. Since the sum of two finite rapidities is another finite rapidity, the composition of any two sublight velocities is necessarily another sublight velocity. No combination of two sublight velocities can exceed or reach c.

If one of the combining velocities is c, we still apply the same rule. Except that one of the combining rapidities is now infinite. The sum of a finite rapidity and an infinite rapidity is just the same infinite rapidity, because all finite rapidities are closer to zero than to infinity. The resulting sum maps back to 1c, and the speed of light is invariant with respect to relative velocity. The counterintuitive behavior of velocity addition is the perfectly logical behavior of infinity.

As a side benefit, the diffeq also explains the origin of the myth of relativistic mass. In analyzing the diffeq with respect to velocity, we paid no attention to the sine projection of the rapidity, and its corresponding velocity. If we scale both sides of the equation by invariant mass, it becomes a diffeq in momentum. And momentum has the additional constraint of conservation laws. They do not allow momentum to be destroyed, so the difference between relativistic momentum and Newtonian momentum, which is the sine or transverse projection of relativistic momentum, must point in a direction perpendicular to the longitudinal vector.

My candidate is toroidal rotation, around the smaller circumference of a torus. This dimension is already perpendicular to ordinary angular momentum and as its longitudinal component is sinusoidal, it has no net effect on forward momentum. It effectively disappears until the object is smashed into a target, and all of its momentum is returned to the surroundings, not just the longitudinal part. This toroidal momentum was mistakenly identified as relativistic mass. An object does not get heavier. The applied force is rotated away from the longitudinal direction to the transverse, toroidal direction, and the added energy is less effective at increasing longitudinal momentum and more and more effective at increasing toroidal momentum.

In the limit of c, the angle of rotation is Pi/2, and none of the input can contribute to forward momentum, because the vector is now at right angles. Of course, at this point, the toroidal component is also infinite, so there can be no further increase. But relativistic mass does not now, nor did it ever, exist. It is a corruption of conservation of momentum, because physicists don’t like imaginary numbers. They reluctantly allow ordinary angular momentum to be represented by an imaginary number, but they overlook the fact that the skin of a torus has two perpendicular imaginary dimensions. It is a hypercomplex surface.

I have tried to show how the math is already used in the physics at every step. It all stems from the same differential equation. This is the basis of special relativity, not some educated guesses. But in the OPINION of mainstream physics, you can’t question special relativity, let alone improve it. That would be heresy.