Much has been made of the fact that Einstein created special relativity out of two postulates. Maybe he should have used more. The fact is, although all coordinate systems are equivalent, the laws of physics take on different forms, depending on the kind of coordinates. The Einstein postulate applies to ordinary coordinate systems that differ by translation, rotation or reflection. It turns out the universe does not prefer these Cartesian coordinate systems, not even the Minkowski version, which is just s silly-putty version of the ordinary kind. The universe is hyperbolic, and it prefers hyperbolic coordinates. It is well-known that the Lorentz transformation is a hyperbolic rotation. It is pervasive through many branches of physics. I propose the reason is that hyperbolic trigonometry is preferred. It is also well known that 4-vectors are intrinsically Lorentz transformable, and that they are associated with a Lorentz invariant. This invariant is the hyperbolic magnitude of hyperbolic coordinates. It is invariant because in normal, orthogonal coordinates, magnitude is unaffected by changes of rotation angle, and the Lorentz transformation is just a hyperbolic rotation. This is not surprising, as the Lorentz matrix has a determinant of 1, a characteristic of a pure rotation matrix. The formula for the Einstein Interval is just the reverse coordinate transformation that maps Minkowski coordinates back to hyperbolic. A consequence of the fact that hyperbolic magnitude is invariant is that the entirety of the Lorentz transformation can be compactly written as w3 = w1+w2.
The universe is also hypercomplex. This is a reference to a family of complex units which begins with the familiar complex numbers, a+ib. These numbers are closed under arithmetic operations, and any combination of them can be reduced to this one basic form. The first extension involves replacing the real coordinates, a and b, with complex scalars. In the complex numbers, this would just reduce to another complex number. But in the hypercomplex number systems, the complex scalars use a different imaginary unit than i. Say, for example, the coordinates are of the form c+jd. If a = c+jd and b = e+jf, then a+ib becomes (c+jd)+i(e+jf) = c+jd+ie+ijf. i and j are different imaginary dimensions, so they do not combine. Their product is yet another dimension, k, so the whole number is c+ie+jd+kf. i, j and k are a 3-cyclic subset of the quaternions. Interestingly, the sum of the squares of these units is c²-e²-d²-f², the signature of a Lorentz invariant. But the universe doesn’t stop there. The real scalars are again replaced by another complex scalar, again different from i, j, and k. Each of these can be interpreted as a rotation angle around a linear dimension. The new hypercomplex unit, h, is perpendicular to each of these three, and represents a rotation away from the axis, as opposed to around it. The Cayley table reveals another interesting property. While each of the lower order imaginaries has a square that is negative, these new imaginary combination units, hi, hj and hk, have squares that are positive, since h² is -1.
In Euclidean geometry, there is a well-known inner product, commonly referred to as the dot product. Given a vector of any length, this product is the term-by-term pairing of elements in their order in the vector. The inner product of a vector with itself is the Euclidean norm. It is always positive definite, unless all coordinates are 0. If we allow complex scalars, this is no longer the case. However, we can define a complex product in a way that behaves the same. If the inner product of a complex vector is with its own conjugate, (a+ib)(a-ib) = (a²+b²)+i0. Technically, this is the same definition as the Euclidean norm, because the conjugate of a real vector is equal to the vector. The rule also works for quaternions, (c+ie+jd+kf)(c-ie-jd-kf) = c²-ice-jcd-kcf+ice+e²-kde+jef+jcd+kde+d²-idf+kcf-jef+idf+f² = c²+e²+d²+f².
When we are working with relativity, only components that are parallel to the relative velocity vector are affected by it. The axes can always be rotated so that the velocity vector is parallel to one axis. The other two, which are normal to velocity are invariant. So we can simplify the description to only two dimensions, one time and one space. Let us define the complex inner product to be <x,y> = xy* = (a+ib)(c-id) = (ac+bd)+i(bc-ad). In polar coordinates, x = |x|cos(ψ)+i|x|sin(ψ) and y = |y|cos(φ)+i|y|sin(φ). The inner product becomes (|x|cos(ψ)|y|cos(φ)+|x|sin(ψ)|y|sin(φ))+i(|x|sin(ψ)|y|cos(φ)-|x|cos(ψ)|y|sin(φ)) =
|x||y|(cos(ψ)cos(φ)+sin(ψ)sin(φ))+i|x||y|(sin(ψ)cos(φ)-cos(ψ)sin(φ)) = |x||y|(cos(ψ-φ)+i sin(ψ-φ)), =
|x||y|e^i(ψ-φ), a standard polar representation. In the special case of the inner product of a complex vector with itself, x = y and ψ = φ. |x||y| = a²+b², and ψ-φ = 0, so <x,x> = |x|² = a²+b². In the extra special case where x is a point on the unit circle, a = cos(ψ) and b = sin(ψ), and a²+b² = 1. So, when x and y are both unit vectors in arbitrary directions, the inner product has a magnitude of 1, making it another unit vector, and the angle is the difference between the phases of the two factors, the included angle. This is the angle which defines the dot and cross product of two vectors, in geometric terms. While the analytic definition involves individual components, the geometric definition, which is equivalent, is simply the product of the scalar magnitudes and the cosine or sine of the included angle. The sum of the squares of the dot product magnitude and the cross product magnitude (it is a vector) is thus the square of the magnitude of the complex vector product. However, this is not helpful for relativity. Circumstances can be easily configured such that the included angle can exceed Pi/2. This is not permitted when combining two velocities, because the complex phase angle is bounded by lightspeed angles.
But it was asserted above that the universe is hypercomplex. This is a small but significant difference. In hypercomplex coordinates, the hyperbolic magnitude is an invariant with respect to hyperbolic rotations, which is what Lorentz transformations are. If we apply the definition of the complex inner product to hypercomplex units, there is an issue. Since the Cartesian representation of a hypercomplex number is of the form, t+hi z, with conjugate, t-hi z, the inner product of a hypercomplex vector with itself is <u,u> =
(t+hi z)(t-hi z) = (t²-(hi)² z²) + hi 0 = t²-z², because (hi)² = +1. This is the subtle difference. And in hyperbolic coordinates, t = s cosh(w) and z = s sinh(w). The inner product becomes s²cosh²(w)-s²sinh²(w) = s², for all values of w, the hyperbolic rotation angle. In the more general case of two different hyperbolic vectors, u and v, u = t+hi z and v = x+hi y. The inner product becomes <u,v> = uv* = (t+hi z)(x-hi y) =
(tx-zy)+hi(zx-ty). If we use ordinary polar numbers, this becomes
(|u|cos(ψ)|v|cos(φ)-|u|sin(ψ)|v|sin(φ))+hi(|u|sin(ψ)|v|cos(φ)-|u|cos(ψ)|v|sin(φ)) =
|u||v|(cos(ψ)cos(φ)-sin(ψ)sin(φ))+hi|u||v|(sin(ψ)cos(φ)-cos(ψ)sin(φ)) =
|u||v|(cos(ψ+φ)+hi sin(ψ-φ)), and the phase angle is not in the same form as the factors. It cannot be expressed as a simple exponential term. But if we use hypercomplex, hyperbolic polar coordinates,
u = s cosh(w)+hi s sinh(w), and v = s’cosh(w’)+hi s’sinh(w’), and <u,v> = uv* =
(s cosh(w)s’cosh(w’)-s sinh(w)s’sinh(w’))+hi(s sinh(w)s’cosh(w’)-s cosh(w)s’sinh(w’)) =
(s s’)(cosh(w)cosh(w’)-sinh(w)sinh(w’))+hi(s s’)(sinh(w)cosh(w’)-cosh(w)sinh(w’)) =
(s s’)(cosh(w-w’)+hi(sinh(w-w’)) = (s s’)e^hi(w-w’), standard form for a hyperbolic, hypercomplex number, just like the factors. If we form the inner product of this general form with itself, the result is
((s s’)e^hi(w-w’))*((s s’)e^hi(w’-w)) = (s s’)² = s²s’², the product of the squares of the hyperbolic magnitudes. In the special case where s = s’ = 1, the hypercomplex inner product of an arbitrary hyperbolic unit vector with itself is 1. Since the exponent is the same for both the real and hyperimaginary coordinates, we do not have a compatibility issue. And since hyperbolic angle composition is by linear addition, and the hyperbolic angle is unbounded, we have no issue with lightspeed limits, either. The hypercomplex inner product of unit vectors gives us the difference in hyperbolic rotation angle between any two factors. Since the magnitude is a scalar, it does not affect the hyperbolic rotation angle, and the formula is valid for any two hypercomplex vectors. The relative hyperbolic angle is known in physics as the relative rapidity between two points, and the gudermannian of the rapidity is the relative phase angle, θ, that determines the measured relative velocity between the same two points, v = c sin(θ). Since <u,v> = uv* = (t+hi z)(x-hi y) =
(tx-zy)+hi(zx-ty), transposing the order of the two factors, which is equivalent to shifting the origin from one frame of reference to the other, results in <v,u> = vu* = (x+hi y)(t-hi z) = (xt-yz)+hi(yt-xz) =
(tx-zy)-hi(zx-ty) = <u,v>*. The real component does not change, which is appropriate for what is the Lorentz factor for the composite velocity, and the difference in rapidity is negated, which is appropriate for the change in origin.
Now, let’s take another look at the inner product expansion. Suppose (x,y) represents the hypercomplex phase shift unit between two frames of reference. Since rapidity combines linearly, we can identify this as the phase shift between the origins of two frames. Then all the points in the phase shifted frame have new coordinates that result from the inner product of the old coordinates with the phase vector, (x,y). In other words, t’ = tx-zy and z’ = zx-ty. We can represent this information in matrix form without changing the values. Then,
│t’│ │tx-zy│ │+tx-zy│ │+x -y││t│
│z’│=│zx-ty│=│-ty+zx│=│- y +x││z│
In case this is not obvious, x = cosh(dw) and y = sinh(dw). In more general units, t is replaced by ct, and to stress that only 1 dimension is affected by relativistic velocity, z is replaced by r. The matrix multiplication is
│ct’│ │cosh(dw) -sinh(dw)││ct│
│ r’│=│-sinh(dw) cosh(dw)││ r│,
the Lorentz transformation. All of special relativity follows from here. I think this dispatches the first Einstein postulate. All coordinate systems are equivalent, but some are more equivalent than others. (A tip of the hat to Orwell) The laws of physics may be the same, but the form is radically different. Consider the inner product. The two vectors, u and v, are arbitrary hypercomplex vectors. The end result of the inner product is a Lorentz factor and a relative rapidity. They are both arbitrary, and are a variation of the Lorentz transformation. But in the transformation, one vector is clearly associated with the frame of reference , and the other one is the transformation matrix. But the end result is that the rapidity of the original coordinates is shifted by dw, a Lorentz boost. In hyperbolic coordinates, the Lorentz transformation is just w’ = w-dw, and its inverse is w = w’+dw. What about Einstein’s other postulate? We haven’t used it at all, yet.
To begin with, it is not clear that it deserves to be called a postulate. Postulates are axioms, and axioms are things that are assumed because the consensus is that they are true, and usually they cannot be proven, either. Sometimes, as with Euclid’s Fifth Postulate, it is not so much a postulate as a filter that selects behaviors that are consistent with it from those that aren’t. Riemann demonstrated that there are whole other geometries that don’t obey Euclid’s Fifth Postulate, and that they are actually better suited to relativity than Euclidean geometry. Einstein was a good guesser. He observed that no one had ever measured the speed of light to be dependent on the relative velocity of the observer or the source. He concluded, correctly that this was due to some fundamental property of nature, and took the leap of asserting his postulate. He had no proof, and the possibility existed that he was wrong. But if that were the case, somewhere in his logic, there would have been a contradiction. He did not find one, so he published special relativity. Since then, there has been no credible evidence that his postulate is incorrect. I do not dispute his claim. Instead, I offer mathematical proof of why his postulate is true. This changes it from an unprovable assumption to a provable theorem.
Much is made of the fact that Einstein developed special relativity from just two assumptions. In fact, it is a logical derivation from a fundamental differential equation, one that has an explicit closed form solution. It is argued that math is not physics, especially by the boys with their toys in the white lab coats. But when an intricate mathematical structure agrees in 100% detail with all the experiments that were ever performed, ostensibly to prove special relativity, the question becomes “Why not?” The differential equation has a history even older than relativity, and its first major application even predates the invention of calculus and differential equations. Where has it been hiding, you might ask. Have you heard the story of the Purloined Letter? Like it, it has been hiding in plain sight, literally for centuries. I am referring to the Mercator Projection Map. The algorithm he used to create the map is an implementation of the differential equation. An even older implementation exists using classical Greek geometry and their tool, straightedge and compass construction. It did not have the global impact of the Mercator map, however. For 4 centuries after its introduction, it was the primary means of navigation on the open seas, until the modern inventions of radar and GPS. Mercator introduced the map centuries before Einstein, before Newton invented calculus, before Newton was even born, when even Galileo was only a child. Although the differential equation did not exist yet, with the benefit of hindsight we can study it that way. The differential equation relates a small change in hyperbolic angle to a small change in circular angle. The relationship to relativity is simple. First, some definitions. Let the hyperbolic rotation angle be w, the circular rotation angle be θ and the Lorentz factor be γ. Then the diffeq is one of two reciprocal forms, either dw/dθ = γ or dθ/dw = 1/γ. Note that this is an analytic definition of the Lorentz factor, not some empirical formula derived from countless experiments.
Rather than dive right into the solution, I want to take the long way round, and illustrate more of its physical implications, first. We begin with some classical Greek geometry. Starting with a couple points, an origin and a unit step, we construct two lines, one a baseline through the two points and the other a perpendicular at the unit step. On top of this grid, we add the unit circle and the unit hyperbola. Now, you can’t construct a unit hyperbola with only ruler and compass, but the Greeks did know how to construct one with another auxiliary. In any case, this isn’t a proof. It’s an illustration. Now, pick any arbitrary point on the perpendicular. Construct a radius through the point and the origin. This cuts through the unit circle, and defines a wedge between the radius, the baseline and an arc of the circle. The area of this wedge is 1/2 the arc length of the piece of the circumference. Using the hypotenuse of the triangle formed by the baseline and the altitude on the perpendicular, construct another arc that swings from the apex of the triangle to the baseline. Construct a line parallel to the base that passes through the apex of the triangle, and another line, perpendicular to the base, that passes through the point where the larger arc intersected the base. Where these two lines intersect is a point which is on the unit hyperbola. Technically a ruler and compass construction must be finished in a finite number of steps. To draw on a piece of paper, this is technically impossible, because distance is an analog function, and there are an infinite number of points. But a computer screen is digital, and has only a finite number of points. And it is technically just an automated ruler and compass. The point is, a computer can repeat the same process a finite number of times, and create as close a simulation of a hyperbola as the pixel density will allow. Now, consider the wedge formed by the radius of this intersection point to the origin, the baseline and a section of the hyperbolic arc. These borders are actually stereographic projections of the same borders that define the circular wedge, and every point inside that wedge maps to a point inside the borders of the hyperbolic wedge. The area of this wedge is 1/2 the hyperbolic rotation angle. Later, I will show that using the other form of unit hyperbola, and a circle that is tangent to it, results in area patches that are equal in size to the actual hyperbolic angle and its gudermannian. In short, the area of a wedge from a circle with radius √2 is the same as the arc length of the border, and therefore, the area is the central angle. The area of the corresponding hyperbolic wedge is the hyperbolic rotation angle. The area of he circular wedge is the gudermannian of the area of the hyperbolic wedge. Clumsy as it may be, this construction implements a transcendental function, faithfully, using ruler and compass. You may not be able to square the circle, but you can find the gudermannian.
This described the exact, closed form solution to the diffeq, but it doesn’t do justice to the trig projections of these two angles. As you will see, a number of the line segments in the drawing can be described in terms of either angle. There are 6 identities which we will look into, next. A drawing might be helpful, but we will look at it from the perspective of coordinate geometry. We start with the unit circle and the unit hyperbola. The equation of the unit circle is x²+y² = 1, while the equation of the unit hyperbola is t²-z² = 1. I use different variables so that we can talk about two arbitrary, unrelated points. Because the hyperbola consists of two disjoint branches that are mirror reflections, we can focus on the right half plane with no loss of generality. We rearrange the equation for the hyperbola by adding z² to both sides, and then dividing both sides by t². We are allowed to do this because t² can never be 0. So, now the equation of the hyperbola is
1 = (1/t)²+(z/t)². The two points, (t,z) and (x,y), have been unrelated up to this point. However, it should be clear that for every point, (t,z), on the hyperbola, there is some point on the circle such that if x = 1/t, then y = z/t. Then, excluding 0’s in the denominator, 1/x = t and 1/y = t/z. Similarly, there is a ratio and its reciprocal, x/y = 1/z and y/x = z. If we substitute circular polar equivalents into x and y, and hyperbolic polar equivalents into t and z, we get 6 identities that relate the same hyperbolic angle to its gudermannian, the circular angle. So, x = cos(θ), y = sin(θ), t = cosh(w) and z = sinh(w). The identities are:
t = cosh(w) = γ = sec(θ) = 1/x
t/z = coth(w) = 1/β = csc(θ) = 1/y
1/z = csch(w) = 1/βγ = cot(θ) = x/y
1/t = sech(w) = 1/γ = cos(θ) = x
z/t = tanh(w) = β = sin(θ) = y
z = sinh(w) = βγ = tan(θ) = y/x
These 6 identities are true for all pairs (w,θ), where θ = gd(w). Their relationship to relativity is expressed by the Greek symbols in the middle. These are all bidirectional mappings, so any one equality defines all the rest. And if we implicitly differentiate any of the 6 trig function identities with respect to either angle, the result is one of the two forms of the diffeq that relates w to θ. We did not need to solve the diffeq to get these identities, but with them, solving it is trivial. The definition of the exponential is e^w = cosh(w)+sinh(w). From the identities, e^w = sec(θ)+tan(θ), and w = ln(sec(θ)+tan(θ)), the closed form solution. As an aside, it is worth noting that e^w and 1/e^w are the eigenvalues of the Lorentz matrix, and k and 1/k of Sir Hermann Bondi’s k-calculus, what he calls a “fundamental ratio”.
Returning now to the diffeq, we have solved it for w. Now we solve it for θ. In terms of θ, the diffeq becomes dθ = dw/γ = sech(w) dw. This can be found in any standard list of integrals, but with the help of the list of identities, the solution is straightforward. As with many diffeq’s the solution is easier with an integrating factor. In this case, the integrating factor is one of the identities already determined, cos(θ) = sech(w). If we multiply the two equations, we get cos(θ) dθ = sech²(w) dw = 1/cosh²(w) dw =
(cosh²(w)-sinh²(w))/cosh²(w) dw = (cosh(w)dsinh(w)/dw-sinh(w)dcosh(w)/dw)/cosh²(w) dw =
d(sinh(w)/cosh(w))/dw dw = d tanh(w). Then the integral is simply sin(θ) = tanh(w), since when θ = w = 0, sin(θ) = tanh(w) = 0. Not surprisingly, this is one of the identities. Then, θ = arcsin(tanh(w)), and we find that another definition of the gudermannian function is arcsin(tanh()). Any one of the other 5 identities could also be used to define the gudermannian.
There is one more identity to explore before we proceed to the “ultimate speed limit”. Referring back to the original graphic, suppose we bisect the angle between the hypotenuse and the baseline. Since the limit of the angle is Pi/2, the limit of its bisector is Pi/4. Since the hyperbola asymptotically approaches the diagonal with slope = 1, the angle bisector, if extended far enough, will intersect the hyperbola, too. Since it must also intersect the perpendicular at the unit step, the two points are on the same line and must have identical slopes. The point on the perpendicular has coordinates, (1,tan(θ/2)). The coordinates of the point on the hyperbola must be of the form (cosh(w’),sinh(w’)), so its slope is tanh(w’). The question is, what is the relationship between θ/2 and w’? We note that tan(θ/2) = sin(θ/2)/cos(θ/2), and that from the half-angle formulas, sin(θ/2) = √((1-cos(θ))/2) and cos(θ/2) = √((1+cos(θ))/2), so tan(θ/2) = √(1-cos(θ))/√(1+cos(θ)). We can rationalize the fraction by multiplying by √(1-cos(θ))/√(1-cos(θ)). The result is (1-cos(θ))/sin(θ) = csc(θ)-cot(θ). From our table of identities, this equals coth(w)-csch(w) = (cosh(w)-1)/sinh(w) = √(cosh(w)-1)²/√(cosh²(w)-1) = √(cosh(w)-1)/√(cosh(w)+1) = √((cosh(w)-1)/2)/√((cosh(w)+1)/2) = sinh(w/2)/cosh(w/2) = tanh(w/2). In other words, w’ = w/2 and tan(θ/2) = tanh(w/2). Graphically, what this means is that no matter how close the hypotenuse gets to vertical, there is always an infinite amount of area between the bisector and the diagonal with slope +1, while the area between the bisector and the original hyperbolic radius vector always equals the area between the bisector and the baseline. The fact that these areas are equal means they represent identical boosts, even though their shapes are radically different. This is a normal property of bivectors that vectors do not have. More on that later.
Returning to the reciprocal of the diffeq, dθ = cos(θ) dw. This tells us that at very small angles, which correspond to very small velocities relative to c, the cos(θ) ≈ 1, and a small change in the hyperbolic angle results in an identical change in the circular angle. Since hyperbolic angles combine linearly, as long as the two velocities can be represented by small circular angles, this means that the circular angles also combine by linear addition. When the angles are that small, another approximation is also true, sin(θ) ≈ θ. Since c sin(θ) represents a relative velocity, when it is small, it adds linearly, which is what we observe in Newtonian physics, where all velocities are small fractions of c. So, although Newtonian velocities do add linearly, that is a special case where a first-order approximation is valid. It is not generally true. When θ becomes a significant fraction of Pi/2, the approximation breaks down, and velocities no longer add linearly. However, the hyperbolic rotation angles add linearly at any value, from -∞ to +∞. Referring, once again, to the table of identities, c sin(θ) = c tanh(w), in general. Physics calls the hyperbolic angle rapidity, and rapidity addition is linear. So, if we have 3 velocities, v1, v2 and v3, then we also have 3 rapidities, w1, w2 and w3. Linear addition means w3 = w1+w2. So, if v3 = c sin(θ3) = c tanh(w3), then v3 = c tanh(w1+w2). According to any list of trig identities, tanh(w1+w2) = (tanh(w1)+tanh(w2))/(1+tanh(w1)tanh(w2)). Substituting this into the formula for v3, v3 = c(tanh(w1)+tanh(w2))/(1+tanh(w1)tanh(w2)) =
((c tanh(w1))+(c tanh(w2))/(1+(c tanh(w1))(c tanh(w2))/c²) = (v1+v2)/(1+v1*v2//c²), the non-linear velocity addition rule of relativity. All finite rapidities generate finite trig projections. The sum of any two finite rapidities is another finite rapidity. Therefore, it is impossible for any combination of velocities, no matter how close to c, to reach or exceed c. Following this thread, there are more correlations. To fully explain, we must bring Proper velocity into the discussion. Physics does not like to acknowledge Proper velocity, because it is unbounded, and its very existence contradicts the lightspeed limit. So they call it hypothetical or non-physical or some other nonsense, despite the fact that all momentum is just invariant mass x Proper velocity. So if you divide relativistic momentum, a physical quantity that is measured, by invariant mass, another physical quantity that can be measured, it equals Proper velocity, a quantity that physics denies is physical. Interesting. In any case, we have made use of one identity to add velocities. The identity for sinh(w) defines Proper velocity, γv = c sinh(w) = c tan(θ). We have already noted that as w approaches ∞, tanh(w) approaches 1. At the same time sinh(w) approaches ∞, same as tan(π/2), while sin(π/2) approaches 1 and v approaches c. This is a functional mapping, and each value is uniquely mapped to one other value. This tells us that a measured velocity of c is uniquely mapped to a Proper velocity of ∞, and an infinite Proper velocity is uniquely mapped to a rapidity of ∞. In other words, all finite values of rapidity map to sublight velocities. So this is why we cannot exceed lightspeed. There is no Proper velocity greater than infinity to project a cosine fraction of itself that is greater than c. In technical terms, c is not an absolute limit. It is the limit of the cosine projections of Proper velocity as Proper velocity approaches ∞. This is the basis of the Einstein postulate. Infinity is the same everywhere, so the limit of its cosine projections is also the same for all observers. This explains why lightspeed is constant. Rapidity addition explains why it is invariant.
We have already covered velocity addition. It is non-linear because rapidity addition is precisely linear. So, it is perfectly logical when combining velocities to first convert each velocity into its corresponding rapidity, and then add the rapidities and convert back to a velocity. When sublight velocities add, the result is always sublight. But what if one, or both, of the combining velocities is already c? According to the algorithm, we first map each velocity to its rapidity. In the case of a lightspeed velocity, its rapidity must be infinite. When we try to add a finite rapidity to an infinite rapidity, the result is the same infinite rapidity, because all finite values are closer to 0 than to infinity. It is essentially like adding nothing. The result is infinity, and it maps back uniquely to lightspeed. And there we have it. Lightspeed is invariant with respect to relative velocity of its source or observer. In the case where both velocities are lightspeed, then both rapidities are infinite. This is different than before, but the result is the same. You can’t multiply infinity by a finite number. After all, an infinitely tall stack of $20 bills is worth the same amount as an infinitely tall stack of $1 bills. So the result of combining two lightspeed velocities is just 1c. These are exactly the same results as the non-linear velocity addition rule gets. It turns out that the lightspeed limit is not the huge number of m/s. After all, in natural units, lightspeed is 1, not a very impressive limit. On the other hand, regardless of the units, lightspeed always maps to infinite Proper velocity, and infinite Proper velocity means infinite momentum, and infinite momentum means infinite energy. That’s an ultimate limit.
I have referred to the linearity of rapidity addition several times. Because the non-linear velocity addition rule is a direct consequence of linear rapidity addition, we have confirmation. But velocity is a measured quantity, and there is no guarantee that at some number of digits of precision, the measurements might disagree with the formula. But there is a more fundamental reason. I mentioned earlier that the area of a wedge from a circle of radius √2 is equal to the central angle. That this is so follows from the fact that the central angle is the fraction of the circumference represented by the arclength. The area of the sector is this same fraction of the area of the circle. Since area is πr², and r = √2, the area of the circle is 2π. The central angle of any whole circle of any size is 2π. So for this particular size circle, the area of a sector is the same as the magnitude of the included angle at its center. The unit hyperbola that is tangent to this circle at its vertex is not the horizontally symmetric hyperbola we have been using. It’s the hyperbola defined by ΣΔ = 1. This hyperbola has a diagonal axis of symmetry, and rapidity is defined as the area of the triangular wedge between the radius vector from the hyperbola to the origin, the axis of symmetry and the arc of the curve. For this hyperbola, the vertex is the point (1,1). Its diagonal length is √2, the radius of the circle. But the invariant is not the semi-major axis, a length. The invariant is the area of the square formed by the vertex and the origin. This is one of those properties of bivectors that I mentioned earlier.
The horizontal hyperbola has a vector invariant. Every point on the curve is chosen because it has the same semi-major axis as every other point. The diagonal hyperbola is defined by the area of the box constructed by dropping perpendiculars to both axes from the point on the hyperbola. The box is only square when one corner is the vertex, but every rectangle formed by every other point has the same area. But while the area is invariant, the length of the diagonal is not. This is the reason that Minkowski grids are silly putty. The magnitude of a bivector is its area, not its diagonal length. Although bivectors are frequently represented by vectors normal to their area, this conceals the fact that they are fundamentally 2 dimensional, the bi- in bivector. Even the Einstein Interval conceals this fact. It is written as c²t²-r² = s². But this can be factored, and (ct+r)(ct-r) = s². This shows its character as a bivector. It turn out that these are the (Σ,Δ) coordinates for an arbitrary point in eigenvector axes. Σ = ct+r and Δ = ct-r. Even the Lorentz transformation looks different. In eigenvectors, the hyperbolic rotation is a squeeze mapping. Because it is the area that is invariant, if one edge is scaled up, the other edge is scaled down by the same factor. That way their product remains constant. The Lorentz matrix is diagonal, with o in the other two corners, and the two eigenvalues on the main diagonal, 1/k and k:
│1/k 0││Σ│ │Σ’│ │Σ/k│
│ 0 k││Δ│=│Δ’│=│Δ*k│
If Σ*Δ = s², then Σ’*Δ’ = (Σ/k)(Δ*k) = (Σ*Δ)(k/k) = Σ*Δ = s². Here, I would just like to add that this makes the invariant the geometric mean between the two eigenvector coordinates, √(Σ*Δ). Σ/√(Σ*Δ) = √(Σ*Δ)/Δ =
√(Σ/Δ).
Returning to rapidity, its geometric definition is the area of an irregular shaped triangular wedge. We want to find this area analytically, and that requires finding an integral. As is commonly done with irregular shapes, we break this one down into smaller pieces and add them together. This is where the properties of the diagonal hyperbola become especially useful. The area of any rectangle formed by a point on the curve is fixed. For this argument, we are using the unit hyperbola, because it is the one that defines rapidity exactly. Any other hyperbola would scale up or scale down the area. But since the area of any rectangle is 1, the area of a triangle formed by bisecting one of these rectangles with its diagonal must be 1/2. If we add one triangle and subtract another, the total area of the shape remains constant. Specifically, if we add the area of the right triangle from the vertex, and subtract the area of the right triangle from the point, the new shape is a quadrilateral with three straight edges and the same section of the hyperbola. More to the point, this is an area that is the definite integral from one point to another under the curve. The height is Δ, and the dummy variable is dΣ. ∫Δ dΣ = ∫dΣ/Σ = ln(Σ), from the vertex to the point. This is backwards from the usual definite integral, so rapidity, w = ln(1)-ln(Σ) = ln(1/Σ). Then, e^w = 1/Σ, and 1/e^w = Σ, and Δ = e^w. These are the coordinates of any point on the curve. They are also the eigenvalues of the Lorentz matrix that boosts a point from the vertex to another location. In other words, every point on the unit hyperbola represents a Lorentz transformation. Rapidity is the area under the curve between any two points. It is a fundamental property of all definite integrals, where one exists, that the limits of integration obey a law: If A and C are the limits of integration, and B is a third point (usually between them, but not necessarily), then the definite integral from A to C is equal to the sum of the definite integral from A to B plus the definite integral from B to C. Since every definite integral is a rapidity, w3 = w1+w2. So the linearity of rapidity addition is a generic property of definite integrals. They are but a single instance.
Before I draw the curtain on this draft, there is 1 more issue I want to cover. That is the myth of relativistic mass. Everybody knew that relativistic particles carried more momentum than Newton predicted. Some clever guys came up with the bright idea that mass increased with velocity, making it harder and harder to accelerate. To his credit, Einstein didn’t buy it. He advised people, who bothered to ask, that they should not use the subject of relativistic mass, but to use relativistic momentum or energy instead. These were at least known to vary with relative velocity in Newtonian physics. He was uncomfortable with the idea of something as fundamental as mass being dependent on the velocity of the observer. Later on, when physics adopted the use of 4-vectors to represent fundamental properties like velocity and momentum, it became clear that mass was a relativistic invariant of the Lorentz transformation of 4-momentum. It could not vary with velocity. There are still schools and teachers who consider this an acceptable concept, more the pity for their students. I wonder, what they would think of a chemistry teacher who still taught phlogiston theory? Most mainstream physics courses just bury the subject. My textbook explicitly said in the Foreward that it would not be mentioned in the text because it did not contribute to the understanding of the subject. I have heard a number of lame excuses for it, but none have any credibility.
I am about to present my explanation, which is based on the time-tested principle of Conservation of Momentum. That and the hyperbolic, hypercomplex nature of the universe, which physics ridicules. We start with the definition of relativistic momentum, invariant mass x Proper velocity. While Proper velocity is a real quantity, it is the magnitude of a hypercomplex vector. It has a real and a hyperimaginary component. (It has an ordinary complex component, too, but there is no rotation involved in this case, so there is no angular momentum component) The real component is longitudinal, and the simple complex component is circumferential, circling the longitudinal component. The hyperimaginary component is perpendicular to the ordinary imaginary component, and can best be visualized as toroidal rotation around the smaller circumference of the torus. This direction is already perpendicular to the ordinary angular momentum, and because it is cyclic it has no net component parallel to the longitudinal component. It is invisible to all attempts to measure it. But we know it is there, because when we smash a relativistic particle into a target, it dumps all of its momentum back into the surroundings, not just the longitudinal, Newtonian part. The division of input energy between real and hyperimaginary is defined by the hypercomplex rotation angle that is the gudermannian of the rapidity. The faster the measured velocity, the greater the angle, and the smaller the cosine projection. The cosine projection is in the longitudinal direction, and is the reason it gets harder to accelerate any mass. It is not the mass which is getting heavier. It is the application of force which becomes less efficient, because the force is being applied at a complex phase angle to the path. Instead of increasing forward velocity and momentum, more and more of the applied force is redirected to increase hypercomplex velocity and momentum. The sum of the squares of all the momentum and velocity components is relativistic momentum squared and Proper velocity squared. Since the gudermannian of infinite rapidity is Pi/2, at that measured speed (which is lightspeed), none of the input energy is directed along the longitudinal vector, and even if there were more than infinite energy available, none of it would contribute to forward velocity. At the same time, hyperimaginary momentum has reached infinity. Of course, this is an asymptote, but it clearly explains that the reason for the myth is just Conservation of Momentum.
At this point, I would like to speculate. While the rest of the text is based on rigorous mathematics, this is unconfirmed material. It is also based on rigorous math, but there are parts which are hypothetical. Not as crazy as the Alcubierre drive, but theoretical, nonetheless. The speculation is based on the fact that it takes no work to change the direction of a velocity vector. Changing the direction does not change the speed, and as the momentum and energy depend on the Proper velocity, not the measured velocity, we have already broken the lightspeed barrier any time measured velocity is more than 71%c. The speculation is that if we can bend the Proper velocity vector without increasing its magnitude, we could apply more, or all, of Proper velocity towards forward motion, without requiring any more energy than it took to get to the starting velocity. In theory then, we could reach multiples of c without getting anywhere near to the infinite momentum barrier usually associated with lightspeed. No matter how carefully we design rockets according to Newton’s laws of motion, we fail to compensate for the invisible hypercomplex rotation. You must remember that Newtons laws of motion are a product of Newton’s first-order approximation physics. That doesn’t work at relativistic speeds, either. Still speculating here, I suspect that some configuration of electromagnetic fields, conductors, dielectrics and geometry might be required to steer the Proper velocity vector. It sounds disturbingly similar to the notorious, 80 year old Philadelphia Experiment. Could the government already know this and have it buried deep in Area 51? I don’t know. Even if they don’t, might it work? I don’t know. But I think a Manhattan-like Project could figure out the answer.