It occurred to me that my exposition of Euclidean Relativity was unintentionally skewed into Cartesian coordinates. There are circles in Euclidean geometry, and when dealing with radially symmetric properties, polar coordinates are more fitting. Using spherical geometry, a point on a shell can be viewed as the sum of a 0-phase linear radius and the sum of circular arcs embedded in the surface. We will associate circular arcs with relative velocity, an arbitrary quantity. The linear radius will be associated with the Galilean coordinates. This vector is independent of the circular component, so it is invariant in magnitude with respect to the relative velocity. But when we try to measure it, it gives the illusion of being shorter. In this post, I explore the nature of multi-dimensional units and the paradoxical situation of multiple different measurements, and how they coexist with resting measurements.

Working in Euclidean geometry does not mean Cartesian, however. Symmetry, especially cyclic symmetry, is better represented by circles. We normally think of a vector as the displacement between two points. It is literally a function of just the endpoints. But the distance between those two points can be anything greater than or equal to the vector magnitude. It is a function of the path itself. We are interested in a set of paths that have very specific symmetry. What they have in common is the magnitude of the radius to the spherical surface that they intersect. The grid on these shells is formed by latitude and longitude lines, and position is specified by two angles. At angles between the two grid lines, the curves are loxodromes, because all these curves on the surface of the sphere map to Mercator's rectangular grid as straight lines.

For the spirals, arclength from pole to pole depends on the tilt angle of the spiral and the corresponding line to the vertical. At 0 tilt, the arclength is πr. When the tilt angle is not 0, the arclength is increased by the secant of the tilt angle. Arclength, in general, is proportional to the radius for two similar curves. Specifically, because of the linearity of addition, 2 spirals of half the diameter will have the same total arclength projections in both directions. Or n spirals of diameter 2r/n will still have the same size projections. In the limit, as n approaches infinity, the arc length from pole to pole is still πr sec(tilt), but the initial diameter of a single spiral at the equator reduces as 1/n, and asymptotically approaches 0. So, what we have is a line which is longer than it looks, longer than it measures, and while the displacement between the endpoints is the same in all cases, the effective radius of the spiral depends on the secant of the tilt angle. All we have to do is identify the relative velocity as c sin(tilt) with an associated Lorentz factor of sec(tilt) = γ, and you've got special relativity's time dilation or length contraction. The arclength remained constant, but the tilt angle changed, causing the spiral to tighten up, and the measured distance from pole to pole to shrink, because the arclength is invariant.

3D rotational symmetry carries a single radius vector to all the other locations on the surface of the unit radius sphere. Since none of these directions is privileged, it is only necessary to model one of them. Before getting into the details of the spirals, let's consider the spherical shells alone. We have a main axis that is horizontal. It represents every radius vector. The sphere between the origin and the shell is the cosine projection of the larger sphere, using the origin as the stereographic projection point. Similarly, any sphere anywhere represents a stereographic projection of a larger sphere along an axis of symmetry. And we can always rotate an axis of symmetry to be parallel to our baseline. Then the stereographic projection ray tilts away from the axis of symmetry and rotates around it. Since the smaller sphere is the cosine projection of the larger one, the larger one is the secant projection along the same ray. The interesting thing is that when we project both curves with the same projection cosine, the smaller shell turns into the larger shell, and the larger shell turns into the tangent plane. And if we treat the tangent plane as the cosine projection of the baseline coordinate, the curve that projected it is the unit hyperbola. The important thing to note is that all these geometries incrementally morph into each other just by using fractional powers of the scale factor, γ.

Applying this geometry to velocity, the point on the hyperbola has the coordinates of 4-velocity. The cosine projection of the Proper velocity in time is the constant velocity of c. The point on the tangent represents Proper velocity in space. This represents total relativistic velocity of an inertial frame. The cosine projection of Proper velocity is Newtonian velocity. The sine projection is responsible for excess relativistic momentum. The point represents an arbitrary, inertial laborarory frame. It is the cosine projection of the vector to the point on the hyperbola: (γc,βγc) cos(θ) = (c,βc) = (c,v). This vector illustrates the gudermannian. The slope of the line through these two points is the same at both of them. At the extreme end, the coordinates are (c*cosh(w),c*sinh(w)), while the coordinates of the lab frame are (c,c sin(θ)) = (c,v). The slope at the endpoint is sinh(w)/cosh(w) = tanh(w), while the slope at the lab point is sin(θ)/1. When θ = gudermannian(w), tanh(w) = sin(θ).

The abscissa here is the rapidity, w, and it is allowed to range from minus infinity to plus infinity. These two extreme points map to minus π/2 to plus π/2. All the finite values of w between those two limits map to tilt angles of less than π/2 in magnitude.

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The complexified unit. It is a curved hyperplane, a cone. In spite of rationalizations about subjective and objective reality, the measurements made by a co-moving observer are different from measurements made by any moving observer. If you believe in the reliability of measurements, then these two observers are simply not measuring the same thing. And yet they are. The resolution of this paradox is that the unit they are trying to measure is more than 1 dimensional. The moving observer is measuring the displacement vector between the two endpoints. The co-moving observer is measuring the distance along the vector. This distance is identical to the displacement projection when relative velocity is << c. But, like Newtonian physics, this statement is only true for those very small velocities.

I started by claiming that the Euclidean unit is actually multi-dimensional.. The hyperplane, to my surprise, doesn't even need to be a plane. It is any n-1 dimensional structure embedded in n dimensions. Like a plane in a volume, or a line in a plane, or even a point on a line. I add another wrinkle by allowing a curved 2D surface. In the embedding space, it takes 3 coordinates to unambiguously locate a point, whereas it only takes two coordinates to locate the point in the hyperplane. The defining equation of the hyperplane behaves, for all practical purposes, like a third coordinate. Which brings us to the cone.

In addition to its magnitude (or altitude), a cone is also characterized by the opening angle, the tilt of the wall of the cone relative to the axis of symmetry. Relative to this vector, a cone is just the surface of rotation of the tilted vector around the axis of symmetry. A point on the cone can be located by a length along the wall of the cone, and a rotation from some reference direction. But, although it is the same for every point on the wall of the cone, it implicitly selects a conical surface on which the other two coordinates locate a point. In 3D, points on the cone have an altitude, and the two dimensions that it takes to locate a point in the base. A word of caution here. This is NOT a light-cone. It is simply a cone.

To illustrate (and there is nothing sacred about the choice of axes or labels) let the axis of symmetry be the z axis. Eventually, this will be the direction of a relative velocity vector, but for this part of the discussion, it is just the label for the axis of symmetry. The corollary is that the normal plane to this axis is the (x,y) or (R,θ) plane. This angle specifies rotation around the axis of symmetry. In terms of these parameters, the opening angle of the cone is the arctan of the ratio of the radius of the base of the cone to the z coordinate of the base, call this φ. Then, in terms of φ and r (r² = x²+y²+z²), z = r cos(φ) and R = r sin(φ) (R² = x²+y²). φ is independent of θ, so we need both of them, but we only need one of r or R. Rotation around the axis is trivial. Thanks to rotational symmetry, the cross-section through the z axis of the cone is the same for any angle, θ.

Although φ is also a rotation angle, it has much richer texture. When φ=0, the cone degenerates into a vector of length r, the altitude of the cone. The radius of the base is 0, so the 0-phase cone is just a 1 dimensional vector. At φ=±π/2, the walls of the cone are perpendicular to the z axis, because r cos(φ) = 0. The fully open cone degenerates into a flat disk of area πr², and circumference 2πr. The area of the cone walls is directly proportional to the ratio of the circumference of the base to 2πr. As a function of φ, the circumference of the base of the cone is
2πR = 2πr sin(φ). The area of the base is πR² = πr²sin²(φ). There is an interesting identity here. The area of the disk is πr². So the area of the annulus that is the difference between these two circles is πr²-πr²sin²(φ) = πr²cos²(φ) = π(r cos(φ))², the area of a circle with radius equal to the altitude of the cone. Then the sum of the area of these two perpendicular circles is just πr², the area of the disk. When the sine and cosine contributions are equal, φ = π/4, and the two circles are both embedded in the unit diameter sphere, which has the same surface area as the disk. So, even though the 0-phase cone has no surface area at all, its complement, the area of a disk with the altitude as a radius is maximum and it equals the area of the flattened cone. The sum of the two areas is constant as the opening angle varies.

In the general case, φ can go beyond the vertical, in fact, all the way around. This is the essence of a toroidal rotation around its circular axis instead of its linear axis. When we restrict the tilt to ±π/2, there is a one-to-one mapping to the hyperbolic rotation. In this mapping, the hyperbolic angle ranges from minus to plus infinity, so there is literally nothing beyond ±π/2. In other applications, like toroidal rotation, there is no reason not to rotate past the vertical.

In any case, the opening angle does correlate with relative velocity, and if we use this multi-dimensional unit for an eigenvector axis, the relativistic effect is distributed equally between the time and space axes. The magnitude of the vector phase is an absolute invariant, so a Galilean geometry is unaffected by relative velocity. But added to the Galilean base are the circular arcs that are proportional to opening angle. These have the effect of subtracting magnitude from the linear projection and increasing the magnitude of the circular projection of the multidimensional unit. Then, if the magnitude of the multi-dimensional unit is constant, the linear projection shrinks according to the cosine of the opening angle of the cone. In other words, the linear projection x Lorentz factor = multi-dimensional magnitude. This applies to time, distance, velocity and momentum and differentials of rapidity and its gudermannian. More to follow.