This interpretation of Special Relativity uses the geometry of the loxodrome instead of the hyperbola. While not immediately obvious, this geometry opens new possibilities in theories of motion faster than light. In a more classical approach, it introduces a coordinate system that preserves the magnitude of the fundamental units of physics - mass, length, and time - independent of the velocity of the frame of reference. This is no small achievement, and, if for no other reason, justifies serious consideration of this approach.
The geometry of the loxodrome spiral needs further explanation. The curve transforms into a straight line on a Mercator projection. In this context it is known as a rhumb line. Although not the shortest arc between two points on a sphere, it has the advantage of being a constant compass course. It was used for the purpose of navigation in Christopher Columbus's time. It requires continuous course correction to travel along a great circle, which would be the shortest distance. Without satellite beacons to steer by, this would be virtually impossible. In any case, it illustrates a basic concept of relativity. The distance between two points depends on the path that separates them. In the case of special relativity, the distance between two points depends on the velocity along the path, because the velocity changes the shape of the path.
It is plain to see that the spiral is longer than the straight line that connects two points, even with the 2-D projection of the 3-D curve. To appreciate the similarity to special relativity, we must examine the precise relationship between the two lengths. The spiral itself never reaches the two endpoints, but approaches both poles asymptotically. This can easily be seen as the result of projecting a helix onto the surface of a sphere. Since the helix can never intersect the axis, no radius vector can quite reach the pole. However, the arc length does converge to a finite value, Pi * Radius * secant(tilt angle). When the tilt angle is 0, the arc degenerates to a longitude half-circle. When the tilt is Pi/2, the spiral collapses into a great circle around the equator. Using simple geometry, the total arc length is resolved into two components, one parallel to longitude lines, the other parallel to latitude circles. For a given radius, the component along the longitude lines is simply Pi * Radius, no matter what the tilt angle is. Using the right triangle formulae, the other component along the latitude circle is Pi * Radius * tangent(tilt angle).
To project the longitude arc onto the axis, we must multiply the arc length by 2/Pi. If we multiply the total arc length by the same factor, the result is 2 * Radius * secant(tilt angle), or Diameter * secant(tilt angle). The formula for the total length tells us that the factor gamma must equal the secant(tilt angle). Obviously, given any legitimate factor gamma, a tilt angle can be found that satisfies the equation. What is not so obvious is the tilt angle so derived also satisfies a different equation: relative velocity of frame = c * sin(tilt angle), where c is the speed of light.
The correlation of speed with tilt angle implies the velocity along the spiral path must be faster than the measured velocity along the straight line. At zero relative velocity the arc length is Pi/2 * Diameter. The value of light speed along the arc is thus Pi/2 * c. Multiplication by the projection cosine along the axis results in precisely the measurable value of c as we know it. However, there is no limit to the amount of circumferential velocity. Since it is perpendicular to the axis it doesn't contribute to the forward velocity, anyway. However, in order to comply with the rules of vector decomposition and to preserve the linearity of the system, we must require the linear velocity along the path (after factoring out the central angle, Pi) to be velocity * secant(tilt angle), or gamma * velocity. The axial component is the cosine projection, which is just velocity. The circumferential component is the sine projection, which is velocity * tangent(tilt angle), as specified above.
This means the extra momentum observed when a relativistic particle is stopped does not come from relativistic mass, but rather from actual velocity along the path of the spiral. This is precisely because the spiral is longer than the axis by the exact same factor. Since the total of units of length is gamma * measured length, the contraction factor can now be seen to be the projection cosine of the spiral onto the axis. The unit of length itself now becomes invariant in magnitude, even as the projection cosine varies. Similarly, the unit of time is also invariant in magnitude as the projection cosine varies. It is only the real projection that varies in size with frame velocity.
By complexifying the units of mass, length, and time, we now have a vector space which has real (axial) and imaginary (circular) projections into the 3-D Cartesian world of measurements. The units now have constant magnitude with respect to velocity, but their cosine projections exhibit all the expected behavior of real measurements. Whenever the tilt angle is greater than Pi/4, the arc length around the circle is greater than the arc length along the axis. More important, the total velocity of the two arcs combined is greater than c. As the circumferential component grows without limit, the projection cosine approaches zero asymptotically. As the tilt angle approaches Pi/2, the velocity along the spiral approaches infinity, as the secant(tilt angle). At infinity, the projection cosine(tilt angle) is zero, and the asymptote for the axial component is c. So our concept of a limiting velocity is really the projection of infinite velocity in a complex vector space.
We can't make anything go faster than light, because all our approaches add energy along the path, with the resulting cosine projections. If we could distort space, we could point the path vector along the axis, and travel at speeds faster than light without requiring infinite energy. Further, since the hidden velocity exists curled around the axis of motion, we would expect angular velocity components to produce some kind of anomalous behavior. This would include both momentum components and circulating currents. It may turn out that there is no way to distort space. However, that is a different problem from trying to break a speed limit.
Another benefit of the tilt angle model is the ability to predict the velocity addition formula. The existence of a speed of light is not in itself a consequence of special relativity. It is a mathematical requirement of the solution of the wave equation. What makes light speed special to relativity is the fact that nothing material appears to be capable of reaching or exceeding that velocity. Lorentz developed the mathematics of relativistic correction before Einstein published Special Relativity, partly in response to failed experiments to detect the ether wind of classical physics. The theory was developed to fit the data. If we examine the tilt angle model we find that the velocity additon formula becomes a mathematical necessity, and the data now supports the theory.
We can represent any observed velocity as c * sin(tilt angle). If we are combining two velocities, each can be represented by a tilt angle. The resultant must also be able to be represented by a tilt angle. To be consistent with observed data, the tilt angle of the result must be less than 90 degrees. The addition of tilt angles is clearly not the correct formula, since that could result in as much as 180 degrees total. To combine two velocities it is necessary to make use of a function called the gudermannian. It is defined as follows: if e^u = sec q + tan q, q is the gudermannian of u, or gd u. Each of the six hyperbolic trig functions of u is matched with one of the six trig functions of q. The sec q = cosh u, and tan q = sinh u, while the sin q = tanh u, and cos q = sech u. Using our tilt angle model, real velocity is c * sin(tilt), and this is equal to c * tanh(TILT), where tilt is the gd TILT. Suppose TILT is made up of two components, A and B. The tanh(A+B) = sinh(A+B)/cosh(A+B) = (sinh(A)cosh(B) + sinh(B)cosh(A))/(cosh(A)cosh(B) + sinh(A)sinh(B)). If we divide both the numerator and the denominator by cosh(A)cosh(B), the result is (sinh(A)/cosh(A) + sinh(B)/cosh(B))/(1 + sinh(A)/cosh(A) * sinh(B)/cosh(B)) = (tanh(A) + tanh(B))/(1 + tanh(A) * tanh(B)). From the definition of gd, this is (sin(a) + sin(b))/(1 + sin(a) * sin(b)). Recalling that this chain of steps began with resultant velocity = c * sin(tilt) = c * (sin(a) + sin(b))/(1 + sin(a) * sin(b)) = (c * sin(a) + c * sin(b))/(1 + (v_a/c * v_b/c)) = (v_a + v_b)/(1 + v_a * v_b / c^2) = v_c. The data confirms the accuracy of this velocity addition formula.
Aside from the geometrical interpretation, there are other points in favor of the spiral approach to relativity. Einstein labored for years on a continuum version of unified field theory. Although special relativity predicts the behavior of particles with mass, there really isn't any place for the point discontinuities themselves in Einstein's theory. While the following suggestions are not as explicit as the discussion of spiral geometry, the ease with which the spiral fits the various domains of particle size is highly encouraging.
At the domain of the very, very small is string theory. These entities are so tiny that there is virtually no chance that they might ever be seen. Yet they are postulated to come in two basic flavors - open and closed. These could just as easily be spirals of tilt angle zero and tilt angle 90 degrees. The zero tilt spiral is a half-circle, with two free ends. The other limit is perpendicular to this, and is an equatorial circle, a closed loop.
At the size range of particles that are very small, but might someday be seen, is the electron, and the magnetic dipole. In the analysis of the standard dipole field, such as a permanent magnet, two models are used. In one the magnetic dipole field is presumed to be caused by a microscopic current loop, i.e. an electron spinning around some axis in a loop. This is just a scaled down version of an ordinary solenoid field. There is an alternate model which presumes the magnetic dipole is the result of a pair of positive and negative magnetic monopoles, in spite of the fact that there is no evidence for the existence of even a single free monopole, unlike the electron. Except for the very near field, the long range behavior of both models is identical. Although it was not proved in my e & m course, it was asserted that these two models did not merely yield the same results. They were two transforms of each other - that it was legitimate to choose whichever model fit the circumstances better.
Once again we have the loxodrome spiral to explain this property. The spiral with a tilt of 45 degrees has a special significance. It is the same degree of rotation to either extreme. The relationship between the latitude and longitude angles is exactly the same as the gudermannian and its hyperbolic equivalent. If we look at the latitude component alone, we see a flux loop. If we look at just the longitude component, we see unclosed lines of a perpendicular flux connecting two poles of opposite polarity. Both views are projections of a common entity, seen from different angles. If we consider the loop flux to be electric, then the open flux is easily seen to be magnetic in nature, being normal to the electric flux direction. If we suppose that flux along the diagonal direction is the usual power flux, we find that its two normal components line up neatly with the latitude circle and the longitude arc, since S = E x H, which is maximum when E and H are perpendicular. In any case, the electric field produced by a time-varying magnetic field is perpendicular, as is the magnetic field produced by a time varying electric field.
In the size range of visible objects, there is a type of device known as a Helmholtz coil. It is an electromagnet that is wound around the surface of a sphere. It is used to create a volume in which the magnetic field is mostly linear. The winding is assumed to have a roughly constant small tilt relative to the horizontal, and its effect is ignored as the winding is represented as a stack of rings. This is just the circumferential component of a loxodrome spiral. The solution to the magnetic field produced by this shape of coil does not depend on the absolute dimensions of the coil. If the coil were of the size of a magnetic dipole, the field measurements would be indistinguishable from either of the two extreme models above from more than a few radii away.>
The previous descriptions rest on fairly solid mathematical ground. The current subject is of a more speculative nature, and should not be taken as a counter argument to spiral relativity. It is merely an attempt to show how easily the loxodrome fits into particle models as well as space models. We base the argument largely on symmetry properties. In the loxodrome, we can easily reckognize the following special "fixed" points: the volume inside the sphere (which may be "empty space", contain mass, or charge; the primary axis of rotationdefined by North and South poles; the great circle defining the equator, which takes three points to map to any other orientation. Spherically symmetric distributions are treated as if they are concentrated at the single point in the center of the volume. The surface area is completely determined by the two poles, and the equator is fixed by three points. If we construct a simple polynomial consisting of one part volume, one half part surface area, and one third part circumference, the result is 4/3pi*r^3 + 2pi*r^2 + 2/3pi*r = pi*atomic number. The energy ordering of the outer filled subshells for the standard periodic table is (in ascending order of energy):
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p 8s
2 2 6 2 6 2 10 6 2 10 6 2 14 10 6 2 14 10 6 2
2 4 10 12 18 20 30 36 38 48 54 56 70 80 86 88 112 120
The first row indicates the number of electrons needed to fill a particular subshell. The second row is the cumulative number of all lower energy filled subshells. Compare these reults with the polynomial above. First, notice that there are four groups of sequences, each of which repeats the same basic sequence. The number of electrons at each group is 4, 20, 56, 120. The polynomial produces the same values (*pi) for inputs 1, 2, 3, and 4. Four points which lie on a quadratric overdetermine the polynomial. The difference between adjacent counts is 4, 16, 36, 64, or 4*1^2, 4*2^2, 4*3^2, 4*4^2. This can be pictured as the sum of the areas of four spheres of radius 1, 2, 3, 4. If we let each hemisphere represent one of the 8 groups in the above sequence, we can then color-code each hemisphere for the order of the subshells within each group. If the first element is located at a pole, and we wind around the hemisphere, the s subshells all fall around the equator (see figure). In fact, the subshells of a given type fall into four concentric sleeves. The number of filled positions in each type of subshell forms another sequence: 16, 36, 40, 28. This sequence also overdetermines a quadratic, this time -8*(q-1/2)*(q-5). If we identify this as a formula for curvature, only those integers associated with positive curvature are associated with an observable subshell type. Pictured in three dimensions with row number along one axis, subshell number along a second axis, and magnetic quantum number a rotation around one of the sleeves, the periodic table is contained between two cones (which are the result of the rotation of a right angle around its bisector, like a light cone). All the integral values of the coordinate system in the interior of this shape correspond to elements, either known or predicted. The shape of this structure suggests that there are no stable elements above number 120. Existing theory suggests that there are possible islands of stability around element number 125, but no elements have yet been made in the laboratory with high enough atomic number to justify either case.
In any case, this whole structure was built out of the symmetry properties of a loxodrome spiral. It has fixed points which are associated with scalar (radially symmetric) properties, dipole (linear potentials), and tripoles (protons and neutrons). To repeat, this section is more speculative than the first, but I hope it illustrates the ease with which we can model the domains of both fields and particles with the same mathematical model. This one model agrees with all the predictions of special relativity, but throws out the absolute light speed rule, and uses fundamental units that are constant in magnitude with relative velocity (but which obey vector projection rules instead).