One of the characteristics of Special Relativity is the notion that space and time are not independent. The Lorentz Transform specifies the degree to which space and time project into each other as a function of the relative velocity of two frames of reference. The aim of this project is to illustrate this  graphically, and ultimately, interactively. See Figure 1. It illustrates the relationship between tilt angle, velocity, rapidity, celerity and Special Relativity. For the purpose of simplifying the graph, the unit of distance was chosen to be light-seconds. Then the numerical value of light speed is exactly 1. Many texts acknowledge the rapidity, and the hyperbolic nature of the Lorentz Transform. The connection to tilt angle is generally overlooked.

For starters, let's embrace the idea that space and time are actually components of a 4-D vector. To illustrate this, it would seem necessary to have sensory faculties that are superhuman. In point of fact, virtually all treatments of Special Relativity specify motion along a single axis. This is a reasonable assumption, since a simple spatial rotation can align the direction of motion with a single axis. The nature of Relativistic effects is such that only the direction parallel to the direction of motion is affected. The other 2 dimensions perpendicular to this are unchanged. In a spherical coordinate system, there is one real coordinate with two angular coordinates. These angles identify the other 2 units that are invariant with respect to a Lorentz Transform. So, to display the behavior of spacetime, instead of 4 dimensions, we really only need 2.

With a simple flat screen, it is straightforward to display a graph with 2 axes. The third axis would project out of the screen. By use of perspective, we can rotate the system of axes so that all 3 are partially visible. Since we can store consecutive images, we can simulate time dependence, and display 4 dimensions. But there are still 2 of the original dimensions hidden. So, what do we define the extra screen dimensions to be? Let's set the stage.

The horizontal axis is assigned to time, and the vertical axis is the one real dimension of space. One choice for the normal direction is relative velocity, specifically, the rapidity, w. The reason for this selection is that rapidity is linear. The rapidity of the addition of two velocities is just the sum of the individual rapidities, unlike the “addition” of the velocities themselves. The invariant, c, times the sinh of the rapidity is the celerity, otherwise known as Proper velocity. As a function of rapidity, it is appropriate to graph celerity on the space axis. In this case, the time component has a velocity of c. Although we don't observe celerity directly (since we have observed a light speed limit for the velocity of any material object) celerity times mass is relativistic momentum, while velocity times mass is Newtonian momentum. So even though we cannot observe celerity directly, it does have a physical meaning.

Since we are looking at time and space components as the projections of a 4-D vector, it is useful to evaluate their vector sum. Using simple Pythagorean math, the hypotenuse of the right triangle formed by the two legs is √c² + c² sinh²(w) = c cosh(w). There is a function which maps the hyperbolic angle, w, to an imaginary central angle, which, for reasons that will become apparent, we will call tilt, and tilt = θ. For every possible value of real rapidity, which is unbounded, there is a corresponding value of tilt angle, which is < π/2. This function is known as the gudermannian, gd, and it has an inverse, ngd. This is an interesting function in that the coefficients of its Taylor series expansion are the Euler numbers, and the only difference between the two expansions is that the gd has alternating signs for the terms, and the ngd has all positive terms. It is the same set of coefficients that are in the Taylor series expansion of the secant, which also equals the instantaneous derivative, d w/d θ. Perhaps more interesting, if the Taylor series is defined as a complex sum, it becomes 4-cyclic. The inverse behaves similarly, but in reverse sequence. Now, why do we care about the tilt angle? The slope of the hypotenuse is just the rise/run, which is space velocity, celerity, divided by time velocity, c. And the slope is just the tangent of the included angle, which turns out to be identical to the tilt angle for all possible values of rapidity. Put another way, the Proper velocity = celerity = c sinh(w) = c tan(θ). Using the Pythagorean identity again, the total 4-D velocity can also be expressed as c secant(θ). In conventional terms, celerity = γ v = βγ c, and total velocity in 4-D is just γ c.

It is trivial to verify that each of the 6 standard trig functions of the circular angle, θ, is uniquely mapped to one of the 6 standard trig functions of the hyperbolic angle, w. Most of these relationships are shown on the graph. From the definitions of β and γ, relative velocity v = β c = c sin(θ). This is the component that is directly observable. Proper velocity = γ v. Having made these definitions, it is worth noting that the Lorentz Transform itself is just a real, hyperbolic rotation as opposed to an imaginary, circular rotation. Some texts refer to hyperbolic rotations as imaginary, but Euler's formula equates a complex number to a complex exponential, e^ix = cos(x) + i sin(x), coordinates of a point on a unit circle in the complex plane. Similarly, e^x = cosh(x) + sinh(x), which are real coordinates on a unit hyperbola in a real plane.

At this point, it becomes helpful to introduce a 4-D vector, called a loxodrome. The loxodrome is a spiral on the surface of a sphere, which intersects all meridians and parallels at the same angles. They correspond to straight lines on a Mercator projection, and were used by sailors for centuries before GPS for navigation. Also known as a rhumb line, or constant compass course, they were convenient because the angle on the map was the same angle that the compass was set to. It isn't the shortest course (which is a great circle) but it is the easiest to steer, since a great circle needs continuous, infinitesimal course correction. The arc length of the spiral is the product of the radius and the central (or latitude) angle, in general. The arc length is linearly proportional to the latitude angle, and a meridian has an arc length of π r, where r is the radius of the sphere. This is the length of a spiral which has zero tilt angle. A spiral with a non-zero tilt angle has a latitude component as well as a longitude component. Regardless of the degree of tilt, the meridian length is invariant. The total length of the spiral obviously depends on the tilt to the axis. The latitude and longitude components are perpendicular, and combine by using the Pythagorean identity. As it turns out, the total length is π r secant(tilt), the same tilt, θ, we already discussed. Then the cosine projection is just π r, and the sine projection is π r tangent(θ). Since the central angle from pole to pole is always π, the effective radius of the latitude component is r tangent(θ), and the effective radius of the whole spiral is r secant(θ). The diameter is just 2 r, or 2/π times the length of the longitude component. The effective diameter of the whole spiral is, similarly, 2/π times the total arc length, or d secant(θ). If we use a loxodrome to represent 4-D velocity, the total length corresponds to the total velocity, and the cosine and sine projections (of the tilt angle) match the time and space velocities.

Although the tilt angle is defined by the ratio of total space velocity to time velocity, its role doesn't stop there. The units for both time and space displacement are scaled copies of the same spiral, in that both have exactly the same tilt as the velocity spiral. In these cases, if the total arc length has a unit diameter, the longitude component is just the cosine projection. This corresponds exactly with the specifications of the Lorentz Transform with respect to time dilation and length contraction of stationary units measured from a moving frame of reference. Only the component of the spiral parallel to the diameter is visible. Since the spiral starts and ends at the same two poles as the diameter, what appears to be shrunk is actually rotated perpendicular to the observable direction, effectively invisible. The total length is unchanged by the rotation, and is what an observer (who is stationary relative to the origin of the moving frame) still sees. So, total length is invariant, but depending on an observer's relative velocity, or tilt angle, some portion becomes invisible. Similarly, total space velocity IS Proper velocity, but only the portion parallel to the diameter is observable. Mass increase with velocity is described in Einstein's book, and it is inferred by the excess momentum that was experimentally observed. But, if total space velocity is actually greater than what is observable, by exactly the same factor, that alone accounts for the relativistic momentum. Thus, there is no such thing as mass increase with velocity. Mass, too, is an invariant. The apparent increase of mass with velocity is an artifact of velocity increase with velocity. As it turns out, if we represent the space component of velocity by a loxodrome of the same tilt angle, the cosine component, which is parallel to the diameter, is exactly the visible portion of the Proper velocity, the Newtonian velocity.

If we multiply each of the components of the 4-D velocity by mass, m, each component becomes the momentum of the invariant mass. This makes total momentum in 4-D, γ m c. And if we multiply this total momentum by c, the result is total 4-D energy, γ m c². When tilt angle = 0, γ = 1, and 4-D velocity = c, momentum = m c, and energy = m c². These are all components in the time dimension, the universal “speed” of time, a momentum in time, and so-called rest energy. When the tilt angle is zero, velocity and Proper velocity are also zero, as are all three spatial quantities. When tilt is non-zero, all three quantities have non-zero values. For more detail, click to see a matrix analysis. Velocity and momentum are vector quantities, so they have sine and cosine projections. The only difference between the resting values and the moving values is the factor, γ. Since γ = secant(tilt), the cosine projection is exactly the same for both moving and stationary scenarios. The sine projection is just βγ times the resting values, or tangent(tilt) times them. That yields celerity (or Proper velocity) and relativistic momentum for the spatial components. The cosine components of each of these terms are simply the Newtonian values of relative velocity and momentum, p = m v. Energy is a scalar quantity, so to find the spatial component, we subtract the rest energy from the total energy to get the energy of motion, or kinetic energy, (γ – 1) m c². For small values of tilt angle, the term (γ – 1) approximates to ½ v²/c², and the kinetic energy approximates to ½ m v², the Newtonian value. So, in all these 4-D quantities, there is only 1 term, common to all, which varies with observer velocity, and that is γ. In 4-D, all velocities are c or greater. There is no light speed limit. And whether we talk about spatial velocity in terms of a tilt angle or a rapidity, the result is Proper velocity, which also has no limit. Once we include the full impact of the tilt angle on observability, it becomes clear that the visible part of infinite celerity is just c, the apparent limiting velocity. It also becomes clear that the addition of two apparent velocities equal to the speed of light does not result in an apparent velocity greater than c, because the sum of their corresponding Proper velocities is infinity plus infinity, which is just infinity.

This does not prove that space is shaped like loxodromes, but it demonstrates the existence of at least one geometry which provides a logical explanation for physical behavior that is predicted by Special Relativity and experimentally confirmed. Length really does measure differently for observers moving in inertial frames with different velocities. But objects do not shrink, any more than a telephone pole gets shorter because it is tipped away from upright. It does require thinking in more dimensions, and accepting that we can not measure correctly with rigid rods and clocks.

The implication of this geometry is that the light speed limit is an artifact of 3 dimensional thinking. Any observed velocity greater than 1/√2 c actually represents a Proper velocity > c. The key to FTL travel lies in understanding this fact. Simply adding energy is not sufficient to break the light speed barrier. It is interesting to note that General Relativity defines the acceleration of a mass (which is the second derivative with respect to time) as being indistinguishable from the curvature of space (which is the second derivative with respect to space) in a closed system. Then Special Relativity defines the velocity of a mass (which is the first derivative with respect to time) as being indistinguishable from the tilt of space (which is the first derivative with respect to space) in an inertial frame. The $64000 question becomes, is it possible to alter the tilt without changing the Proper velocity? Because if the tilt can be reduced without changing the celerity, FTL travel can be achieved, without prohibitive inputs of energy.

The math would seem to indicate that it is not possible to alter the relationship of w and θ, but we must remember that Special Relativity imposes some severe restrictions: constant velocity in a straight line in the absence of external forces. If any of these conditions is broken, the problem enters the realm of General Relativity instead. One possibility to explore is the differential equation which defines γ, dw/dθ = γ. If we integrate this equation, it introduces an arbitrary constant which may change the relationship between w and θ.