Visualizing Constant Inertia

Pictures first, then explanations.

Constant inertial signals from a body are either
timelike,	lightlike,	or spacelike.

Newton's First Law of Inertia states that bodies in motion tend to stay in motion. It is easy enough to generate an animation that does this: an event appears in one spot, then the subsequent events appear in a consistent pattern. The law of inertia is about mass, which is not an event. How does one make a link between events in spacetime and energy, momentum and mass? Energy and momentum form the tangent space of spacetime, with mass as a Lorentz invariant distance in that tangent space. One can calculate the relevant tangents using special relativity. The ratio of energy to mass is called gamma, the ratio of a change in time over an interval. For momentum/mass, the ratio of change in space to the interval is the key:

(E/m, P/m) = (gamma, gamma beta) = (dt/dtau, dR/dtau)

A different way to state the inertia law is that the ratios of dt/dtau and dR/dtau are constant in spacetime. Given one pair of events, dt/dtau and dR/dtau can be calculated directly. This ratio can be used to find the next event in spacetime. The program constant_linear_motion, when given a pair of events, does exactly that calculation (tarball here). The output of that program is fed into another, eq2svg (tarball here), to generate a Scalable Vector Graphic, an open source version of flash. You may need Adobe's SVG Viewer available for PCs and Macs, an older version works on Linux.

Constant inertia only happens if both E/m and P/m are constant. This does not make an exciting animation on the surface. It is important because of the new perspectives about inertia are opened. The animation was made with 200 discrete events, adding E/m and P/m at each step. If that amount had been subtracted, the animation would show the path in the past, not the future. The choice of function to use changes the meaning of the animation.

The two events used to generate the first animation were timelike separated, meaning that dt² - dR² > 0. This is the stuff of classical physics, where dt >>> |dR|. There are two other cases to consider.

What happens if dt = |dR|? The two events are said to be lightlike separated, because only a particle traveling at the speed of light could possibly link the two events. Since dtau is equal to zero, one cannot calculate the ratio dt/dtau because that would involve dividing by zero, a mathematical sin. The program constant_linear_motion uses dt and dR directly to calculate the next step in spacetime. The first two animations look similar. Inertia is inertia is inertia, whether the events that record it are timelike or lightlike separated. What is intriguing is the mathematical machinery for generating the animation is simpler for the inertia of light. Only massless particles can reach the speed of light. The more direct math for massless particles may be one manifestation of the difference between massless and massive particles.

The third case starts with a body that has inertia, but the two events generated by that body are spacelike separated. Such a body would have to take up a volume of spacetime instead of being possibly a point particle like the timelike and lightlike bodies may have been. A distributed body could generate events that are spacelike separated without breaking the rule that nothing travels faster than light. Because dtau is not zero, the ratios of E/m and P/m may both be calculated. In the previous two examples, how much father in spacetime the next event was known, |(E/m, P/m)| and |(dt, dR)|, respectively. The direction of the 3-vector was also known. The behavior of the bodies were determined.

Because dtau² < 0, when the square root is calculated, an imaginary unit vector is required. If there is one real number and one imaginary number like there is for the complex numbers, there is only one choice for the imaginary basis vector, i. In spacetime, there is an i, a j, and a k basis vector, along with linear combinations of the three. There is no way to choose, so no direction should be chosen deterministically. Instead, the body with this constant inertia can be described by some sort of probability distribution. A chance event consistent with the probability distribution is how the direction is chosen. The hops in spacetime for the third animation are exactly the same size as the first, but the direction is random. This makes the animation look like a Mexican jumping bean, never going too far from where it just was. If the last events was in the upper left hand corner, then the next event cannot appear in to lower left. The body has a certain amount of inertia, and the size of the jumps reflect the inertia.

There are two important technical comments here. First is that this description is not about the hypothetical particle known as a tachyon, which is proposed to move faster than the speed of light. Because the system under study takes up a volume of spacetime (ie is not a point particle), it is possible for such a distributed system to create two independent events that happen to be spacelike separated. Second, the imaginary velocity is not directly measurable, although its norm is. This is similar to what happens for measurements in quantum mechanics. By asserting that only the norm of the constant velocity for spacelike events can be observed, I may dodge the problems of an imaginary ratio of energy to mass which has logical difficulties.

What is this an image of? It may be a first step toward visualizing quantum mechanics. Such bodies have inertia, but are characterized by probability, not determinism. The same machinery of inertia was used to drive this animation, but the direction of the 3-vector could not be known, and that was the source of the jumping behavior. Nothing is actually jumping. The body under question takes up a volume of spacetime. The signals arise from different parts of that body. The pattern of those signals may be what is studied in quantum mechanics.

Conclusions

A body with constant inertia in spacetime will broadcast its next location a constant distance away from were it currently is. For a body whose signals are timelike separated, the ratios of energy to mass and momentum to mass can be calculated and used for each subsequent step. For a body with lightlike separated events, those ratios cannot be calculated. In a certain way, information about particles traveling at the speed of light is more direct. For a body that is distributed in spacetime, it is possible to generate events about the inertia that are spacelike separated. Those events will be a constant distance away in spacetime, but the direction is not knowable. This may be a door into visualizing quantum mechanics.