Visualizing Constant Inertia
Pictures first, then explanations.
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 dt^{2}  dR^{ 2} >
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 3vector was
also known. The behavior of the bodies were determined.
Because dtau^{2} < 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 3vector 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.
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