The Relativity of Linear Momentum

Let’s start by stating as axioms two propositions that we have already deduced, proven, and verified as standing true to Reality. First we assert that the speed of light has the inherent character of a universal constant, which means that any two observers will measure a given pulse of light passing them each at exactly the same speed, regardless of how those observers move relative to each other. And second we assert that linear momentum obeys a conservation law, which means that in a system of bodies isolated from outside forces the linear momenta of the bodies always add up to a number that remains constant, regardless of how the bodies interact with each other.

If we imagine two observers making their measurements on Cartesian coordinate grids that have a common x-axis, then each of those observers can determine the speed of light through the ratio of two actual measurements, those of distance and duration. The big observer (who uses upper-case letters to represent their measurements) moves at a speed V in the positive x-direction relative to the little observer (who uses lower-case letters to represent their measurements). Both observers measure the distance between two markers on their common x-axis and measure the time that a pulse of light traveling in the positive x-direction takes in going from one marker to the other. They then calculate the speed of light from their measurements,

(Eq’n 1)

In order to compare those two ratios the big observer writes

(Eq’n 2)

in which the gamma factor acknowledges the possibility of the two inertial frames having different scales. That result turns Equation 1 into

(Eq’n 3)

which the big observer can solve for dt to get

(Eq’n 4)

The little observer writes Equation 2 as

(Eq’n 5)

which raises two questions.

In writing Equation 5 the little observer has tacitly assumed that the big observer moves at a speed equal to the speed at which the little observer moves relative to the big observer (though in the opposite direction) and that the scale factor (gamma) has the same value for the little observer as it has for the big observer. Those assumptions come under the principle of relativity (Einstein’s first postulate) and, thus, stand true to Reality. But can we justify them without invoking that principle?

To answer that question we begin by noting that the constancy and sameness of the speed of light for all unaccelerated observers necessitates that each observer occupy and mark the center of an inertial frame of reference identical to all of the other inertial frames with which it unites in constituting space and time. The fundamental property of the speed of light requires that the geometry of the Universe appear the same to all observers (note that this statement does not give us Einstein’s first postulate, the principle of relativity, because it does not refer to nor does it apply directly to the laws of physics). Each and every inertial frame necessarily has the same hyperspherical symmetry, so if two observers can communicate or otherwise interact with each other, their respective inertial frames have the same age and the same spatial extent.

From any point in a given inertial frame all other inertial frames, an infinite set of them, pass at velocities that range from arbitrarily close to zero up to, but not beyond, the speed of light. Because the frames constitute an infinite set, any observer sees as many frames moving in one direction at a given speed as they see moving in the opposite direction at the same speed. That symmetry requires that if the big observer sees the little observer moving at a certain speed, then the little observer must see the big observer moving in the opposite direction at the same speed: the observers see each other moving at the same fraction of the speed of light. If that statement did not stand true to Reality, then one of the frames would require a distortion relative to the other in addition to the distortion needed to make the boundary of each frame, expanding away from each point at the speed of light, merge with the common boundary of all inertial frames, the boundary of space. But such a privileged distortion cannot occur in a set of uniformly expanding, perfectly congruent geometric objects. Thus we infer that the statement |V|=|v| must stand true to Reality.

Further, the scale factor between the two frames must have the same value for both observers. Because the only difference between the two frames consists solely of the relative velocity between them, the scale factor must conform to the description given by a function of that variable only. But the relative velocity between the frames has the same value for both observers, so the scale factor must also have the same value for both observers.

Next, consider two straight line segments drawn parallel to each other and parallel to the direction of relative motion between the frames. The points constituting those lines came into existence at the instant in which the Universe blossomed into existence. As space expanded with the elapse of cosmic time the lines grew apart at the same rate in both frames and, thus, lie the same distance apart in both frames for all time. We may then infer that anything that touches a point on one of the lines and a point on the other line extends the same distance perpendicular to the direction of relative motion in both frames. Mathematically we write

(Eq’ns 6)

The little observer writes Equation 1 in the forms dX=cdT, dx=cdt, and dt=dx/c and then writes Equation 5 in the form

(Eq’n 7)

Dividing that equation by the speed of light yields

(Eq’n 8)

Although the above derivations involve distances and durations whose ratios equal the speed of light, the spatial and temporal distortion equations apply to all distances and durations that the observers measure, regardless of the value of their ratios. That statement must stand true to Reality because the equations merely transform one observer’s measurements of a geometric entity into the other observer’s measurements of the same entity: calculation of a ratio doesn’t affect the transformations themselves, regardless of what value the ratio takes.

Now the two observers can rewrite Equation 2 as

(Eq’n 9)

which necessitates that

(Eq’n 10)

the Lorentz factor. Our observers thus have the Lorentz Transformation as

(Eq’ns 11)

With that transformation or its inverse

(Eq’ns 12)

the observers can calculate the magnitude of the distance-duration between two events by way of Minkowski’s version of the Pythagorean theorem,

(Eq’n 13)

That number has the same value for all observers, so we identify it as a Lorentz invariant, even though the quartets of differences between the location-instants of a pair of events have different values for all observers.

Now consider the linear momentum, **p**=m**w**, of a
simple body or particle as measured by our little observer. We can represent
that momentum as a vector, (p_{x}, p_{y}, p_{z}), and
work out the transformation of the momentum as our two observers would measure
it. As above, the observers move relative to each other in the x-direction only,
so we can start by writing

(Eq’ns 14)

We know that those equations stand true to Reality because we can imagine accelerating from one observer’s frame to the other’s: that x-ward acceleration causes no force to act in the y- and z-directions, which necessitates that the components of the linear momentum in those directions not differ between the frames.

In accordance with the definition of force encoded in Newton’s second law of motion, the big observer writes

(Eq’n 15)

The little observer measures the same force as **f** and calculates the
body’s linear momentum as

(Eq’n 16)

We note that **V**, the velocity between the inertial frames, participates
in the integration as a constant, so we have

(Eq’n 17)

in which we define

(Eq’n 18)

The number e (not to be confused with the base of the
natural logarithms in this case) represents what the little observer calls
potential momentum (and what we call energy). We notice that e/c^{2} has
the units of mass and plays the role of mass in Equation 17, so we make the full
identification

(Eq’n 19)

for the little observer and

(Eq’n 20)

for the big observer. Of course we have, in accordance with Equation 18,

(Eq’n 21)

which gives us

(Eq’n 22)

Equations 17 and 22 look like parts of a Lorentz
Transformation of a four-vector (p_{x}, p_{y}, p_{z},
e/c^{2}) in which the energy provides the temporal component. We can
test that proposition by applying Minkowski’s theorem to those equations.
Squaring **P**’ and E’/c, subtracting the latter square from the former, then
multiplying by -c^{2} for convenience gives us

(Eq’n 23)

which means that the magnitude of the momentum-energy four-vector has the character of a Lorentz invariant, a number that remains unchanged when the Lorentz Transformation alters the four-vector itself.

Equation 19 and 20 give us the famous mass-energy equivalence relation, so we can rewrite Equation 23 as

(Eq’n 24)

If we were to divide that equation by -c^{2} and compare the result
with Equation 13, we would see that the mass-energy term plays the same role as
does the temporal displacement term. From that comparison we infer that the
mass-energy of a particle or body represents the temporal component of the
body’s linear momentum, the quantity of motion that the object carries in the
timeward direction. That idea may seem absurd, but we gained it from a proper
mathematical derivation and interpretation with no countervailing rationale to
negate it. Thus, we must accept the idea that simply possessing mass gives a
particle momentum through time.

The mass that we used in the derivation above comes from work done upon a pre-existing particle. But every particle also possesses an inherent mass, a quantity of inertia that seems to have nothing to do with any work done on the particle by an outside influence. Dynamically, in accordance with Newton’s laws of motion, both kinds of mass have the same effect on motion, producing the same relation between force and acceleration. That fact necessitates that both kinds of mass participate in the particle’s linear momentum in the same way.

Just by virtue of existing a particle possesses a timeward
component of linear momentum. If we represent the particle’s inherent mass by m_{0}
and set p=0, then we can rewrite Equation 24 as

(Eq’n 25)

If we make the substitution P=MV, then we can rearrange that equation into the familiar

(Eq’n 26)

Thus we identify m_{0} as the rest mass of the particle, the mass
that the particle has in the inertial frame in which it does not move. Equation
26 also tells us that the rest mass is the minimum mass that the particle
possesses.

But not only the squares of four-vectors remain Lorentz invariant: in accordance with a theorem of matrix algebra any inner product of two four-vectors exists as a Lorentz invariant. For example, we can multiply together the momentum-energy of a particle by the differential distance-duration the particle crosses between two neighboring events. We thus obtain

(Eq’n 27)

We call that number the differential of the action played out by the particle
as it goes from one event to the next. Using the fact that d**x**=**w**dt,
we have the action played out between two widely separated events as

(Eq’n 28)

in which we define, for convenience, the Lagrangian function L=**p**A**w**-e.
Because the one coordinate we do not control is the elapse of time, we want to
make time our independent variable with all the other coordinates dependent upon
it through equations of evolution.

That number, the action, seems to give us a figure of merit pertaining to the path that the particle follows between the given events. Can we gain anything beneficial from that number?

For any two events that a single particle can touch, there exists an inertial frame in which both events occupy the same point in space. In that frame a particle following a straight line between the two events will enact

(Eq’n 29)

because in that frame **p**=0, so the action comes entirely from the
minimum energy carried by the particle. In accordance with the theorem stated
above, that action will have the same value in all inertial frames. If we
express the integral in terms of measured time instead of proper time, we get

(Eq’n 30)

Returning to the proper frame, we consider the possibility
that, in the absence of an applied force, the particle follows an alternate
path. Mathematically, we say that the alternate path differs from the true path
by δ**x**
with δ**x**=0
at the events that define the endpoints of the path that we have under
consideration. In order to follow the alternate path the particle must have a
linear momentum to take it away from and bring it back to the point where the
boundary events occur. Multiplying that momentum by the differential of
δ**x**
yields only a positive number, so integrating that product over the full extent
of the alternate path always adds a positive number to the action calculated for
the true path.

That fact remains true to Reality even when an applied force acts on the particle. In our proper frame that force will make the particle trace a loop in the space surrounding the point at which both of our boundary events occur. Again, if the particle follows an alternative path, it requires a change in its linear momentum without a corresponding change in the applied force and the product of that change with the differential of the variations in the path always comes out positive, thus adding a number to the action played out when the particle follows the true path.

Given the fact that the particle following a path that varies from the true path that the particle actually follows always adds a positive number to the action associated with the true path, we infer that the action associated with the true path takes the minimum possible value relative to all possible alternate paths. So, even though geometry allows us to describe a particle following any path between two events, dynamics doesn’t. We thus obtain the principle of least action (Hamilton’s principle), which we express mathematically as

(Eq’n 31)

in which the operation of variation, having the character of a differential, commutes with the integration. From that equation we can derive, by the standard method, the Euler-Lagrange equations, which produce the equations of motion governing the particle.

Again, because the action remains invariant under a Lorentz Transformation, the mathematical form of the laws of motion must also remain invariant. Because we can only observe the motions of bodies and particles (either directly or indirectly), we can only encode the fundamental laws of physics in equations of motion. Thus we can state, as Einstein postulated, that the laws of physics all have the same mathematical form for all observers, regardless of how those observers move relative to each other.

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