Relativistic Potential Energy
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When we re-deduced Einstein's mass-energy equivalence equation we used kinetic energy to change the mass of a moving body and we determined that the rest mass of a body corresponds to kinetic energy. Now we need to look at the other kind of energy, the potential energy manifested in forcefields. If an observer occupying a certain inertial frame measures the potential energy of a body at a point in a forcefield, what potential energy would an observer occupying a different inertial frame measure? As has become our Einsteinian habit, we shall deduce the answer to that question by way of a suitable imaginary experiment.
Imagine that a thin, flat plate made of some suitable substance floats motionless before us. And imagine that the x-axis that we share with other observers passes through the plate in such a way that it makes a right angle with any straight line we can draw on the plate. Two photons fly along that axis toward the plate, one moving from left to right and the other moving from right to left. Each of the photons carries an amount of energy that conforms to the statement that
which we have made just sufficient to free one electron from the plate and leave it floating motionless next to the plate (at least temporarily). If we identify the potential energy U that binds the electron to the plate's substance, then that energy conforms to the statement that
Now an alien observer moves along the x-axis at speed V in our frame, proceeding from right to left. In that observer's frame the plate moves from left to right at the speed V and the photons have energies manifesting the Doppler shift: the photon coming from the left has an energy that conforms to the description
and the photon coming from the right has an energy that conforms to the description
On first impression, those equations appear to us to describe one photon carrying too much energy for the bare liberation of the electron and the other photon carrying not enough energy to accomplish the task, even though both photons carry just enough energy in our frame. How can we resolve that apparent paradox?
Let's look at the linear momenta of the bodies in question. We describe the linear momentum of a photon through the statement that P=E/c, so in our alien's frame the photon coming from the left carries a linear momentum that conforms to the statement that
and the photon coming from the right carries a linear momentum that conforms to the description
In the case of the electron we don't have such a clear picture. After the electron has absorbed one of the photons and has come free of the plate, it has a linear momentum that conforms to the statement that
in which statement m0 represents the electron's rest mass. When the electron lay bound to the plate it had the same velocity that it has after the photon liberates it, but it had less energy, which corresponds to less mass in accordance with the statement that
in which U' represents the potential energy binding the electron to the plate. We describe linear momentum as the product of a body's mass and its velocity, so for the electron bound to the plate we calculate a linear momentum of
The linear momentum that the electron gains as it goes from the state described by Equation 9 to the state described by Equation 7 must come from the photon that liberates the electron from the plate. The remainder of the photon's linear momentum goes into the plate, just as the photons' linear momenta go into the plate in our inertial frame, either as
if the photon strikes the plate from the left or
if the photon strikes the plate from the right. If we compare those two equations with Equations 5 and 6, we can reasonably infer that in Equations 5 and 6 the first term on the right side of the equality sign represents the part of each photon's linear momentum that goes directly into the plate. That analysis leaves the second term on the right side of the equality sign to represent the part of the photon's linear momentum that goes into changing the linear momentum of the electron, thereby freeing the electron from the plate, so we have
Referring to Equation 2, we can rewrite that equation as
Now let's look at how the plate and the electron absorb the energy that the photons carry, the energy that I described in Equations 3 and 4. Because a photon carries linear momentum, it exerts a force upon any body that absorbs it. If the absorption of a photon by our plate takes a time interval dt, then we can calculate the magnitude of the average force as
in which I have used only the part of the photon's momentum that goes into the plate as inferred above. Moving at the uniform velocity V, the plate moves a distance dx=Vdt, so the photon does work upon the plate in the amount
If the photon comes from the left, the plate removes that much energy from its original amount (Equation 3) via the force in Equation 14 and if the photon comes from the right, the plate adds that much energy to its original amount (Equation 4). In both cases the photon ends up with an amount of energy that conforms to the statement that
which provides just enough to liberate the electron from the plate if U' conforms to the description in Equation 13.
But now our alien observer seems to have devised a minor contradiction. In the momentum representation of the liberation of the electron from the plate the hνV term in the algebraic description of the photon's linear momentum represents the momentum absorbed by the electron and the hν term represents the momentum absorbed by the plate. In the energy representation of the process, on the other hand, the hν term in the description of the photon's energy represents the energy that liberates the electron from its bound state and the hνV term represents the work done on the plate by the photon (either positive or negative, depending on the direction whence the photon approaches the plate). Those statements tempt us to ask which term actually represents the liberation of the electron and which represents the photon's impact on the plate.
To that question we answer neither and/or both. Although our mathematical descriptions of each photon's linear momentum and energy each break into two parts, the photon itself remains an unbroken entity and its interaction with the electron and the plate remains an unbroken phenomenon. Here we may want to remind ourselves that, to paraphrase S. I. Hayakawa's dictum in "Language in Thought and Action", the Map of Physics is not the territory it represents.
Just as we use rulers to divide up continuous distances to make them countable and comprehensible, so we use our algebraic descriptions to make indivisible phenomena countable and comprehensible. But often those descriptions, our maps, display features that do not actually exist on the territory that they represent. On certain maps of Western Montana we find a line labeled "Continental Divide". If we go to the indicated place, we do not find a line on the ground, only rocks and soil. But standing on that site, we know that rain falling on one side of the ridge will flow toward the drainage of the Missouri River and rain falling on the other side will flow toward the drainage of the Columbia River, thereby manifesting the meaning of the words "Continental Divide". In much the same way our description of photons' momenta and energies splitting into pairs of algebraic terms does not represent any real dividing line: it merely allows us to determine that, even though the two photons carry different linear momenta and energies in the alien observer's frame, they nonetheless just barely liberate an electron from the plate.
We require from any scientific theory no more and no less than that it "save the appearances"; that is, where the theory describes a measurement or observation that we could conceivably make, that description must match the corresponding measurement or observation that we or someone else actually make. Any features of the mathematical description or logical analysis that do not represent measurements or observations that someone could actually make thus lack any scientific meaning and we can ignore them as irrelevant to our understanding of Reality. Thus we may catch a glimpse of how we may understand the wave-particle duality of the quantum theory, seeing it as an ignorible artifact of the mathematical description of matter and nothing more.
And I will note here, as long
as I have the subject up, that in employing references to the quantum theory in
this essay I have cheated on the purely axiomatic-deductive process that I have
supposed will generate the Map of Physics. At this stage in our development of
the Map we should not know about the photoelectric effect or the wave-particle
nature of light and so we should not use that knowledge to infer the rule for
the relativistic transformation of potential energy. However, these essays, as I
have noted, comprise the preliminary plats of the Map of Physics and do not
constitute or present a finished product. I believe that someday when someone
completes this version of the Map of Physics they will have eliminated all of
these look-aheads and will present Humanity with a proper axiomatic-deductive
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