Energy and Its
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So far I have shown you how we have obtained two of the big three conservation laws of physics, the conservation laws pertaining to linear momentum and to angular momentum. Deduced from the relation between Reality and Absolute Nothingness, they stand as absolute laws; that is, they admit of no exceptions. In deducing those laws we found that we had natural definitions of the momenta: they simply represent the quantity of motion, either linear or rotary, contained in systems of bodies.
Now we come to energy. To what can the word "energy" refer? We will discover that it does not name a kind of motion, like momentum. Instead it gives us a quantity that seems to constitute a kind of bookkeeping device that we apply to the momenta, a device whose conservation law seems more contrived than deduced and admits of at least one grand exception.
Consider a number of pairs of identical bodies. Bodies A and B float motionless relative to each other. Bodies C and D move away from each other at some speed V1 (that is, we see Body C move at the speed V1 in one direction and we see Body D move at the speed V1 in the opposite direction); Bodies E and F move away from each other at some speed V2; and so on. We know that each of those pairs has a net linear momentum of zero, yet the pairs all move differently. Can we devise some quantity that will tell us something useful about that difference?
Let's start to answer that question by asking why each pair of bodies has zero net linear momentum. We answer that in setting up the experiment we applied equal and oppositely directed forces to the members of each pair for some interval of time. The two forces that we applied to the bodies in each pair cancel each other out when summed, so the net momentum they produce comes out equal to zero, regardless of how long we apply them. We know that linear momentum comes from applying a force over some interval of time, but the Lorentz Transformation tells us that relative motion creates an intimate connection between distance and duration: a purely temporal interval in one inertial frame has, in part, a spatial aspect in another frame. If you had flown past me when I was setting up those pairs of bodies and had sought to calculate their momenta, you would have found that your calculation, controlled by the fourth equation of the Lorentz Transformation (the equation describing the transformation of temporal intervals), involved multiplying the applied forces by both temporal intervals (the durations over which the forces were applied) and spatial intervals (the distances the bodies moved while the forces were applied to them). That multiplication of force by distance gives us what we want.
You gained that multiplication of force by distance through application of the Lorentz Transformation to a collection of moving bodies. But now, in order to maintain a consistent picture of Reality, I must take the cue from you and also multiply the forces that I apply to the bodies in my frame by the distances that those bodies move while I apply the force. In fact, we both apply the process called integration, from the calculus, to carry out the actual calculations; that is, we multiply the applied force acting at a certain place by an extremely short span of distance that the body moves across that place while that particular force acts and then add up all of the little pieces that we create in that way for all the places that the body occupied while the force acted.
We give the name work to the product of multiplying a force by the distance that a body moves while under the influence of that force (and please note that only the part of the force oriented parallel to the motion of the body does any actual work). When the application of the force ceases, the work that the force has done upon the body stays with the body as energy. Now we discover that energy presents two aspects for us to contemplate.
Imagine that I, like the mythical Sisyphus, roll a heavy rock up a hill. I exert a force upon the rock and the rock moves, so I certainly do work upon the rock. But only the part of the rock's motion oriented straight up lies parallel to the force pushing the rock down, so only the rock's motion in that direction adds to the work I do: the part of the rock's motion oriented horizontally makes no contribution to the work, though it makes my job easier by spreading the work that I must do over a greater distance. Now if I pause in my exertion and relax my grip on the rock too much, the rock, pulled by Earth's gravity, will roll back down the hill with increasing speed.
We can see that the rock carries two kinds of energy. We call the energy that the rock gains as a result of my pushing it to a certain position within Earth's gravitational forcefield potential energy and we call the energy that the rock gained when Earth's gravity accelerated it down the hill kinetic energy. If we calculate the amount of potential energy and the amount of kinetic energy that the rock possesses at any given time, we will discover that their sum remains unchanged (at least until friction begins to slow the rock down). We call the sum of a body's potential energy and its kinetic energy its total energy and we assert that total energy obeys a conservation law: in any isolated system of bodies the sum of the total energies of all the bodies remains unchanged, regardless of how the bodies interact with each other. In another, more common, way of stating that law we say that energy can be neither created nor destroyed, but only transferred or transformed.
We may have based that statement upon a too simplistic application of Newton's second law of motion. But we can get around that objection by way of a clever trick involving the calculus, one that physicists have used to define energy as a conserved quantity, albeit with some apparent loopholes.
As I said above, we use the process called integration to create our conservation law. In integration we take a function of some coordinate, multiply it by an extremely small (nearly infinitesimal) increment of that coordinate, and then use the tricks that mathematicians have gathered together for over three centuries to add up all of those little products and produce a new function of the coordinate. It turns out that if the function in an integral equals zero, then the indefinite integration yields an indefinite constant, a number that does not change when the coordinate changes. Conservation laws give us formal statements that certain things remain constant when the coordinates involved in their mathematical descriptions change, so if we can find some suitable zeros in our description of Reality, then we may contrive conservation laws by integrating them.
Newton's second law of motion gives us the textbook example of a suitable zero. As we usually express that law, we equate the force applied to a body to the rate at which the body's linear momentum changes in consequence of that application, the kinetic reaction to the force's action. For the purpose of devising our conservation law we subtract the applied force from the rate at which the body's momentum changes and get zero. Then we multiply that equation by a minuscule part of the distance that the body crosses while the force acts upon it, subject to the proviso that we multiply the element of distance only by the component of force that lies parallel to it, and add up all of the minuscule elements of work that we have thus set up in the process of integration.
That process gives us a new equation. On one side of the equality sign we have a constant, which we identify as the body's total energy. On the other side of the equality sign we have two algebraic formulae. We identify the formula that comes from the negative of the applied force as the body's potential energy vis-a-vis the body exerting the force. And we identify the formula that comes from the rate at which the body's linear momentum changes as representing the body's kinetic energy. Our derivation of the new equation tells us, then, that the sum of the body's kinetic and potential energies does not change as the body moves.
Does that equation represent a true conservation law? We should entertain some skepticism here. After all, we know that we don't actually describe any given body's energy with a number that never changes. Does that new equation, then, give us merely the illusion of a conservation law? We have only one way to find out the answer to that question: we must try to devise a way to break that putative conservation law.
In devising that new equation I made a tacit assumption that we should re-examine. I assumed that the body exerting the force on the body whose energy I calculated doesn't move as it interacts with that body. If that forcing body has a big enough mass that Newton's equal and opposite reaction gives it a very small increment of velocity, then we can take that assumption as effectively true to Reality. But now I want to see what I get if I let the forcing body have a smaller mass, so that both it and the forced body suffer sizable accelerations.
We know from Newton's third law that the bodies (call them Body-A and Body-B for convenience) must exert equal and oppositely directed forces upon each other along the straight line passing through their centers of mass. We also know that we can treat the two bodies as if they comprised one body that we have not subjected to a force from any other body, so we know that this composite body's center of mass does not accelerate. We now want to consider the situation in which the common center of mass of the Body-A/Body-B system does not move.
In that situation the net linear momentum of the system equals zero. From that datum we can deduce the location of the system's center of mass along the line between the bodies as a proportion related to the ratio of the bodies' masses. We can then devise an equation that tells us the total energy of the system as a fixed multiple of that ratio and thereby infer that the total energy of our two-body system does not change with the elapse of time.
But an equation that we consider all by itself tells us very little different from nothing. Let's consider, then, a more explicit description of the two-body encounter: we start with both bodies moving toward their common center of mass and, in this particular example, each body exerts a force that repels the other body. Now I say that at any given instant each body's kinetic energy and its potential energy vis-a-vis the other body add up to a total energy that does not change throughout the encounter. A closer examination of the encounter gives us the necessary proof and verification of that proposition.
In any interval of time elapsing while the bodies move toward each other each body's potential energy increases by some amount and the body's kinetic energy decreases by the same amount. At some instant, the same instant for both bodies, the bodies' kinetic energies come equal to zero, the bodies become motionless, and then the bodies begin to move away from each other. At any amount of time after that instant each body possesses the same kinetic energy and the same potential energy that it possessed the same amount of time prior to that instant, so the distribution of the bodies' energies forms a perfect symmetry about that instant.
We could conceivably spoil that symmetry (and thus possibly enable a violation of the conservation law) in one of two ways: either one body can give some of its energy to the other body or the force between the bodies may undergo an inherent change, thereby changing the temporal profile of the energies possessed by the bodies. If one body gives some of its energy to the other, then at some given time after the instant of motionlessness one body moves slower than it did and the other body moves faster than it did at the same time prior to the instant of motionlessness. But that outcome violates the law mandating conservation of linear momentum, so we may say with certainty that energy transfer cannot happen in this situation. As for an inherent change in the force acting between the bodies, that also cannot happen, but the proof and verification of that proposition must wait for the next section.
We can also spoil that symmetry by asking how our two-body encounter looks to an observer who moves past the system's center of mass at some speed. How would that observer calculate the system's energy?
First, in that observer's frame of reference the system has an overall kinetic energy effectively due to the motion of its center of mass. That energy, which we calculate as if the system comprised one body whose mass equals the sum of the masses of Body-A and Body-B, will not change until the system interacts with another system of bodies. So we can ignore this energy.
Second, if that moving observer measures the kinetic energy of, say, Body-A over a suitable length of time, they will discover that the body leaves the encounter with Body-B with a different amount of kinetic energy from what it carried into the encounter. But in describing the encounter as we see it, I implicitly claimed that the encounter cannot happen in a way that has one body leaving the encounter with more kinetic energy than it had when it entered the encounter and the other body exiting the encounter with less. So long as Reality upholds that claim the system conserves energy. But that claim does not hold true to Reality for our moving observer. Did I go wrong, then, in making the original claim? And, if so, can we manipulate this example in a way that increases or decreases the total energy of the two-body system, thereby invalidating my presumed conservation law? If we can answer yes to that question, then we have serious logical trouble on hand, because this example merely shows us how a moving observe sees what we see.
Let's look at this encounter as the moving observer sees it in more detail. Suppose that the center of mass of the two-body system appears to the moving observer (if we could somehow mark it) to move in the same direction in which Body-B initially moves. Assume also that the we and the moving observer begin to observe this system when enough distance separates the two bodies from each other that they have negligibly small potential energies; that is, the potential energies are so small relative to the bodies' total energies that we can ignore them as if they were equal to zero. Thus the bodies start out with energies that are almost entirely kinetic.
After some time elapses the moving observer sees Body-B come to rest and begin to move in the opposite direction, even though they see its distance from Body-A continuing to diminish. Shortly thereafter the distance between the bodies stops decreasing and begins to increase. Then the moving observer sees Body-A come to rest and begin to move in the opposite direction. Later yet, when the bodies have gone as far apart as they were at the beginning of our observation of them they have kinetic energies that differ from what they had originally.
You may find in that fact, as I once did, a temptation to see a potential for the creation or destruction of energy, but that temptation evaporates when you recall that in this imaginary experiment, as I have described it, the bodies have equal (though oppositely directed) linear momenta in our frame at all times. In the moving observer's frame the bodies still have those momenta plus a linear momentum, due to the motion of the frame, that remains constant for each body. When we express that fact in the mathematical description of the encounter we discover that Body-B gains exactly as much kinetic energy as Body-A loses in the encounter.
We can easily see how the transfer of energy took place. At first Body-B moves against the force acting to drive Body-A and it apart, thereby converting its kinetic energy into potential energy. After the instant when it comes to rest, Body-B moves with the force, converting potential energy into kinetic energy, while Body-A continues to move against the force, still converting kinetic energy into potential energy. After Body-A comes to rest and begins to move with the force the amount of potential energy available to it for conversion to kinetic energy has been so depleted by Body-B that Body-A has less energy available to it than it would have had if both bodies had come to rest simultaneously.
In that analysis you may see reflected a clear image of the process that NASA uses to send space probes deep into the outer solar system by bouncing them off Jupiter's gravitational field. That refection illuminates the contrast between the two situations that I have described above. For an observer occupying a non-rotating jovocentric frame of reference Voyager 1, for example, left the Jupiter system with as much kinetic energy as it had when it entered the system, albeit expressed in a different direction. On the other hand, for an observer occupying a non-rotating heliocentric frame of reference the probe left the Jupiter system with more kinetic energy than it had when it entered the system, enough more to carry it out to Saturn and beyond.
Thus it seems as if our energy equation does, indeed, express a conservation law. But before we can claim that statement as fully true to Reality we must address a possibility that I have not yet considered - what happens when the force changes with the elapse of time? As we contemplate that question we will find that, unlike the other two fundamental conservation laws, we cannot call this one absolute. I can only assert it as fundamental by way of the mysteries of time travel as described by Herbert George Wells in his famous 1895 novel.
On Timeward Force
In the forgoing discussion I have assumed that we cannot legitimately express the force exerted between two bodies as an explicit function of the elapse of time. Of course, the force that acts upon a body can vary as time goes by, but that variation comes entirely from the motion of the body through a region of space in which the exerted force varies with location, even in the case, as we saw above, in which the motion of the force generator makes the force at any given stationary point change with the elapse of time. I take as my proposition, then, the following statement: If a body spreads out a forcefield that will act on other bodies and if that body occupies an inertial frame of reference, then the force exerted by that body will not change at any point in that inertial frame. We know, for example, that Earth's gravity diminishes with altitude, but at any given altitude the force that Earth's gravity exerts upon a given body will not change with the elapse of time.
In order to prove and to verify that proposition I must make a digression. I want to take a quick look at the basic theory of time travel and see how it relates to the conservation of energy.
Imagine looking down on a symmetrical smooth-sided hill whose apex lies high above a broad plain. Our man Sisyphus rolls a large spherical rock up the hill going due east. Interaction between the rock's weight and resistance from the hillside generates a force that pushes the rock parallel to the hill's slope. Working against that force as he rolls the rock up the hill, Sisyphus produces a potential energy that we must describe mathematically as an explicit function of the longitude of the rock. Just as Sisyphus gets the rock to the top of the hill a wedge on the ground gives the rock a slight shove to the left and the rock rolls north, gaining speed as the force acting on it converts potential energy into kinetic energy. With a sigh Sisyphus begins the long walk downhill to retrieve his rock.
Our knowledge that work done in one direction could yield a force oriented in another direction comes from the shape of the hill. We can devise a formula that lets us calculate the rock's potential energy from the rock's horizontal distance from the apex of the hill. In our example we must have the rock's potential energy as a function of the rock's altitude, of its height above some arbitrarily determined level in the landscape surrounding the hill. But we describe the rock's altitude on the hill with a function of its latitude and longitude, so, while the fundamental force coming from the rock's potential energy acts in the vertical direction, the shape of the hill gives it expressions toward all the points of the compass. So Sisyphus can give the rock some potential energy by pushing it eastward and the expression of the force northward can then confound him just when he believes that he has finally brought his burden to the top of the hill. But if, instead of a hill for this example, I had used a ridge of uniform height whose crest ran due north-south, then we would have had no changes of altitude to the north; the potential energy function for the rock would have had no explicit dependence upon the latitude; and no force would have acted toward the north.
Now suppose that we have a body with a potential energy whose mathematical expression contains, in addition to other coordinates, an explicit (that is, a direct) reference to elapsed time. In that case we would have a force pushing the body in the temporal direction. But what does it mean to say that we have a force exerted in the direction of time? Can we make something undergo a timeward acceleration?
In 1895 H.G. Wells published a story about a man who did just that. He called his story "The Time Machine" and it has become a classic of the science-fiction genre, though, as we will see, we must rightly regard it as science fantasy.
When we exert a force in a certain direction, the body upon which we exert the force accelerates in that chosen direction. But we know that acceleration changes the body's velocity, the ratio of distance crossed to time elapsed. Timeward force should impose a temporal acceleration upon a body, but what could we possibly mean by the ratio of time crossed to time elapsed?
We often describe time as a river, one whose flow carries us past events as the flow of water carries us past places. We even have a way of measuring distances between places on the banks of the River Chronos: we use events, other places on the banks, that we have contrived to position with uniform separations and count those as distance; that is, we use clocks. Now, to mix metaphors very badly, suppose that I sit on my raft with a clock that counts one hour for every sixty minute markers that I pass on the riverbank. With reference to that image it makes sense for me to say that I move through time at a speed of sixty minutes per hour. I rescue that statement from utter silliness by pointing out that in the same image a person going down the river in a motorboat would pass more than sixty minute markers for every hour elapsed on their clock. We thus discover that we can indeed define a timeward velocity if we take the ratio of the times measured on two different clocks: we would measure time crossed on a clock set up on a stationary base and measure time elapsed on a clock mounted on a time machine.
And a time machine, of the kind described by H. G. Wells, is exactly what the timeward force that I described above would enable us to build. Once we have built such a vehicle, we need only flip a few switches and push the velocity bar forward in order to embark on a grand adventure into ages yet to come. Like Wells's Time Traveler, we would watch future history flash by in a hyperfrenetic blur.
On the other hand, if we were observing from outside the time machine, we would see everything aboard the time machine slow down drastically. If our Time Traveler were to move at a speed of 100,000 years per hour (about 8.766 x 108 times our normal rate of travel), a little slower than Wells's Time Traveler did, then the simple act of taking their hand off the velocity bar and leaning back in their seat would appear to us to take as long as a century. And that observation exposes the error in my presumption.
In both of the movies made from Wells's novel a major component of the time machine is a large spinning disc. For simplicity's sake let's assume that the time machine must spin up the disc, giving it its full, rather large, angular momentum, before it can engage the time drive. We wave good-bye to the Time Traveler and watch as they and the time machine appear to freeze into utter motionlessness. We watch as the spinning disc seems to stop rotating altogether, its angular momentum dropping close to zero though nothing exerts a torque upon it. Likewise the linear momenta of any parts in reciprocating or other straight-line motion diminish, though nothing exerts the forces necessary to mediate the obligatory equal and oppositely directed reactions. To our astonishment our time machine seems to violate two absolute conservation laws.
Perhaps in our observation we have missed something? Perhaps the time machine obeys the conservation laws by increasing its mass to compensate the slowed motions? Such an increase would certainly save the momenta, but in order to effect it the time machine would have to violate the law pertaining to conservation of mass. We now have an unavoidable contradiction: our time machine must necessarily violate at least one of the conservation laws that we have already deduced. Logic does not allow us to incorporate contradictions into the construction of formal systems and, thus, obliges us to discard at least one of the premises that led to the contradiction. At this point I cannot falsify the conservation laws that I have in hand, so I have only one premise to discard as false to Reality - my assertion that timeward force exists. I must assert instead that the temporal force acting on any body necessarily equals zero forever and always. That proposition necessitates in turn that whatever potential energy that a body or system of bodies possesses cannot be such that we can describe it by an explicit function of the elapse of time.
If I add that proposition to my previous statements pertaining to kinetic energy, then I seem to have come up with a firm law of conservation of energy; that is, I think that I can state with full confidence that no phenomenon can create energy out of nothing or destroy energy completely out of existence. But I have a nagging suspicion that I have still neglected something in this derivation, that we have not yet blocked all ways to circumvent the conservation of energy. Our definition of energy in terms of work done seems to offer some possibilities for circumventions if we can manipulate the forces in suitable ways, two in particular.
Since we can freely manipulate the velocities of bodies, we may wonder whether we can also manipulate forces that we must describe as functions of bodies' velocities to violate the conservation law pertaining to energy. Suppose that we have two bodies that exert forces upon each other and that those forces have such a nature that we must describe them with algebraic formulae that include the bodies' velocities as factors determining the magnitude and direction of those forces. We have two basic possibilities in such descriptions: either the force may act in a direction perpendicular to the direction of the forced body's velocity or it may act in the direction parallel to the forced body's velocity. Because I can dismiss the first possibility quickly and easily, I will consider it first.
In fact, we do know a force that acts in a direction perpendicular to the direction in which a forced body moves. We call it the Lorentz force, the force that a magnetic field exerts upon a moving, electrically charged body. Because the Lorentz force acts in a direction perpendicular to the forced body's velocity, it does no work upon the body, adding nothing to and subtracting nothing from the body's energy. This kind of force, then, has no effect upon conservation of the body's total energy and thus has no effect upon the validity of our conservation law. That fact remains true to Reality even if we confine our electrically-charged particles inside a wire and constrain the wire to turn the armature of a motor.
Now consider the second possibility. If a force acts in the direction parallel to the velocity of the forced body, then it certainly does work upon the body, converting kinetic energy into potential energy and vice versa. Such a force certainly has relevance to any conservation law pertaining to energy, but what makes that force non-conservative also ensures that it cannot exist.
Like any other force, a velocity-dependent force must automatically obey the conservation laws pertaining to linear and angular momenta. We take the fact that the force conserves linear momentum, for example, and deduce that our mathematical description of the force must incorporate the velocities of both bodies involved in the interaction; that is, we must necessarily have as true to Reality some statement that the force that each body exerts upon the other involves the velocity of Body-1 and the velocity of Body-2 in the formula that we use to calculate the force. Only such a statement ensures that the force that we calculate obeys the equal-and-oppositely-directed rule of Newton's third law of motion.
If two bodies exert such a force each upon the other, then we can calculate for one of them, let's say Body-2, a description of its potential energy due to its interaction with the other body. We don't know the actual mathematical form of that description, but that form does not concern us here: we need only know the fact that the mathematical formula depends on both the velocity of Body-1 and the velocity of Body-2. You see, the importance of that dependency lies in the fact that someone might manipulate Body-1 and change its velocity and in so doing change the magnitude of Body-2's potential energy without having done any actual work on Body-2. But that change does not look like the change of potential energy that a body acquires when it moves toward or away from the crest of a potential energy hill or, as seen by some observers, when the crest of a potential energy hill moves toward or away from the body. No, this change gives us a change in the height and shape of the hill while the body lies motionless on it. But that would make Body-2 subject to a timeward force and the no time travel theorem that we deduced above forbids such a thing from existing, so any force, however oriented, that produces a potential energy that conforms to such a description cannot exist.
Thus we may infer that velocity-dependent forces necessarily validate our proposed conservation of energy law, either by obeying the law or by not existing.
One Other Possibility
In testing the validity of the proposition that Reality always conserves energy I have considered and dismissed the possible existence of non-conservative forces, those forces whose mathematical descriptions show us how to use them to create or destroy energy gratuitously. But our definition of energy gives us one other factor that we must consider. Recalling that energy comes from the product of both force and distance, we now ask whether we can contrive ways to create or destroy energy by means of non-conservative geometries.
We know that we can make the energy that some body possesses grow or diminish in a non-conservative way if the force exerted upon the body changes according to the direction in which the body moves as it goes over a closed path. Consider the simplest example: the body goes from Point A to Point B against a uniform force acting to push it back to Point A. In order to move the body to Point B we must do work upon it equal to the product of the force and the distance between Points A and B. If we let the body return to Point A under the influence of a force of strength different from that of the first force, then we will gain back an amount of work greater or less than the work we put into the body by an amount equal to the product of the distance between the points and of the difference between the forces. We know, of course, that we cannot create any such cycle, because a force that would change in the right way to make it possible would also violate the no time travel theorem.
But what would we get if we could make the distance between the Points A and B change as the body makes its circuit? Suppose that the body goes a certain distance against a given force from Point A to Point B: because the force acts to oppose the body's motion, we must do a certain amount of work to move the body. As the body dwells at Point B, we assume, the distance between Points A and B grows by some amount. We then allow the force to move the body from Point B back to Point A, harnessing the work that the force does upon the body during the transit. In this case we expect to gain back via the body more work than we put into moving it to Point B, an amount that we calculate by multiplying the force by the extra distance the body moved in the second step.
We thus seem to have gained purely created energy, coming ex nihilo and not from any other body or collection of bodies, the body generating the force included. But in devising and presenting that imaginary experiment I have made a tacit assumption that deserves some scrutiny. I have assumed into my premises the proposition that when the space in I have embedded my imaginary experiment expands, Point B carries the body along with it. Can we assert that Existence makes that assumption true to Reality? The answer to that question depends upon the nature of space and the nature of the expansion. Our own experience of the space in which we find ourselves embedded predisposes us to believe that space has such a nature as to make my assumption false to Reality. That experience and that belief have led us tacitly to postulate that space has the simple nature of an immutable and inert vessel in which matter carries out its existence and motions. Until we find reasons that compel us to believe otherwise, we should accept that postulate as true to Reality for all of space except those regions, however small and however temporarily, must have manifested a non-conservative geometry in order to enable the creation of matter when the Universe blossomed into existence.
Encoding Conservation of Energy
Over the past two and a half millennia philosophers and scientists have conceived physics at its most fundamental as the science of motion. For more than a century and a half now the discoveries of self-sustaining forcefields and a zoo of elementary particles have obscured that fact. Those discoveries, though, came as unintended side effects of studies aimed at clarifying what makes bodies move as they do. Hans Christian ěrsted's discovery of the magnetic effect of an electric current, which led ultimately to James Clerk Maxwell's electromagnetic theory of light, came from ěrsted's effort to explain the queer motions of compass needles in the presence of thunderstorms. Twentieth-Century physicists' discovery of the neutrino came from their efforts to understand the motions of nuclei and their emitted electrons or positrons coming out of certain beta decays. Behind the shimmering veil of those discoveries and all others in physics stands the Platonic Idea of bodies in motion.
In taking motion as the fundamental idea of physics we automatically assume into our premises the idea that anything that sums up the legitimate motions of a system of bodies also sums up the laws of physics that apply to that system. Nicely enough, the concept of energy gives us just such a summation by way of Hamilton's Principle.
William Rowan Hamilton (1805-1865), an Irish mathematician, constructed his eponymous principle on the concept of virtual work, a concept that the Swiss mathematician Jean Bernoulli (1667-1748) had conceived in 1717 and that the French mathematician Jean Le Rond D'Alembert (1717-1783) had expanded in 1743. What Hamilton did looks very much like the definition of energy that I showed you above and in its raw form Hamilton's Principle looks like a mathematical statement of the law of conservation of energy. But it goes deeper than that.
Bernoulli stated that if the applied forces acting on a body add up to zero, then the virtual work done upon the body by any minuscule imaginary (or virtual) movement of the body away from its true position must equal zero. D'Alembert extended that concept by pointing out that the body's kinetic reaction, the rate at which its linear momentum changes, also constitutes a force and, thus, that the forces acting on a body always add up to net zero under Newton's laws of motion. So we calculate the work that the forces do upon the body as if the body had moved and find, as we expect, that it adds up to a constant. We can do this for all possible virtual motions that the body could follow.
We come again to physics in the subjunctive mood, the physics of as if. Every body follows a path from one point to another and we can calculate the kinetic and potential energies that the body has at every point on that path. We can also calculate the virtual kinetic and potential energies that the body would have if it followed a slightly different path between the same two points. When physicists compared those energies mathematically, they discovered that the body follows a path that follows a straightforward rule. If we subtract a body's potential energy from its kinetic energy, we get a mathematical description that physicists call the body's Lagrangian function. In going from one point to another the body follows a path that makes the average value of that body's Lagrangian function over the time the body takes to go from one point to the other a minimum.
If the body floats in deep space, for example, it has only kinetic energy and, so, follows a straight line between the two points. But if we have a body near Earth's surface, the body has both kinetic energy and a potential energy due to its immersion in Earth's gravitational field. In that case the body can gain a little altitude on its path and obtain some additional potential energy to subtract from its Lagrangian function; the higher the body rises, the more potential energy diminishes the Lagrangian function. But if the body rises too high, it must travel faster; thus, it gains more kinetic energy and re-enlarges the Lagrangian function. The body will end up following a path that balances those contributions and minimizes the average value of the Lagrangian function over the time it takes in going from one point to the other. We can describe the path that conforms to that balancing act by way of the calculus of variations and when the chalk dust clears we will see that we have described the parabolic arc that thrown bodies follow near Earth's surface.
That example gives us a brief description of the Euler-Lagrange version of the principle of least action. In using that version physicists use the Euler-Lagrange Equation to transform a description of the distribution of energy in a system of bodies into equations of motion that describe how that system evolves with the elapse of time, giving a complete accounting of the system's motions. So now we know that a description of a body's energy distribution encodes those laws of physics that apply to the body in a given situation.
Let me also note in passing that my fondness for the particular example that I gave above comes from how I got it. One day in 1968 I joined several other students from the UCLA Physics Department in going to the California Institute of Technology to attend a lecture given by Richard P. Feynman, one of the greatest physicists of the Twentieth Century. In the course of his lecture (on the subject of General Relativity) Professor Feynman gave that example as a means of introducing the idea of geodesics, paths that conform to some minimizing principle.
At last, then, we have obtained two major results. We know that energy can be neither created nor destroyed, but can only be transformed and transferred; that it is subject to a conservation law, albeit one that admits exceptions under circumstances that do not occur in normal space. And we also know that a description of the distribution of energy in a system of bodies encodes the laws of physics that govern that system of bodies.
Oh, By The Way
I noted above that I obtained the inspiration for an important part of my derivation of the conservation of energy theorem from a contemplation of Herbert George Wells's 1895 novel about a man who travels to the year A.D. 802,701 and beyond. I showed you that the machine that Wells described could not possibly work because its operation would violate the absolute conservation laws pertaining to linear and angular momenta, but some years ago I conceived three other reasons why Wells's Time Traveler could not have made his journey into Futurity. Shortly after beginning his journey the Time Traveler guesses that the machine is carrying him through time at the rate of about one year per minute, about half a million times faster than normal. I'll use that speed in what follows here:
I. From the Time Traveler's point of view everything in the world beyond the time warp surrounding his machine seems to speed up. That would include the vibration rate of photons. Because we can relate a photon's vibration rate to its energy content by way of Planck's formula, we can calculate that the Time Traveler would see visible light, whose photons have energies ranging from 1.6 electron-volts to 3.2 electron-volts, transformed into gamma rays with energies in the range from 800,000 electron-volts to 1,600,000 electron-volts. Further, he would see the energy in sunlight, which normally strikes Earth at a rate of about one kilowatt per square meter, intensified to 500 megawatts per square meter. Such a bombardment would strip every atom in the Time Traveler, his time machine, and the air in the machine down to its nucleus and pervade the resulting plasma with a thick electron-positron haze.
Looking at that phenomenon, we can see that the time machine seems to be enveloped in a remarkably stiff gravitational field. That means that not only does it subject light coming into the machine from outside to a strong blue shift, but it also subjects light originating in the incandescent bulbs inside the machine to a strong red shift. Because the time machine allows the operator to control that effect, it allows the operator to create energy ex nihilo or destroy it in a way that we cannot do with true gravity, which does not change. Energy carried in the time machine's batteries and converted into light in the bulbs would disappear from Reality as that light crossed the boundary of the time warp. This gives us another example of how a time machine would violate the law of conservation of energy.
II. Everything in the world seems to speed up in the Time Traveler's view. That would include the motions of the molecules comprising the laboratory and the air within it. Tragically for the Time Traveler, the temperature of a body is proportional to the average kinetic energy of its molecules, which energy is, in turn, proportional to the square of the relevant speed of molecular motion. If the speeds of the molecules increase by a factor of half a million, then the temperature of the body they comprise increases by a factor of one-quarter trillion. We can only guess at what the temperature was in the Time Traveler's laboratory, but a good guess would be 293 Kelvin (20 Celsius or 68 Fahrenheit). Boosted by the time machine's operation, that temperature would appear to the Time Traveler to have gone to 73.25 trillion Kelvin, a temperature typically found near the center of a supernova in the first instants of its explosion. He would have had little time to contemplate that phenomenon, because the pseudo-gravitational effect of the time warp would almost instantly bring a large quantity of the superhot air into the time machine.
III. Sitting in the saddle of the time machine the Time Traveler saw Mrs. Watchett, walking sedately, cross his laboratory "like a rocket". Had she knocked something off a table, he would have seen that something leap to the floor as if shot from a gun. Gravity would appear greatly stiffened, by a factor of one-quarter trillion because we measure it in accordance with the inverse square of the elapse of time. Unless the time machine included a suitable amount of adjustable Cavorite, that gravity would have pervaded the machine with tragic result.
Blasted, burned, and smashed flat, Wells's Time Traveler would not have gone far. But that's only a criticism of "The Time Machine" taken as science fiction. I still enjoy the story as science fantasy. When I'm not exploring the physics implicit in the story I quietly contemplate the grand vista of Time that Wells laid out and I find myself haunted by the spectral images of the effete Eloi and the sinister Morlocks, ghosts of futures past and, I hope, not of futures yet to come.
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