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The ability to translate velocities from one inertial frame to another gives us the ability to incorporate dynamics into Relativity. Up to now I have been showing you only the kinematics of Relativity, the geometry of space, time, and motion. Now I want to show you the dynamics, the actual physics that governs how bodies interact in relativistic situations.
We have already seen how to translate a velocity oriented parallel to the relative motion between two inertial frames from one frame to the other. Now let's look at a velocity that we have contrived to orient perpendicular to the relative motion between those frames. Of course, that velocity is truly perpendicular to the direction of relative frame motion in only one of the frames; in the other frame it is slanted relative to the frame's axes because it's combined with the motion of the frame relative to the other frame. However, we can break that compound motion into two components, one oriented parallel to the interframe motion and the other oriented perpendicular to it. It's that perpendicular component that we wish to describe.
Let's go back to our fantasy world, in which the speed of light is 100 miles per hour, and assume that we are aboard a train that's traveling eastward on an east-west oriented track at 86.6 miles per hour. Assume that a track oriented due north-south crosses our east-west track and that a track speeder moves north on that track at a steady thirty miles per hour as measured in the stationary frame, the frame occupied and marked by an observer standing beside our railroad track. The implied measurement of the speeder's motion is, of course, carried out by that observer, who has set up two synchronized clocks at opposite ends of a measured length of the north-south track. That observer notes the readings on the clocks when the speeder passes each of them, subtracts one reading from the other to get the elapsed time, and then divides the length of the section of track by that elapsed time to calculate the speed of the speeder.
We can, in theory, make the same measurement and calculation from our eastward moving train. Our two synchronized clocks can be imagined as flying on paths parallel to the rails on which our train moves, perhaps held on long outriggers cantilevered off a heavy freight car. We take the readings on those clocks as they pass over the speeder (and they do that as the speeder passes one or the other of the stationary clocks), subtract the smaller reading from the larger, and divide the result into the length of the section of track between the two stationary clocks. Our result will differ from the stationary observer's result and we know how it will differ. First, we know that the length of the section of track will be the same in both frames because that track is oriented perpendicular to the direction in which our train is moving: we and the stationary observer will use the same number of meters or feet. Second, we know that because of the orientation of that section of track the distance between the stationary observer's clocks is also perpendicular to the direction of relative motion, so the temporal offset between those clocks, as seen from our frame, is zero. And third, we know that the time elapsed on those clocks is dilated relative to the time elapsed on our clocks by the Lorentz factor between our two inertial frames. When we carry out the calculation, we find that the northward component of the track speeder's velocity in our frame is equal to the speeder's northward speed as measured in the stationary frame divided by the Lorentz factor. In our frame, then, the speeder appears to move westward at almost eighty-seven miles per hour and northward at fifteen miles per hour.
That last fact raises a question. What is the track speeder's northward linear momentum in the two frames? In the stationary frame we simply multiply the speeder's mass by thirty miles per hour, so shall we then in our frame multiply the speeder's mass by fifteen miles per hour? Our problem here becomes clearer if we imagine that our train is halted in a station when we first become aware of the track speeder and begin measuring its northward speed. As our train leaves the station and gains speed eastward, we will notice that the northward speed of the track speeder diminishes: the speeder appears to be decelerating in the north-south direction. Now we have no difficulty with the observation that the speeder seems to be gaining speed in the westward direction: that's simply a consequence of the fact that our train is accelerating eastward. But how are we to understand the observation that the speeder seems to be suffering a southward acceleration?
We know that the eastward acceleration of our train cannot lead to a southward force being exerted upon the speeder, because such a force would lead to a violation of the conservation laws. But if no force is being exerted in the north-south direction, then as our train shifts through the continuum of inertial frames from the stationary one to the one moving eastward at almost eighty-seven miles per hour, the northward linear momentum of the speeder remains unchanged. But linear momentum is equal to the product of the body's velocity and mass, so if we do something that changes the velocity without changing the linear momentum, then the mass must change in a way that leaves the product unchanged. In this case the velocity of the speeder is equal to the "proper" velocity of the speeder (the velocity measured in the frame that we regard as stationary) divided by the Lorentz factor between our frame and the "proper" frame, so the mass of the speeder must be equal to the "proper" mass multiplied by that same Lorentz factor.
But previously we have deduced the proposition that mass is conserved; that is, that the mass of a body cannot change spontaneously. Doesn't the increase of mass deduced above violate that conservation law? Not at all, because what we have deduced is not a statement that the mass of the track speeder increases spontaneously. What we have deduced is a statement that the mass of an object is different in different inertial frames and that it appears to increase or decrease as we shift ourselves from one frame to another.
We have also deduced, albeit implicitly, the proposition that the mass of a body is at its minimum in the inertial frame in which the body is motionless. The mass of a body in any inertial frame is thus equal to that minimum mass multiplied by the Lorentz factor between the given inertial frame and the frame in which the body is at rest.
That proposition gives us one more way in which to understand our deduction that nothing can move faster than light. If we were able to accelerate an object continuously, then as it shifted into inertial frames progressively closer to the speed of light its mass would tend toward infinite. We have already deduced that the Universe cannot contain any body with infinite mass, so no body can ever reach the speed of light, much less surpass it.
But is it true to Reality? Do we have any evidence to support this weird claim that bodies are heavier when they move than they are at rest? Indeed we do, most of it from experiments aimed at studying subatomic particles.
The first evidence came from the discovery of a major design flaw in the first of the electromagnetic particle accelerators that physicists use, the cyclotron. Invented by Ernest O. Lawrence in the early 1930's, the cyclotron is a relatively simple machine. The foundation is an electromagnet with broad pole faces only a few inches apart, a design intended to create a magnetic field of uniform stiffness within the space between the poles. Into that space the builders put a pair of copper dees (think of a motion-picture film can that's been cut in half), connected them to a radio transmitter as if they were opposite ends of its antenna, and then so enclosed the space that the air could be pumped out to create a vacuum. The beauty of the cyclotron lies in the fact that the frequency at which an electrically charged particle, typically a proton (the nucleus of a hydrogen atom), revolves within the device depends only upon the mass of the particle and the stiffness of the magnetic field: when the radio transmitter is tuned to that frequency, protons injected into the center of the machine are accelerated and thus gain kinetic energy every time they cross the gap between the dees. Even though the cyclotron operates with moderate voltages, in a fraction of a second a proton gains as much energy as it would if it had passed through a voltage difference of several million volts (Mev, pronounced em-ee-vee, standing for Million electron volts in physicists' jargon).
As they explored the structures of the atomic nucleus and of the particles that comprise it, physicists sought to use protons carrying increasingly large amounts of energy. The design flaw in the cyclotron became clear when the operators discovered that after gaining a certain amount of energy the protons would stall in the machine: they simply would not absorb any more energy from the vibrating electric field that the radio transmitter was creating in the gap between the dees. What was happening in the machine was that the increasing mass of the protons was making the protons swing into orbits that were too wide for the velocities at which they were flying. The frequency at which the protons were crossing the gap between the dees was diminishing, so the protons were losing synchrony with the electric field in the gap until the field actually began to decelerate them. The protons were thus locked into a vicious cycle, in which they would accelerate up to some speed, then decelerate to some lower speed until they regained their synchrony with the electric field, reaccelerate to their maximum speed, and so on. If not for the relativistic increase of mass, the amount of energy that a cyclotron could give a proton would be limited only by the size of the cyclotron itself.
Rather than modify the design of the cyclotron to compensate the relativistic mass increase, physicists chose to use a new kind of machine, called a synchrotron, that is designed from the beginning to accommodate relativistic effects. Since 1952 synchrotrons of increasing size and power have been built and operated successfully. The prime example in this country is the Tevatron, whose mile-wide main ring is buried under a patch of Illinois prairie some thirty miles southwest of Chicago: protons circulating in this device gain as much energy as they would by passing through a potential difference of one trillion volts. And in the forty-six years that such machines have been operating, pushing protons to Lorentz factors of one thousand and more and slamming them into both stationary targets and counter-rotating beams of protons, no one has seen any evidence to suggest that the conservation laws pertaining to mass, momentum, or energy are being violated.
Of course you are
not astonished at that last fact. We deduced Relativity from the conservation
laws pertaining to linear and angular momentum, so any aspect of Relativity,
properly interpreted and applied, must conform to those laws. However, as I
shall confess more fully later, I fudged the deduction in one place and within
the gap that I left we may find the answer to the dilemma posed by the fact that
we have compelling evidence that the law of conservation of energy has been
violated in at least one instance. So now we must consider energy.
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