Relativistic Mass and Linear Momentum

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    Up to now, in the previous essays, I have shown you only the kinematics of Relativity, the geometry of space, time, and motion. In effect, I have shown you the layout of the stage and now I want to bring the actors out onto that stage and examine how they interact in the play of Reality. Now I want to show you the dynamics of Relativity, the actual physics that governs how bodies interact in relativistic situations. To do that I must make a connection between relativistic descriptions of the movements of bodies and the forces that alter those movements. The ability to translate velocities from one inertial frame to another will allow us to make that connection, to incorporate dynamics into Relativity.

    Let's go back to our fantasy world, in which the speed of light is 100 miles per hour, and board 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 in which the railroad tracks do not move). Of course, an observer (let's call him Stan) who has set up two synchronized clocks at opposite ends of a measured length of the north-south track carries out that measurement. 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 get the speed of the track speeder.

    We can, in theory, make the same measurement and calculation from our eastward moving train. We can imagine our two synchronized clocks as flying on paths parallel to the rails on which our train moves. 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 we use the same length for the section of track as Stan does because that section of track makes a right angle with the track on which our train moves: we and Stan will use the same number of meters or feet. Second, we know that because of the orientation of that section of track (perpendicular to the track on which our train runs) the distance between Stan's clocks also makes a line perpendicular to the direction of our 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, ours and Stan's. When we carry out the calculation, we find that the northward component of the track speeder's velocity in our frame equals 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.

    Algebraically let's represent the eastward speed of our train in the stationary frame (the frame occupied and marked by the railroad tracks) by v and the speed of the track speeder in the stationary frame by w. We have so oriented our coordinate grid that east defines the positive x-direction and north defines the positive y-direction, so we have v = dx/dt and w = dy/dt. The Lorentz factor between our train's frame and the stationary frame is given by

(Eq'n 1)

time elapsed in the stationary frame dilates relative to time elapsed in the train's frame by that factor, so the northward velocity of the track speeder transforms to

(Eq'n 2)


    That calculation now raises a question. How much northward linear momentum does the track speeder have in the two frames? In order to answer that question I must make a modest feint in the direction of General Relativity, the relativity of accelerated motion.

    I want to clarify our problem here by imagining that our train is halted in a station when we first become aware of the track speeder and begin measuring its northward speed. Equation 2 tells us that as our train leaves the station and gains speed eastward (that is, as v goes from zero to 86.6 miles per hour), we will notice that the northward speed of the track speeder diminishes: the speeder appears to be decelerating in the north-south direction, slowing from thirty miles per hour to fifteen miles per hour. Now we have no difficulty with the observation that in our train's frame the speeder seems to be gaining speed in the westward direction: that simply comes to us as 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?

    Our two Cosmic Theorems, equivalent to Einstein's postulates of Relativity, do not involve accelerated motion, so I can't invoke them as I have done before. However, Existence makes the conservation laws true to Reality without regard to the state of motion of an observer or an observed body, so I will exploit conservation of linear momentum in solving the present problem. We certainly expect the east-west component of the track speeder's linear momentum to change as our train accelerates, because we can understand the force accelerating the train eastward as being, in some manner, equivalent to a force accelerating the track speeder westward. We can do that because linear momentum is not a relation between a body and an absolute space, but is, rather, a relation between a body and any observer. Clearly there's a force accelerating the train, but as far as calculating linear momenta is concerned we could equally well claim that there is no force acting on the train and that it is the rest of the Universe that is being accelerated past us.

    I now infer that the eastward acceleration of our train cannot lead to a southward force being exerted upon the track speeder, because such a force would lead to a violation of the conservation law. If the exertion of a force upon one body led immediately (that is, without mediation, as would be the case with electrically charged particles in a magnetic field) to a change in another body's linear momentum in a direction perpendicular to the applied force, then that action would fail to be matched with an equal and oppositely directed reaction, as required by Newton's third law of motion (the more common form in which we state the conservation law). In our example the force that accelerates our train eastward is matched by the westward force that the locomotive's drive wheels exert upon the rails, but there's no equal and oppositely directed force to match the presumed force accelerating the track speeder.

    Now if the acceleration of our train exerts no force 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 86.6 miles per hour, the northward linear momentum of the speeder remains unchanged. But we calculate linear momentum as 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 equals 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 stationary frame, so the mass of the speeder must equal the "proper" mass multiplied by that same Lorentz factor. If the mass of the speeder in the stationary frame is represented by m and in the train's frame by m', then we have

(Eq'n 3)


    But we have deduced the proposition that Reality conserves mass; 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 we have not deduced a statement that the mass of the track speeder increases spontaneously. We have deduced, rather, a statement that the mass of an object differs in different inertial frames and that it merely appears to increase or decrease as we shift ourselves from one frame to another. Indeed, we have deduced the relativistic increase of mass from the conservation law pertaining to linear momentum just as we deduced the conservation of mass from it, so we should expect that the relativistic mass formula conforms to the requirement that Reality conserve mass.

    We have also deduced, albeit implicitly, the proposition that the mass of a body has a minimum in the inertial frame in which the body is motionless. Now let's make that proposition explicit. We know that the Lorentz factor has a value of one when the velocity between two observers equals zero. As the magnitude of the velocity between the observers increases, in either the positive or negative direction, the value of the Lorentz factor increases, tending toward infinity as the relative speed between the observers tends toward the speed of light. We can see clearly that the Lorentz factor has its minimum value when the observers have zero relative velocity between them, so, in accordance with Equation 3, we can see that the mass of a body has its minimum value when it has zero velocity relative to the observer who measures that mass. The mass of a body in any inertial frame thus equals that minimum mass, the proper 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, the number describing its mass would tend toward infinity. 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 does Existence make that statement true to Reality? Do we have any evidence to support this weird claim that bodies ponder more mass when they move than they do when 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 (technically "magnetic resonance accelerator"; "cyclotron" is a laboratory slang term for the device). In 1929 Ernest O. Lawrence conceived the cyclotron as a relatively simple machine. He founded the machine upon 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 Lawrence and his assistants put a pair of copper dees (think of a motion-picture film can that's been cut in half along one of its diameters), connected them to a radio transmitter as if they were opposite ends of its antenna, and then so enclosed the space occupied by the dees 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 experimenter tunes the radio transmitter to that frequency, the electric field that oscillates between the dees makes protons injected into the center of the machine gain kinetic energy every time they cross the gap between the dees. Even though the cyclotron operates with relatively 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, in physicists' jargon, for "million electron volts"). The first proof-of-principle model, with 4-inch wide dees, used a gap voltage of 2000 volts and was able to give protons 80,000 electron-volts of energy. The first practical cyclotron, made in 1932 with 10-inch wide dees, could give protons one Mev of energy.

    We can calculate the cyclotron frequency quite easily. Any given body revolves about a circle at a frequency that equals the body's speed divided by the circumference of the circle (two pi times the radius). We determine the speed at which an electrically charged particle revolves in a circle in a uniform magnetic field by equating the centrifugal force with which the particle's inertia acts to make the particle fly in a straight line with the centripetal force with which the magnetic field strives to pull the particle toward the center of the circle. If a particle of mass m and charge q moves at the yet to be determined speed v in a direction at right angles to a magnetic induction field of strength B, then we have the force equation

(Eq'n 4)

in which r represents the radius of the circle. After solving that equation for v, we can then calculate the cyclotron frequency as

(Eq'n 5)


    The elegance of the cyclotron, expressed in that equation, lies in the fact that the cyclotron frequency depends only upon the properties of the accelerated particles and the strength of the magnetic field in the machine. For every thousand gauss of magnetic field in the dees the frequency needed to accelerate protons increases by 1.52 mega-Hertz (million cycles per second) and the frequency needed to accelerate deuterons (the nuclei of heavy hydrogen) and doubly ionized helium increases by 760 kilo-Hertz. Once a particle has been injected into the machine no further manipulation of the machine is necessary to boost the particle to high kinetic energy. As the particle gains energy (and speed) it revolves in progressively wider orbits, but it does so at progressively increasing speeds that keep the frequency of the revolution unchanged, thereby ensuring that the particle returns to the gap in the dees only when the vibrating electric field is oriented to accelerate the particles. It works beautifully...up to a point.

    In their explorations of the atomic nucleus and of the particles that comprise it, physicists sought to use protons carrying increasingly large amounts of energy. That quest required bigger, more powerful cyclotrons, machines that could take full advantage of the subtleties of cyclotron design. In 1937 and 1938 Bethe and Rose and then Rose and Wilson worked out the complete theory of the cyclotron, the theory that accounted for every relevant effect known to physicists at the time. Among those effects they included the relativistic increase of the mass of a moving body. That effect, the designers of cyclotrons saw, would make protons and other particles stall in the machine: once they had acquired a certain amount of kinetic energy, the particles simply would not absorb any more energy from the vibrating electric field that the radio transmitter created in the gap between the dees. As the particles gained large amounts of energy and, consequently, high speeds in the cyclotron the increasing mass of the particles would make the particles swing into orbits that were too wide for the speeds at which they were flying. The frequency at which the particles crossed the gap between the dees would diminish, so the particles would lose synchrony with the electric field in the gap, falling out of step with it, until the field actually began to decelerate them. The particles 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 particle would be limited only by the size of the cyclotron itself.

    Physicists have a number of tricks that can enhance the performance of a cyclotron, but they add only small increments to the final energy gained by the particles from the machine. The practical limit to the cyclotron, due mostly to the relativistic increase of the particles' masses, is a machine with 60-inch wide dees giving protons or deuterons 25 Mev of energy. But physicists discovered one trick that simply transforms the cyclotron into a more powerful machine with what, in theory, is a minor modification.

    In 1945 E.M. McMillan and V. Veksler, working independently, conceived the idea of the synchrocyclotron or frequency-modulated cyclotron. In order to compensate the decrease in the frequency of orbital motion of the accelerated particles that comes from the relativistic mass increase, they suggested building the synchrocyclotron to provide a vibrating electric field whose own frequency shifts downward as the particles accelerate: the device is simply a cyclotron whose electronic characteristics have been altered by the addition of a variable capacitor. Attached to the ends of the copper tubes that feed the radio-frequency voltage to one of the machine's dees, the variable capacitors are either flat toothed wheels that spin between the plates of a flat-plate capacitor (inspired, certainly, by the variable capacitors used to tune radios at the time) or longish flat-plate capacitors whose plates were made to vibrate like the reed in an oboe; only such devices can shift the cyclotron's frequency rapidly enough to track the decrease in the frequency of the particles' orbital motions. Built in 1946, a synchrocyclotron with 184-inch wide dees was able to boost deuterons to 190 Mev and doubly ionized helium to 380 Mev. By 1957 the same machine was able to boost protons to 720 Mev, nearly doubling their masses (the rest mass of a proton is equivalent to 938 Mev, according to Einstein's mass-energy equivalence theorem). That a machine three times the size of the best feasible cyclotron can surpass the cyclotron's limitation and give protons nearly thirty times the energy that the cyclotron can give offers us strong evidence that the primary difference between the two machines, the assumption of the relativistic increase of particles' masses, is true to Reality.

    Now we need to consider relativistic momentum and forces. If one observer tracks a number of bodies with various measured linear momenta and certain forces are being exerted upon those bodies, how would another observer, moving relative to the first, calculate the corresponding momenta and forces in their frame?

    We can transform linear momentum in a relatively straightforward way. If someone gives us the mass and velocity of a body in one inertial frame, we simply apply the velocity addition formula to obtain the body's velocity in a second inertial frame and multiply the body's rest mass by the Lorentz factor calculated with that compound velocity. If we restrict the bodies' and observers' motions to go only parallel to one axis, and if a body moves with speed v past one observer who moves with speed -u past a second observer, then the body will move past the second observer with a speed w give by

(Eq'n 6)

We then have the Lorentz factor that the second observer must apply to the body as

(Eq'n 7)

We then obtain the body's linear momentum as the product of multiplying the body's mass in the given frame by its velocity in that frame.

    The transformation of forces won't come to us as easily as did the transformation of mass and linear momentum. Yes, we can deduce the transformation of force from the transformation of linear momenta because we have defined force as the derivative of linear momentum with respect to time. But we must take care here; the derivatives with respect to space and time have their own transformations. For that reason I have put the transformation of forces into a separate essay.



Livingston, M. Stanley, and John P. Blewitt, "Particle Accelerators", McGraw-Hill Book Co., Inc., New York, 1962, LCCCN 61-12960.

This is a comprehensive and technically detailed text on the first thirty years of particle accelerator development. It is, especially in its discussions of the history of the subject, quite readable even by those who are unfamiliar with the more mathematical expressions of electronic engineering.


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