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Mass is Relative |
The phenomenon of mass manifests itself in so many different ways in the theory of relativity that terminology often becomes cumbersome and concepts often become confusing. The following table lists just some of the many ways mass could be defined in relativistic physics.
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Type of Mass |
Value |
Definition |
|
Proper mass (rest mass) |
mo |
The mass of an object measured in its own proper frame of reference. |
|
Longitudinal mass |
g3 mo |
The ratio of force to acceleration in the direction of motion. |
|
Transverse mass |
g mo |
The ratio of force to acceleration perpendicular to the direction of motion. |
|
Gravitational mass |
-- |
The characteristic of mass that produces gravity. Not applicable in special relativity. |
|
Energy mass |
g mo |
The mass equivalent of the total energy of the object. |
|
Momentum mass |
g mo |
The ratio of the momentum of an object to its velocity. |
Rest Mass
Because there are so many different ways to define mass in relativity, many scientists prefer to restrict the use of the word “mass” to a single concept, namely, that of the “rest mass” or “proper mass” of an object. This practice is equivalent to retaining the classical definition of mass in the proper frame and leaving mass completely undefined in all other frames of reference. Einstein recommended this practice during the later years of his life. And many modern physics textbooks today follow his recommendation (including both Serway pp. 31-33 and Tipler pp. 69-73).
By defining mass only in its proper frame, mass becomes an unique, absolute, invariant scalar quantity associated with any object or system that can be observed from its own center-of-mass frame. And the distinction between mass and energy becomes very clear. In the proper frame, we talk about the object’s mass. In any other frame, we talk about the object’s energy and momentum.
But in my opinion, emphasizing the distinction between mass and energy tends to blur the connection. And leaving mass undefined in every frame except the proper frame obscures the conceptual meaning of mass. And treating mass as an invariant quantity in a relativistic environment can easily lead to puzzling questions and erroneous conclusions. For example, if mass is invariant, how can it possibly be created or destroyed? And if the mass of a flywheel is independent of its motion, then a spinning flywheel should have the same mass as one at rest. (False conclusion.)
For these and other reasons I prefer to define mass in a manner suitable for any frame of reference, and show that that mass reduces to the proper mass in the proper frame. Then the concept of mass becomes more general, its connection with energy becomes more obvious, the puzzling questions become less confusing, and the erroneous conclusions become less numerous.
Inertial Mass
In this course, we define mass to be the ratio of an object’s momentum to its velocity. Since this definition becomes indeterminate in the proper reference frame (where both momentum and velocity are zero), we define mass in the proper frame to be the limit of the momentum/velocity ratio as the velocity approaches zero. This definition is clearly consistent with the classical definition of mass, not only in the proper frame, but also in every other classical frame of reference. This generalized concept of mass is also consistent with the usual conceptual definition of mass in classical physics. In all frames of reference under all conditions (including relativistic conditions) mass is pictured as a quantitative measure of inertia. Specifically, mass is the scalar conversion factor between velocity and momentum. Since momentum quantifies the tendency an object has to keep on moving in a certain direction and mass quantifies the momentum for a given velocity, then we can say in simple conceptual terms that “mass quantifies inertia”.
But before we can apply this definition of mass to determine the properties
of mass at high speeds, we need a relativistic definition of momentum. This is
accomplished through a generalization of the law of conservation of momentum.
Specifically, we define relativistic momentum as that quantity conserved in
high speed collisions. Both Serway p. 39 and Tipler p. 69-71 verify
and/or derive the expression for the momentum of a particle moving at a
relativistic speed. From that expression, the relativistic mass is easily
identified.
Relativistic Mass |
|
|
mo = proper mass of particle
in its own frame m = relative mass of
particle in the lab frame v = velocity of particle
relative to the lab frame g = g(v) = (1-v2/c2)-1/2 |
m
= gmo |
According to this equation, the mass of a moving object increases by a factor of gamma.
It is interesting to note that both Serway and Tipler briefly mention relativistic mass – probably because Einstein and others used the concept extensively during the earlier years of relativistic physics – but deliberately avoid that terminology in the rest of their book. Nevertheless, the authors of Tipler do utilize the concept of relative mass several times, calling it by a different name, the “measured mass.” Apparently they agree that the mass of a moving object does have meaning, and that its measured value will equal the relativistic mass defined above
Energy Mass
According to Einstein’s famous mass-energy equation, E = mc2, mass and energy are equivalent. The energy mass m in this equation is mathematically the same as the relativistic mass m given above. Therefore, relativistic mass not only expresses the inertia of an object, it also expresses the total energy associated with the object. So the rest mass of a system of particles is equal to the mass equivalent of the total energy of the system when viewed from its center-of-mass frame. (The center-of-mass frame is defined to be that frame in which the total vector momentum of the system is zero.)
Therefore, relativistic mass could equally well be defined conceptually as “total energy” and rest mass as “energy at rest.”
Acceleration
Mass
From the table at the top of this page you can see that the relativistic mass m is also equal to the transverse mass. Therefore, a third way to define relativistic mass is as the ratio of force to acceleration perpendicular to the direction of motion. However, the longitudinal mass is not equal to m but g2m.
Because the value of the mass defined through the equation F = ma depends upon the direction of acceleration, this method of defining mass is not very satisfactory. Furthermore, it is difficult to conceptualize the meaning of mass defined through this equation. Acceleration mass certainly has something to do with inertia, but only the transverse mass is the same as the inertial mass defined through the momentum/velocity relationship. Because the longitudinal mass is different from the transverse mass, we will avoid using either term in the future.
Context
In this course, when I use the word “mass” without any qualifier, I will usually mean the relativistic mass, momentum mass, inertial mass, and energy mass of the system as measured in the laboratory frame of reference. But when the context of the discussion seems clear or when the terms “proper mass” and “rest mass” seem too wordy, I will sometimes drop the qualifiers hoping you will understand my meaning.
In general, if the focus of the discussion deals with mass comparisons in
different frames I will retain the qualifiers. When it deals with mass-energy
comparisons in a single reference frame, I will generally drop the qualifiers
and use the mass appropriate to that frame.
And when it deals with mass in one frame and energy in another, I will
generally drop the qualifiers and use proper mass in the mass frame and
relative energy in the energy frame.
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1. Massless Particles – Particles traveling at the speed of light having
zero proper mass. |
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ModPhy1/Unit1/SpecialRelativity/RelativeView/