Comments about the FAQ: What is relativistic mass?

Following is a discussion about this Faq
In the last paragraph I explain my own opinion.

[Physics FAQ] [Copyright]

Updated by Don Koks, 2008
Original by Philip Gibbs and Jim Carr, late 1990s.

What is relativistic mass?

The concept of mass has always held been fundamental in physics. It was present in the earliest days of the subject, and its importance has only grown as physics has diversified over the centuries. Its definition goes back to Galileo and Newton, who essentially defined mass as that property of a body that governs its acceleration when acted on by a force.
This is too simple. Infact two issues are at stake: A mass at rest and a moving mass.
In fact we have here two situations: An object at earth and an object in space.

This definition of mass could be applied in a straightforward way for almost two centuries until Einstein arrived on the scene. In Einstein's theory of motion known as special relativity, the situation became more complicated.
It is not very scientific to claim that it is more complicated. In stead you should describe what mass is in SR. The above definition of mass still holds for a body at rest, and so is also called rest mass.
Under Newton's Law mass is constant. As such you cannot use that definition to define rest mass in SR.

When a body is moving, we find that its force–acceleration relationship is no longer constant, but depends on two quantities: its speed, and the angle between its direction of motion and the applied force.
Why this rather vaque sentence ? Why not give first more details what this relationship is en how you find it.

If we relate the force to the resulting acceleration along each of the three mutually perpendicular spatial axes, we find that in each of the three expressions a factor of γ m appears, where the gamma factor γ = (1–v2/c2)–1/2 is a common quantity in special relativity, and m is the body's rest mass. The new quantity γ m is traditionally called the body's relativistic mass. While rest mass is routinely used in many areas of physics, relativistic mass is mainly restricted to the dynamics of special relativity. Because of this, a body's rest mass tends to be called simply its "mass".
This requires clarification. The text is very slippery. What means in many areas of physics ?

The idea of relativistic mass actually dates back to Lorentz's work. His 1904 paper Electromagnetic Phenomena in a System Moving With Any Velocity Less Than That of Light introduced the "longitudinal" and "transverse" electromagnetic masses of the electron. With these he could write the equations of motion for an electron in an electromagnetic field in the newtonian form, provided the electron's mass increased with its speed. Between 1905 and 1909, the relativistic theory of force, momentum, and energy was developed by Planck, Lewis, and Tolman. A single mass dependence could be used for any acceleration—thus enabling mass to be now defined independently of direction—if F = d(mv)/dt (where m is relativistic mass) were to replace F = ma. It seems to have been Lewis who introduced the appropriate speed dependence of mass in 1908, but the term "relativistic mass" appeared later. (Gilbert Lewis was a chemist whose other claim to fame in physics was naming the photon in 1926.)
This whole section IMO describes Newton Law.

Relativistic mass came into common usage in the relativity text books of the early 1920s written by Pauli, Eddington, and Born.
That is okay, but how ? All in the same matter ?

The quantities that a moving observer measures as scaled by γ in special relativity are not confined to mass. Two others commonly encountered in the subject are a body's length in the direction of motion and its ageing rate, both of which get reduced by a factor of γ when measured by a passing observer. So, a ruler has a rest length, being the length it was given on the production line, and a relativistic or contracted length in the direction of its motion, which is the length we measure it to have as it moves past us. Likewise, a stationary clock ages normally, but when it moves it ages slowly by the gamma factor (so that its "factory tick rate" is reduced by γ). Lastly, an object has a rest mass, being the mass it "came off the production line with", and a relativistic mass, being defined as above. When at rest, the object's rest mass equals its relativistic mass. When it moves, its acceleration is determined by both its relativistic mass (or its rest mass, of course) and its velocity.

The use of these γ-scaled quantities is governed only by the extent to which they are useful.
Laws are not useful. Laws are the tools to describe and understand the physical reality. The same for the γ-scaled quantities. At least that is the way it should be.

While contracted length and time intervals are used—or not—insofar as they simplify special relativity analyses,
This sentence requires cleaning up. When moving clocks are used in experiments the fact that their rate decreases has to be taken into account. This is a must. Length contraction is the same. To call this field of study "special relativity" is of no importance.

relativistic mass has found itself at the centre of much debate in recent years about whether it is necessary in a physics curriculum. All physicists use rest mass, but not all physicists would have relativistic mass appear in textbooks, preferring instead always to write it in terms of rest mass when it is used. So, if all physicists agree that rest mass is a very fundamental concept, then why use relativistic mass at all?
This paragraph only makes sense if you write down in detail how different groups of physicists describe the same experiment.

When particles are moving, relativistic mass provides a very economical description that absorbs the particles' motion naturally. For example, suppose we put an object on a set of scales that are capable of measuring incredibly small increases in weight. Now heat the object. As its temperature rises causing its constituents' thermal motion to increase, the reading on the scales will increase. If we prefer to maintain the usual idea that mass is proportional to weight—assuming we don't step into an elevator or change planets midway through the experiment—then it follows that the object's mass has increased. If we define mass in such a way that the object's mass does not increase as it heats up, then we will have to give up the idea that mass is proportional to weight.
This whole section is not clear. What has temperature to do with this?

Another many-particle example occurs in pre-relativistic physics, in which the centre of mass of an object is calculated by "weighting" the position vector ri of each of its particles by their mass mi:

i miri
Centre of mass = ————————
                   ∑i mi
The same expression will hold relativistically if each of the above masses is now a particle's relativistic mass. If we prefer to use only rest mass then we must replace the mi in the above expression by γi mi where mi is rest mass, but now the expression has lost a certain economy. Similarly, if two objects with relativistic masses m1 and m2 collide and stick together in such a way that the resulting object is at rest, then its (rest = relativistic) mass will be m1 + m2. This accords with our intuition, and intuition is mostly what good conventions are about. In contrast, a rest-mass-only analysis describes the interaction by saying that the objects have (rest) masses of M1 and M1, with a combined (rest) mass of γ1M1 + γ2M2. Whether our intuition has anything to gain from this new expression is not clear.
Intuition has nothing to do with physical laws, in casu the factor γ.

Another place where the idea of relativistic mass surfaces is when describing the cyclotron, a device that accelerates charged particles in circles within a constant magnetic field. The cyclotron works by applying a varying electric field to the particles, and the frequency of this variation must be tuned to the natural orbital frequency that the particles acquire as they move in the magnetic field. But in practice we find that as the particles accelerate, they begin to get out of step with the applied electric field and can no longer be accelerated further. This can be described as a consequence of their masses increasing, which changes their orbital frequency in the magnetic field.

Lastly, the energy E of an object, whether moving or at rest, is given by Einstein's famous relation E = mc2, where m is its relativistic mass. Because, for example, the photon has no rest mass but does have relativistic mass, the use of relativistic mass makes it much easier to describe the mass changes that happen when light interacts with matter. See the FAQ article What is the mass of a photon?

While relativistic mass is useful in the context of special relativity, it is rest mass that appears most often in the modern language of relativity, which centres on "invariant quantities" to build a geometrical description of relativity. Geometrical objects are useful for unifying scenarios that can be described in different coordinate systems. Because there are multiple ways of describing scenarios in relativity depending on which frame we are in, it is useful to focus on whatever invariances we can find. This is, for example, one reason why vectors (i.e. arrows) are so useful in maths and physics; everyone can use the same arrow to express e.g. a velocity, even though they might each quantify the arrow using different components because each observer is using different coordinates. So the reason rest mass, rest length, and proper time find their way into the tensor language of relativity is that all observers agree on their values. (These invariants then join with other quantities in relativity: thus, for example, the four-force acting on a body equals its rest mass times its four-acceleration.) This is one reason why some physicists prefer to say that rest mass is the only way in which mass should be understood.

While the use of relativistic mass is purely a matter of taste, it appears that at least some physicists who oppose the use of relativistic mass believe, mistakenly, that all physicists who use relativistic mass are against the idea of rest mass. It's not clear just why there should be this perennial confusion about preferences.

A debate of the subject surfaced in Physics Today in 1989 when Lev Okun wrote an article urging that relativistic mass should no longer be taught (42, June 1989, pg 31).

Wolfgang Rindler responded with a letter to the editors defending its continued use (43, May 1990, pgs 13 and 115). In 1991 Tom Sandin wrote an article in the American Journal of Physics that argued in favour of relativistic mass (59, November 1991, pg 1032). (Links are provided here, but the articles cannot be downloaded for free.)

A commonly heard argument against the use of relativistic mass runs as follows: "The equation E = mc2 says that a body's relativistic mass is proportional to its total energy, so why should we use two terms for what is essentially the same quantity? We should just stay with energy, and use the word 'mass' to refer only to rest mass." The first difficulty with this line of reasoning is that it is quite selective; after all, it should surely rule out the use of rest mass as well, since that's proportional to a body's rest energy. The second difficulty of the line of reasoning is that, in the interests of consistency, it should surely be applied to rule out either the "momentum density" or the "energy flux density" of light, since these also are simply related by a factor of c2. Yet, and quite rightly, these last two terms co-exist in modern literature; no one ever suggests that either of these terms should be dropped in favour of the other, because they both have their uses and are fundamentally different quantities: a spatial density and an areal density.

So likewise do the concepts of mass and energy have their uses. The above argument that E = mc2 demotes mass in favour of energy—or rather, that it selectively demotes relativistic mass, but not rest mass—also neglects the very definitions of mass and energy. Mass is a property of a body that we have an intuitive feel for; its definition as a resistance to acceleration is very fundamental. Energy, on the other hand, is defined in physics in a technical way that involves the concept of a system's time evolution; this is not something that bears any obvious similarity to the concept of an object's resistance to being accelerated. If the concept of mass exists in some sense "prior" to that of energy, and if energy itself is defined in a different way to mass, then it does not seem reasonable to drop the idea of mass in favour of energy. Rather, E = mc2 becomes an expression that tells us how much energy a given mass has; it also tells us how much a body will resist being accelerated depending on its energy content. And, perhaps best of all, it reminds us that Einstein's equation is a triumph of relating two disparate quantities—and this is one of the great aims of physics.

Another argument sometimes put forward for dropping the use of relativistic mass is that since e.g. all electrons have the same rest mass (whereas their relativistic masses depend on their speeds), then their rest mass is the only quantity able to be tabulated, and so we should discard the very idea of relativistic mass. However, when we say without qualification that "the height of the Eiffel Tower is 324 metres", we clearly mean its rest length; but that doesn't mean the idea of contracted length should be discarded. Similarly, it's okay to say that the mass of an electron is about 10–30 kg without having to specify that we are referring to the rest mass; everyone knows we mean rest mass when we tabulate a particle's mass. That's purely a useful linguistic convention, and it does not imply that we have discarded the idea of relativistic mass, or that it should be discarded at all.

An optimistic view would hold that it's a measure of the richness of physics that focussing on different aspects of concepts like mass produces different insights: intuition in the case of relativistic mass in special relativity, and the notion of invariance and geometrical quantities in the case of tensor language in special and general relativity. The two aspects do not contradict each other, and there is room enough in the world of physics to accommodate them both.
Why this sentence ?

Abandoning the use of relativistic mass is sometimes validated by quoting select physicists who are or were against the term, or by exhaustively tabulating which textbooks use the term. But real science isn't done this way. In the final analysis, the history of relativity, with its quotations from those in favour of relativistic mass and those against, has no real bearing on whether the idea itself has value. The question to ask is not whether relativistic mass is fashionable or not, or who likes the idea and who doesn't; rather, as in any area of physics notation and language, we should always ask "Is it useful?" And relativistic mass is certainly a useful concept.

What is the relativistic version of F = ma ?




Created: 21 september 2008

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