Updated by Don Koks, 2008.
Original by Philip Gibbs and Jim Carr, late 1990s.
The origin of mass—where an object gets its mass from, and precisely what mass is—is not understood by physicists. Its fundamental definition goes back to Newton, who essentially defined mass as that property of a body that governs what acceleration it will undergo when acted on by a force. This definition of mass could be applied in a relatively straightforward way for almost two centuries, until Einstein arrived on the scene. In Einstein's theory of motion known as special relativity, the situation is more complicated. Here we find that not only does a body's resistance to acceleration increase with its speed, but that resistance also has a directional dependence, as discussed in the addendum below. If we write down the body's resistance to acceleration along each of the three mutually perpendicular spatial axes, we see that the three expressions have a "greatest common factor" of γ m, where the gamma factor γ = (1–v2/c2)–1/2 is a common quantity in special relativity, and m is the body's mass that would have been measured by Newton if it had been at rest when the force was applied. In the terminology of the subject, this mass m is called the body's rest mass, while γ m is known as its relativistic mass.
On the one hand, the fact that these two terms γ and m appear together indicates that they might be fundamentally related. On the other hand, the fact that they are not mixed together in some more complicated and inextricable way suggests that they are really two quite different entities that are fundamentally unrelated. Until physicists succeed in finding an origin of mass, the extent to which these two terms should or need be treated as "belonging" together can only be a matter of taste, dependent on the application in which they are being used. Currently, rest mass is essential in many areas of physics, whereas 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".
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, Planck, Lewis and Tolman developed the relativistic theory of force, momentum and energy. A single mass dependence could be used for any acceleration 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.) Relativistic mass came into common usage in the relativity text books of the early 1920s written by Pauli, Eddington and Born.
The quantities that a moving observer measures as scaled by γ in special relativity are not restricted 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. 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 governed 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. While contracted length and time intervals are used—or not—to the extent that they simplify special relativity analyses, 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 make extensive use of 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. While this is purely a matter of taste and is not especially deep, it appears that at least some physicists who oppose the use of relativistic mass believe, mistakenly, that all pro relativistic mass physicists are against the idea of rest mass. Just why there should be this perennial confusion about beliefs is a mystery that this FAQ entry doesn't discuss. The question addressed here is: what are some uses of relativistic mass?
When particles are moving, relativistic mass can provide a very economical description that absorbs the particles' motion naturally. For example, in pre-relativistic physics, 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 γ
mi where mi is rest mass; the economy of
expression provided by relativistic mass is then lost. 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
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 M2, with a combined (rest) mass
of γ1M1+
γ2M2. The economy of expression has been lost,
and with it perhaps the hope of any gain in our intuition.
Another place where the idea of relativistic mass surfaces is when describing the cyclotron, an instrument 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 that rest mass, rest length, and proper time find their way into the tensor language of relativity is because all observers agree on their values. This is a fundamental reason for the importance of rest mass, and why some physicists prefer to say that rest mass is the only way in which mass should be understood.
A debate of the idea of relativistic mass 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 favor 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.
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 rather ad hoc ways, albeit ways that work beautifully. Neither concept is even remotely understood by modern physics. (See The Feynman Lectures, Vol. 1 Section 4.1 for a discussion of the definition of energy.) The surprising thing about Nature is that she conserves energy, but there is no obvious reason why this should be so; nor is there any obvious reason for why we should be able to quantify anything, or even be able to speak of conservation principles at all. (The very fact that we can describe Nature using mathematics is a deep and mysterious thing.) If the concept of mass exists in some sense "prior" to that of energy, and if energy itself is defined in an ad hoc way while mass is not, 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.
Another argument sometimes put forward to drop 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.
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.
In this last section we'll write down the relativistic version of Newton's second law, F = ma. In Newton's mechanics, this equation relates vectors F and a via the mass m of the object being accelerated, which is invariant in Newton's theory. Because m is just a number, in Newton's theory the force on a mass is always parallel to the resulting acceleration.
The corresponding equation in special relativity is a little more complicated. It turns out that the force F is not always parallel to the acceleration a. We can express this fact using matrix notation. Let m be the rest mass, and v be the velocity as a column vector, whose entries are expressed as fractions of c and whose magnitude v is the speed as a fraction of c. Let vt be the velocity as a row vector, and let 1 be the 3 × 3 identity matrix. Also, as usual, let γ = (1 – v2) –1/2. The relativistic version of F = ma turns out to be
F = (1 + γ2 v vt) γ m a
and
a = (1 – v vt) F
——————————
γ m
These matrix expressions show that a force is generally not parallel to the acceleration
it produces, so that defining mass in this way isn't as simple as it was for Newton.
Nevertheless, the three components of the first expression have a highest common factor of
γm, and the rest mass m only ever appears in both expressions
accompanied by γ. The acceleration is not necessarily parallel to the force
that produced it, and it's not hard to see from the above equations that it's easier to
accelerate a mass sideways to its motion than it is to accelerate it in the direction of
its motion. This is how relativity reproduces Lorentz's original concepts of
longitudinal and transverse masses; they are actually contained in these equations.
The directional dependence that the newtonian meaning of mass has now taken on is neatly
contained in the matrices 1 + γ2 v
vt and 1 – v vt, and we can
simply define the relativistic mass to be the remaining factor, γm. This
idea of taking our cue from the equations is a powerful tool in mathematical physics.