Updated by Don Koks, 2012.

Original by Philip Gibbs and Jim Carr, late 1990s.

The concept of mass has always been fundamental to 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, for whom mass was that property of a body that enables it to
resist externally imposed changes to its motion. Newton used mass to define momentum and force vectors: he
defined a body's momentum as * p = mv* (where

This definition of mass was applied in a straightforward way for almost two centuries. Then Einstein
arrived on the scene and, in his theory of motion known as *special relativity*, the situation
became more complicated. The above definition of mass still holds for a body at rest, and so has come to
be called the body's *rest mass*, denoted *m _{0}* if we wish to stress that we're
dealing with rest mass. But when the body is moving we find that its force–acceleration
relationship now depends on two quantities: the body's speed, and the angle between its direction of motion
and the applied force. When we relate the force to the resulting acceleration along each of three
mutually perpendicular spatial axes, we find that in each of the three expressions a factor of
γ

The idea of a speed-dependent 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 was allowed to increase with its speed. Between 1905 and 1909, the
relativistic theory of force, momentum, and energy was developed by Planck, Lewis, and Tolman. It turned
out that a single mass dependence could be used for any acceleration, thus enabling mass to retain its
independence of the body's direction of acceleration, if a speed-dependent "relativistic mass" *m* was
understood as present in Newton's original expression * p = mv*.

So a body moving with speed *v* and whose momentum has magnitude *p* has a relativistic mass
given by *m = p/v*, and (it turns out) a total energy of *mc ^{2}*. A body with
rest mass

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 texts of the early 1920s written by Pauli, Eddington, and Born. But whereas rest mass is routinely used in many areas of physics, relativistic mass is mostly restricted to the dynamics of special relativity. Because of this, a body's rest mass tends to be called simply its "mass".

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. While contracted length and time intervals are used—or not—insofar as 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 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 (although this can't be done for photons). So, if all physicists agree that rest mass is a very fundamental concept, then why use relativistic mass at all?

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 onto an elevator or change our home planet 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.

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 * r_{i}* of each of its particles by
their mass

∑The same expression will hold relativistically_{i}m_{i}r_{i}Centre of mass = ———————— ∑_{i}m_{i}

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 = mc ^{2}*, where

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*.) Some physicists cite this view to maintain that rest mass is the only
way in which mass should be understood.

As with many things, the use of relativistic mass can be a matter of taste, but it seems that at least some
physicists who vehemently oppose the use of relativistic mass believe, mistakenly, that pro-relativistic mass
physicists are against the idea of *rest* mass. It's not clear just why there should be this
perennial confusion about preferences, and why some of those who dislike the idea of relativistic mass show
such fundamentalist opposition to a choice of formalism that can never produce wrong results. The world
of physics and its language is full of useful alternative notations and ways of approaching things, and
different choices of notation and language can shed light on the physics involved. Selecting one of the
other of relativistic versus rest mass will never lead to problems for practitioners of the subject.

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 [1]. Wolfgang Rindler responded
with a letter to the editor defending its continued use
[2]. In 1991 Tom Sandin wrote an article in the *American Journal of Physics*
that argued in favour of relativistic mass [3].

A commonly heard argument against the use of relativistic mass runs as follows: "The equation *E =
mc ^{2}* 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

So likewise can the concepts of mass and energy coexist. The above argument that *E =
mc ^{2}* demotes mass in favour of energy—or rather, that it selectively
demotes

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. But 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.

Everyone agrees that a moving train's rest mass is a fixed property inherent to it, just as its rest length
is a fixed property inherent to it. And yet, strangely, many of the same physicists who insist that a
moving train's mass does not scale by γ are quite happy to say that its length *does* scale by
γ. There is no argument in the literature about the uses of rest length versus moving length, so
why should there be any argument about the uses of rest mass versus moving mass?

Another mass concept that everyone agrees on is the idea of *reduced mass* in non-relativistic
mechanics. When the mechanics of e.g. a sun–satellite system or a mass oscillating on a spring is
analysed, a mass term appears that combines the two masses in a particular way. As far as the maths
goes, it's *as if* we are replacing the two original bodies by two new ones: the first new body has
*infinite* mass, and the second new body has a mass equal to the system's reduced mass, which has this
name because it's smaller than either of the two original masses that gave rise to it. This is a
fruitful way to view the original system, and it's completely standard. No one gets confused into
thinking that we actually have an infinite mass and a reduced mass in our system. No one worries that
the new, infinite, mass is somehow going to become a black hole, or that the reduced mass lost some of its
atoms somewhere. Everyone knows the realm of applicability of the concept of reduced mass and how useful
it is. Why then, do so many physicists criticise relativistic mass by squeezing it into realms where it
was never intended to be used? They presumably don't do the same thing with reduced mass.

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 also-intuitive notion of invariance and geometrical quantities in the case of rest mass within the tensor language of 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. There can be little
doubt that *some* of its vocal opponents even use it to gain intuition when analysing a scenario in
special relativity.

The concept of relativistic mass is neatly encapsulated in the expression * F
= *d

Besides this definition and use of relativistic mass, we wish here to write down the relativistic version
of Newton's second law, * F = ma*. In Newton's mechanics, this equation relates
vectors

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

So defining mass via force and acceleration isn't as simple as it was for Newton (although itF= (1+ γ^{2}vv^{t}) γ maanda= (1–vv^{t})F—————————— γ m

[1] *The Concept
of Mass*, Physics Today, **42** June 1989, pg 31

[2]
*Putting to Rest Mass Misconceptions*, Physics Today **43**, May
1990, pgs 13 and 115

[3]
*In Defense of Relativistic Mass*, Am. J. Phys. **59**, November 1991, pg 1032

Some historical details can be found in *Concepts of Mass* by Max Jammer and
*Einstein's Revolution* by Elie Zahar.