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
velocity 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 resistance to acceleration—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 = γm _{0}* was understood as present in Newton's original expression

So, a body with rest mass *m _{0}* that is moving with speed

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 in our frame, it ages slowly by the gamma factor: we measure its tick rate to be the tick rate of
our own clocks divided 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'll have to give up the idea that mass is proportional to weight. But if we watch the mass pressing down on the weighing scales while we maintain that its mass is not changing, we can well be accused of ignoring what our eyes and measuring instruments are telling us.

Consider a many-particle example of 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
or the other of relativistic versus rest mass will never lead to problems for practitioners of the subject. In
calculations, because relativistic mass *does* factor neatly and trivially into a constant *m _{0}* and
the gamma factor (which depends only on

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, the concepts of mass and energy can 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 "factory-built" property, just as its rest length is a fixed
"factory-built" property. 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?

A mass concept that no one feels it necessary to argue about 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, useful 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 quietly in their own minds, 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

andF= (1+ γ^{2}vv^{t}) γ m_{0}a,

Defining mass via force and acceleration clearly isn't as straightfoward as it was for Newton (although ita= (1–vv^{t})F—————————— . γ m_{0}

[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.