Does mass change with velocity?
There is sometimes confusion surrounding the subject
of mass in relativity. This is because there are
two separate uses of the term. Sometimes people say "mass"
when they mean "relativistic mass", mr
but at other times they say "mass" when they mean "invariant
mass", m0. These two meanings are not
the same. The invariant mass of a particle is independent of its
velocity v, whereas relativistic mass increases with
velocity and tends to infinity as the velocity approaches the speed
of light c. They can be defined as follows,
mr = E/c2
m0 = sqrt(E2/c4 - p2/c2)
Where E is energy, p is momentum and c
is the speed of light in vacuum. The velocity dependent relation between
the two is,
mr = m0 /sqrt(1 - v2/c2)
Of the two, the definition of invariant mass is much
preferred over the definition of relativistic mass. These days when
physicists talk about mass in their research they always
mean invariant mass. The symbol m for invariant mass is
used without the suffix 0. Although relativistic mass is not wrong it
often leads to confusion and is less useful in advanced applications
such as quantum field theory and general relativity. Using
the word "mass" unqualified to mean relativistic
mass is wrong because the word on its own will
usually be taken to mean invariant mass. For example, when
physicists quote a value for "the mass of the electron"
they mean invariant mass.
At zero velocity the relativistic mass is equal to the invariant mass.
The invariant mass is therefore often called the "rest mass".
This latter terminology reflects the fact that historically it was
relativistic mass which was often regarded as the correct concept of mass
in the early years of relativity. In 1905 Einstein wrote a paper
entitled "Does the inertia of a body depend upon its
energy content?", to which his answer was "yes".
The first record of the relationship of mass and energy explicitly in the form
E = mc2 was written by Einstein in a review of relativity
in 1907. If this formula is taken to include kinetic energy then
it is only valid for relativistic mass, but it can also be taken as
valid in the rest frame for invariant mass. Einstein's conventions and
interpretations were sometimes ambivalent and varied a little over the years,
however examination of Einstein's papers and books on relativity
show that he almost never used relativistic mass himself. Whenever the
symbol m for mass appears in his equations it is always
invariant mass. He did not introduce the notion that the mass of a body
increases with velocity, just that it increases with energy content. The
equation E = mc2 was only meant to be applied in the
rest frame of the particle. Perhaps Einstein's only definite reference
to mass increasing with kinetic energy is in his "autobiographical
To find the real origin of the concept of relativistic mass
you have to look back to the earlier papers of Lorentz.
In 1904 Lorentz wrote a paper "Electromagnetic
Phenomena in a System Moving With Any Velocity Less Than That
of Light." There he 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
F = ma where m increases with mass. 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/dt(mv)
is used instead of F = ma. This introduced the concept of
relativistic mass which can be used in the equation
E = mc2 even for moving objects. It seems
to have been Lewis who introduced the appropriate velocity
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 became common usage in the relativity
text books of the early 1920's written by Pauli, Eddington and Born.
As particle physics became more important to physicists in the
1950's the invariant mass of particles became
more significant and inevitably people started to use the term
"mass" to mean invariant mass. Gradually this took over as the
normal convention and the concept of relativistic mass increasing
with velocity was played down.
The case of photons and other particles which move at the speed of
light is special. From the formula relating relativistic mass
to invariant mass, it follows that the invariant mass of a photon
must be zero but the relativistic mass need not be. The phrase
"The rest mass of a photon is zero" sounds nonsensical because
the photon can never be at rest but this is just a misfortunate
accident of terminology. In modern physics texts the term
mass when unqualified means invariant mass and photons are said to
be "massless" (see Physics FAQ What is the mass of the photon?). Teaching experience
shows that this avoids most sources of confusion.
Despite the general usage of an invariant mass in the
scientific literature, the
use of the word mass to mean relativistic mass is still
found in many popular science books. For example, Stephen
Hawking in "A Brief History of Time"
writes "Because of the equivalence of energy and mass,
the energy which an object has due to its motion will
add to its mass." and Richard Feynman in "The
Character of Physical Law" wrote "the energy
associated with motion appears as an extra mass, so things
get heavier when they move." Evidently, Hawking and Feynman
and many others use this terminology because it is intuitive and is
useful when you want to explain things without using too much
mathematics. The standard convention followed by some physicists
seems to be: use invariant mass when doing research and
writing papers for other physicists but use relativistic mass
when writing for non-physicists. It is a curious dichotomy
of terminology which inevitably leads to confusion. A common
example is the mistaken belief
that a fast moving particle must form a black hole because of its
increase in mass ( see relativity FAQ article
If you go too fast do you become a
black hole? )
Looking more deeply into what is going on we find
that there are two equivalent ways of formulating special
relativity. Einstein's original mechanical formalism is described in
terms of inertial reference frames, velocities, forces, length
contraction and time dilation. Relativistic mass fits naturally into
this mechanical framework but it is not essential. If relativistic mass
is used it is easier to form a correspondence
with Newtonian mechanics since some Newtonian equations
F = dp/dt
p = mrv
Also, in this picture mass is conserved along with energy.
The second formulation is the more mathematical one introduced a year
later by Minkowski. It is described in terms of space-time, energy-momentum
four vectors, world lines, light cones, proper time and invariant
mass. This version is harder to relate to ordinary intuition because
force and velocity are less useful in their 4-vector forms. On the
other hand, it is much easier to generalise this formalism to
the curved space-time of general relativity where global inertial
frames do not usually exist.
It may seem that Einstein's original mechanical formalism
should be easier to learn because it retains many equations
from the familiar Newtonian mechanics. In Minkowski's
geometric formalism simple concepts such as velocity and force
are replaced with worldlines and four-vectors. Yet the
mechanical formalism often proves harder to swallow and is at the root
of many peoples failure to get over the paradoxes which are
so often discussed. Once students have been taught about
Minkowski space they invariably see things more clearly.
The paradoxes are revealed for what they are and calculations
also become simpler. It is debatable whether or not the relativistic
mechanical formalism should be avoided altogether. It can still
provide the correspondence between the new physics and the
old which is important to grasp at the early stages.
The step from the mechanical formalism to the geometric can
then be easier. The alternative modern teaching method is to
translate Newtonian mechanics into a geometric formalism using
Galilean relativity in 4 dimensional space-time then modify the
geometric picture to Minkowski space.
The preference for invariant mass
is stressed and justified in the classic relativity textbook
"Spacetime Physics" by Taylor and Wheeler who
"Ouch! The concept of 'relativistic mass' is subject to
misunderstanding. That's why we don't use it. First, it applies
the name mass - belonging to the magnitude of a 4-vector - to
a very different concept, the time component of a 4-vector. Second,
it makes increase of energy of an object with velocity or momentum
appear to be connected with some change in internal structure of the
object. In reality, the increase of energy with velocity originates
not in the object but in the geometric properties of space-time
In the final analysis the issue is a debate over whether
or not relativistic mass should be used is a matter of
semantics and teaching methods. The concept of relativistic
mass is not wrong. It could have its uses in special relativity
at an elementary level. This debate surfaced in "Physics
Today" in 1989 when Lev Okun wrote an article urging
that relativistic mass should no longer be taught
(42, #6 June, 1989 p. 31). Wolfgang Rindler
responded with a letter to the editors to defend its
continued use. (43, #5 May, 1990 p. 13).
The experience of answering confused questions on usenet suggest that
its use in popular books and elementary texts is not helpful.
The fact that relativistic mass is virtually never used in contemporary
scientific research literature is a strong argument against teaching it to
students who will go on to more advanced levels.
Invariant mass proves to be more fundamental in Minkowski's
geometric approach to special relativity and relativistic mass
is of no use at all in general relativity. It is possible
to avoid relativistic mass from the outset by talking of
energy instead. Judging by usage in modern text books the
consensus is that relativistic mass is an outdated concept
which is best avoided. There are people who still want to use
relativistic mass and it is not easy to settle an argument
over semantic issues because there is no absolute right or wrong,
just conventions of terminology. It is hard to impose
conventions on usenet and there will always be people who
post questions using terms in which mass increases
with velocity. It is unhelpful to just tell
them that what they read or heard on cable TV is wrong but it
will reduce confusion for them in the longer term if people
can be persuaded to think in terms of invariant mass instead
of relativistic mass.
In a 1948 letter to Lincoln Barnett Einstein wrote
"It is not good to introduce the concept of the mass
M = m/(1-v2/c2)1/2
of a body for which no clear definition
can be given. It is better to introduce no other mass than
'the rest mass' m. Instead of introducing M,
it is better to mention the expression for the momentum and
energy of a body in motion."
The viewpoint above, emphasising the distinction between mass,
momentum, and energy, is certainly the "modern" view.
Fifty years later, can relativistic mass be laid to rest?
Arguments against the term
"relativistic mass" are given in the
classic relativity text book "Space-Time Physics"
by Taylor and Wheeler, 2nd edition, Freeman Press (1992).
The article "Does mass really depend on velocity,
dad?" by Carl E Adler, American Journal of Physics 55, 739 (1987)
also discusses this subject and includes the above quote from
Einstein against the use of relativistic mass
Einstein's original papers can be found
in English translation in "The Principle
of Relativity" by Einstein and others, Dover Press
Some other historical details can be found in
"Concepts of mass" by Max jammer
and "Einstein's Revolution" by Elie Zahar.