Addendum added by DK 2002.
Updated by Jim Carr 1998.
Original by Philip Gibbs 1997.

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", m_r 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:

    m_r = E/c²
    m₀ = sqrt(E²/c⁴ - p²/c²)

where E is energy, p is momentum and c is the speed of light in a vacuum. The velocity dependent relation between the two is

   m_r = 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 subscript 0. Although the idea of 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 its invariant mass.

At zero speed, 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 an examination of his papers and books on relativity shows 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 notes".

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 came into common usage in the relativity text books of the early 1920s written by Pauli, Eddington and Born. As particle physics became more important to physicists in the 1950s, 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 that 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 its relativistic mass need not be. The phrase "The rest mass of a photon is zero" might sound nonsensical because the photon can never be at rest; but this is just a side effect of the terminology, since by making this statement, we can bring photons into the same mathematical formalism as the everyday particles that do have rest mass. In modern physics texts, the term mass when unqualified means invariant mass in most cases, 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 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 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 remain valid:

           F = dp / dt

           p = m_r v

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 spacetime, 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 four-vector forms. On the other hand, it is much easier to generalise this formalism to the curved spacetime 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 world lines and four vectors. Yet the mechanical formalism often proves harder to swallow, and is at the root of many people's failure to get over the paradoxes that 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. But 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. An alternative modern teaching method is to translate Newtonian mechanics into a geometric formalism, using Galileian relativity in four dimensional spacetime, and 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 write

"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 four-vector--to a very different concept, the time component of a four-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 itself."

In the final analysis the issue is a debate over whether or not relativistic mass should be used, and is a matter of semantics and teaching methods. The concept of relativistic mass is not wrong: it can 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, pg 31). Wolfgang Rindler responded with a letter to the editors to defend its continued use. (43 #5, May 1990, pg 13).

The experience of answering confused questions in the news groups suggests that the use of relativistic mass 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. There will always be those 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 might reduce confusion for them in the longer term if they 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?

Addendum: What is the relativistic version of F = ma ?

For 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 (hence the bold script) 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! To express this fact, we need to use matrix notation. Letting m be the invariant mass, v be the velocity as a column vector, and 1 be the 3 x 3 identity matrix, the actual result turns out to be

    F = gamma m (1 + gamma² v v^t) a

    and

    a = (1 - v v^t) F / (gamma m)

(As an aside, there's a nice correspondence to the one dimensional case here. Just as gamma² can be written as 1 + gamma² v², and its inverse (i.e. reciprocal) is 1 - v², so the matrix 1 + gamma² v v^t has inverse 1 - v v^t, as well as determinant gamma².)

Looking at this relativistic version of F = ma, we might say that when the (invariant) mass m appears, it's accompanied by a factor of gamma, so that really it is the relativistic mass that's appearing. Isn't this then, a good reason why we might want to give the notion of relativistic mass more credence? Not really. Notice that now the acceleration is not necessarily parallel to the force that produced it. 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. So now, if we still want to maintain some meaning for relativistic mass, then we'll have to realise that it has a directional dependence--as if the object somehow has more mass in the direction of its motion, than it has sideways. Evidently the idea of relativistic mass is becoming a little more complicated than at first we might have hoped! And this is another reason why, in the end, it's so much easier to just take the mass to be the invariant quantity m, and to put any directional information into a separate, matrix, factor.

References:
Arguments against the term "relativistic mass" are given in the classic relativity text book Space-Time Physics by Taylor and Wheeler, 2^nd edition, Freeman Press (1992).

The article Does mass really depend on velocity, dad? by Carl 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.