Original by Philip Gibbs 12-Apr-1997

Does mass change with velocity?

There is often 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 = p/v
    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,


   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. 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.

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 when the theory of relativity originated. 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. However examination of Einstein's papers and books on relativity show that he did not use relativistic mass himself. Whenever the symbol m for mass appears in his equations it is always invariant mass. He never said 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.

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. His transverse mass is what we now call relativistic mass and the same quantity could also have been used for longitudinal acceleration if he had used F = d/dt(mv).

Some early texts on special relativity such as the ones by Born and Pauli used relativistic mass as well as rest mass. 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 became 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 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. 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. This usage 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 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 itself.";

Despite the general acceptance of this usage 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 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_rv

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 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 is 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 seen 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 is then 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. This works too but not everyone finds it such an easy route.

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. Experience of teaching relativity suggests that it is best dropped to avoid confusion at more advanced levels or avoided from the beginning but it can still be found in popular science books where the author may be less concerned about his reader's later development. 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 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.

references:
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).
Einstein's original papers can be found in English translation in "The Principle of Relativity" by Einstein and others, Dover Press