• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

Updated by Don Koks, 2008
Original by Philip Gibbs and Jim Carr, late 1990s.

### What is relativistic mass?

The origin of mass--where an object gets its mass from, and precisely what mass is--is not understood by physicists. Because of this, we might not be surprised to find that some physicists use the word "mass" differently to others. Whatever mass might actually be, if we quantify it as a measure of how much an object will be accelerated by a force (as Newton did), then yes, mass does increase with speed, because the faster an object moves, the more resistant it becomes to being accelerated. But some physicists insist on imposing an extra requirement on the definition of mass: that it not change with speed. In that case, not surprisingly, their mass does not change with speed. The question is: which concept is more useful?

Relativity shows us that when an object moves with speed v, three of its properties are affected through the famous "gamma factor": γ = (1-v2/c2)-1/2. These three properties are: its length in the direction of motion, which gets reduced by γ, its ageing rate, which also gets reduced by γ, and the extent to which a force can accelerate it; this contains a factor of γ, although the acceleration does have a directional dependence. So, a ruler has a rest length, being the length it was given on the production line, and a relativistic or contracted length, which is the length we measure it to have as it moves past us. The ratio of the two is the gamma factor. Likewise, a stationary clock ages normally, but when it moves, it ages slowly by the gamma factor. Lastly, an object has a rest mass, being the mass it "came off the production line with", and a relativistic mass, which we can define through studying the object's response to a force. When at rest, the object's rest mass equals its relativistic mass. When the object moves, it will accelerate in response to a force in inverse proportion to its relativistic mass, as discussed below.

Physicists who are "anti" relativistic mass do not disagree with the subject of relativity itself. Rather, they might argue 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 there is nothing deep about this. When we say 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; but that does not imply that we mean to discard the idea of relativistic mass.

When particles are moving, relativistic mass is useful and intuitive in ways that rest mass is not. For example, in pre-relativistic physics, the centre of mass of an object is calculated by "weighting" the position vector ri of each of its particles by their mass mi:

```                  ∑i miri
Centre of mass = ————————
∑i mi
```

The same expression will hold relativistically if each of the above masses is now a particle's relativistic mass. Those who refuse to use relativistic mass must replace the mi in the above expression by γ mi where mi is rest mass; but surely the idea of relativistic mass is staring us right in the face in such an expression. Similarly, if two objects with relativistic masses m1 and m2 collide and stick together in such a way that the resulting object is at rest, then its mass will be m1+ m2, which accords with our intuition, and intuition is mostly what good conventions are about.

In stark contrast, those who refuse to use relativistic mass are forced to describe this interaction by saying that the objects have (rest) masses of M1 and M2, then their combined (rest) mass will be γ1M1+ γ2M2. This is rather artificial, and there is no intuition to be gained from such a statement.

Another place where the idea of relativistic mass surfaces is when describing the cyclotron, an instrument 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 = mc2, where m is the relativistic mass. "Whether moving or at rest". If that is the case what does the equation E=mc2 realy mean? What situation does it describe ?

And physicists who refuse to use relativistic mass need to resort to somewhat artificial lengths to describe the mass changes that happen when light interacts with matter. This is because the photon has no rest mass, but it does have relativistic mass. See the FAQ article What is the mass of a photon?

The idea of relativistic 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 F = ma, where m increases with the electron's speed. 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(mv)/dt is used (where m is relativistic mass) instead of F = ma. 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 text books of the early 1920s written by Pauli, Eddington and Born.

Despite the usefulness of relativistic mass, it is rest mass that appears most often in the modern language of relativity, because this language is centred on "invariant quantities" such as scalars and vectors, since these 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 the whole point of using vectors (i.e. arrows) in maths and physics; everyone can use the same arrow to express e.g. a velocity, even though they might each give the arrow different components because each is using different coordinates. So rest mass, rest length, and proper time find their way into the tensor language of relativity because all observers agree on them, and this is probably the fundamental reason why some physicists prefer to say that there is only one kind of mass, the rest mass. But none of this requires us to discard the ideas of relativistic mass, contracted length, and contracted time.

A common argument against the use of relativistic mass is the fact that the equation E=mc2 says that a body's relativistic mass equals 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. But this argument neglects the definitions of the words mass and energy. Mass is a property of a body that we have an intuitive feel for; its definition as a resistance to acceleration is very fundamental. Energy, on the other hand, is defined in physics in rather ad hoc ways. Neither concept is even remotely understood by modern physics. (See The Feynman Lectures, Vol. 1 Section 4.1 for a discussion of the ad hoc definition of energy.) The surprising thing about Nature is that she conserves energy, but there is no reason why this should be so; nor is there any reason why we should be able to quantify anything, or even be able to speak of conservation principles at all. The very fact that we can describe Nature using mathematics is a deep and mysterious thing. If the concept of mass exists in some sense "prior" to that of energy, and if energy itself is defined in an ad hoc way while mass is not, then it does not seem reasonable to drop the idea of mass in favour of energy. Rather, E=mc2 becomes an expression that tells us how much energy a given mass has; it also tells us how much a body will resist being accelerated depending on its energy content.

In calculations that use relativistic mass, it often makes sense to write it as γ m where m is the object's rest mass, since that way the speed dependence is wholly contained in the γ. This makes relativistic mass easy to analyse. But although the rest mass has been singled out here by being given a symbol, that does not imply that rest mass is being preferred over relativistic mass; we need to be clear about the difference between the mathematical simplicity of the individual factors versus the physical simplicity of the whole. Because it's mathematically useful to single out the rest mass in this way, a separate symbol for relativistic mass might not appear in some calculation, even though only relativistic mass is being used. Opponents of relativistic mass might interpret this to mean that relativistic mass is not found in the literature and therefore has become unfashionable. But that's not valid; the absence of a symbol for relativistic mass in papers on the subject simply means that a useful way to write relativistic mass is by using an invariant (i.e. rest mass), along with a gamma to encapsulate the speed dependence. (It probably also means that there aren't many papers published in relativistic dynamics anymore.) This is all no different to writing a relativistic (i.e. contracted) length as L/γ, where L is a rest length; giving a symbol only to rest length does not imply that we have abandoned the idea of relativistic length.

A debate of the idea of relativistic mass surfaced in Physics Today in 1989 when Lev Okun wrote an article urging that relativistic mass should no longer be taught (42, June 1989, pg 31). Wolfgang Rindler responded with a letter to the editors defending its continued use (43, May 1990, pgs 13 and 115). In 1991 Tom Sandin wrote an article in the American Journal of Physics arguing very persuasively in favor of relativistic mass (59, November 1991, pg 1032). (Links are provided here, but the articles cannot be downloaded for free.)

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 for the case of relativistic mass in special relativity, and the notion of invariance for the case of tensor language in special and general relativity. But it's also unfortunate that whereas "pro relativistic mass" physicists will happily live with both ideas, "anti relativistic mass" physicists spend a lot of time trying to have relativistic mass outlawed.

Abandoning the use of relativistic mass is often validated by quoting select physicists who are or were against the term. But real science isn't done that 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 be asked 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 c

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Created: 21 september 2008