## Does Mass Increase With Speed Or Not?

It is almost hopeless to think that the confusions over the term “mass”, which have entered and become ingrained in popular culture, will ever be entirely straightened out; there are so many books and websites that insist on completely contradictory points of view, many of them containing wrong or inconsistent information. But among theoretical particle physicists, there is no debate about the question: **mass, as theoretical particle physicists define it, does not increase with speed.** That’s because, for us, mass (without a modifier) implicitly means “rest mass”, and we view the quantity called “relativistic mass” is conceptually inconsistent.

There are two simple ways to understand why we view relativistic mass so negatively. The first is that the best arguments in favor of “relativistic mass” are incomplete and inconsistent. The second is that Einstein’s formulation of space and time has an underlying mathematical structure, with respect to which rest mass is consistent, while relativistic mass is not. (A somewhat more technical discussion of these issues, written for *Physics Today* in 1989 by Professor Lev Okun, is reprinted here.)

At the end of this commentary, I’ll add just a few words about gravitational mass, which, although it does increase with speed too, is not the same as relativistic mass. Particle physicists never need it and never use it. My friends who are expert in Einstein’s theory of gravity, general relativity, tell me they rarely use it either, as it cannot be defined in a general way. But some physicists and astronomers do seem to like to focus on this quantity in those situations where it can be defined. Since I’m not fully satisfied with my understanding of who uses it, and why, across scientific fields, I will leave gravitational mass as a somewhat open issue, which I’ll try to return to at a later date.

### Why it’s No good Trying to View Einstein as Newton 2.0

The reason many people love relativistic mass is that **at first** it makes it seem as though Einstein’s advance can be viewed as a simple update of Newton. Simply by defining mass using Einstein’s equation

*M = E/c*^{2}

(where M is relativistic mass and E is total energy), it seems as first as though one can then** leave Newton’s other equations unchanged**. That’s because two other important Newtonian equations, involving force F, acceleration a, momentum ** p** and velocity

*v**F = M a**p = M v*

seem **at first** to survive to Einstein’s physics.

But this is wrong.

As Okun was fond of pointing out, this tendency to take these Newtonian equations from a first-semester physics course, and try to define relativistic mass ** M** so that they would remain true,

**is doomed unless the object’s motion, acceleration and force are all pointed along exactly the same direction.**In our three-dimensional world, where an object moving to the north might be pushed by a force to the west or to the northeast, there simply is no definition of relativistic mass that is consistent with the laws of Newtonian physics, or even internally consistent with itself.

Specifically, while the second of these equations (** p = M v**) remains true for Einstein,

**the first formula,**. It does not apply, for example, to the motion of an electron spiraling around in a magnetic field, or to the force which holds atoms together in a spinning molecule. Intransigence, in other words, is

*F=M a*is only true if the direction of the force is the same as the object’s direction of motion**always equal to total energy divided by c**

*not*^{2}; it is often equal to something else. This makes the reasoning in favor of relativistic mass inconsistent. Its touted properties simply doesn’t hold in all circumstances.

More generally, if you try to view Einstein as Newton 2.0, consisting of Newton plus the idea of relativistic mass, you will end up with all sorts of mistakes.

By contrast, rest mass is unambiguous, as a perspective-independent quantity. It never causes any such trouble.

(I myself was once witness to an event where Okun, planning to give an advanced seminar, first gave his audience a brief test on rest mass and relativistic mass. When the audience failed it, he dropped his planned topic and instead taught us a remedial lesson on the subject of mass.)

### The Old Symmetries of Newton

Here’s a more profound way, albeit a more mathematical one, of looking at the problem with relativistic mass.

In Newton’s way of looking at the world, every quantity used by physicists is either a **scalar** or a **vector**… (or, as learned later on, a simple “**tensor**.”) I’ll give you a few examples.

- The temperature outside is a
**scalar**: it is given by a single number: it is, say, 10 degrees. So is the air pressure: 980 millibarns. - But the wind is given by a
**vector**— an arrow — which has a direction as well as a strength, perhaps 20 miles per hour from the east. - The force holding the Earth around the Sun is a
**vector**: it is a pull of a certain strength toward the Sun. Indeed, all forces are vectors.

In the three spatial dimensions we live in, to specify a vector requires three numbers. A location, specified relative to a certain point, is given by a vector. To specify the location of the top of a mountain, relative to the Earth’s center, requires three numbers: its longitude, its latitude, and its altitude. For instance, this is true of Denali, the highest mountain in North America: latitude 63.0692° N, longitude 151.0070° W, and an altitude of 6190 meters (20,310 feet). When I lived in New York City a decade ago, I would travel to my favorite restaurant by going downstairs one flight, going half a city block west, and four city blocks south. Those three pieces of information would tell you how to find my restaurant starting from my apartment.

To keep my point simple, though, let’s just consider a vector in two spatial dimensions, such as the black arrow shown below. The figure shows four ways to specify the black arrow, using two coordinate systems, one associated with the intersecting red lines at right (the “red coordinate system”), one specified by the intersection blue lines (the “blue coordinate system”).

In the red coordinate system, we can specify the arrow in two ways:

- an arrow of length
that makes an arrow alpha with the horizontal red axis*L* - an arrow specified by first going a distance
along the horizontal red axis and then a distance*x*along the vertical red axis*y*

We call “** x**” and “

**” the coordinates or**

*y***components**of the black arrow in the red coordinate system.

Since these are two ways of describing the same arrow, they must some how be related; and indeed they are, by high school trigonometry

*x = L cos a**y = L sin a*

with x and y related by the Pythagorean theorem: ** x^{2} + y^{2} = L^{2}**.

In the blue coordinate system, it is much the same: we can specify the arrow as:

- an arrow of length
that makes an arrow alpha’ with the near-horizontal blue axis*L* - an arrow specified by first going a distance
along the near-horizontal blue axis and then a distance*x’*along the near-vertical blue axis*y’*

As before,

*x’ = L cos a’**y’ = L sin a’*

with x’ and y’ related by the Pythagorean theorem: ** x’^{2} + y’^{2} = L^{2}**.

Now here is the crucial point. The vector is a fixed object, but the way you describe it depends on your coordinate system. It makes a qualitative difference to the vector’s components. In the red coordinate system, the components ** x** and

**of the vector are equal, while in the blue coordinate system,**

*y***is smaller than**

*x’***; in fact,**

*y’***. That means that**

*x’ < x = y < x’***a single component of a vector is meaningless by itself**; it is insufficient, in any coordinate system, for a reconstruction of the entire vector. In any particular coordinate system, you need both components to reconstruct the vector, and thus obtain something that is meaningful outside of the specific coordinate system used.

Meanwhile the temperature or air pressure in the room are the same no matter whether you use the red coordinate system or the blue one. They simply don’t care. They are scalars, and don’t have components.

What I now want to convince you is that in Einstein’s theory of relativity, **rest mass acts like a scalar, while relativistic mass acts like one component of a vector.** This makes the latter meaningless by itself; it needs more components to become a sensible quantity.

### The New Symmetries of Einstein

For Newton, **the distance between two objects was a vector in three dimensions, and the time between two events was a scalar**. Everyone agrees on how long a movie is, right? Newton would have said “yes.”

But Einstein said, “no, that’s not actually true.” And this changed everything.

Einstein suggested that, instead, **the space-time distance between two events **(say, the start and end of a movie that you are watching on a moving train) **is** **a vector in four dimensions**. In other words, the space and time between two events are to be viewed as four numbers that act as a vector in four-dimensional space-time — usually called a ** four-vector** to distinguish it from the usual Newtonian

**. When you change your perspective, specifically by changing your speed and direction of motion, both the distance and the time between two events appears to change; they make up four-components of a four-vector, and so they depend on the coordinate system you use, which in turn depends on your speed as well as your orientation.**

*three-vector*From the point of view of someone on the ground, the length of a movie played in a passing airplane is slightly longer than the length of a movie played in a movie theater (and the reverse is true from the perspective of someone in the plane.) We cannot separate space and time, according to Einstein, any more than we could separate * x* components from

*components in our picture above; not can we separate longitude, latitude and altitude in Newton’s view of the world. All of them are needed.*

**y**For Newton, the energy of an object is a scalar, as is its mass. Meanwhile its momentum (which has a direction as well as an amount) is a vector.

But for Einstein, **energy and momentum together are a four-vector**! Only rest mass is a scalar.

### Relativistic Mass: A Mere Component

And relativistic mass? It is neither a scalar nor a vector. It is relative — it depends on your perspective — but unlike energy, which combines naturally with momentum to make a four-vector, there is no natural meaningful quantity with which relativistic mass combines to make a four-vector. Thus it simply falls outside the categories that all quantities ought to satisfy in a theory with Einstein’s relativity. This is why particle physicists don’t use it.

The reason there is so much confusion about mass and how to define it is that back in the late 19th and early 20th century, physicists, including Einstein, weren’t sure of the best ways to think about it. But by now, the dust has settled. It’s clear that one should use the symmetries of relativity to classify objects. In Newton’s equations it would not be a good idea to use quantities which are neither scalars, vectors or some generalization thereof. And similarly, it is not a good idea, in Einstein’s approach to the world, to use objects that are neither scalars, four-vectors, or some other consistent generalization thereof.

### Einstein’s Own Viewpoint

Einstein’s views on this subject were not steady in the early days. But certainly by 1948, a few years before his death, Einstein had clearly come to the same conclusion as described above. As he wrote in a letter to Lincoln Barnett, who was a big fan of relativistic mass,

- “It is not good to introduce the concept of the mass
of a moving body for which no clear definition can be given . It is better to introduce no other mass concept than the “rest mass”**M= m / (1 – v**^{2}/c^{2})^{1/2}. Instead of introducing*m*, it is better to mention the expression for the momentum and energy of a body in motion.”*M*

In other words, said Einstein, use the scalar ** m** , and use the four-vector that combines momentum and energy; but do not use relativistic mass, because it is neither fish nor fowl.

All of which is to say: you’re free to define quantities however you wish, but you should not fool yourself that defining relativistic mass actually simplifies anything. It neither gives you Newton 2.0 nor respects the essential structure of spacetime.

### Gravitational Mass

What about gravitational mass, which determines how gravity acts on objects, and is something Einstein also sometimes referred to? Like relativistic mass, gravitational mass is relative. Sometimes the two types of mass are the same, but in general, they are not. As Okun points out, a photon heading toward Earth has gravitational mass equal to ** E/c^{2}** , while a photon passing by the Earth has gravitational mass equal to

**. There are similar problems, less simply explained, for particles that have non-zero rest mass.**

*2 E /c*^{2}Again, you’re free to use whatever definitions you prefer, as long as you yourself avoid giving incorrect or inconsistent reasons for why you like them, are clear about how they work, and explain clearly to others what definitions you are using. But one thing’s for sure: blithe and imprecise statements such as “mass increases with speed” should be avoided, both because the word “mass” is ambiguous and because the statement is completely false for “rest mass”, a well-defined scalar quantity that plays a central role in Einstein’s relativity.