# Why is there a controversy on whether mass increases with speed?

Some people say that mass increases with speed, some people say that the mass of an object is independent of its speed.

I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be 'true' is any sense of the word.

Either mass increases or it doesn't, right?

Can't we just measure it, and find out which 'interpretation' is right? e.g.: (in some sophisticated way) heat up some particles in a box and measure their weight?

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UPDATE

Right, so I've got two identical containers, each with identical amounts of water, each on identical weighing scales, and each in the same g field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and g is constant, does this mean that the m in w=mg has got bigger?

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Comment to the question (v2): Note that there are various notions of mass, cf. e.g. physics.stackexchange.com/q/8610/2451 – Qmechanic Aug 31 '14 at 20:13
Lev Okun: "there is only one mass" (itep.ru/theor/persons/lab180/okun/em_3.pdf). – Johannes Aug 31 '14 at 21:06
There is no controversy whatsoever on physics, only on the best way to assign vocabulary. Parties on both side of the divide (such as it is) can agree on the results of any calculation. – dmckee Aug 31 '14 at 22:33
@RonanDejhero This is in the context of special relativity, which states that mass actually does (effectively) increase with speed. You're talking about an increase in momentum, which isn't the same thing. – Ajedi32 Sep 2 '14 at 13:06
@User17670 The Answer is yes, and it does means that the mass of the water got bigger, but it does not mean that the mass of the particles that make up the water got bigger. Internal energy is mass. Nor is this trivial most of the mass of atoms comes from the internal energy of the protons and neutrons not from the bare mass of their valence quarks. But this doesn't have much affect on the naming argument. – dmckee Sep 6 '14 at 15:36

There is no controversy or ambiguity. It is possible to define mass in two different ways, but: (1) the choice of definition doesn't change anything about predictions of the results of experiment, and (2) the definition has been standardized for about 50 years. All relativists today use invariant mass. If you encounter a treatment of relativity that discusses variation in mass with velocity, then it's not wrong in the sense of making wrong predictions, but it's 50 years out of date.

As an example, the momentum of a massive particle is given according to the invariant mass definition as

$$p=m\gamma v,$$

where $m$ is a fixed property of the particle not depending on velocity. In a book from the Roosevelt administration, you might find, for one-dimensional motion,

$$p=mv,$$

where $m=\gamma m_0$, and $m_0$ is the invariant quantity that we today refer to just as mass. Both equations give the same result for the momentum.

Although the definition of "mass" as invariant mass has been universal among professional relativists for many decades, the modern usage was very slow to filter its way into the survey textbooks used by high school and freshman physics courses. These books are written by people who aren't specialists in every field they write about, so often when the authors write about a topic outside their area of expertise, they parrot whatever treatment they learned when they were students. A survey[Oas 2005] finds that from about 1970 to 2005, most "introductory and modern physics textbooks" went from using relativistic mass to using invariant mass (fig. 2). Relativistic mass is still extremely common in popularizations, however (fig. 4). Some further discussion of the history is given in [Okun 1989].

Oas doesn't specifically address the question of whether relativistic mass is commonly used anymore by texts meant for an upper-division undergraduate course in special relativity. I got interested enough in this question to try to figure out the answer. Digging around on various universities' web sites, I found that quite a few schools are still using old books. MIT is still using French (1968), and some other schools are also still using 20th-century books like Rindler or Taylor and Wheeler. Some 21st-century books that people seem to be talking about are Helliwell, Woodhouse, Hartle, Steane, and Tsamparlis. Of these, Steane, Tsamparlis, and Helliwell come out strongly against relativistic mass. (Tsamparlis appropriates the term "relativistic mass" to mean the invariant mass, and advocates abandoning the "misleading" term "rest mass.") Woodhouse sits on the fence, using the terms "rest mass" and "inertial mass" for the invariant and frame-dependent quantities, but never defining "mass." I haven't found out yet what Hartle does. But anyway from this unscientific sample, it looks like invariant mass has almost completely taken over in books written at this level.

Oas, "On the Abuse and Use of Relativistic Mass," 2005, http://arxiv.org/abs/physics/0504110

Okun, "The concept of mass," 1989, http://www.itep.ru/theor/persons/lab180/okun/em_3.pdf

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We have two answers giving different statements of what the convention is. Is it feasible to acquire and post statistics backing up a claim of one usage or the other being the convention? – user2357112 Sep 1 '14 at 5:13
There's also an excellent reason why we use mass to mean the invariant mass. The 'relativistic mass' has a much better name: it's called the energy. Suddenly that odd statement "mass increases with speed" becomes the very obvious "energy increases with speed". – Holographer Sep 1 '14 at 15:55
@VladimirF: Unfortunately, I have been thought the relativistic mass in the high school around the year 2000 and I am not sure if that changed. You can still meet that in books that are not high level. Right, I don't claim that high school and freshman college books follow the modern convention, only that the convention is universal among professional relativists. It's common for lower-level textbooks to be out of date, especially because they're survey texts that treat subjects in which the authors are not specialists. – Ben Crowell Sep 1 '14 at 16:52
"In a book from the Roosevelt administration [...]" ::chortle:: – dmckee Sep 1 '14 at 23:42
@ScottCentoni - An even more revealing ngram is "relativistic mass" vs "invariant mass" vs "rest mass": books.google.com/ngrams/… . Two things pop out: "Rest mass" dwarfs the other two terms, and it's use peaked 50 years ago. That's when using just "mass" to mean "intrinsic mass" started becoming popular. – David Hammen Sep 3 '14 at 1:51

There's no controversy about whether mass increases or not, there's controversy about what you call mass. One possible definition is that you consider some object's rest frame, and call the $\tfrac{F}{a}$ you measure there (for small accelerations) the mass. This notion of mass can't change with speed because, by definition, it's always measured in a frame where the speed is zero.

There's nothing wrong about this way of thinking, it's basically a question of mathematical axiom. Only, it's not really useful to require the rest frame, because we're constantly dealing with moving objects1. Therefore, the (I believe) more mainstream opinion is that that quantity should only be called rest mass $m_0$. The actual ("dynamic") mass is defined by what we can directly measure on moving objects, and, again simply going by Newtons law, if you e.g. observe an electron moving with an electric field at $0.8\:\mathrm{c}$, you'll notice it is accelerated not with $a = \tfrac{F}{m_0}$ but significantly slower, namely as fast as a nonrelativistic electron with mass $m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$ would. It is therefore reasonably to say this is the actual mass of the electron, as seen from laboratory frame.

1 Indeed, you can argue it's never possible to really enter the rest frame. In macroscopic objects you'll have thermal motion you can't track, and yet more fundamentally there's always quantum fluctuations.

Edit as noted in the comments, amongst physicists there will of course not really be controversy about what mass definition is meant: they'll properly specify theirs, usually just following the convention of invariant mass. That can easily be calculated for any given system, from the total energy and momentum rather than the actual movements of components (which, again, you can't track). That still leaves scope for confusion to the unacquainted though, because whether the invariant mass increases or not when accelerating an object depends on whether you consider the mass of some bigger system, say with some much heavier stationary target, or the accelerated object on its own. This may seem counterintuitive, so when hearing accounts of the same experiment based on either of these "system" definitions you think there's a controversy, when really the accounts are just talking about different things.

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Not only is there not a controversy about whether mass changes, there's also no controversy anymore about how to define mass. It's been 50 years since the invariant mass convention became universal among relativists. – Ben Crowell Aug 31 '14 at 21:08
@BenCrowell that still leaves ambiguity if it's not clear what system you consider. For two particles, if you accelerate them in opposite directions, the invariant mass increases. If you only consider one of the particles, it stays constant. – leftaroundabout Aug 31 '14 at 21:19
You emphatically don't need to enter any particular frame to get the (rest) mass because it is a relativistic invariant and is defined perfectly well in any frame: $m = \sqrt{E^2 - \mathbf{p}^2}$ (in $c = 1$ units and with a $\mathrm{Tr} = -2$ metric). It's that property of invariance which makes this definition so useful. – dmckee Aug 31 '14 at 22:40
@BenCrowell: it is an ambiguity if the problem is phrased ambiguously (duh), i.e. if it wasn't properly specified whether both particles should be considered. Yet to the layman it's not obvious that the ambiguity carries over, because in the nonrelativistic case mass is simply additive. Hence the apparent controversy, when really you're just comparing two different situations with results that seem incompatible (but aren't). – leftaroundabout Aug 31 '14 at 23:30
Actually, $dv/dt$ (lab frame) will depend on the angle between force and velocity. From the point of view of relativistic dynamics, comparison of collinear and perpendicular forces by their 3-vectors in the lab frame is a wrong idea at all. While response of the body to perpendicular forces supports “relativistic mass” thing, collinear forces cannot be conveniently described with 3-force vectors because there a transfer of “energy”, not only 3-momentum. (Please point out if Ī use a confusing or non-standard terminology.) – Incnis Mrsi Oct 29 '14 at 9:13

Some people say that 'mass increases with speed'. Some people say that the mass of an object is independent of its speed. I understand how some things in physics are a matter of ... definitions. But, I can't get my head around how both be 'true' is any sense of the word. Either mass increases or it doesn't, right? Can't we just measure it... heat up some particles in a box and measure their weight.

The technicalities of the issue have been masterly presented. I'll try to give you a more simple 'user-friendly' explanation.You make some confusion in your own post, between mass and weight, and if/when you clarify that it can help you bring correctly into focus the problem.

Suppose you can count literally up the (electrons/protons atoms) of your body considering as an average an atom of carbon 12. That number is dimensionless, absolute (instead of weighing it, which is relative). Suppose you ascertain that the mass of your body is made up by $10^{27}$ atoms. That mass is the real mass of your body and it can/will never increase.

Now, suppose you weigh your body on the Earth then on the Moon and then on Jupiter, what do you get? that your 'mass' apparently increases and decreases. You seem to have accepted that, forgetting that your body sill has the same number of atoms.

You have accepted so far that the same mass can be 'observed' to have different values in different circumstances, in this case: gravity.

Now, try to apply the same logical mechanism that made you accept this apparent contradiction to another situation in which what varies is speed: when a body acquires kinetic energy it acquires (temporarily, as long as it conserves that KE) the same property that your body acquired on Jupiter. Your body at 0.8 c weighs much more than when it is travelling at 0.01 c, yet its 'true mass' is still made up by $10^{27}$ atoms.

In this case, besides gravity, you might find a more simple, 'rational', explanation that can make it easier for you to understand and to accept it: energy (kinetic, thermal etc) bound in a body has a tiny 'mass/weight' attached to it, which temporarily increases its 'weight'

Can't we just measure it... heat up some particles in a box and measure their weight.

It is not clear what you are trying to prove with that, but if you heat matter its weight will change, due also to 'gravity'

If you have absolutely identical objects that have the same weight exactly when they are at the same temperature, then when one object is heated, it will weigh more. This is because the gravitational force depends on the stress energy tensor in general relativity. The stress energy tensor 00 component is the total energy of the body, which includes the rest mass plus the kinetic energy of the object. Temperature differences means that there is a different amount of kinetic energy in the motion of the atoms of the two bodies.

For example, if you start with two identical kilograms of water at 0 Celsius, and if you then heat one of them to 100 Celsius, then the kilogram at 100 Celsius would be heavier by an amount equivalent to 4.6 nanograms of additional water weight (see 100*1000 calories / c^2 ).

Now 4.6 nanograms is not very much, but it is equivalent to 154 trillion molecules of water (see 4.6 10^-9 gm water in molecules ). Just imagine - the energy used to heat the water is equivalent to the weight of 154 trillion additional water molecules if they could be converted completely into energy (remember E=mc^2)!

This extra mass/weigth is temporarily added to your body, and when it slows down or cools off, it loses energy and consequently its 'weight' attached to it and returns to its 'true' value. Does this help you clarify your doubts?

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But while an object's weight changes when moved from one gravity field to another, its inertia does not. My understanding is that an object's velocity does affect its inertia, thus the whole "can't reach light speed" thing. – user20846 Jan 22 at 0:27

As in Ben Crowell's Answer, the concept of "Relativistic Mass" is not wrong, but it is awkward. There are several things a loose usage of the word "mass" could imply, all different and thus it becomes a strong convention to talk about the meaning of the word "mass" that is Lorentz invariant - namely the rest mass, which is the square Minkowski "norm" of the momentum 4-vector. Given its invariance, you don't have to specify too much to specify it fully, and so it's the least likely one to beget confusion.

Here's a glimpse of the confusion that might arise from the usage of the word mass. To most physicists when they learn this stuff, the first time they see "mass" they think of the constant in Newton's second law. So, what's wrong with broadening this definition? Can't we define define mass as the constant linking an acceleration with a force? You can, but it depends on the angle between the force and the velocity! The body's "inertia" is higher if you try to shove it along the direction of its motion than when you try to introduce a transverse acceleration. Along the body's motion, the relevant constant is $f_z=\gamma^3\,m_0\,a_z$, where $m_0$ is the rest mass, $f_z$ the component of the force along the body's motion and $a_z$ the acceleration begotten by this force. At right angles to the motion, however, the "inertia" becomes $\gamma\,m_0$ (the term called relativistic mass in older literature), i.e. we have $f_x=\gamma\,m_0\,a_x$ and $f_y=\gamma\,m_0\,a_y$. In the very early days people spoke of "transverse mass" $\gamma\,m_0$ and "longitudinal mass" $\gamma^3\,m_0$. Next, we could define it as the constant relating momentum and velocity. As in Ben's answer, we'd get $\gamma\,m_0$. We can calculate $\vec{f}=\mathrm{d}_t\,(\gamma\,m_0\,v)$ correctly, but not $\vec{f}=\gamma\,m_0\,\vec{a}$, it fails not only because $\gamma$ is variable but also because the "inertia" depends on the direction between the force and velocity.

So, in summary, "inertia" (resistance to change of motion state by forces) indeed changes with relative speed. You can describe this phenomenon with relativistic mass, but it is awkward, complicated particularly by the fact that the "inertia" depends on the angle between the force and motion. It is much less messy to describe dynamical phenomena Lorentz covariantlt, i.e. through relating four-forces and four-momentums and one uses the Lorentz invariant rest mass to see these calculations through.

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I think it is a question of the reference frame. You pick a reference frame tied to your object (the rest frame) then the mass is always the same in that frame and it is the rest mass of the object, $m_0$.

If you pick another reference frame in which your object can move, its mass will be different and will definitely depend on its velocity. Its expression will be $m=\gamma m_0$, where $\gamma=1/\sqrt{1-v^2/c^2}$.

The reason for this is that the energy of the moving object will be seen from your reference frame as kinetic plus the rest energy of that object. The total energy of the object is still $E=m c^2$, only this time $m$ will depend on the velocity of the object in the reference frame you chose.

In my limited experience with relativity theory (special and general), it appears that most of the confusion in understanding its workings come from not understanding the role of reference frame. Whenever you want to calculate anything you must first set the reference frame (a ruler, a watch and origins for both space and time axes). Once you do that you can make statements about the system you study.

Sometimes you may have 2 objects moving relative to each other. Usually you can compute everything about those isolate objects much easier in their respective rest frames. Then you must worry about the whole system and you need to set a common reference frame for the system of two objects and calculate whatever you need to calculate (distances, velocities, electromagnetic fields) in that frame. For this, you need to use the transformations (Lorentz or Poincaré transformations, for example) to transform the quantities you calculated in those objects' rest frames to the common reference frame.

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The whole point of the question is that there are two camps on the usage of the word "mass" You've related the description used by the camp that speaks of "relativistic mass", but totally ignored the (rather larger among working rather than teaching physicists) camp that assigns the symbol $m$ to the invariant mass and don't give any name to $\gamma m$ at all. In other words you haven't even shown that you understood the question much less addressed it. – dmckee Sep 1 '14 at 23:40
The "controversy" between the two "camps" is just a mere technicality. In other words is not that important (I'd side with the workers, but I don't do relativity for a living). What is important is what I wrote below. In other words, whatever physical quantity one calculates depends on the reference frame one chooses. Once that is clear, there is no more confusion, "controversy" and "camps". Sorry for not addressing directly the question especially since OP seems to be just learning the subject. – user3653831 Sep 2 '14 at 5:18