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I'm reading Nano: The Essentials by T. Pradeep and I came upon this statement in the section explaining the basics of scanning electron microscopy.

However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. At such velocities, one needs to do relativistic correction to the mass so that it becomes[...]

We all know about the famous theory of relativity, but I couldn't quite grasp the "why" of its concepts yet. This might shed new light on what I already know about time slowing down for me if I move faster.

Why does the (relativistic) mass of an object increase when its speed approaches that of light?

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13 Answers 13

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The complete relevant text in the book is

The de Broglie wave equation relates the velocity of the electron with its wavelength, $\lambda = h/mv$ ... However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. ...

Actually, the de Broglie wavelength should be $$ \lambda = \frac hp, $$ where $p$ is the momentum. While $p = mv$ in classical mechanics, in special relativity the actual relation is $$ \mathbf p = \gamma m \mathbf v = \frac{m\mathbf v}{\sqrt{1-\frac{v^2}{c^2}}} $$ where $m$ is the rest mass. If we still need to make the equation $p = mv$ correct, we introduce the concept of "relativistic mass" $M = \gamma m$ which increases with $v$.

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The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".

What actually changes at relativistic speeds is the dynamical law that relates momentum and energy depend with the velocity (which was already written). Let me put it this way: trying to ascribe the modification of the dynamical law to a changing mass is the same as trying to explain non-Euclidean geometry by redefining $\pi$!

Why this law changes is the correct question, and it is discussed in the answers here.

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    $\begingroup$ This is the correct answer. $\endgroup$ – MoonKnight May 8 '13 at 17:38
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    $\begingroup$ @Killercam, yep. +1 for Okun $\endgroup$ – Peter Kravchuk May 8 '13 at 20:41
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    $\begingroup$ This doesn't answer the question. It just advises the OP to ask the question in different language. $\endgroup$ – Ben Crowell May 8 '13 at 23:18
  • $\begingroup$ @Ben It's not an issue of asking the same question in different language. Suppose I ask 'why is the moon made of blue cheese'. The question as stated makes no physical sense. If you change the language so that it makes sense, it becomes a different question. $\endgroup$ – Anton Tykhyy Mar 28 '14 at 11:01
  • $\begingroup$ Yeah, it would be foolish to try explaining non-Euclidean geometry by redefining pi factorial. $\endgroup$ – kingfrito_5005 Oct 18 '17 at 21:30
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There is a point of view, that under the term "the mass" one must mean "the rest mass".

From that point of view there is obviously no dependence of the (rest) mass on the speed of an object. And, therefore, the mass of an object does not increase when its speed increases.

The correct (from that point of view) way to talk about the phenomenon is to say that with increase of the speed of an object you need more and more energy in order to make it move faster.

Of course there is no fundamental controversy between this point of view and that of many books and articles. But the usage of the concept of "relativistic mass" makes things much more complicated, even if it was introduced in pursuit of simplicity.

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  • $\begingroup$ The phenomenon you are talking about (one in the italics) is actually quite similar to definition of inertial mass. Inertial mass measures how hard it is to move an object. And this is why the concept of relativistic mass is useful. It is similar to inertial mass. On the other hand, invariant mass is just a number that characterizes the particle but has nothing to do with dynamics. I think properly distinguishing between various concepts of masses takes some time and thinking (and there is also gravitational mass and related equivalence principle but let's leave that for another time). $\endgroup$ – Marek Dec 6 '10 at 18:02
  • $\begingroup$ Inertial mass measures how hard it is to move an object. Yes. But note that you can change the speed of an object in different directions. Do deal with that you will need to introduce "logitudinal" and "transverse inertial masses". I support people, saying that talking about "inertial mass" is an overcomplication leading to mistakes and "terminological swamp". $\endgroup$ – Kostya Dec 6 '10 at 18:24
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    $\begingroup$ Point taken Kostya. These past few days I've been shown quite a huge heap of evidence that any concept of mass different from invariant mass is really too messy to be worth even talking about :-) And I really have to wonder why I was introduced to all these things when I was learning SR myself (from books and also at our uni SR course). They seem just an unnecessary baggage now. But perhaps it is still useful to know these concepts exist? I am not sure. $\endgroup$ – Marek Dec 6 '10 at 19:57
  • $\begingroup$ I had the same confusion about this unecessary baggage after I dscovered all this mess. The person, who promotes the "only rest mass" point of view is Lev Okun. The great, very basic and free book on that topic: "ENERGY AND MASS IN RELATIVITY THEORY" by Lev B Okun $\endgroup$ – Kostya Dec 7 '10 at 9:45
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In special relativity the actual invariant is the magnitude of the covariant energy momentum 4-vector $(E_0/c_0, p_x,p_y,p_z)$, not the apparent mass itself. See also the section on "momentum in 4 Dimensions", here. The apparent mass in a moving frame is just a projection.

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Sometimes the same word "mass" is used with different meanings. There are two different quantities associated with the word "mass":

  • A quantity that physicists usually call "mass", which is an intrinsic property of the object and does not depend on how fast it is moving. I'll use the symbol $m$ for this quantity.

  • A synonym for the object's energy $E$, but expressed in mass-like units as $E/c^2$. This is sometimes called the object's "relativistic mass" and it does depend on how fast the object is moving (because the object's energy does). I'll use the symbol $m_R$ for this quantity.

We are already familiar with the fact that the kinetic energy of an object is higher when the object is moving faster. "Relativistic mass" is just a synonym for the object's total energy, expressed in mass-like units. From this perspective, consider the question again:

Why does the (relativistic) mass of an object increase when its speed approaches that of light?

Answer: Because the object's energy increases. "Relativistic mass" is just a synonym for the object's energy, expressed in mass-like units. Why did people ever start using the name "relativistic mass" for the object's energy? I don't know. In my experience, most physicists just call it energy.

Here are some equations to help clarify things:

The energy $E$, momentum $p$, speed $v$, and mass $m$ of an object are related to each other according to these equations: $$ E^2 - (pc)^2 = (mc^2)^2 \hskip2cm v = \frac{pc^2}{E} $$ where $c$ is the speed of light. The $m$ in the first equation is what physicists usually mean when they use the word "mass". It is an intrinsic property of the object and does not depend on the object's speed. The object's energy $E$ and momentum $p$ do depend on the speed, and they do so in such a way that the combination $E^2-(pc)^2$ does not depend on the speed. That's why this particular combination is interesting, and that's why the $m$ on the right-hand side of the equation deserves a special name: mass.

To relate this to the "relativistic mass" $m_R$ (which, again, is not used by the majority of physicists in my experience), re-arrange the second equation shown above to get $$ p = \frac{E}{c^2}v. $$ If we use $m_R$ as an abbreviation for $E/c^2$, then this becomes $$ p = m_R v, $$ which looks superficially like the more familiar low-speed approximation $p=mv$. This resemblance is also misleading, though, because the energy $E$ (and therefore $m_R$) is a function of $v$. The momentum $p$ is not really proportional to the velocity $v$, except approximately when $v\ll c$.

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  • $\begingroup$ I'm afraid we have to blame Einstein for relativistic mass. Before Einstein, people noticed that something funny happens with the momentum & kinetic energy of high speed electrons. Special relativity solved the mystery via relativistic mass; it took a few decades for people to realise that it's better to work with invariant mass and that relativistic mass is more trouble than it's worth. $\endgroup$ – PM 2Ring Nov 18 '18 at 3:04
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Fundamentally, mass and energy are the same thing. They are two "points of view" of the same reality.

From the "point of view" (inertial frame) of an electron, its mass does not increase, its speed is always zero.

From the "point of view" (inertial frame) of a stationary observer, the electron has a very high kinetic energy (some in the mass term and some in the speed term)

From the "point of view" (inertial frame) of a moving observer, the electron has a different kinetic energy (some in the mass term and some in the speed term)

And so on.

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Keeping it simple (with a link):

Special Relativity

"The relativistic increase of mass happens in a way that makes it impossible to accelerate an object to light speed: The faster the object already is, the more difficult any further acceleration becomes. The closer the object's speed is to light speed, the greater the increase in inertial mass; to reach light speed exactly would require an infinitely strong force acting on the body. This enforces special relativity's speed limit: No material object can be accelerated to light speed.

The increase in inertial mass is part of a more general phenomenon, the relativistic equivalence of mass and energy: If one adds energy to a body, one automatically increases its mass; if one takes energy away from it, one decreases its mass. In the case of acceleration, the object in question gains kinetic energy ("movement energy"), and this increase in energy automatically means an increase in mass."

See http://www.einstein-online.info/elementary/specialRT/emc

This, to most, helps clear things up without adding complexity. You are, of course, welcome to delve deeper.

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  • $\begingroup$ As many comments & answers on this page mention, relativistic mass is not used in modern treatments of special relativity because it's often awkward to work with, confusing, and potentially misleading. $\endgroup$ – PM 2Ring Nov 18 '18 at 3:08
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If you want to intuitively see why the mass increases, consider the following.

  • Firstly, nothing can travel faster than the speed of light (this is the premise on which Special Relativity is based)

  • Secondly, applying a force to an object will increase its kinetic energy (assuming the force acts in the same direction as the object's motion)

Since kinetic energy $K.E.$ = $m v^2/2$, if $v$ is limited to $c$, then as $v$ approaches $c$ the only way for $K.E.$ to increase is for $m$ to increase.

This isn't a fully mathematical answer, but may help you to intuit why the mass increases.

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    $\begingroup$ I have never before heard that special relativity is based on the premise that nothing can move faster than light. Elementary SR usually says 1) The speed of light is constant in all frames. 2) All physical laws are the same in inertial frames. $\endgroup$ – Mark Eichenlaub Dec 6 '10 at 20:58
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    $\begingroup$ @Mark: correct. SR says no such thing and actually admits existence of tachyons. The sectors "faster than light" and "slower than light" are obviously dual to each other and SR doesn't distinguish between them at all. The reason why we don't want tachyons is an additionally assumption of causality. $\endgroup$ – Marek Dec 6 '10 at 21:11
  • $\begingroup$ +1 @Sam, I was also thinking along the same line you did, relating mass to speed and energy. $\endgroup$ – Kit Dec 7 '10 at 0:10
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    $\begingroup$ This argument is wrong. Relativistic kinetic energy doesn't equal $(1/2)mv^2$, and doesn't approach a limit as $v$ approaches $c$. $\endgroup$ – Ben Crowell May 8 '13 at 23:14
  • $\begingroup$ This is simply wrong. Basically beeing limited to c does not mean that you have to push on a equation that contain c. Just go to c and stop. Second , c emerged as a limit , It is not taken as a postulate. $\endgroup$ – Alchimista Oct 24 '17 at 22:36
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The reason why you are having this confusion is because you think that mass should not change. As many have said above, and I would reiterate, REST MASS is the property that does not change for any particle, ever. For eg, the rest mass of a photon is zero. So, basically, when einstein put forward the very famous equation, $E = M.C^2$, he meant very clearly that mass IS energy, and energy IS mass. They are just one and the same thing!.

Now, tell me, if energy increases, would the mass not increase? And why not in daily life, the answer is because $ \delta M = \frac{\delta E}{c^2}$...and so, if your energy changes by an amount comparable to $c^2$, only then would you be able to observe a change in mass.

Hope it helps...if any more doubts arise, please comment!

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Relativistic mass, by definition, is the quantity $m_\mathrm{rel}(v) := \gamma(v)\ m$, where $m$ is the intrinsic, or "rest", mass. Mathematically, it increases because the Lorentz factor $\gamma(v)$ increases with increasing speed $v$.

However, a more physical reason is that this "relativistic mass" is really just the total energy $E_\mathrm{tot}$, consisting of the combination of the rest energy and kinetic energy of a material body, interpreted in units of mass, by the mass-energy equivalence relation $E = mc^2$: $m_\mathrm{rel}(v) = \frac{E_\mathrm{tot}(v)}{c^2}$. It is thus seen to increase because objects moving at higher speeds have more kinetic energy, and thus also, total energy.

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The mass of object changes when its speed approaches zero because according to Einstein postulates of theory of relativity all the laws are same in all inertial frames and speed of light remains constant in inertial frame in vacuum. All the concepts of relativity are based on these two postulates. As one can not add any speed in speed of light, the Lorentz transformation equations are derived and using these variation of mass with velocity relation. Almost every concept of Physics changes at a speed comparative to speed of light.

One can see the derivation here

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There are plenty of misinformation here.

"The mass of a body is not constant; it varies with changes in its energy."

[Einstein, A. The Meaning of Relativity, Princeton University Press, 1988]

See also Section 10, Dynamics of the Slowly Accelerated Electron, of the paper 'On the Electrodynamics of moving Bodies' [Einstein, A. Annalen der Physik, 17, 1905]. Also see Section 29, Ponderomotive forces. Dynamics of the electron, in the book 'Theory of Relativity' [Pauli, W. Dover Publications Inc., 1981, (first published in 1921 in German, first published in English in 1958)]

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    $\begingroup$ You should probably be aware that the modern take on relativity does not group $\gamma$ with $m$ and call $\gamma m$ "the relativistic mass", but rather takes the invariant square of the energy-momentum four-vector (what in the old language would have been called the "rest mass") to be the definition of the (only!) mass of the object. The math is the same, but (1) it engenders less confusion and (2) the emphasis on invariants help to make problems easier. $\endgroup$ – dmckee Nov 4 '13 at 15:20
  • $\begingroup$ @dmckee I personally feel that relativistic mass confused and brake my understanding along years. I am a chemist and always felt that, besides observables, mass should intimately reletad to the baryonic constitution of a body. However, or exactly for that, I still have trouble on seeing why m does increase with T. Perhaps the concept of relativistic mass is useful for that (in a microscopic view).... $\endgroup$ – Alchimista Feb 17 '18 at 12:39
  • $\begingroup$ @dmckee ..... not sure that I cleary convey what my concern is. Not very easy to formulate. .. I tried to follow debates on mass energy and found EVEN three four schools of thinking! Convertible but not the same / the same / not the same and not convertibles / .... I got lost as I found myself fine with multiple but different leading logics. Can you point me to something on this? $\endgroup$ – Alchimista Feb 17 '18 at 12:45
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If I accept true mass as being constant then defining increases to such a given mass is impossible (see Newton) because I can sit and watch it forever at rest or in motion: a given mass is a given mass, and a given quantity does not change, regardless of speed or energy applied. The effect(s) then, either perception or theoretical, that creates the illusion that mass changes, is better described by a more accurate theory than Einstein put forth or modifying Einstein to a more accurate level.

On another note, the arbitrary Einsteinian speed of light limit for velocity is similar to the ceiling of the 4 minute mile. There is no proof that such a limit exists. Until we get past this arbitrary barrier and explain these phenomena in a more useful way we will never achieve deep space travel.

To make such travel possible we must get beyond mass particles into non-mass particles that pass easily through the light barrier. They might or might not exist at this point. In the future these particles will be the building blocks and the carriers of information necessary to transfer life to habitable planets throughout the universe. Granted, your body will not beam up but your current conscionsness and DNA sequence might: A quick resurrection and you are you three galaxies away.

Granted that I am not a physicist and I am an iconoclast when it comes to accepting dogma. To me, devolving honest perception into theoretical soup does not make you any more informed on the true nature of mass vis a vis energy and velocity. You simply have learned what someone else learned and thought it to be true, i.e. the appeal to authority that Einstein believed and said it therefore it is true.

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  • $\begingroup$ "There is no proof that such a limit exists." There's tons of proof. Particle accelerators routinely accelerate particles to speeds higher than 0.99c, but they've never managed to break the lightspeed barrier. $\endgroup$ – PM 2Ring Nov 18 '18 at 3:35

protected by Qmechanic Nov 4 '13 at 17:14

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