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Why is mass a very concentrated form of energy? Does it have to do something with photons, phonons or nucleus?

Qmechanic
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Omer Farooq
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Mass is not a "concentrated" form of energy. I'll come to this later but first let's understand the relation between mass and energy.

In relativity, energy and momentum are parts of a unified object called the four momentum vector $(E, \mathbf{p}$). Now, just like usual vectors in Euclidean geometry, there is a magnitude attached to this vector which is given by $\sqrt{E^2-|\mathbf{p}|^2}$, this is what is mass $m$. Just like the length of a displacement vector doesn't change when you rotate your coordinate system in Euclidean geometry, the magnitude of the four momentum vector doesn't change among different inertial frames in relativity. This magnitude, the mass $m$, thus, is a coordinate independent way to characterize the energy-monentum content of a system. Now, if you go to a reference frame where the momentum $\mathbf{p}$ vanishes, you get that the energy $E_0$ in such a frame which we call rest frame is equal to $m$.

Thus, mass is the energy of a system in its rest frame, i.e., in the reference frame where the momentum of the system is zero. In other words, $E_0 = m$.

So far I've done everything in natural units where $c=1$. If you restore the factors of $c$, you'll write $E_0 = mc^2$. The factor of $c^2$ gives the illusion that mass is a concentrated form of (rest) energy because it looks like a small amount of mass corresponds to a large value of rest energy. However, this is purely an illusion created out of our traditional choices of units which are different for energy and mass. In the natural system of units, as I wrote, the rest energy and mass are exactly equal. Saying that mass is a concentrated form of (rest) energy would be like saying length of my arm in meters is a concentrated form of length of my arm in nanometers.

Sociological edit: Why do people like to believe mass/matter is a concentrated form of energy?

From comments on this and other answers, it seems necessary to address this point. OK, so mass is what I described above. Now, what does it imply about the mass of a system of particles? Very simple, you go to frame of reference where the spatial momentum $\mathbf{p}$ of the system is zero and the energy of the system in this frame (i.e., the rest energy of the system) is what the mass of the system is. If you are trapped in Newtonian intuition, you'd expect this mass to be the sum of the masses of individual particles. This is obviously not true in relativity, just like other Newtonian things. For example, if you have a neutron decaying to a proton, an electron, and an anti-neutrino, the mass of the the system of the proton, the neutron, and the anti-neutrino would be exactly the same as the mass of the neutron (because mass is conserved) but this total mass would not be equal to the naive summation of the individual masses of the three particles. See, a related answer of mine. So, since the mass of the neutron would not be the same as the sum of the masses of the proton, the electron, and the neutrino, people like to say that some mass of the neutron is now converted to kinetic energy of these particles. Since the difference of the sum of the masses of the resulting particles and the mass of the neutron is a "small" number in the units in which we measure mass and the kinetic energy of the resultant particles would be a relatively "big" number in the units in which we measure energy, people like to say that mass is the concentrated form of energy. But particle physicists use natural units where they measure both mass and energy in electron volts. This would make sure that the difference of the sum of the masses of the resulting particles and the mass of the neutron is exactly the same as the kinetic energy of the resulting particles. So in conclusion, all of this confusion arises because people like to talk using Newtonian intuition and they also like to use unnatural units. :)

PM 2Ring
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  • Is it correct to say that "things" with no rest frame must have 0 mass but that they still can carry energy? – undefined May 26 '20 at 12:04
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    @undefined Yes, that's exactly the right thing to say. :) –  May 26 '20 at 12:11
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    I disagree with your implicit assumption that the only reason someone might believe mass is concentrated energy is due to the $E = mc^2$ formula and its choice of units. Whenever mass is converted to energy the result is always an event who's magnitude and effects are felt across a much, much larger region of space than the original mass ever influenced. The resulting energy of such a conversion quickly disperses. These facts make it a very reasonable view in my opinion to view mass as concentrated energy. – orlp May 26 '20 at 13:26
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    @orlp I think it would be more appropriate to say "matter is a concentrated form of energy". The point of that was that mass is a unit of measurement, and energy is a also a unit of measurement. Neither of those units of measurements have anything to do with how concentrated energy is. – Bryce Wagner May 26 '20 at 13:44
  • @orlp I've added an addendum to my answer to address your and similar comments. –  May 26 '20 at 15:14
  • Part of the problem is that "mass" and "matter" are often treated as synonyms. I agree with Bryce: matter is a fairly concentrated form of energy compared to other forms of energy that we encounter in low velocity (& low gravity) regimes. Of course, when we deal with ultra-relativistic particles, the rest mass can be negligible compared to the kinetic energy. – PM 2Ring May 26 '20 at 15:25
  • @PM2Ring I didn't address the matter part because it's even more ill-defined statement as you'd agree I suppose. Matter is an old concept that we only use metaphorically in modern physics. What is matter? Everything other than gauge particles? If so, massless vector bosons and/or fermions would have no mass so they'd be matter but not concentrated energy? Also, what about composite bound states which involve gauge bosons? Is proton matter? It's composed a lot more of gluons than quarks in terms of its mass. –  May 26 '20 at 17:10
  • Fair enough. In the spirit of the traditional idea of matter, I'd say that massless gauge bosons are concentrated energy, but not matter. A proton is matter, if we don't look to closely, but on closer examination we find that most of its mass is due to the kinetic & potential energy of its component quarks & gluons. – PM 2Ring May 26 '20 at 17:26
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    This answer is a bit misleading. While saying mass is a 'concentrated' form of energy is nebulous and not quite correct, one can think of mass as an emergent property of confined energy (and this is, in fact, the standard view in QCD). – J... May 26 '20 at 18:43
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    @J... I completely agree with your latter statement and I didn't intend to given an otherwise impression in my answer. If you can suggest me what parts of my answer imply that I'd disagree with your second statement, I'd be more than happy to edit that part. –  May 26 '20 at 19:09
  • @DvijD.C. Well, your answer starts out saying that OP is flat out wrong - effectively that they are starting from the false premise that mass is 'concentrated' energy. In reality, their 'misconception' is actually not so far from standard views and only really needs precision in terminology. I think the answer is missing that big "but...". – J... May 27 '20 at 12:09
  • @J... I disagree with the "concentrated form of energy" description because it's not like energy is being put into a smaller volume and it suddenly becomes mass. So, I don't think it is truly related to the fact that massless or nearly massless particles can form a bound state which is massive. That's why I didn't append it with a but because I don't see the second statement to be related to the first. The first is just a pure misconception, the second is perfectly correct and unrelated to the first except for sharing the common words such as mass and energy. –  May 27 '20 at 12:14
  • However, the point that you bring up is a nice one and I'd try to edit the question to add that to avoid any potential confusion. –  May 27 '20 at 12:16
  • because it's not like energy is being put into a smaller volume and it suddenly becomes mass. Except that does happen, more or less. Photons in a massless, mirrored box behave as though the box has mass equal to the sum of the energy of the photons. So, yeah, putting energy into a box does produce mass - this is what I mean, I think you're not properly capturing how much correctness there is in OP's concept. – J... May 27 '20 at 12:20
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    @J... No, the volume of the box has nothing to do with it. Putting them in a small volume is irrelevant. Making sure that their spatial momenta cancel out is important. –  May 27 '20 at 12:44
  • @DvijD.C. Yes, I know, which is why I started this with making the distinction between 'concentrated' and 'confined'. Still, it's a subtle distinction, especially for a non-scientist. Shrinking the box does increase mass density, of course, so that even goes to make your above statement even weaker. The more you localize the energy confinement the more mass dense the system appears. The answer starts with the statement that Mass is not a "concentrated" form of energy. But kind of, in many ways, it really is. – J... May 27 '20 at 12:59
  • @J... It doesn't even need to be confined really. The system of two photons one in New Jersey and other in Nebraska will also have a non-zero mass if they are moving in the opposite direction, for example. –  May 27 '20 at 13:05
  • @DvijD.C. Yes, but you can't act upon them as a system, so it doesn't really matter. In any case, it digresses from the point. – J... May 27 '20 at 13:32
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The fact that mass and energy are two realisations of the same thing is a direct consequence of the theory of special relativity. Nothing more is required to find this other than Einstein postulates.

The fact that mass and energy are the same thing, although was considered a very strange result at the time, has been countlessly proven experimentally. In accelerators physicists do it all the time: initial particles are given a lot of energy, $17$ TeV in the center of mass frame at CERN for example, smashed together to create countless other particles whose masses can be many times higher than the ones of the particles we started from.

In a general sense it has nothing to do with specific particles you mentioned, but more with the way our universe is built.

Davide Morgante
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    If I could ask a follow up to this, you mentioned that particles can decay to heavier ones if they have high enough energy. If we instead move into a frame in which the particles are at rest the original particles appear to have lower energy and so that reaction becomes forbidden, what is wrong with this reasoning? – Charlie May 25 '20 at 17:53
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    You misunderstood what I said! Particles with a certain mass cannot decay in particles with an heavier mass (they can if only the particle in which they decay is a virtual one, but I'm getting ahed of myself). What I said is that if we collide two particles with one another, or one with a target at rest, we can create in the collision heavier particles. – Davide Morgante May 25 '20 at 17:56
  • So particle decay can't create particles with greater total rest mass, but in collisions there does not exist a frame in which both particles are at rest so we inescapably see the system as a whole having more energy than the sum of the two rest masses? – Charlie May 25 '20 at 17:58
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    @Charlie Yes, you are correct. Then the creation of certain particles from collisions of given particles is governed by the standard model which says what particles can be created (from a specific collision) and what can't. – Davide Morgante May 25 '20 at 18:02
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    Thank you very much, that answers a huge question I've had for a while but hadn't gotten around to asking. – Charlie May 25 '20 at 18:02
  • Can energy be converted into matter? – Omer Farooq May 25 '20 at 18:06
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    Yes, It can, and again is what we do many times at particle accelerators, for example. – Davide Morgante May 25 '20 at 18:09
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I can't agree with the affirmation "mass a very concentrated form of energy", I would agree with "matter is a very concentrated form of energy".

Mass is a property of energy. Of every kind of energy, be it trapped inside the atoms and molecules or not.

Where the mass of the matter comes from? Almost all comes from the binding energy between quarks, inside the protons and neutron in atoms' nucleus. A little comes from the bindings between protons and neutrons making up the nucleus, a little from the bindings between electrons and nucleus, maybe a little from the bindings between atoms to form molecules, and a little from electrons and quarks themselves (I may have missed some sources of energy here).

Why is it so? If by "why", you mean how it was discovered, as Davide Morgante said in his answer, it follows directly from Einstein's special relativity. If by "why" you mean in some grand philosophical sense, you may consider this as physical law, as good as any other.

EDIT about the photon's mass: Despite pragmatic physicists saying photon has zero mass, photons distort spacetime, like all other masses, so much that the whole concept of kugelblitz depends on it. Photons also have momentum and inertia, and the working of light sails depends on it. If you put light inside a perfectly mirror walled box, and weight the box, it will be heavier than without light inside. So please understand mass as "the thing that the scale shows", or "the thing that has inertia" or "the thing that bends spacetime", and photons do have it.

lvella
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    It's not true that mass is a property of energy. Proof: Photons have energy, but no mass. –  May 26 '20 at 11:44
  • Of course they have mass, and it is proportional to their inverse wavelength. They don't have rest mass, but they are never at rest, so... – lvella May 26 '20 at 11:46
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    In mainstream physics, i.e, physics, there is only one mass, the invariant mass. Rest mass was a confusion that arose in history and physicists eventually realized that it was a confusion. There is only mass which is equal to rest energy for massive particles and is zero for massless particles. Photons have energy which is proportional to inverse wavelength, their mass is zero. –  May 26 '20 at 12:11
  • Either you claim the $m = E/c^2$ is valid for photon's energy or not. One of the alternatives is wrong, and $m = E/c^2$ is valid for all kinds of energy. – lvella May 26 '20 at 12:20
  • You are essentially saying solar sails are impossible. If photons don't have mass, they can't provide propulsion, as it would violate conservation of momentum. You are also saying kugelblitz are impossible, as photons without mass can not distort spacetime to form a black hole. These are all mainstream physics. – lvella May 26 '20 at 12:28
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    Yes, that's the wrong thing to write. That's exactly the point. The valid expression is $E_0 = mc^2$ not $E=mc^2$. In other words $m=E/c^2$ is always wrong unless $E$ happens to be the rest energy $E_0$. –  May 26 '20 at 12:39
  • Let us continue this discussion in chat. –  May 26 '20 at 12:39
  • Relativistic mass isn't wrong, but it can be a misleading or confusing concept (as evidenced by numerous questions on this site), so we prefer to avoid it here on Physics.SE, as explained here. OTOH, it can be a handy concept, but you do need to be careful when you use it; see this old Usenet Physics FAQ article. – PM 2Ring May 26 '20 at 15:37
  • @PM2Ring If on one hand it is misleading as the mass of a relativistic moving object, who would have different masses depending on the observer, on the other hand, without it, it is very difficult to explain that "the mass of this battery is the mass of its components plus this stuff dependent on the stored energy that has mass unit but is totally not a mass". The answer you linked (and Okun's paper that started all this in the 80s) only deals with the first problem, and left the second problem unanswered (and several other problems raised by the rebuttal paper by Sandin). – lvella May 26 '20 at 16:14
  • Hey, I agreed that relativistic mass can be convenient, and the Usenet FAQ gives several examples of that. But I've also seen plenty of occasions where it's misleading. A common case is people asking if a body with high speed turns into a black hole, eg https://physics.stackexchange.com/q/3436/123208 – PM 2Ring May 26 '20 at 16:20
  • Your "second problem" is resolved by realizing that relativity is under no obligation to respect one's Newtonian expectations. The mass of the battery is obviously different than the sum of the masses of its components because when the momentum of the battery is zero, its energy (i.e. rest energy) does include potential energy and since mass is simply rest energy, mass would also come from this potential energy which is non-zero in the rest frame. –  May 26 '20 at 17:05
  • What Newtonian expectation? That mass is proportional to energy (and vice-versa)? – lvella May 26 '20 at 17:13
  • The Newtonian expectation that the mass of a bunch of particles is summation of their individual masses. –  May 26 '20 at 17:29
  • So, besides the particles mass, you add their "rest energy"? What "rest energy" would you add to a hot water tank to make up for the mass difference to a cold water tank? – lvella May 26 '20 at 18:09