Three Meanings of E=mc²
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It's a lot more fun if we go one level down though: protons are far HEAVIER than the sum of their constituents. Apparently almost 99% of the rest mass of a proton is the binding energy of the zero-rest-mass gluons that hold their three quarks together.
I think it's fun that the vast majority of the mass of ordinary matter, of which we are all constituted, is best explained as gluon binding energy via E=mc^2.
Would like to understand that why does having constituents with binding energy reduce the total energy of the system instead of adding to it?
Example: If you have a proton and an electron, making a hydrogen atom will release about 13.6 eV (if it goes to the lowest energy state). You have to spend that 13.6eV to ionize the hydrogen, i.e. to get the electron far away from the proton again.
What is certainly true is that, writing down the QCD Lagrangian, there is nothing which tells you the scale of the proton mass, nor the somewhat different scale of the pion mass. This is in sharp contrast to the Hydrogen case, where the masses which appear in the Lagrangian are very close to the total mass of the Hydrogen.
The thing is, you don't read E = mc² but rather (as Einstein wrote in his surprisingly easy to understand 2-pages paper [0]), m = E / c². The direct interpretation of this formulation is that inertial mass is actually just a side effect of the energy of a particle. Put into a catchy phrase, mass is energy at rest.
Edit :
[0]: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf, translated in english. Note the understatement of the sentence before the last one: "It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test."
(Still, the "relativistic mass" can be useful to make a few back of the envelope calculations, but you must be careful.)
More details: https://en.wikipedia.org/wiki/Mass_in_special_relativity#The...
Also interesting is how conservation laws for collisions are handled much better in relativistic setting.
The best interpretation is as part of the full momentum-energy equation (letting c=1): E^2 = p^2 + m^2. This simply says that the energy E of a system is a combination of energy due to movement (p) and energy due to mass (m). At rest (p=0) this reduces to E=m, or E=mc^2 if you kept track of units.
> Even masses at rest have an energy inherent to them.
This is a real insight.
> Mass can be converted into pure energy. This is the second meaning of the equation, where E = mc² tells us exactly how much energy you get from converting mass
This is a pop-sci explanation, but it falls apart when you dig a bit. Does F=ma tell you that "force can be converted into acceleration?" Of course not; it tells you that force implies acceleration, and vice versa.
Or: if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold. It can't be both an equivalence and an exchange ratio.
> If you take a photon and and electron and smash them together, you get a photon and an electron out. But if you smash them together with enough energy, you’ll get a photon, and electron, and a new matter-antimatter pair of particles out. In other words, you will have created two new massive particles.
Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too.
I thought that F=ma is actually a definition of the concept of force. I mean, I may be wrong but it seems to me that before Newton, people only had a vague notion of what a force is. I suppose people considered it to be, in modern terms, a vector with a magnitude, and the direction of the vector was obvious, but I doubt they had any meaningful idea of what the magnitude was. When Newton stated that F=ma, he defined the concept of force precisely.
Honestly, I suspect something similar is true about E=mc^2, except Einstein discovered that formula instead of positing it. After all, we know what Energy is, at least from quantum mechanics (E=ihd/dt), but do we know what mass is? We can't say it's the ratio between force and acceleration, since that would be circular.
I suspect one can say mass is just a very dense form of energy. So dense that its different order of magnitude makes it look like it's a different thing.
As for the distinction between impulsion and mass, well can't it be said that Energy (or mass, since I'm arguing it's the same thing) is actually a four-vector, and as such it has different projections on time and space depending on the frame of reference?
Newton wasn't the first to define the concept of force, nor the first to formulate the basic laws of motion. E.g. Galilei had formulated both the first and second laws in some sense, restricted to cases such as straight line motion on a plane. The laws came out of experiments.
Newton's brilliant work was in collecting the three laws, generalizing them and realizing their combination applied to celestial mechanics as well as the mundane brick sliding down a plane.
Mass should rightly be considered more inherent in SR that energy. Energy in relativity is coordinate dependent, but mass is not: all observers agree on a system’s mass, because it is invariant.
Re: the last paragraph, yes you can project the four vector onto time and space, and you get energy and momentum respectively. So energy is not the four vector itself but its projection onto the time axis (and mass is of course its magnitude).
So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.
Yes they do and it's precisely the point of GR.
GR is fine with most reference frames as long as it's translation. Rotation is however not ok.
GR deals with curved spacetime, but the only way to work in rotating spacetime is by changing how you calculate what's going on. Which means your calculations must be mapped and don't generalize.
EX: Try and do an actual calculation for say two electrons hitting each other at say 0.5c. In rotating, non rotating inertial, and non rotating non-inertial reference frames of your choice.
That's what the Christoffel symbols already do - it's part of the formalism, there's nothing to add!
Seen from a given reference frame, the temporal component of P is the energy E, its spatial components are the momentum 3-vector p, and its magnitude is the rest mass |P| = sqrt(E^2 - p^2) = m. The minus sign rather than plus sign is due to the Lorentzian signature. It is very helpful to visualize these quantities geometrically as 4-vectors in 4-dimensional space.
Yes, that is part of what millstone was getting at.
The root of the problem is that for a long time, people were taught since elementary school that the equality symbol was not symmetrical, but was something more like a => symbol; you take the things on the left and can produce the things on the right, usually via some vague idea of simplification. In other words, 2 + 2 => 4; "from 2 + 2 I can produce 4", but people with this misunderstanding (which is a lot of people) will resist 4 = 2 + 2, reading it as 4 => 2 + 2. They'll often have a hard time articulating the problem, but it'll boil down to the fact that you're not "allowed" to "unsimplify" like that. It turns out you can get quite far in the standard math curriculum with that misconception, somewhere round about "factoring equations by completing the square"[1], and even farther if you just learn those handful of exceptions of one-off tricks and never go into a field of work where one way or another you need to deeply understand what is going on.
(Some people complain that you can see this misconception in the standard programming definition of =, which runs the other direction, and isn't a statement that "these two things are equal" that may be true or false, but is a command to assign this value into that slot. It's a natural outgrowth of the incorrect = understanding.)
I remember in physics that it took us collectively quite a while to get that it was not F => ma, or ma => F, that it was not that there was a "force" and a "mass acceleration", but that they were the simply the same thing. There is no free-floating "force" that can be "converted" into "mass acceleration", they are so inextricably the same thing that there was no difference. And I was in a fairly advanced and educated class at the time.
While I tend to dislike the common core in many ways, one admirable thing about it is that it is attempting to correct this mistake; as soon as = is introduced for simple arithmetic, children are given ____ = 2 + 2 as often as 2 + 2 = ____, and only slightly later than that they are getting 2 + ___ = 4.
[1]: http://www.purplemath.com/modules/solvquad3.htm It's been a while since I did this stuff, and you may not have had the exact same order as me anyhow, but this is the first time I can think of where we had to deliberately "complexify" an equation in order to solve it. And I can still recall the gears skipping in my head as I tried to deal with this, even as to me now it's nothing at all and I do an analogous thing quite often.
I'm confused why this wouldn't hold? IANAP so I don't know if E=mc2 comes into play in a "non-nuclear" mass -> force conversion. Say a rocket with x kg of fuel mass lifting off with y J of energy spent. Naively I would say that before liftoff there was x kg of fuel which is equivalent to xc^2 energy and no(?) energy y = 0. At a later time (b) When the rocket has spent half of the fuel, the energy yb spent this far could be calculated as (x/2)c^2 and energy left in mass (xb) would (x/2)*c^2. So that the total amount of energy/mass is the same at start and at time b.
Hmm, this got really confused, so... I just hit "reply".
But really this never happens, rockets work by ejecting mass at a tremendous speed. Thus not all the mass of the fuel is not converted into energy.
But otherwise, yeah, converting mass to energy completely works. This is what happens in stars and nuclear bombs, actually.
Maybe it's hard to explain from a real world example like a rocket but if mass isn't converted to energy in that case, in the E=mc2 sense, where does the energy come from?
The first one is the action-reaction principle: fuel goes in one direction, rocket goes in the opposite one.
The second one is indeed the E=mc^2 equation: as the fuel burns, some of its mass (a negligible part, but still results in some decent amount of energy) is converted into energy and this energy can be used to propel the rocket. The rest of the mass is ejected, see first point.
So if you compare the mass of the rocket + fuel before the take-off and the mass of the rocket + burned fuel after the take-off, you will indeed see a delta, which comes from a partial conversion of mass into energy of the fuel during combustion.
Actually, as pointed out in an other comment, every reaction (or most of) leads to a decrease of the energy of the objects: an empty battery has less energy than a charged battery (yes I know I'm a genius for pointing that out). The difference of energy results, according to m = E/c², to a loss of mass. In the case of a battery, the delta is waaaay to small to be detected, but it exists.
E=mc² is not about nuclear reactions, it's about every reaction. A chemical reaction of combustion transforms some fuel molecules into new molecules. If you compute the mass of the products of the combustion and compare it to the mass of the fuel before burning, you will notice a sliiiiight difference (probably negligible, barely detectable). The disappearing mass has been converted to kinetic energy (and heat, but who cares).
If I heat 100 molecules of water ice to melting, the 100 molecules of liquid water aren't more massive. The energy went largely into kinetic energy, not changing the atomic number of the atoms. Ok, that's a phase change, not a chemical reaction. I've changed the momentum, not the mass. Are you saying this would be encompassed in the full form E^2=m^2 c^4 + p^2 c^2 ?
[1] https://www.huffingtonpost.com/quora/is-a-hot-object-heavier...
EDIT> Is there any way to derive this outside of GR?
EDIT2> I notice that you were careful to say "heavier", not more "massive". Does this point to the effect being part of the gravitational interaction specifically?
Not in a meaningful sense - the square of the total momentum 4-vector is of course conserved (and even frame-invariant), but it's not equal to the sum of the masses of the constituents:
M² = (E₁ + E₂)² - (p₁ + p₂)²
= E₁² - p₁² + E₂² - p₂² + 2E₁E₂ - 2p₁p₂
= m₁² + m₂² + 2(E₁E₂ - p₁p₂)I think people use this to mean "insubstantial" and to contrast versus matter. But that still leaves the problem of light pressure. I.e. a flashlight is like a very very weak rocket engine.
Maybe I'm being pedantic, but not only does classical mechanics already tell us that the energy of a system different for different for observers, but this by itself doesn't mean it's not conserved, merely that it's not a universal quantity.
Furthermore E = mc^2 doesn't break conservation of mass in any way, it's more accurate to say that it equates energy with mass, meaning conservation of energy and conservation of mass are one and the same. And in fact both still hold, when the energy and mass are replaced with their relativistic counterparts (E = mc^2 is obviously false using the classical notions of mass and energy).
https://www.youtube.com/watch?v=gSKzgpt4HBU
This sentence is so typical of sloppy so-called science writing in the US. What is "plain mass?" What is "old mass?" Or maybe he put the comma in the wrong place and he meant "plain old regular mass?" If so what is "plain old regular mass?" There is no such thing.
But more importantly, nothing is ever at rest in the known world. It is absurd to discuss the properties of something when it is at rest. It will never be at rest. There is no absolute rest.
Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.
What are you talking about? Motion is definitely not absolute, and only Galilean invariance is needed to see that.
An object is at rest with respect to its own reference frame.
> The academic descendants of the same academics who believed in absolute rest now call themselves physicists and make the same mistake and assume that absolute rest exists.
In other words, you think all physicists are wrong and you're right?
Motion is absolute means that all is in motion. This is an axiom. When you say motion is not absolute, can you give an example of an object which is at rest, which does not move with the earth and does not move with the galaxy? There is no such object.
That's not a usual use of “absolute”; “motion is universal” would be the normal way of expressing that idea.
> This is an axiom.
Well, you could have an axiomatic system in which that was an axiom, but I don't see why you would want to.
This is a straw man. No one is making that assumption. Something is in motion or at rest with respect to a reference frame. For some reason, you keep misinterpreting this statement and ranting about "absolute motion" and "absolute rest".
The ship is at rest with respect to its own reference frame.
> So do you deny that the earth is moving? I assume not.
The earth is at rest with respect to its own reference frame.
It is moving with respect to other things, such as the sun, or the center of the galaxy. Motion and rest are relative. This is the key concept, and I don't understand why you're having such trouble with it.
This is nonsensical, given that you can construct an inertial reference frame at rest with respect to any massive particle, and that reference frame is just as valid as any other inertial reference frame. That is one of the primary features of Relativity, so stating what you did implies you do not understand it very well.
It is not moving with respect to the reference frame.
> Do you have motion when you are in a plane? Or do you say you are at rest because the passenger next to you is moving with you?
You are having a hard time understanding reference frames. You are at rest with respect to the plane. You are at rest with respect to the passenger beside you (assuming they're sitting still, relative to the plane). You are not at rest with respect to the ground below.
The whole point of relativity is that no inertial frame is in an absolute state of "motion" or "rest". The laws of physics are the same in all inertial frames, whether it be your plane or the earth itself.
I always felt that E=mc² is essentially the Euler's identity of physics: a basic, simple, beautiful equation relating some core components of the entire field.
Our universe cares about conservation globally - but as scattering shows, at the quantum scale, our universe could care less about you switching a left and a right for an up and a down.