One corollary of E=mc^2 is that the rest mass of a bound state of fundamental particles will not equal the sum of the rest masses of the individual constituents. For example, the rest mass of a Helium-4 nucleus turns out to be less than the sum of the rest masses of the four nucleons of which it is composed. The difference is only about 0.7% but is famously quite a lot of energy when interpreted using E=mc^2.
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.
I think you've got that last part backwards. When you have constituents with binding energy, that reduces the total energy of the system and reduces the mass. So it seems that the mass of the proton is quark mass + gluon kinetic energy - bonding energy, and most of the mass comes from gluon kinetic energy.
You have to think about it the other way around: The constituents don't have binding energy. That energy becomes available when you let them bind, and you have to pay it back if you want to unbind them.
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.
It's a little bit tricky with the protons, as free quarks don't exist. One could argue that the mass of free quarks is infinite, which explains hadronization: it is energetically cheaper to produce quarks out of nothing so that the quarks are not free.
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.
I am not a physicist, but isn't all mass just energy if you just dig deep enough? As far as I know there is nothing that has [some] mass that is not ultimately explained as kinetic or potential energy.
As I explained in an other article here on HN, another meaning (and the most important one imo) is the one Einstein actually wrote about: the inertia of a particle depends on its energy.
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."
The problem is that the "mass" in the direction of the movement is different than the "mass" in the perpendicular direction. So most modern books try to avoid the relativistic mass "m" and use only the rest mass of the particle "m0".
(Still, the "relativistic mass" can be useful to make a few back of the envelope calculations, but you must be careful.)
What isnt usually mentioned is that this equation is a special case valid in some circumstances. The more general relation is E^2=(mc^2)^2+(pc)^2 which includes momentum term, and applies to photons which dont have a mass. Here's a simple video about this from Fermi Lab, https://www.youtube.com/watch?v=eOCKNH0zaho
Also interesting is how conservation laws for collisions are handled much better in relativistic setting.
E=mc^2 is surely one of the most misunderstood equations in physics.
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.
> 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.
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?
Yes, it is not immediately obvious how mass ought to be undershood in the context of relativity. First attempts treated mass as a directed vector, with transverse and longitudinal components for an accelerated particle (now we would say that the mass is fixed, it is the momentum that changes non-linearly and just use F=dp/dt).
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).
This. The quantity that “truly” exists is the energy-momentum 4-vector P. A vector exists independently of any particular reference frame. Its components, on the other hand, depend on such a reference frame.
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.
(Relativistic) mass is totally frame dependent in SR. That's why SR normally talks about the rest or invariant mass, which is the mass in a special frame, the rest frame of the object.
You can construct a reference frame such that a rotating object is rotating but not translating. You can also construct a non-inertial reference frame such that it's not even rotating.
Not quite, the laws of physics don't apply in a rotating reference frame. For example the speed of light stops being a constant.
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.
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.
That's what the Christoffel symbols already do - it's part of the formalism, there's nothing to add!
Consider a reference frame rotating at say 1 RPM. At 1 light year from the center a 'stationary' particle in a non rotation reference frame is now moving in a circle at 2 * π light years per minute. At 2 light years from the center it's moving 4 * π light years each minute and that continues the further from the center you get.
GR is fine with most reference frames as long as it's translation. Rotation is however not ok.
GR is fine with that - what will happen is that there'll be a sign change in the metric (the 'time' coordinate will become spacelike, whereas the 'angular' coordinate will become timelike), making it impossible to sit at a fixed value of the rotating frame's angular coordinate (which would correspond to faster-than-light motion in the non-rotating, inertial frame).
That's just a notation for doing exactly what I am describing.
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.
> It seems to me that before Newton, people only had a vague notion of what a force is.
> When Newton stated that F=ma, he defined the concept of force precisely.
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.
"I thought that F=ma is actually a definition of the concept of force."
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.
> if mass can be converted into energy, then you would have more energy and less mass, so E=mc^2 would no longer hold
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".
The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency.
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.
> The conversion you are talking about is true only if you annihilate your fuel (which creates photons), and then you convert photons into speed in some way with 100% efficiency.
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?
Well, in case of a rocket, there are two phenomenon resulting into thrust.
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.
While it might be true that the temperatures of rocket fuel burning are enough to create plasma and some reactions that lead to mass turning into energy, I think Jabavu is right in that the energy comes from the chemical reaction, not from E=mc2.
Well, yes, but it's the same thing, really. It's what my last paragraph above is about.
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).
From chemical reactions (breaking chemical bonds), as opposed to nuclear reactions. There's a certain amount of energy required to shove a bunch of stable molecules together into a (more) unstable molecule (fuel, explosive). When that unstable molecule (fuel) is burned, it breaks apart into more stable products and releases the energy that was required to combine them in the first place.
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 ?
Indeed, in the m term, not in the p, which would be the total momentum of the water system in your frame.
For more mind-boggling numbers see for example [1]
Holy sh*t! Is this effect typically taught / introduced in a GR course? Why did I not know this!?
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?
Later we learn that mass is determined by energy and momentum, both of which are conserved, so mass must be conserved too.
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:
The use of the term "pure energy" always grates me; it gives the impression that energy is an entity unto itself, which is not the case. I realize that it's used as a proxy for photons most of the time, but I still always think it's misleading.
Oh, this is a pet peeve of mine. Jim Butcher does this constantly in the Dresden Files series. I keep coming up with lampoons like "the witch countered with a blast of pure angular momentum that sent him spinning" or "after a visit to the bank, my wallet was bursting with pure price".
Yes! I just volunteered as a parent helper to a scientist in the school program. I was gritting my teeth as the "scientist" explained that light was "pure energy".
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.
>But under the laws of special relativity, mass simply couldn’t be the ultimate conserved quantity, since different observers would disagree about what the energy of a system was.
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).
The YouTube channel PBS SpaceTime gives an excellent overview of these same ideas. The presenter is great at making things as simple as possible, but not simpler.
" But even plain, old, regular mass at rest has energy inherent to it..."
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.
You may be right about my comment about his positioning of the comma but my statement that all is motion and nothing but motion is not pedantic or trivial.
No. Motion is absolute. Rest is relative. That is, rest is only an appearance. Since you are observing from earth everything you see is in motion since the earth is moving. So there is no absolute rest. And relative rest is just another name for motion.
Therefore, no object can have a property that exists only when the object is at rest. Because there is no absolute rest.
What is Galilean invariance? You are below deck in a ship moving uniformly on a smooth sea. No experiment can tell you if the ship is docked or if it has motion relative to the see. Assume that the ship is docked. In that case you say that the ship is at rest. And you attribute to the ship some properties because it is at rest. But is the ship at rest absolutely? Not at all. The ship is moving with the earth. So do you deny that the earth is moving? I assume not. If so how can you claim that the ship is at rest?
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.
> If so how can you claim that the ship is at 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.
Can you give an example of a system which is not based on axioms? There is none. There may be hidden assumptions but there will always be axioms. Axioms are simply definitions that are kept constant. In your case you make the assumption that absolute rest exists. The axiom of the universality of motion is a good one because it agrees with the observations.
> In your case you make the assumption that absolute rest exists.
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".
Yeah, I agree, that sounds like word games. The whole point of it being relative is that rest and motion are indistinguishable without a point of reference. That point is the relative part.
But this is not word games. The statement is: "Nothing is absolutely at rest." Where is the word game in this statement? Or "all is in motion." These are clear statements. If people heeded Heraclitus who said this first thousand years ago astronomy would have advanced much faster. Astronomers built a whole geocentric system by assuming that the earth was stationary. They assumed stars were stationary. These turned out to be wrong. There is no absolute rest. 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.
> The statement is: "Nothing is absolutely at rest."
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?
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.
You say "inertial reference frame at rest with respect to any massive particle..." Is your particle moving with the reference frame? I believe so. How is it at rest then? 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?
> Is your particle moving with the reference frame? I believe so.
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.
That is indeed the one million dollar question. Physicist search for a reaction which breaks this balance to explain the matter/anti-matter imbalance of the observed universe.
Fascinating. After first taking a crack at learning about this stuff decades ago, I still have trouble retaining it to the degree that I can succinctly explain its profoundness. This article does a great job at doing just that, and reigniting the sense of wonder and amazement at Einstein's achievements!
> one of the simplest but most powerful equations ever to be written down
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.
The most important part is the conversion of a vector quantity into a scalar. Energy packets, photons, always have a prescribed velocity or direction. When mass is created, this direction is lost. It shows us a fundamental anisotropy of our universe. Total momentum is conserved..but since 0 momentum is degenerate and has no direction, we get the mixing of isotropic emissions, that should in theory, cancel out any biases that may have once existed.
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.
We are talking about a net momentum of zero. It can be in all kinds of different forms without violating conservation. Exchanging a left and right moving particle for up and down ones, changes the system but retains global conservation
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.