
Three Meanings of E=mc² - LisaDziuba
https://medium.com/starts-with-a-bang/the-three-meanings-of-e-mc%C2%B2-einsteins-most-famous-equation-a0ec1549b4cd
======
ocfnash
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.

~~~
cohomologo
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.

~~~
hjaved
Thank you.

Would like to understand that why does having constituents with binding energy
reduce the total energy of the system instead of adding to it?

~~~
davrosthedalek
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.

------
plopilop
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](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."

~~~
gus_massa
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.)

More details:
[https://en.wikipedia.org/wiki/Mass_in_special_relativity#The...](https://en.wikipedia.org/wiki/Mass_in_special_relativity#The_relativistic_mass_concept)

~~~
plopilop
I did not know about this relativistic mass, but doesn't wikipedia says it is
only true if you consider the mass as m = F/a rather than m = p/v?

------
enugu
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](https://www.youtube.com/watch?v=eOCKNH0zaho)

Also interesting is how conservation laws for collisions are handled much
better in relativistic setting.

------
millstone
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.

~~~
grondilu
> 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?

~~~
millstone
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).

~~~
cgmg
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.

~~~
cgmg
Also, the magnitude of the momentum 4-vector is a Lorentz scalar, meaning it
is invariant. Hence the rest mass m = |P| is invariant.

------
akubera
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.

~~~
cousin_it
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".

------
contravariant
>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).

------
doctoboggan
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.

[https://www.youtube.com/watch?v=gSKzgpt4HBU](https://www.youtube.com/watch?v=gSKzgpt4HBU)

[https://www.youtube.com/watch?v=fHRqibyNMpw](https://www.youtube.com/watch?v=fHRqibyNMpw)

[https://www.youtube.com/watch?v=kixAljyfdqU](https://www.youtube.com/watch?v=kixAljyfdqU)

------
nyc111
" 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.

~~~
ythn
This comment reads like the ravings of a pedant who is unable to relate to the
layman.

~~~
nyc111
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.

~~~
betenoire
Motion is relative, no? I see things at rest from where I'm standing.

~~~
nyc111
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.

~~~
cgmg
> Motion is absolute.

What are you talking about? Motion is definitely _not_ absolute, and only
Galilean invariance is needed to see that.

~~~
nyc111
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.

~~~
dragonwriter
> Motion is absolute means that all is in motion

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.

~~~
nyc111
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.

~~~
cgmg
> 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".

------
stcredzero
"E = mc^2 is Wrong" (Actually, just incomplete.)

[https://www.youtube.com/watch?v=eOCKNH0zaho](https://www.youtube.com/watch?v=eOCKNH0zaho)

------
ythn
If mass created from "pure energy" is always in matter/antimatter pairs, why
is the ratio of matter/antimatter so skewed in favor of matter?

~~~
davrosthedalek
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.

~~~
100ideas
Most of the antimatter ended up as early supermassive black holes.

------
ramzyo
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!

------
ucaetano
> 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.

------
goldenkey
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.

~~~
millstone
Huh? If you have a system with nonzero momentum, it can't later have zero
momentum. The "prescribed velocity" can't be lost, it must be conserved.

~~~
goldenkey
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

