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A hydrogen atom has less mass than the combined
masses of the proton and the electron that make it up.
That's right, less.
How can something weigh less than the sum of its parts?
Because of this.
And today, we're going to clarify
what the most famous equation in physics really says.
E equals mc squared is probably the most famous equation in all
of physics, but in his original 1905 paper,
Einstein actually wrote it down differently,
as m equals E divided by c squared.
That's because at its core, this cornerstone of physics
is really a lesson in how to think about what mass is.
You'll often see statements like "mass is a form of energy"
or "mass is frozen energy" or "mass
can be converted to energy."
That's the worst one.
Unfortunately, none of these statements is quite correct,
so trying to make sense of them can be frustrating.
I think instead we can get a better sense of what
m equals E over c squared means if we start with some things
that it implies that seem at odds
with our everyday experience of mass.
Here's a pretty mind blowing one.
Even if two objects are made up of identical constituents,
those objects will not in general have equal masses.
The mass of something that's made out of smaller parts
is not just the sum of the masses of those parts.
Instead, the total mass of the composite object
also depends on, one, how it's parts are arranged,
and two, how those parts move within the bigger object.
Here's a concrete example.
Imagine two windup watches that are identical atom for atom
except that one of them is fully wound up and running,
but the other one has stopped.
According to Einstein, the watch that's running
has a greater mass.
Well, the hands and gears in the running watch are moving,
so they have some kinetic energy.
There are also wound up springs in the running watch that
have potential energy, and there's
a little bit of friction between the moving
parts of that watch that heats them up ever so slightly
so that its atoms start jiggling a little bit.
That's thermal energy, or equivalently,
randomized kinetic energy on a more microscopic level.
OK, got it?
Now, what M equals E over c squared says
is that all of that kinetic energy and potential energy
and thermal energy that resides in the watch's parts
manifests itself as part of the watch's mass.
You just add up all that energy, divide it
by the speed of light squared, and that's
how much extra mass the kinetic and
potential and thermal energies of the parts
contribute to the whole.
Now since the speed of light is so huge,
this extra mass is tiny, only about a billionth
of a billionth of a percent of the total mass of the watch.
That's why, according to Einstein, most of us
have always incorrectly believed that mass
is an indicator of the amount of matter in an object.
In everyday life, we just don't notice the discrepancy
because it's so small, but it's not zero.
And if you had perfectly sensitive scales,
you could measure it.
So wait a second.
Am I saying that individually, the mass of the minute hand
is bigger because the minute hand is moving?
That's an outdated viewpoint.
Most contemporary physicists mean mass while at rest,
or "rest mass," when they talk about mass.
In modern parlance, the phrase "rest mass" is redundant.
There are lots of good reasons for talking
this way, among them that rest mass is
a property all observers agree about,
much like the space-time interval
that we discussed in a previous episode.
This all gets a little bit more complicated
in general relativity, but we'll deal with that another time.
For us, today, the m in m equals E over c squared is rest mass.
You can think of it as an indicator
of how hard it is to accelerate an object or an indicator
of how much gravitational force an object will feel.
But either way, a ticking watch simply has more of it
than an otherwise identical stop watch.
So more examples might help to clarify what's going on here.
Whenever you turn on a flashlight,
its math starts to drop immediately.
Think about it.
The light carries energy, and that energy
was previously stored as electrochemical energy
inside the battery, and thus manifesting as part
of the flashlight's total mass.
Once that energy escapes, you're not weighing it anymore.
And yes, since the sun is basically
an enormous flashlight, its mass drops
just by virtue of the fact that it shines by about 4
billion kilograms every second.
Don't worry, Earth's orbit is going to be fine.
That's just a billionth of a trillionth of the sun's mass,
and only 0.07% of the sun's mass over its entire 10 billion
year lifespan.
So does this mean that the sun converts mass to energy?
This isn't alchemy.
All the energy in sunlight came at the expense of other energy,
kinetic and potential energy, of the particles that
make up the sun.
Before that light was emitted, there was simply more kinetic
and potential energy contained within the volume of the sun
manifesting as part of the sun's mass.
Those 4 billion kilograms that the sun loses every second
is really a reduction in the kinetic and potential energies
of its constituent particles.
What we've been weighing is the energies
of the particles in objects all along.
We just never noticed it.
Another example.
Suppose that I stand with a flashlight
inside a closed box that has mirrored walls
and is resting on a scale.
Will the reading on the scale change
if I turn on the flashlight?
Interestingly, no.
The flashlight alone will lose mass,
but the mass of the whole box and its contents
will stay fixed.
Yes, it's true that the scale is registering
less electrochemical energy, but it's also
registering an exactly equal amount of extra light energy
that we're not allowing to escape this time.
That's right, even though light itself is massless,
if you confine it in a box, its energy
still contributes to the total mass of that box via m
equals E over c squared.
That's why the reading on the scale doesn't change.
OK, here's the really fun part.
In every example we've done so far,
things have weighed more than the sum
of the parts that make it up.
But at the top of the episode, I stated
that the mass of a hydrogen atom is less than
the combined masses of the electron and the proton
that make it up.
How does that work?
It's because potential energy can be negative.
Suppose we call the potential energy of a proton
and electron zero when they're infinitely far apart.
Since they attract each other, their electric potential energy
will drop when they get closer together, just
like your gravitational potential energy drops when
you get closer to the surface of Earth, which
is also attracting you.
So the potential energy of the electron and proton
in a hydrogen atom is negative.
Now the electron in hydrogen also has
kinetic energy, which is always positive, due to its movement
around the product proton.
But as it turns out, the potential energy
is negative enough that the sum of the kinetic and potential
energies still comes out negative,
and therefore m equals E over c squared also
comes out negative, and a hydrogen atom
weighs less than the combined masses of its parts.
In fact, barring weird circumstances,
all atoms on the periodic table weigh less than the combined
masses of the protons, neutrons, and electrons
that make them up.
The same is true for molecules.
An oxygen molecule weighs less than two oxygen atoms
because the combined kinetic and potential energies
of those atoms once they form a chemical bond is negative.
What about protons and neutrons themselves?
They're made of particles called quarks, whose combine mass
is about 2,000 to 3,000 times smaller than a proton's
or neutron's mass.
So where does the proton's mass come from?
Basically, quark potential energy.
Veritasium did a nice episode on this
that you can click over here to view.
Every time he says "gluons" in that video,
just substitute "quark potential energy,"
and you'll have a roughly correct picture
of what's going on.
All right, what about the masses of electrons and quarks?
At least in the standard model of particle physics,
they're not made up of smaller parts,
so where does their mass come from?
Is it some kind of baseline mass in the pre-Einstein sense
of the word?
Well, that's a subtle question, but crudely speaking, you
can think even of this mass as being
a reflection of various kinds of potential energies.
For instance, there's the potential energy
associated with the interactions of electrons and quarks
with the Higgs field.
And there's also potential energy
that electrons and quarks have from interacting
with the electric fields that they themselves produce,
or in the case of quarks, also with the gluon fields
that they themselves produce.
OK, what about matter-antimatter annihilation?
Doesn't that have to be thought of as mass
being converted into energy?
Interestingly, no.
There's a way to conceptualize even this process
as simple conversions of one kind of energy
to another-- kinetic, potential, light, and so forth.
You never need mass to energy alchemy.
But please take my word for it, you
don't actually have to talk about converting mass to energy
Instead, the punchline of this episode
has been that mass isn't really anything at all.
It's a property, a property that all energy exhibits.
And in that sense, even though it's not
correct to think of mass is an indicator of amount of stuff
in the material sense, you can think of it
as an indicator of amount of energy.
So without realizing it, you've really
been measuring the cumulative energy content of objects
every time you've ever used a scale.
I'm going to wrap up with two comments.
First, Einstein's original paper on this topic
is only three pages long and not that hard to read.
We've linked to an English translation of it down
in the description, and I strongly
encourage you to check it out.
Second, I want to leave you with a challenge question
to test your understanding.
First, some background.
Suppose you put two identical blocks side by side on a scale
and weigh the combo, then stack them one on top of each other
and weigh them again.
The second configuration has more gravitational
potential energy than the first because the second block
is higher up, so it will have more mass than the first.
Keep that in mind for the following challenge question.
Suppose that every person on Earth
simultaneously picks up a hammer from the ground.
Would the total mass of the planet
increase, and if so by how much?
Do not put answers in the comments section.
That, as always, is for your questions.
Instead, submit your responses by email
to with the subject line "E=MC2
Submit your answers no earlier than 5:00 PM New York City
local time on this date.
I want to give everyone a chance to think about it and everyone
a chance to respond, because we're going to shout out
the first five correct answers, which must also
have correct explanations to count,
on the next episode of "Space Time."
Last week we talked about NASA spinoffs.
First off, you guys mentioned some things,
to set the record straight, that are not
NASA spinoffs-- microwave ovens, Tang, Velcro, cordless power
tools, the space pen, MRI machines-- none of those
are NASA spinoffs.
But you did mention some NASA spinoffs that we missed.
Ryan Brown brought up space blankets,
David Shi brought up oxygen permeable contact lenses,
and UndamagedLama2 brought up robotic endoscopic surgery.
Nice finds.
Jay Perrin, who's an airline dispatcher
and former firefighter, vouched for the importance
of being able to see through smoke and fog.
Great to hear from someone with first hand experience
with NASA tech.
jancultis, or "yawn"-cultis, points out the NASA is great
but inefficient and has lots of room for improvement.
I agree, and I think NASA does, too.
And finally, to Ms. Croco's fourth grade class
at Dunbar Hill Elementary in Hamden, Connecticut,
thanks a lot for watching the show.
And yes, Ms. Croco and I really are friends.
Stop saying she's making it up.