Mass That ClicksEpisode 1 / Starting from "what is mass"

"Can stop = heavy" — now, properly, as an equation

Being able to stop, that is mass Light can't stop, so its mass is zero. Mass means "being able to have a rest frame," and it is
"the energy that remains once you strip out all the momentum." A single equation explains light and matter at once.

Tools you need: multiplication, speed = distance ÷ time, square roots Hook equation: $E^2=(mc^2)^2+(pc)^2$

In the intro version we said "weight is being able to stop." This time we'll verify that properly, as an equation. The key is a single equation, $E^2=(mc^2)^2+(pc)^2$. Read it, and "what mass is" comes into astonishingly sharp focus ── mass is the "floor" of energy that remains even after you strip out all the momentum. And light can't stop because that floor is zero.

01"Heavy" means hard to move (inertia)

Recall Newton's equation of motion, $F=ma$. Apply the same force $F$, and the bigger the mass $m$, the smaller the acceleration $a$ ── that is, harder to move. An empty cart pushes easily; a fully loaded one won't budge under the same force. This "resistance to being moved" is inertial mass, the star of today's episode.

A connecting voice Mass has another face too: "how easily it is pulled by gravity (gravitational mass)," and experimentally the two match exactly (the equivalence principle). This time we focus on the resistance-to-being-moved = inertia face. The gravity face will return later in the series, when we tie the size of the universe to mass.

02Light can't stop ── meaning "there is no rest frame"

A massless particle (the photon) moves at the speed of light $c$ from every point of view. No matter how fast you chase it, you can never overtake it. So for a photon there is no "vantage point from which it looks stationary (a rest frame)." A massive particle, on the other hand, can be overtaken and brought into a rest frame ── you can build a viewpoint from which it looks stationary.

The plainest definition of mass

Has mass　⇔　can have a rest frame (can stop)
Zero mass　⇔　light speed in every frame (can't stop)

A connecting voice (toward Episode 2) "Being able to stop" actually also means gaining one extra degree of freedom. Light, which can't stop (spin 1), has two polarizations (transverse only). The $W,Z$ particles, which can stop, have three (a longitudinal mode sprouts). This "one longitudinal mode sprouting" is the fingerprint of mass, and why light gets to stay at zero is tackled head-on in Episode 2, "symmetry protects mass."

03Mass and energy are the same ledger ── $E=mc^2$

Even an object at rest actually holds energy. That is its rest energy.

The energy when at rest

$$E_0 = mc^2$$

Since $c^2$ is a staggeringly large number, a tiny bit of mass has an enormous amount of energy sleeping inside it. Let's actually work out how much.

Let's try it ── turning 1 gram into energy

Substitute $m=1$ g $=10^{-3}$ kg, $c=3\times10^8$ m/s

$$E=mc^2=(10^{-3})\times(3\times10^8)^2=9\times10^{13}\ \text{J}$$

$9\times10^{13}$ joules is more energy than the Hiroshima-type atomic bomb (roughly $6\times10^{13}$ J). The mass of a single sugar cube has the power to move a city compressed inside it ── mass is a lump of energy that dense.

When an object is moving, its energy increases by the amount of its momentum $p$. The equation that writes rest, motion, and light all in one line is today's hook equation.

The energy–momentum–mass relation (the one line that bundles it all)

$$E^2=(mc^2)^2+(pc)^2$$

•At rest ($p=0$) → $E=mc^2$ (rest energy = the face of mass)
•Zero mass ($m=0$) → $E=pc$ (photon; momentum alone is energy)

This single line explains matter ($m>0$) and light ($m=0$) at once. They were just two ways of reading the same equation.

◇ ◇ ◇

04Mass is "the energy that remains once you strip out the momentum"

Here is the heart of this episode. In the equation $E^2=(mc^2)^2+(pc)^2$, shrink the momentum $p$ step by step and $E$ goes down. But ── however far you push it down, it can never go below $mc^2$. At $p=0$ it hits bottom exactly.

What mass really is (in equation form)

Mass $m$ = the floor of energy that remains even when you set the momentum to zero ($E$ at $p=0$).
The photon's floor is zero ($m=0$), so stripping out its momentum makes it vanish ── which is why it can't stop.

Figure: the relation between energy $E$ and momentum $p$. With mass (blue), an $mc^2$ "floor" remains even at $p=0$. Zero mass (amber) passes through the origin and vanishes when it stops. Move $p$ with the slider.

Momentum p (faster to the right; in units of mc)1.50

Move momentum p and E changes.

With mass E=√((mc²)²+(pc)²) Zero mass E=pc (light)

The blue curve stops at a height of $mc^2$ even when you strip out the momentum. The amber line (light) slides all the way down to the origin. This "floor you can't erase" is exactly mass. And this viewpoint ── the lowest energy that remains no matter what you strip out ── returns later in the series, in a deeper form, as "the smallest mass the universe allows" and "the mass gap." Mass is a kind of floor you can't erase.

05So mass is not "the sum of the parts"

Finally, the fact that amazed you in the intro, now as an equation. The proton's mass is about $938$ MeV. Yet even adding up the rest masses of the three quarks inside it (two up, one down) gives only about $9$ MeV.

Let's try it ── the breakdown of the proton's weight $$\frac{\text{rest mass of 3 quarks}}{\text{proton mass}}\approx\frac{9\ \text{MeV}}{938\ \text{MeV}}\approx 1\%$$

The remaining ~99% is the energy of the strong force (the gluon field) that confines the quarks, turned into mass by $E=mc^2$. Mass was not "the sum of the parts" but trapped energy.

So mass wells up in at least two ways ── the mass received directly from the Higgs, like the electron's, and the mass born from the energy of confinement, like the proton's. In Episode 3 we'll carefully tell these two apart.

The honest line

The "mass" in $E=mc^2$ refers to rest mass (invariant mass). The phrasing "you get heavier as you move faster" (relativistic mass) is not standard usage anymore ── what increases is energy, while the mass $m$ is constant regardless of speed. We untangle this misconception in detail in Bonus ②.

Also, this time we deliberately keep the watchword $c\cdot t=\text{constant}$ out of the spotlight. That mass is "being able to have a rest frame" or "a floor of energy" is standard physics that holds without invoking $c\cdot t$. Where this lens legitimately works is in Episode 4, where a scale emerges, and Episode 6, where the size of the universe sets the smallest mass. Not forcing it to work where it doesn't ── that was the lesson of the sister series' Bonus ③.

Practice problems (solvable with today's equation alone)

Explain in one line why a massless photon "can't stop," using $E^2=(mc^2)^2+(pc)^2$.
See the answer
If $m=0$ then $E=pc$. To stop (to minimize $E$) you need $p=0$, but then $E=0$ ── which is as good as not existing. Stripping out the momentum makes it vanish, so light has no choice but to keep moving forever (= no rest frame can be defined).
How many joules is the rest energy of a 1 kg object ($c=3\times10^8$ m/s)?
See the answer
$E=mc^2=1\times(3\times10^8)^2=9\times10^{16}$ J. That's 1000 times the value for 1 gram ($9\times10^{13}$ J). Enormously larger.
In one line, where does about 99% of the proton's mass come from?
See the answer
The energy of the strong force (the gluon field) that confines the quarks, turned into mass by $E=mc^2$. The total rest mass of the quarks — the "parts" — is only about 1%.

Episode 1 summaryMass is "an energy floor you can't erase"

Mass means "being able to stop" = being able to have a rest frame. In equation form, it's the floor of energy that remains even after stripping out the momentum, $E=mc^2$, in $E^2=(mc^2)^2+(pc)^2$. Light has a floor of zero ($m=0$), so it can't stop and always runs at light speed. And mass is not the sum of the parts but also trapped energy (99% of the proton).

"Mass as a floor you can't erase" ── this one phrase is the backbone of the whole series. Why is the zero floor (light) protected? How does a finite floor come to be? And what sets the minimum value of the floor? From the next episode on, we'll turn over the "origin" of that floor one sheet at a time.

Next time ── Episode 2 "Zero mass is protected by symmetry." Why does the photon get to stay at a "zero floor"? The mechanism by which gauge symmetry forbids mass, the story of one longitudinal mode sprouting on particles that can stop, and the moment mass switches on (when symmetry breaks). Zero is not obvious — there was a watchman standing guard.

This document is Episode 1 of the "Mass That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The relativistic energy–momentum relation $E^2=(mc^2)^2+(pc)^2$, the rest energy $E_0=mc^2$, the fact that massless particles are fixed at light speed by $E=pc$, and the fact that the bulk of the proton's mass (about 99%) comes not from the quarks' rest masses but from the binding energy of quantum chromodynamics (the gluon field) are all established physics. The "mass" in this piece refers to rest mass (invariant mass); so-called relativistic mass is not used. The figure is a schematic drawing of the energy–momentum relation in units where $c=1,\,mc^2=1$. ── To print, use "Print" in your browser and choose "Save as PDF" (in the printed version the slider and answers are frozen and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, moving p with the slider reveals the "floor" of mass. "See the answer" opens the solutions.

Being able to stop, that is mass Light can't stop, so its mass is zero. Mass means "being able to have a rest frame," and it is"the energy that remains once you strip out all the momentum." A single equation explains light and matter at once.

01"Heavy" means hard to move (inertia)

02Light can't stop ── meaning "there is no rest frame"

03Mass and energy are the same ledger ── \(E=mc^2\)

04Mass is "the energy that remains once you strip out the momentum"

05So mass is not "the sum of the parts"

Episode 1 summaryMass is "an energy floor you can't erase"

Being able to stop, that is mass Light can't stop, so its mass is zero. Mass means "being able to have a rest frame," and it is
"the energy that remains once you strip out all the momentum." A single equation explains light and matter at once.