Neither "you get heavier the faster you move" nor "mass is conserved" is quite accurate
In Episode 1, mass was "the floor of energy that remains, that you can't erase, once you strip away momentum." This "floor" has an important property ── it's the same for everyone (invariant). Yet out in the world, two misconceptions that conflict with this property stubbornly survive ── "you get heavier the faster you move" and "mass is conserved." This time we correct both, going back to E=mc² and the Episode 1 formula. Where we end up is a little surprising ── the sum of rest masses is not conserved, and you can build a massive system out of nothing but massless particles.
Old textbooks had a phrasing: "relativistic mass \(m_{\rm rel}=\gamma m\) (larger the faster you go)." But we don't use it anymore. Go back to the Episode 1 formula and the reason is clear.
Move fast and both \(E\) and \(p\) increase. But \(m\) (the floor, the intercept) doesn't budge. What increases is energy, not mass.
The reason we dropped the phrase "relativistic mass" is that it fools you into thinking \(m\) changes with speed or direction, and it muddles the \(m\) in \(E=mc^2\). Mass is a "frame-independent property" of a thing ── speed it up, turn it around, and it doesn't change.
This is the crux. The mass \(m\) is a Lorentz invariant. Viewed at rest or viewed while flying by at tremendous speed, it's the same value. This is Episode 1's "floor" not changing its height when you change frame.
By contrast, energy \(E\) is frame-dependent ── at rest it's \(mc^2\), viewed in motion it's \(\gamma mc^2\). The same object gives a different \(E\) to different observers, but the same \(m\) to everyone. The "you get heavier when moving" misconception mixed up this distinction. That's why the accurate term today is "invariant mass" rather than "rest mass."
The "conservation of mass" you learn in chemistry is actually an approximation. In a reaction, the total rest mass is not conserved. The most vivid example is the annihilation of an electron and a positron.
electron + positron → 2 photons
$$\underbrace{0.511+0.511}_{\text{has rest mass}}\ \text{MeV}\ \longrightarrow\ \underbrace{2\ \text{photons}}_{\text{zero rest mass}}\ (\text{total energy }1.022\ \text{MeV})$$Before the reaction there's \(1.022\) MeV worth of rest mass. After, only zero-rest-mass photons. The sum of rest masses vanished, \(1.022\to0\) ── but the energy stays \(1.022\) MeV, conserved as light. The "mass defect" of nuclear fusion (the Sun) and fission (reactors) is the same story.
Conserved: total energy and total momentum (four-momentum).
Not conserved: the "sum" of rest masses (\(\sum m_i\)).
This is the most interesting part today. The mass of a system made of several particles (the system's invariant mass \(M\)) is not the sum of the parts' rest masses. It's determined by the total energy and total momentum.
And then something astonishing happens ── even two zero-rest-mass photons can, as a system, have a mass \(M\neq0\). Two photons flying in opposite directions have their momenta cancel (\(\sum\vec p=0\)), yet the total energy remains (\(\sum E\neq0\)), so \(M\neq0\). Check it in the figure below by changing the opening angle.
This is the purest form of Episode 3's "99% of the proton is the energy of the gluon field." Mass is not a "property of the parts" but a "combination of the system's energy and momentum." Episode 1's floor \((mc^2)^2=E^2-(pc)^2\) holds for the whole system, not just a single particle ── that is the true meaning of "invariant mass."
What "mass is not conserved" refers to is strictly that the "sum" of individual rest masses \(\sum m_i\) is not conserved. The total energy and total momentum of a closed, isolated system (= the system's invariant mass \(M\)) are properly conserved. Everyday "conservation of mass" is just a low-energy approximation that holds well enough in practice, because only a tiny fraction of the energy shows up as a mass difference.
Also, relativistic mass \(\gamma m\) is less "wrong" than old notation; the modern standard calls only invariant mass "mass." This time we're doing standard special relativity, unrelated to the watchword \(c\cdot t=\text{constant}\).
The mass \(m\) is Lorentz invariant (the same for everyone) ── so "you get heavier the faster you move" is a misconception; what increases is energy and momentum. Meanwhile, in a reaction the "sum" of rest masses is not conserved (annihilation, mass defect); what's conserved is total energy and total momentum. And the mass of a system is not the sum of the parts but \((Mc^2)^2=(\sum E)^2-(\sum \vec p c)^2\) ── even two massless photons can build a massive system.
Episode 1's floor \((mc^2)^2=E^2-(pc)^2\) is an "invariant length" that holds for one particle and for a whole system. That is what "invariant mass" is, the same story as Episode 3's 99%-of-the-proton. Invariant when you set it in motion, but a different mass once a reaction changes the system ── with this you can tell two easily-confused things apart.
Print / PDF: Ctrl+P (Cmd+P on Mac). On screen, change the opening angle with the slider and a system's mass emerges from two massless photons. Click "See the answer" to open each solution.