Mass shell of one particle
Every particle's energy and momentum sit on this hyperbola in every frame; photons sit on the light-like lines.
Special relativity · Energy and momentum
Collisions and Invariant Mass
Why does every frame agree on a collision's invariant mass when energy and momentum change? Energy and momentum both change when you switch frames, but one combination of the totals does not: the invariant mass, equal to the total energy in the frame where the total momentum is zero. It decides what a collision can make. A kaon at 0.80 GeV/c hitting a resting proton gives 1.70 GeV, and making an antiproton needs a proton beam with six proton masses of kinetic energy.
Collide particles along one line, slide to other frames and watch energy and momentum change while the invariant mass stays fixed and decides what can be made.
Total energy in this frame
Total momentum in this frame
Invariant mass
Energy left after making the products
White and blue change with the frame; the green invariant mass does not.
Four-momentum table and the saved record
Particle
kaon · in
proton · in
kaon · out
proton · out
The JSON record lists every particle's energy and momentum in the laboratory and in the viewing frame.
Physics tutorial
One number every frame agrees on
BackgroundIn a collision, observers moving at different speeds record different energies and momenta for the same particles. Accelerator physicists still quote one number for the whole collision.
Why it mattersThat number decides what particles a collision can make, so it must not depend on who is watching.
Start with the essentials
- Focus question
- Why does every frame agree on a collision's invariant mass when energy and momentum change?
- One-sentence intuition
- Energy and momentum are components of one four-momentum. A boost mixes the components, but the total's invariant mass, its length in spacetime, stays fixed and sets the energy available for making particles.
Core mathematical model
Invariant mass of the total
Add the four-momenta first, then take the length. Adding the masses instead gives the wrong answer.
Changing frame
In the frame moving at of the total, the total momentum vanishes and the total energy equals .
A beam on a resting target
For a proton beam on a resting proton, making three protons and an antiproton needs a total mass of four proton masses, so the beam energy must reach seven proton masses: a kinetic energy of six.
Common difficulties
Mass is not additive
Typical misconceptionThe mass of a system is the sum of its parts' masses.
Better mental modelThe invariant mass of the system includes the kinetic energy in its centre-of-momentum frame. Two photons with no mass can have a nonzero invariant mass together.
Energy is conserved, not invariant
Typical misconceptionIf energy is conserved, every observer must agree on it.
Better mental modelEach observer finds the same total before and after the collision, but different observers find different totals. The invariant mass is both conserved and the same for everyone.
What the model leaves out
Typical misconceptionThe scene shows where real detector tracks would go.
Better mental modelMotion along one line only, with c = 1 and energies in GeV. Masses of the proton, charged kaon and neutral pion are PDG 2024 values; the two sticky lumps are an idealized 1 GeV example. Products are placed back to back in the centre-of-momentum frame (equal shares for the four-proton final state) and boosted to the chosen frame; real angular distributions, spin, cross-sections and detector effects are left out. A forbidden reaction is shown as a notice, not as invented products.
Run the experiment
- 01
Check a published number
Start with the kaon preset: a kaon at 0.80 GeV/c hits a resting proton.
What to observe: The invariant mass reads 1.70 GeV, the centre-of-momentum energy in the Particle Data Group's worked example. - 02
Slide to other frames
Drag the round tip of the total along its green curve, or move the frame slider.
What to observe: Total energy and momentum change; the invariant mass does not. Where the total momentum crosses zero, the total energy equals the invariant mass. - 03
Turn motion into mass
Choose Motion turns into mass and raise the beam momentum.
What to observe: The merged lump is heavier than the two lumps that made it. The extra mass is the kinetic energy of the centre-of-momentum frame. - 04
Cross a threshold
Choose the antiproton preset, then lower the beam momentum slightly.
What to observe: The energy left over turns negative and the reaction is forbidden: below threshold no frame can supply four proton masses.
Related experiments and articles
Relativity as measurement · Reading path
Relativity as measurement: clocks, light signals and four-momentum
Articles on this path that use this experiment, followed by the other experiments they suggest.