RELATIVISTIC COLLISIONS

Newton gave us two separate conservation laws for collisions: momentum is conserved, and kinetic energy is conserved (in elastic collisions). They feel like independent facts because in Newtonian mechanics momentum $\mathbf{p} = m\mathbf{v}$ and kinetic energy $K = \frac{1}{2}mv^2$ are completely different objects — one is a vector, one is a scalar, and the second only holds in elastic collisions.

Relativity unifies them. The correct object is the Four-momentum $p^\mu = (E/c,\, p_x,\, p_y,\, p_z)$, the four-vector whose time component is total energy divided by $c$ and whose spatial components are the relativistic 3-momentum $\mathbf{p} = \gamma m \mathbf{v}$. At every interaction vertex — elastic, inelastic, decay, creation — the single law is:

All four components simultaneously. That one equation carries both Newtonian momentum conservation (spatial components) and energy conservation (time component) as a single geometric statement. It holds because the laws of physics are translation-invariant in spacetime — Noether's theorem applied to a four-dimensional symmetry.

What changes between elastic and inelastic collisions is not whether four-momentum is conserved — it always is — but whether the individual rest masses $m_i$ are preserved. In an elastic collision each particle's Lorentz-invariant norm $p^\mu p_\mu = m_i^2 c^2$ is unchanged: the particle emerges with the same rest mass it entered with, so kinetic energy is also conserved. In an inelastic collision the total Four-momentum is still exactly conserved, but the individual particles can recombine into particles with different rest masses: kinetic energy converts to rest mass or vice versa. The law does not care. Only $\Sigma p^\mu$ is sacred.

saw this coming in 1905. The equation $E = \gamma mc^2$ already encoded the fact that energy and mass are the same currency. A system of particles has a total four-momentum, and the invariant mass of that total — its Minkowski norm — is the "mass" of the whole system regardless of how the internal particles are arranged. When particles collide and rearrange themselves, that total norm is preserved even while individual rest masses are not. This is the §04 module's central lesson, and it runs from the simplest 1D elastic scattering all the way to Compton scattering, pair production, and the Large Hadron Collider.

The simplest test: two equal-mass particles approaching head-on along the $x$-axis at velocities $\pm\beta c$. In the Newtonian limit the outcome of an elastic 1D collision between equal masses is a velocity swap — the particles pass through each other's momenta. Remarkably, the relativistic equal-mass elastic 1D collision has exactly the same outcome: the particles exchange velocities. The incoming four-momenta are:

Their sum has zero spatial momentum — by symmetry, the CoM frame is the collision frame itself. After the velocity swap, $p_1^\mu \leftrightarrow p_2^\mu$, and the total is unchanged. This is a special property of equal-mass elastic 1D scattering; for unequal masses or off-axis angles, the relativistic corrections begin at order $\beta^2$ and grow rapidly.

The key observation in the HUD: the time component $\Sigma p^0 = 2\gamma mc$ is strictly greater than the Newtonian sum $2mc$ whenever $\beta > 0$. That surplus energy is the kinetic energy, now written into the same component as rest energy. Energy conservation is not a separate law — it falls out automatically from the time component of four-momentum conservation.

For unequal masses or masses with different velocities, the elastic outcome is no longer a simple swap. But the constraint $\Sigma p^\mu_\text{in} = \Sigma p^\mu_\text{out}$ with both rest masses fixed still uniquely determines the outcome given the CoM scatter angle. The relativistic corrections to the deflection angles become visible as soon as $\beta \gtrsim 0.3$.

Now turn the collision completely inelastic: the two particles of mass $m$ collide head-on at $\pm\beta c$ and merge into a single particle at rest in the CoM frame. Four-momentum conservation fixes the final state:

The spatial components cancel by symmetry. The time component is $2\gamma mc$. The rest mass of the merged particle follows from the Minkowski norm:

At $\beta = 0.5$, $\gamma = 1/\sqrt{1-0.25} \approx 1.1547$, so $M_\text{final} \approx 2.309\,m$. The merged particle weighs 15.5% more than the two incoming particles combined. That 15.5% surplus came from the kinetic energy of the two incoming particles — it has been converted, irreversibly, into rest mass.

At $\beta \ll 1$, the mass excess is $(M_\text{final} - 2m) \approx \beta^2 m$ — the leading-order kinetic energy per particle, exactly as Newtonian mechanics would predict if you knew that kinetic energy has inertia. The relativistic formula is the exact version of that intuition, valid all the way to $\beta \to 1$.

In two dimensions the constraint $\Sigma p^\mu_\text{in} = \Sigma p^\mu_\text{out}$ fixes a relationship between the outgoing angles; it does not fix them uniquely (the CoM scatter angle is a free parameter set by the interaction dynamics). But it imposes a rigid relativistic correction that diverges from the Newtonian billiard prediction.

In the Newtonian equal-mass elastic 2D case the outgoing particles always scatter at a 90° total angle: $\theta_1 + \theta_2 = 90°$. This is the famous billiard-ball right angle — you can verify it at any pool table. Relativistically, the sum $\theta_1 + \theta_2 < 90°$ for any $\beta > 0$, with the deficit growing monotonically with $\beta$.

The mechanism is straightforward. In the CoM frame the scatter is always symmetric: particle 1 scatters at angle $\phi$ above the beam axis, particle 2 at angle $\phi$ below. Boosting back to the lab frame, the angle $\theta_1$ that particle 1 makes in the lab is compressed forward because the boost pushes both particles in the +x direction. At low $\beta$ this compression is negligible and the angles sum to 90°; at high $\beta$ the forward-focusing is severe and both angles are pushed toward the beam axis, reducing their sum well below 90°.

This is directly observable in particle accelerators. At the Large Hadron Collider, $\beta \approx 1 - 10^{-8}$, and the Newtonian 90° rule is wrong by nearly 90°: most outgoing particles from high-energy collisions are focused tightly forward along the beam pipe. The detectors (ATLAS, CMS) are specifically engineered to catch the rare wide-angle scatter events; the forward calorimeters handle the beam-focused majority.

The three scenes in this topic — elastic swap, inelastic merger, 2D scattering — are the same law applied three times. $\Sigma p^\mu_\text{in} = \Sigma p^\mu_\text{out}$. Every particle interaction in nature is a vertex where this equation holds, supplemented by whatever other conservation laws (charge, lepton number, baryon number) the specific interaction respects.

The next two topics in §04 are both applications of the same principle to more exotic kinematic regimes. §04.4 watches Compton scattering enforce four-momentum balance on a photon-electron collision: the photon has a Four-momentum too — null, with $p^\mu p_\mu = 0$ — and the shift in its wavelength after scattering from an electron follows uniquely from $\Sigma p^\mu$ conservation and the electron's rest mass. The Compton shift formula is not a separate postulate; it falls directly out of four-momentum algebra, as computed in 1923.

§04.5 watches the same equation forbid pair production from a single photon in vacuum: if you try to write $\gamma \to e^+ + e^-$ with the photon alone as the initial state, four-momentum conservation cannot be satisfied because the photon's null norm cannot equal the timelike norm of the massive pair. You need a second photon or a nucleus to supply the missing four-momentum — the conservation law itself tells you the interaction is kinematically forbidden without a recoil partner. The forbidden vertex is as physically powerful as the allowed one.

Mass-energy equivalence is not merely a formula on a blackboard. It is the exchange rate between the kinetic energy of incoming particles and the rest mass of whatever they create. Every particle collider in the world is, at bottom, a machine for converting kinetic energy into rest mass via $\Sigma p^\mu$ conservation. The heavier the particle you want to create, the higher the invariant mass of the initial state you need to provide — which is why the LHC runs at 14 TeV. The kinematics of §04.3 tell you exactly what collision energy a machine needs; the rest is engineering.