Physics · Gravitation and the universe · Grade 10

Relativistic dynamics

Why can't a proton be accelerated to the speed of light, while its energy grows without limit?

The situation

In an accelerator a proton gains the same energy each time it passes through a gap. According to Newtonian mechanics, giving it just 469 MeV of kinetic energy should have been enough to reach the speed of light. But in the Large Hadron Collider protons have an energy of 6.8 TeV — more than 14 000 times as much — and still travel slower than light. On top of that, two events that happen at the same time for one observer are not simultaneous for another. Where does the energy go, and why can nobody exceed $c$ ?

Topic

Relativistic dynamics, Minkowski diagram. Proton in an accelerator: E = mc², momentum and the relativity of simultaneity

Key ideas

Minkowski diagram
E² = p²c² + m²c⁴
Relativity of simultaneity

Formulas

x^{'} = γ (x - u t), t^{'} = γ (t - \frac{u x}{c^{2}}), γ = \frac{1}{\sqrt{1 - u^{2} / c^{2}}}

The Lorentz transformations: the coordinates of an observer moving at speed $u$ . The term $u x / c^{2}$ in $t^{'}$ is the relativity of simultaneity.

s^{2} = c^{2} Δ t^{2} - Δ x^{2} = c^{2} Δ t^{' 2} - Δ x^{' 2} = c^{2} Δ τ^{2}

The interval is the same for all inertial observers; for timelike events it gives the proper time $τ$ .

w = \frac{u^{'} + v}{1 + u^{'} v / c^{2}}

Adding velocities: $0.6 c$ and $0.6 c$ together give $0.882 c$ , never more than $c$ .

E = γ m c^{2}, \vec{p} = γ m \vec{v}, E^{2} = {(p c)}^{2} + {(m c^{2})}^{2}

Relativistic energy and momentum; $K = (γ - 1) m c^{2}$ . For a body at rest, $E_{0} = m c^{2}$ .

f = f_{0} \sqrt{\frac{1 + β}{1 - β}}, \cos θ^{'} = \frac{\cos θ + β}{1 + β \cos θ}

The relativistic Doppler effect (source approaching) and the aberration of light; $β = v / c$ .

Theory

Postulates and simultaneity

Galileo's principle of relativity held for mechanics, but Maxwell's electrodynamics predicts that the speed of light equals $c$ — relative to which frame of reference? Einstein adopted two postulates: the laws of physics are the same in all inertial frames, and the speed of light does not depend on the motion of the source. The Lorentz transformations follow from these. In the “Minkowski” scene the flash occurs in the middle of the ship: for the ship's crew the light reaches both ends at the same time (R and F), but for the station the rear end moves towards the light and is lit up first: $t_{F} - t_{R} = γ v L_{0} / c^{2}$ . Time dilation and length contraction are covered in detail in the “Special relativity” experiment.

The Minkowski diagram, the interval and four-vectors

In the diagram every body is a worldline, and light is a line at 45°. When you switch to another observer (a boost) the axes close up like scissors, while the light cone stays where it is. The interval $s^{2}$ does not change: between R and F, $s^{2} = - L_{0}^{2}$ (spacelike) — so their order depends on the observer (for $u > v$ , F comes first!). Between O and R, $s^{2} = 0$ , and between O and the clock marks, $s^{2} > 0$ (timelike) — the order of such events never changes: causality is preserved. $(c t, x, y, z)$ is a four-vector; the four-velocity is $u^{μ} = γ (c, \vec{v})$ and the four-momentum is $p^{μ} = (E / c, \vec{p})$ , whose “length” is $E^{2} / c^{2} - p^{2} = m^{2} c^{2}$ . The four-acceleration is always “perpendicular” to the four-velocity: $a_{μ} u^{μ} = 0$ .

Velocities, the Doppler effect, aberration and the Fizeau experiment

If a probe is launched forward from the ship at speed $u^{'}$ , the station measures $w = (u^{'} + v) / (1 + u^{'} v / c^{2})$ . For light ( $u^{'} = c$ ) the result is again $c$ . A receiver in front of the ship's beacon measures a frequency $\sqrt{(1 + β) / (1 - β)}$ times higher, and one behind it the same factor lower; one at the side measures it $1 / γ$ times lower — in classical physics this transverse Doppler effect does not exist. From the cabin the stars gather ahead (aberration) and turn bluer. Because of the Earth's orbital speed (30 km/s), the stars also shift by about 20.5″ over the year — this was discovered by J. Bradley in the 18th century. In 1851 H. Fizeau measured the speed of light in flowing water and found $c / n + v (1 - 1 / n^{2})$ — this too is a consequence of the law of addition of velocities.

Relativistic dynamics: $E = mc^2$

The momentum is $p = γ m v$ and the energy is $E = γ m c^{2}$ . As $v \to c$ , $γ \to \infty$ : accelerating a proton to $c$ would take infinite energy. At low speeds $K = (γ - 1) m c^{2} \approx m v^{2} / 2$ — Newtonian mechanics is correct in the limit $v ≪ c$ . Older textbooks write $m = γ m_{0}$ and call it the “dependence of mass on speed”; in modern physics $m$ is the invariant (rest) mass, and $γ$ appears in the energy and the momentum. The energy of a body at rest is $E_{0} = m c^{2}$ : in nuclear reactions the decrease in mass is released as energy. In theoretical mechanics everything follows from the Lagrangian of a free particle, $L = - m c^{2} \sqrt{1 - v^{2} / c^{2}}$ (the action $S = - m c^{2} \int d τ$ is proportional to the proper time): the momentum is $\vec{p} = \partial L / \partial \vec{v} = γ m \vec{v}$ and the energy is $E = \vec{p} \cdot \vec{v} - L = γ m c^{2}$ ; for $v ≪ c$ , $L \approx - m c^{2} + m v^{2} / 2$ — the Newtonian kinetic energy. For a charge in an electromagnetic field, $L = - m c^{2} / γ + q \vec{v} \cdot \vec{A} - q φ$ . In the “Accelerator” scene the length of the tubes is $L = β λ / 2$ : they get longer as the proton speeds up, and at $v \to c$ they become almost equal. A charge accelerated along a circle emits synchrotron radiation; its power is proportional to $γ^{4}$ , so energy losses in electron rings are large.

Decays and collisions

In a closed (isolated) system the four-momentum is conserved: the sum of the energies and the sum of the momenta separately. Mass, however, is not conserved — what is conserved is the invariant mass of the system, $M = \sqrt{E^{2} - p^{2} c^{2}} / c^{2}$ . A $π^{0}$ at rest decays into two photons, each with $m_{π} c^{2} / 2 = 67.5$ MeV. In the decay $π^{+} \to μ^{+} ν$ the muon gets only 4.12 MeV of kinetic energy. In an elastic collision of particles of equal mass (with the target at rest), in Newtonian mechanics the two particles scatter at 90° to each other, while in the relativistic case the angle is less than 90°. The reaction $p + p \to p + p + π^{0}$ on a stationary target begins only at $K \geq 2 m_{π} c^{2} + m_{π}^{2} c^{2} / 2 m_{p} \approx 279.7$ MeV: part of the energy goes into the motion of the centre of mass.

Fields, quanta and gravity — the limits of the model

The electric and magnetic fields are parts of a single electromagnetic field tensor: when you move to another observer they mix ( ${\vec{E}}_{⟂}^{'} = γ {(\vec{E} + \vec{v} \times \vec{B})}_{⟂}$ ), but $\vec{E} \cdot \vec{B}$ and $E^{2} - c^{2} B^{2}$ are invariant. A charge at rest produces only an electric field; in a frame moving at velocity $\vec{v}$ there is also a current, and a magnetic field $\vec{B} = \vec{v} \times \vec{E} / c^{2}$ appears. Two electrons flying side by side: in the frame moving with them there is only the Coulomb repulsion $F_{0}$ , while in the laboratory a magnetic attraction is added as well, and the result is $F_{0} / γ$ — which field is “electric” and which “magnetic” depends on the observer. Einstein's 1905 paper begins with exactly this kind of example — the relative motion of a magnet and a conductor. The potentials $(φ / c, \vec{A})$ form a four-vector. If in $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ we replace $E \to i ℏ \partial_{t}$ and $\vec{p} \to - i ℏ \nabla$ , we get the Klein–Gordon equation; Dirac turned it into a first-order equation — from it come the spin of the electron and antiparticles (the positron was discovered in 1932). The equivalence principle (acceleration and gravity cannot be told apart) leads to general relativity (see “Black holes”). The model: motion along a single axis, ideal clocks, no magnetic field in the detector, and time slowed down so that it can be followed.

Did you know?

In 1932 J. Cockcroft and E. Walton used accelerated protons to split lithium-7 nuclei into two helium nuclei. The energy released (about 17 MeV) matched the decrease in the mass of the nuclei according to $E = m c^{2}$ .
Since 2022 the protons in the Large Hadron Collider have had an energy of 6.8 TeV. Here $γ \approx 7 250$ , and the proton travels slower than light by only about 3 m/s.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The Solar System — Grade 7
Scales of the universe — Grade 9
Newton's cannon: an artificial satellite — Grade 10
Special relativity — Grade 10
Law of universal gravitation — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
The expanding universe — Year 1
Black holes — Year 4