Coaching Centre
Relativistic momentum and E = mc²
Why Newtonian momentum fails at high speeds, relativistic momentum and the cosmic speed limit, evidence from particle accelerators, and mass–energy equivalence E = mc² applied to rest energy, annihilation, nuclear reactions and the Sun.
At the Australian Synchrotron in Melbourne, electrons are boosted to an energy of 3 billion electronvolts and sent around a 216 m ring, producing intense X-rays used to study everything from proteins to ancient fossils. Each lap, they gain no measurable speed. They already travel at more than 99.999 99% of the speed of light, and adding more energy only adds momentum. Newton's laws say that should be impossible. Relativity says it's the only possibility.
When Newton's momentum fails
In Newtonian physics, momentum is $p = mv$ and kinetic energy is $K = \tfrac{1}{2}mv^2$. A constant force keeps adding momentum at a constant rate, so in principle anything can be pushed to any speed, including faster than light.
In relativity, $c$ is the same for all observers, so nothing can be seen to catch up with or overtake a light beam. If conservation of momentum is to hold in every inertial frame (postulate 1) with this speed limit, momentum must be redefined. Einstein's analysis gives the relativistic momentum:
$$p_v = \frac{m_0 v}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \gamma m_0 v$$
where $m_0$ is the rest mass, the mass measured by an observer at rest relative to the object. At low speeds $\gamma \approx 1$ and this is just $m_0v$. As $v$ approaches $c$, $\gamma$, and with it the momentum, grows without limit.
The consequence: a force on a fast object keeps adding momentum, but the speed creeps closer and closer to $c$ without reaching it. An infinite amount of energy would be needed to reach $c$, so no object with mass can reach or exceed the speed of light.
(Older books describe this as a "relativistic mass" that increases with speed. Physicists now prefer to keep the mass as $m_0$, a property of the object, and put the factor $\gamma$ in the momentum and energy. The predictions are the same.)
Find the momentum of an electron ($m_0 = 9.109 \times 10^{-31}$ kg) at 0.90$c$, and compare it with the Newtonian value.
Evidence for relativistic momentum
- Bucherer, 1908. Fast electrons (beta particles) passing through electric and magnetic fields were deflected less than Newtonian physics predicted, exactly as if their momentum included the factor $\gamma$.
- The ultimate speed, 1964. William Bertozzi at MIT accelerated electrons to kinetic energies from 0.5 to 15 MeV, timed them over an 8.4 m flight path and measured the energy they deposited in a target. Newton's $\tfrac{1}{2}mv^2$ predicted that 15 MeV electrons should travel at 7.7$c$. They were measured at 0.99$c$ or more; the speed approached but never reached $c$.
- Particle accelerators. In a circular accelerator, the magnetic field needed to bend a particle is set by its momentum, $r = p/(qB)$. Early cyclotrons stopped working at high energies because particles fell out of step as relativistic momentum grew. Modern synchrotrons, such as the Australian Synchrotron and CERN's Large Hadron Collider, ramp up their magnetic fields to match the relativistic momentum. Every working accelerator is a daily test of $p_v = \gamma m_0 v$.
This is also the limitation of particle accelerators: beyond a certain point, extra energy produces almost no extra speed, and the magnets must get ever stronger, or the ring ever larger, to hold the particles on course. The LHC's 27 km ring holds protons with 6.8 TeV of energy, travelling at about 0.999 999 99$c$; to go further would need stronger magnets or a bigger ring.
Mass–energy equivalence
Later in 1905, Einstein showed that the postulates have another consequence: mass and energy are equivalent. An object at rest has a rest energy
$$E = mc^2$$
Because $c^2 = 9.0 \times 10^{16}$ m² s⁻², a tiny mass corresponds to an enormous energy. Whenever a system gives out energy, its mass decreases by $\Delta m = \Delta E/c^2$; when it gains energy, its mass increases. Mass is not separately conserved; mass–energy is.
A moving object has total energy $E = \gamma m_0c^2$, so its kinetic energy is $K = (\gamma - 1)m_0c^2$. At low speeds this reduces to $\tfrac{1}{2}m_0v^2$.
Calculate the rest energy of an electron in joules and in MeV.
Mass–energy in action
Annihilation. When a particle meets its antiparticle, both vanish and their entire rest energy becomes photons. An electron and positron produce two gamma rays of 0.511 MeV each, travelling in opposite directions. PET (positron emission tomography) scanners detect these pairs to map where a radioactive tracer collects in the body.
Nuclear reactions. In fission and fusion, the products have slightly less mass than the reactants. The missing mass, the mass defect, is released as kinetic energy and radiation, $E = \Delta mc^2$. Fission of one uranium-235 nucleus releases about 200 MeV, about 0.09% of its mass. (You'll analyse nuclear reactions fully in Module 8.)
The Sun. In the Sun's core, hydrogen nuclei fuse to form helium. Converting four protons to one helium-4 nucleus releases 26.7 MeV, about 0.7% of the mass. The Sun's power output of $3.83 \times 10^{26}$ W means it converts about 4 million tonnes of mass into energy every second, and it has enough hydrogen to keep doing so for billions of years.
Chemistry too. A chemical reaction that releases energy also loses mass, but the change is around one part in $10^{10}$, far too small to measure on a balance. That is why "conservation of mass" works so well in chemistry.
The Sun radiates $3.83 \times 10^{26}$ W. How much mass does it convert to energy each second?
A proton ($m_0 = 1.673 \times 10^{-27}$ kg) travels at 0.60$c$. Find its kinetic energy using relativity, and compare with $\tfrac{1}{2}mv^2$.
- At the default 1 MV, read the electron's γ and speed. Newton's ½mv² gives 1.978c. What does relativity give?
- Slide down to about 100 kV. How close are Newton and relativity now? Why do old TV picture tubes (about 25 kV) barely need relativity?
- Slide up to the maximum. How does the speed change as the energy rises by factors of ten? What happens to the momentum?
- Switch to Mass into energy. Check the annihilation energy of 1.022 MeV using $E = mc^2$.
- Compare the mass converted by fission of one uranium nucleus with burning a whole kilogram of coal. Why does nobody notice that chemical reactions change mass?
Test yourself
A constant force acts on a particle for a long time. According to special relativity, the particle's
$p_v = \gamma m_0v$ can grow forever while $v$ stays below $c$.
How much energy is released when 1.0 g of mass is completely converted into energy?
$E = (1.0 \times 10^{-3})(3.00 \times 10^8)^2 = 9.0 \times 10^{13}$ J.
The ratio of the relativistic momentum to the Newtonian momentum of a particle at 0.80$c$ is
The ratio is $\gamma = 1/0.60 = 1.67$.
Which observation is evidence for relativistic momentum?
The magnetic field needed depends on momentum, which grows by $\gamma$.
Label the momentum graph
Order these processes by the energy released per kilogram of fuel, smallest first.
Sort the evidence.
The mass of an object measured at rest relative to it is its mass. No object with mass can reach the speed of . The difference in mass between reactants and products in a nuclear reaction is the mass . When an electron meets a positron they undergo . The Sun's energy comes from nuclear .
Electrons in a linear accelerator gain 15 MeV of kinetic energy. (a) Show that Newtonian physics predicts a speed greater than $c$. (b) Explain, using relativistic momentum, why the electrons are actually observed to travel slower than $c$. (c) Explain why this is a limitation for the design of particle accelerators.
(a) $K = 15 \times 1.602 \times 10^{-13} = 2.40 \times 10^{-12}$ J. Newtonian: $v = \sqrt{\dfrac{2K}{m}} = \sqrt{\dfrac{2 \times 2.40 \times 10^{-12}}{9.109 \times 10^{-31}}} = 2.3 \times 10^9$ m/s, about 7.7 times the speed of light.
(b) In relativity, momentum is $p_v = \dfrac{m_0v}{\sqrt{1 - v^2/c^2}}$. As $v$ approaches $c$ the denominator approaches zero and the momentum (and energy) grow without limit. So work done on the electron increases its momentum and energy, but its speed only creeps closer to $c$. Bertozzi's 1964 experiment confirmed this: 15 MeV electrons were timed at just under $c$.
(c) Each further increase in energy gives a smaller and smaller increase in speed, so accelerators must supply enormous energies. In circular machines the radius is $r = p/(qB)$, and because momentum keeps growing, the bending magnets must become ever stronger or the ring ever larger. This sets practical and cost limits on the energies accelerators can reach.