3.12.3.5 - Mass and Energy

Rest energy and changes of system mass

An object has rest energy even when its centre of mass is stationary. Its rest mass $m_0$ and rest energy $E_0$ are related by

This is an equivalence of mass and energy, with $c^2$ supplying the conversion between kilograms and joules. It does not mean that every object's entire rest energy is available in an ordinary reaction.

If a stationary system loses energy to its surroundings, its mass decreases by $\Delta m=\Delta E/c^2$, where the deltas here denote positive amounts lost. Energy retained as internal energy contributes to the mass of the system. Heating therefore increases a sealed system's rest mass slightly; choosing a frame in which the whole system moves faster is a different change and does not change its rest mass.

For example, energy $9.00\times10^{10}\,\mathrm{J}$ leaving a stationary system corresponds to a mass loss $9.00\times10^{10}/(3.00\times10^8)^2=1.00\times10^{-6}\,\mathrm{kg}$. The energy and mass account must refer to the same system. A reaction's lower product rest masses can supply kinetic energy and radiation; total energy is conserved when all products and transfers are included.

AQA's relativistic-mass convention

The current AQA specification uses $E=mc^2$, $E=m_0c^2/\sqrt{1-v^2/c^2}$, and graphs of mass against speed. In this convention $m$ is relativistic mass, defined by total energy divided by $c^2$; $m_0$ is the rest mass.

For the same unchanged particle, rest mass is invariant: observers in relative motion agree on $m_0$. They can disagree about its energy and therefore about $m$ in the AQA convention. Many modern accounts use “mass” to mean invariant rest mass throughout; the physical predictions are the same once the notation is identified. Do not say the particle's rest mass increases just because it is accelerated.

At $v=0.60c$, $\gamma=1.25$, so $m=1.25m_0$. At $0.990c$, $\gamma=7.089$: a proton with $m_0=1.67\times10^{-27}\,\mathrm{kg}$ has $m=1.18\times10^{-26}\,\mathrm{kg}$ in this convention, while its rest mass stays $1.67\times10^{-27}\,\mathrm{kg}$.

Avoid inserting this speed-dependent mass into $F=ma$ as though all classical dynamics remained unchanged. The energy equations give the required conclusions directly.

Separating total and kinetic energy

The total energy of a free massive particle is its rest energy plus its kinetic energy, all referred to the same inertial frame.

The subtraction matters: at rest, $\gamma=1$, so $E_k=0$ but $E=m_0c^2$. Calling $\gamma m_0c^2$ the kinetic energy would wrongly give a stationary particle a non-zero kinetic energy.

An electron at $0.800c$

With $m_0=9.11\times10^{-31}\,\mathrm{kg}$, $E_0=8.199\times10^{-14}\,\mathrm{J}$. Here $\gamma=1/0.600=1.667$, giving

The classical calculation gives $\tfrac12m_0v^2=2.62\times10^{-14}\,\mathrm{J}$. Its percentage underestimate is measured relative to the relativistic value:

At $v\ll c$, the approximation $\gamma\approx1+v^2/(2c^2)$ yields $E_k\approx\tfrac12m_0v^2$. Classical mechanics is an excellent low-speed approximation, with accuracy determined by the measurement precision required.

If kinetic energy is given instead, first find $\gamma=1+E_k/(m_0c^2)$, then $v/c=\sqrt{1-1/\gamma^2}$. For example, $E_k=3E_0$ gives $\gamma=4$, not 3, and $v=0.968c$.

Reading the graphs and the speed limit

Graphs distinguish the non-zero rest energy from the zero kinetic energy at rest and show how rapidly relativistic effects grow near $c$. Normalised axes allow the same curves to represent particles with different rest masses.

[DIAGRAM: asset_name: D.3.5 - Mass and Energy - Diagram 1; asset_slug: D.3.5 - Mass and Energy - Diagram 1; recommended_method: matplotlib; caption: Relativistic mass rises from its rest value as speed increases; description: Exact graph of relativistic mass ratio m/m0 against speed ratio v/c, with rest-mass reference at 1 and vertical asymptote at v/c=1.]

Relativistic mass starts at $m/m_0=1$, remains close to 1 at low speed and rises without bound as $v/c\to1$. The dashed horizontal line represents invariant rest mass.