Energy, Rest

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ENERGY, REST

Rest energy is the energy associated with a particle's mass. A free particle of mass m has rest energy E₀ = mc², and its total energy is the sum of its rest energy and its energy of motion (kinetic energy, T ): E = E₀ + T . If you imagine traveling in a spaceship along with a particle and measuring the total energy of the particle in your spaceship laboratory, then in that laboratory the particle will be at rest, and the energy you measure will be the rest energy. There is no energy associated with motion in that case, only energy associated with the particle's mass.

Rest Energy and Special Relativity

Albert Einstein discovered the relationship between energy and mass when he formulated the special theory of relativity. Two fundamental equations of relativity for a free particle are with total energy E, mass m, momentum p , velocity v , and where c is the speed of light. When the particle is at rest, v = 0, so p = 0 and E = E₀ = mc², the rest energy.

The relation between energy and mass follows from the two postulates of special relativity: (1) the principle of relativity: the laws of physics have the same form in all reference frames moving at constant speed with respect to one another; (2) the speed of light is the same in all such reference frames.

Rest Energy and Elementary Particle Physics

Elementary particle physics seeks to produce and study fundamental particles that make up all the matter in our universe and study their interactions. This is done by producing and observing collisions between particles at high energies. Some of the kinetic energy of the particles in such collisions can be transformed into the rest energy of new particles, subject to the conservation of energy and momentum.

Energy and momentum conservation mean that the total energy, including rest energy, and total momentum are the same after the collision as they were before that event. In the high-energy collisions of elementary particle physics, rest energy is an essential element of the conservation bookkeeping; energy will appear not to be conserved in most cases if rest energy is not explicitly accounted for.

For example, consider the process in which the antiproton was discovered in 1955: a high-energy proton was fired at a stationary proton target producing a proton-antiproton pair in addition to the two original protons. With the notation p for protons, p̄ for antiprotons, we write this reaction as p + p → p + p + p + p̄. There are two additional particles on the right-hand-side, after the collision, than were there before the collision, a proton and an antiproton. These two additional particles each have rest energies equal to the proton rest energy: 938 MeV. Where did the additional 2 × 938 = 1,876 MeV of rest energy come from? It came from converting some of the initial kinetic energy of the incoming proton to the rest energy (and kinetic energy) of the two new particles. In this discovery experiment, the proton beam energy was 6.2 GeV (that is, 6,200 MeV), while we can calculate that the minimum kinetic energy of the incoming proton must be 5.6 GeV to supply the rest energies of the outgoing particles and still conserve momentum. The extra kinetic energy of the incoming particle (6.2 - 5.6 GeV = 600 MeV) was distributed in the motion of the outgoing particles as kinetic energy.

In particle decay processes, some of a particle's rest energy can be converted into kinetic energy, the converse of the antiproton experiment referenced above, in which kinetic energy was converted into rest energy. An example of this is the decay of a negative pion at rest: π¯ → μ¯ + v̄_μ. The rest energy of the pion is 139.6 MeV, while that of the muon is 105.7 MeV, and the neutrino has very small rest energy, negligible relative to the other two. The difference in rest energies before and after the decay shows up as kinetic energy of the muon and neutrino.

It is possible to convert all of a particle's rest energy to kinetic energy in particle-antiparticle annihilation. An electron and a positron will annihilate if they approach closely enough. If they start at rest so they have only rest energy, but close enough to interact, then in the annihilation, two photons will be produced with no rest energy. All the rest energy in this case will be converted to light energy.

Rest Energy as an Invariant

A relativistic invariant is a quantity that has the same value in all reference frames related by a constant relative velocity. The rest energy, E₀ = mc², is an invariant quantity, whether for a single particle or a system of particles. For a system of particles, the total rest energy of all particles in the system is an invariant.

An invariant does not necessarily have the same value before and after a collision as observed in one particular reference frame. In the antiproton experiment discussed above, the total rest energy before the collision was 1.876 GeV (two protons), while after the collision it was 3.752 GeV (three protons and one antiproton). The collision transformed some kinetic energy into rest energy. However, at any instant during this experiment, the rest energy of the system of particles in existence at that instant would be the same whether measured in the laboratory, or in a spaceship moving with the incoming particle, or in a spaceship moving with the center of mass, or in any other reference frame moving at constant velocity.

Examples of Rest Energies of Elementary Particles

The rest energies of elementary particles range from zero (photons, gluons, gravitons according to current theory) to 174 GeV for the t quark. Some important rest energies, as now known, are as follows (given in MeV, except as noted, with ranges shown in cases where current experiments still leave large uncertainties):

photon	0	electron	0.511	u quark	1-5
d quark	3-9	muon	105.7	pion (π⁰)	135
pion (π^±)	139.6	s quark	75-170	proton	938.3
neutron	939.6	c quark	1.15-1.35 GeV	tau	1.78 GeV
b quark	4-4.4 GeV	W boson	80 GeV	Z boson	91 GeV
t quark	174 GeV