bei48482_FM

Hence

E^2  (mc^2 )^2  p^2 c^2 (1.24)

which is the formula we want. We note that, because mc^2 is invariant, so is E^2 p^2 c^2 :

this quantity for a particle has the same value in all frames of reference.

For a system of particles rather than a single particle, Eq. (1.24) holds provided

that the rest energy mc^2 —and hence mass m—is that of the entire system. If the

particles in the system are moving with respect to one another, the sum of their

individual rest energies may not equal the rest energy of the system. We saw this in

Example 1.7 when a stationary body of mass 2.5 kg exploded into two smaller bodies,

each of mass 1.0 kg, that then moved apart. If we were inside the system, we would

interpret the difference of 0.5 kg of mass as representing its conversion into kinetic

energy of the smaller bodies. But seen as a whole, the system is at rest both before

and after the explosion, so the systemdid not gain kinetic energy. Therefore the rest

energy of the system includes the kinetic energies of its internal motions and it cor-

responds to a mass of 2.5 kg both before and after the explosion.

In a given situation, the rest energy of an isolated system may be greater than, the

same as, or less than the sum of the rest energies of its members. An important case

in which the system rest energy is less than the rest energies of its members is that of

a system of particles held together by attractive forces, such as the neutrons and pro-

tons in an atomic nucleus. The rest energy of a nucleus (except that of ordinary

hydrogen, which is a single proton) is less than the total of the rest energies of its

constituent particles. The difference is called the binding energyof the nucleus. To break

a nucleus up completely calls for an amount of energy at least equal to its binding

energy. This topic will be explored in detail in Sec. 11.4. For the moment it is inter-

esting to note how large nuclear binding energies are—nearly 10^12 kJ per kg of

nuclear matter is typical. By comparison, the binding energy of water molecules in liq-

uid water is only 2260 kJ/kg; this is the energy needed to turn 1 kg of water at 100°C

to steam at the same temperature.

Massless Particles

Can a massless particle exist? To be more precise, can a particle exist which has no rest

mass but which nevertheless exhibits such particlelike properties as energy and mo-

mentum? In classical mechanics, a particle must have rest mass in order to have en-

ergy and momentum, but in relativistic mechanics this requirement does not hold.

From Eqs. (1.17) and (1.23), when m0 and c, it is clear that Ep0.

A massless particle with a speed less than that of light can have neither energy nor mo-

mentum. However, when m0 and c, E 0 0 and p 0 0, which are inde-

terminate: Eand pcan have any values. Thus Eqs. (1.17) and (1.23) are consistent

with the existence of massless particles that possess energy and momentum provided

that they travel with the speed of light.

Equation (1.24) gives us the relationship between Eand pfor a particle with m0:

Massless particle Epc (1.25)

The conclusion is not that massless particles necessarily occur, only that the laws

of physics do not exclude the possibility as long as cand Epcfor them. In fact,

Energy and

momentum

Relativity 31

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