Appendix 2: Mass in Multi-Particle Systems

In Special Relativity Theory the mass is given by the length of the energy-momentum

4-vector, so in natural units (i.e. taking c = 1):

m2 = E2 – p2

Consider two particles travelling in the same direction, taken as the x-axis.

The energy-momentum 4-vectors along the x-axis for the particles are:

p1 = (E,  px,  0,  0)

p2 = (E,  px,  0,  0)

Each of the particles has an individual mass given by:

m2 = E2 – px2

The total energy-momentum of this system (call it system 'A') is then:

PA = (2E,  2px,  0,  0)

and since the velocity 4-vector has a squared length of 1 (in natural units),

the total mass of this system (squared) is given by:

mA2 = PA2
       = (2E)2 – (2px)2

       = 4(E2 – px2)

       = 4m2

or mA = 2m, just as in Newtonian physics.

Now consider a second system (call it system 'B') in which the two particles are moving in opposite directions along the x-axis, so we now have:

p1 = (E,   px,  0,  0)

p2 = (E, –px,  0,  0)

Once again each particle has an individual mass given by:

m2 = E2 – px2

but now the total energy-momentum of this system is:

PB = (2E,  0,  0,  0)

The total mass (squared) for the system is given by:

mB2 = PB2

       = (2E)2

       = 4(E2 – px2 + px2)

       = 4m2 + 4px2

- i.e. by simply making the particles travel in opposite directions,

the mass of the two-particle system has increased.

When the particles are photons, px2 = E2, so m = 0 and we get:

mA = 0 and

mB = 2px

- i.e. a system of photons has zero mass when the photons are all travelling in the same direction, but non-zero mass when the photons are not all travelling in the same direction.