Alpha Institute for Advanced Studies

E = T + E0 = T + m c squared

so the Newtonian kinetic energy is defined as the limit of E – E0 when v << c. The E0 was of course unknown to Newton. The energy associated with relativistic momentum is always E, the total energy. The famous equation:

E0 = m c squared

means that mass at rest or in motion has an energy m c squared. The relativistic momentum gamma m v is necessary for conservation of momentum in special relativity (see Marion and Thornton). The four momentum is
p sup mu = (E / c, p)

where E is the total relativistic energy and where p is the relativistic momentum. If the velocity of the particle is zero, then:

p sup mu = (E0, 0)

The rest energy E0 is so called because it exists when v is zero. In Newtonian physics a particle at rest has no kinetic energy. So E0 is more precisely “the kinetic rest energy”.