The time component of the four-momentum, , times gives , the energy of relativity. Accelerating a body from rest to speed , the work done by the force is , which is the kinetic energy.
is when , so a body at rest still has the energy . The curve in the figure never drops below the height of the rest energy, even at zero speed.
For small , is well approximated by . Putting this into gives
The second term is Newtonian kinetic energy. The first, , is a constant independent of speed, and it never appeared in mechanics, which only deals with differences in energy.
is called the rest energy. Because of the factor it is very large: the rest energy of g is J.
That is enough to keep a W bulb lit for seconds, nearly years.
Chemical reactions reduce the mass slightly too, but when hydrogen burns to water the fraction lost is only about . What can be taken out as energy is only the of the mass that is lost.
Eliminating from and gives .
It has the form of the Pythagorean theorem, with hypotenuse and sides and . The triangle in the figure stops at , , and , where is , , and times the rest energy.
For a particle of zero mass one side vanishes and only remains. Light is such a case: it has momentum and energy without mass.
The mass of a nucleus is slightly less than the total mass of its protons and neutrons taken apart. This difference is called the mass defect, and its is the energy that holds the nucleus together.
When a nuclear reaction reduces the mass, the of the mass lost goes out as energy. In the centre of the Sun four hydrogen nuclei become one helium nucleus, and percent of the mass becomes energy.
The Sun emits J of light each second; dividing by , it loses about million tonnes of mass per second. What is conserved in a reaction is the total energy, with the rest energy counted in.