Conservation of mass and energy

The concept of mass–energy equivalence unites the concepts of conservation of mass and conservation of energy, allowing rest mass to be converted to other forms of energy, like kinetic energy, heat, or light. Kinetic energy or light can also be converted to particles which have rest mass. The total amount of mass-energy in a closed system remains constant because energy cannot be created or destroyed and, in all of its forms, trapped energy has mass. According to the theory of relativity, mass and energy as commonly understood are two names for the same thing, and neither one appears without the other.

Fast-moving objects and systems of objects

If you push on an object in the direction of motion, it gains momentum and it gains energy. But if the object is already travelling near the speed of light, it can't move much faster, no matter how much energy it absorbs. Its momentum and energy continue to increase, but its speed approaches a constant value—the speed of light. This means that in relativity the momentum of an object cannot be a constant times the velocity, nor is the kinetic energy given by ½mv2.

The relativistic mass is defined as the ratio of the momentum of an object to its velocity, and it depends on the motion of the object. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass. As the object approaches the speed of light, the relativistic mass becomes infinite, because the momentum becomes infinite.

The relativistic mass is always equal to the total energy divided by c2. Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. If length and time are measured in natural units, the speed of light is equal to 1, and even this difference disappears. Then mass and energy have the same units and are always equal, so it is redundant to speak about relativistic mass, because it is just another name for the energy. This is why physicists usually reserve the useful short word "mass" to mean rest-mass.

For things made up of many parts, like a nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses of the parts, because energy adds up. In some cases, however, the parts include fields of force, and if the fields are attractive, they contribute a negative amount to the mass-energy. For example, the mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up. The amount by which it is smaller is the energy required to break up the nucleus into individual protons and neutrons. Similarly, the mass of the solar system is slightly less than the masses of sun and planets individually, since the gravitational field is attractive.

The relativistic mass of a moving object is bigger than the relativistic mass of an object that isn't moving, because a moving object has extra kinetic energy. The rest mass of an object is defined as the mass of an object when it is at rest, so that the rest mass is always the same independent of the motion of the observer: it is the same in all inertial frames.

For a system of particles going off in different directions, the invariant mass is the analog of the rest mass, defined as the total energy (divided by c2) in the center of mass frame, where the total momentum is zero.