Background
E = mc2 where m stands for rest mass The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the (invariant mass The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the) m0, applies most simply to single particles viewed in an inertial frame where they have no momentum In classical mechanics, momentum is the product of the mass and velocity of an object (p = mv). For more accurate measures of momentum, see the section "modern definitions of momentum" on this page. It is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum. Linear momentum is a vector. But it also applies to ordinary objects composed of many particles so long as the particles are moving in different directions so the "net" or total momentum is zero. The rest mass of the object includes contributions from heat and sound, chemical binding energies, and trapped radiation. Familiar examples are a tank of gas, or a hot poker. The kinetic energy of their particles, the heat motion and radiation, contribute to their weight on a scale according to E = mc2.
The formula is the special case of the relativistic energy-momentum relationship:
This equation gives the rest mass of an object which has an arbitrary amount of momentum and energy. The interpretation of this equation is that the rest mass is the relativistic length of the energy-momentum four-vector In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis. The components transform between these bases as.
If the equation E = mc2 is used with the rest mass The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the or invariant mass The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the of the object, the E given by the equation will be the rest energy The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the of the object, and will change according to the object's internal energy, heat and sound and chemical binding energies (all of which must be added or subtracted from the object), but will not change with the object's overall motion (in the case of systems, the motion of its center of mass). However, if a system is closed, its invariant mass does not vary between different inertial observers (different inertial frames In physics, an inertial frame of reference is a reference frame, tied to the state of motion of an observer, with the property that each physical law portrays itself in the same form in every inertial frame. The contrasting case is the set of non-inertial frames, in which the laws of physics change from frame to frame, and the usual forces), and is also constant, and conserved.
If the equation E = mc2 is used with the relativistic mass The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass may also be applied to systems of particles in relative motion, of the object, the energy will be the total energy of the object, which is also conserved so long as no energy is added to or subtracted from the object, However, like the kinetic energy, this total energy will depend on the velocity of the object, and is different in different inertial frames. Thus, this quantity is not invariant between different inertial observers, even though it is constant over time for any single observer. As in the case of rest energy The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c2, which is equal to the mass of the, these relationships for total energy are also true for systems of objects, so long as the system is closed.
- Mass-Velocity Relationship
In developing special relativity Special relativity (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies", Einstein found that the kinetic energy of a moving body is
He included the second term on the right to make sure that for small velocities, the energy would be the same as in classical mechanics:
Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einstein found that the total momentum of a moving particle is:
and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass may also be applied to systems of particles in relative motion,, m.
And the relativistic mass and the relativistic kinetic energy are related by the formula:
Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy which obeys:
which is a sum of the rest energy m0c2 and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning.
In a collision process where all the rest-masses are the same at the beginning as at the end, either expression for the energy is conserved. The two expressions only differ by a constant which is the same at the beginning and at the end of the collision. Still, by analyzing the situation where particles are thrown off a heavy central particle, it is easy to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits.
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