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quote:Mass, momentum, and energyIn addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.Given an object of invariant mass m0 traveling at velocity v the energy and momentum are given (and even defined) by where #947; (the Lorentz factor) is given by and c is the speed of light. The term #947; occurs frequently in relativity, and comes from the Lorentz transformation equations.Relativistic energy and momentum can be related through the formula which is referred to as the relativistic energy-momentum equation.For velocities much smaller than those of light, #947; can be approximated using a Taylor series expansion and one finds that Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and #947; = 1) there is a non-zero energy remaining: This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful.Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy. This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life have made it one of the most famous equations in all of science.
quote:Originally posted by thebrain13I guess im missing something, what is the equation? shouldn't this equation start with v?