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SR_67 - ttlt Ex 66 Particlesol Zerc Rest Mass equal to the...

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ttlt Ex. 66 Particles ol Zerc Rest Mass C. PHOTONS 66. Particles of zero rost mass On what arguments does the derivation of the im- portant relation E - p': rnr depend?Derive from this formula a relation between momentum and enerry that is valid in the case of zero rest mass (photons, gravitons, neutrinos). What does the re- sulting relation say about the slope of the world line for such a particle and hence about its speed? How does your result depend, as a limiting $, on the equality of sinhd and coshd for large d? Is there a "rest frame" for particles of zero rest mass? 67. Einstein's derivation of the equivalence of energy and rest mass-a worked oxamplo Problem: From the fact that light exerts pressure and carries energy, show that this energy is equivalent to mass and hence-by extension-show the equiva- lence of all energy to mass. Commentary: The equivalence of energy and mass is such an important consequenc that Einstein very early, after his rela- tivistic derivation of this result, sought and found an alternative elementary physical line of reasoningt that leads to the same conclusion. He envisaged a closed box of massM initially at rest (Fig. 106). A directed burst of electromagnetic enerry is emitted from the hft-hand wall. It travels down the length I, of the box and is absorbed at the other end. The radiation carries an enerry E. But it also carries momentum. This one sces from the following reasoning. The radiation ercrts a pressure on the left-hand wall during the ernbsion. In consequenc of this pressure the box rcceivcsa push to the left, and a momentum -p. But Flg. 105. Transfer of mass without net transferof particles or radia- tion. the momentum of the system as a whole was zero initially. Therefore the radiation carries a momentum p opposite to the momentum of the box. How can one use his knowledge of the transport of enerry and momentum by the radiation to deduce the raass equivaknt of the radiation? Einstein got his answer from the argument that the center of mass of the sys- tem was not moving before the transport process and therefore cannot be in motion during the transport proc ss.But the box obviously carries mass to the ldt. Therefore the radiation mwt cany mass to the right. So much for Einstein's reasoning in broad out- line. Now for the details. From relativity Einstein knew that the momentum p of a directed beam of radiation is equal to the energt E of that beam (both p and E measured in units of mass; Section l0). However, to secure a derivatjon free of all direct reference to relativity principles, he based the conclusion p: E on the following ele- mentary argument. The pressure exerted on an ideal emitter or absorber by a directed beam of radiation is Fk. 106. Transfer of mass by radiation.
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