Simple Nature - Light and Matter

these lines. For example, a car sitting in a parking lot has p = 0and E = mc 2 .Now what happens to such a graph when we change to a differentframe or reference that is in motion relative to the originalframe? A massless particle still has to act like a massless particle,so the diagonals are simply stretched or contracted along their ownlengths. In fact the transformation must be linear (p. 387), becauseconservation of energy and momentum involve addition, andwe need these laws to be valid in all frames of reference. By thesame reasoning as in figure j on p. 389, the transformation must bearea-preserving. We then have the same three cases to consider asin figure g on p. 388. Case I is ruled out because it would implythat particles keep the same energy when we change frames. (Thisis what would happen if c were infinite, so that the mass-equivalentE/c 2 of a given energy was zero, and therefore E would be interpretedpurely as the mass.) Case II can’t be right because it doesn’tpreserve the E = |p| diagonals. We are left with case III, which establishesthe fact that the p-E plane transforms according to exactlythe same kind of Lorentz transformation as the x-t plane. That is,(E, p x , p y , p z ) is a four-vector.The only remaining issue to settle is whether the choice of unitsthat gives invariant 45-degree diagonals in the x-t plane is the sameas the choice of units that gives such diagonals in the p-E plane.That is, we need to establish that the c that applies to x and t isequal to the c ′ needed for p and E, i.e., that the velocity scales of thetwo graphs are matched up. This is true because in the Newtonianlimit, the total mass-energy E is essentially just the particle’s mass,and then p/E ≈ p/m ≈ v. This establishes that the velocity scalesare matched at small velocities, which implies that they coincide forall velocities, since a large velocity, even one approaching c, can bebuilt up from many small increments. (This also establishes thatthe exponent n defined on p. 421 equals 1 as claimed.)Since m 2 = E 2 − p 2 , it follows that for a material particle, p =mγv.422 Chapter 7 Relativity