4. (25 pts total) The SLAC linear electron accelerator can accelerate electrons to energies of 50 GeV. Consider shining an intense ruby laser (photon energy 1.8 eV) directly into the electron beam, so that photons from the laser collide essentially head-on with electrons. This will produce many elastic scattering events, e + e +y. In some of these events, the photon will be back-scattered, meaning that the final photon is moving in the same direction as the initial electron. (a) [10 pts] Let p and p denote, respectively, the initial and final four-momenta of the electron, and likewise let k and k denote the initial and final photon four-momenta. Prove that 0 = pi ki (pi + ki) k. . (b) [10 pts] Use the relation in part (a) to express the final energy E', of the back-scattered photon in terms of the energies Ee and E, of the initial electron and photon. (c) [5 pts] Evaluate E', using the given numerical information. By what factor does the photon energy change?