Physics 515 Problem Set #3 April 15, 2004
Due Thursday, April 22, 2004

1. (10 pts) Let $\hat n$ be a spatial unit vector (with components $n_i$). Show that a boost with rapidity $\chi$ in the direction $\hat n$ is represented by the Lorentz transformation $l^\mu_{\;\nu}$ with $l^0_{\;0} = \cosh\chi$, $l^0_{\;i} = l^i_{\;0} = -n_i \, \sinh\chi$, and $l^i_{\;j} = \delta_{ij} + n_i \, n_j \, (-1 + \cosh \chi)$. (The rapidity $\chi$ is related to the boost velocity by $v/c = \tanh \chi$.) Show that rapidities of multiple boosts along the same direction add, so that $l^\mu_{\;\nu}(\chi_1{+}\chi_2) = l^\mu_{\;\lambda}(\chi_2) \, l^\lambda_{\;\nu}(\chi_1)$.
2. (10 pts) An infinite straight wire of negligible thickness is uncharged and carries a current $I$ (when viewed in its rest frame). Choose coordinates so that the wire sits on the $z$-axis. What is the 4-current density $j^\mu(x)$ in the rest frame of the wire? What is the 4-current density $\bar \jmath^\mu(\bar x)$ in the rest frame of an observer moving at velocity $v \hat z$ relative to the stationary wire?
3. (25 pts)
   
   A photon is emitted at point $A$ inside a centrifuge which is rotating at angular velocity $\omega$, and absorbed at another point $B$ inside the centrifuge. If $\nu_e$ denotes the frequency of emission (as measured by an observer at $A$ at rest in the rotating frame of the centrifuge), and $\nu_o$ denotes the frequency of absorption (as measured by an observer at $B$ at rest in the rotating frame of the centrifuge), what is the ratio $\nu_o/\nu_e$? In other words, is any Doppler effect observed in this experiment?
   
4. (10 pts) Two particles with 4-momenta $p_a$ and $p_b$ collide. Their rest masses, $m_a$ and $m_b$, may each be either zero or non-zero. Explain why the total 4-momentum $P_{\rm tot} = p_a+p_b$ is timelike, and hence there exists a frame, called the CM (center-of-mass) frame, in which the total spatial momentum vanishes, $\vec P_{\rm tot} = 0$. [The name ``center-of-mass'' is really a misnomer, even though it is universally used. It should really be called the ``zero-momentum'' frame.] Show that the Lorentz invariant quantity $s \equiv -(p_a+p_b)^2$ equals the square of the total center-of-mass energy, $(E^{\rm c.m.}_{\rm tot})^2$.

   Suppose that the particles, viewed in the lab frame, collide head-on. Re-evaluate $s$ in terms of the lab-frame energies $E_a$ and $E_b$ of the two particles (and their masses), in order to relate the total center-of-mass energy to the individual lab frame energies.

5. (50 pts) A laser emits photons of energy $\omega$ which scatter from electrons of energy $E$. The collision is head-on (i.e., the photons and electrons are collinear and moving toward each other). For numerical estimates, assume that $\hbar \omega = 1$ eV and $E = 40$ GeV. Let $\theta$ be the angle between the initial and final photon directions.
   1. Compute the energy $\omega'$ of the scattered photon as a function of the scattering angle $\theta$ (as well as $\omega$ and $E$). Sketch this function and show that it has a peak for backward scattering, $\theta = \pi$. How high is this peak? How wide is the peak? Express your answers in terms of $\omega$ and $E$ assuming that $E \gg m_e c^2$, and also numerically using the values given above.
   2. As a crude approximation, suppose that the photon is scattered isotropically in the center-of-mass frame [the frame in which the total spatial momentum vanishes] with a total cross section $\sigma_0$. Denote the initial and final photon 4-momenta by $k$ and $k'$, respectively, and compute the invariant momentum transfer squared $t \equiv (k-k')^2$ in the center-of-mass frame. Explain why isotropy in the center-of-mass frame is equivalent to having the differential cross section with respect to momentum transfer, $d\sigma/dt$ be independent of $t$. This cross section is a Lorentz scalar with respect to boosts along the beam direction -- the initial direction of the collinear photons and electrons. Use this Lorentz invariance to compute the angular distribution of the differential cross section $d\sigma/d\Omega$ observed in the laboratory [that is, the cross section as a function of $\theta$]. Sketch the result and indicate the width of the back-scattering peak for the given values of $\omega$ and $E$.