Physics 515 Problem Set #2 April 8, 2004
Due Thursday, April 15, 2004

1. (30 pts) Consider an empty spherical cavity of radius $R$ with perfectly conducting walls. Suppose the (radiation gauge) vector potential has the form $\A(\r,t) = \Grad \times i\L\, \psi(\r,t)$, where $i\L\equiv \r \times \Grad$. Recall from last quarter (or basic quantum), that $[L_m,L_n] = i\, \epsilon_{mnp} \, L_p$, that $\Grad \cdot \L = \r \cdot \L = 0$, and that $\L$ commutes with anything which is spherically symmetric. Find $\B(\r,t)$ (in terms of $\psi$, simplified as much as possible), and show that $\B$ has no radial component, $\hat\r \cdot \B = 0$. Find $\E(\r,t)$, and show that all four vacuum Maxwell's equations are satisfied provided $\psi(\r,t)$ obeys the wave equation. Derive the required boundary condition on $\psi(\r,t)$ assuming that the cavity has perfectly conducting walls. Let $\{ \lambda_\alpha \}$ denote the eigenvalues of $-\Grad^2$ with your choice of boundary condition at $r = R$. Express the possible resonant frequencies in terms of $\lambda_\alpha$. Extra credit: Find corresponding modes for which $\hat \r \cdot \E = 0$.
2. (30 pts) Draw a spacetime diagram which clearly resolves the pole-in-the-barn ``paradox'': a relativistic runner carrying a pole of length $L'$ approaches a barn of length $L$ (whose front and back doors are both open). The barn is shorter than the pole, $L \lt L'$, but in the rest frame of the barn, the Lorentz contracted length of the pole equals $L$, so the pole is seen to just fit within the barn. In the rest frame of the runner, it is the barn, not the pole which is Lorentz contracted, so the pole cannot possibly fit inside the barn. In your diagram, show the worldlines of the front and back of the barn, and the front and back of the pole. Clearly identify event `A' where the front of the pole enters the barn, event `B' where the front of the pole exits the barn, event `C' where the back of the pole enters the barn, and event `D' where the back of the pole exits the barn. Show a surface of simultaneity for observers in the rest frame of the barn, and a surface of simultaneity for observers in the rest frame of the pole, with both surfaces passing through event A. Mark on these surfaces those events which are an interval $L$, or $L'$, away from event A.
3. (40 pts) Lorentz Group Properties. Starting from the defining condition for a Lorentz transformation, $g_{\alpha\beta} = l^\mu_{\;\alpha} \, l^\nu_{\;\beta} \, g_{\mu\nu}$, prove that:
   1. $(l^0_{\;0})^2 \ge 1$. Therefore either $l^0_{\;0} \ge 1$ or $l^0_{\;0} \le -1$.
   2. $\det l = \pm 1$, where the matrix $l = \Vert l^{\mu}_{\;\nu}\Vert$. Hint: Write the defining condition as a matrix equation involving $l$ and $l^T$.
   3. These results imply that the Lorentz group has (at least) four disconnected components:
      1. proper, orthochronous: $l^0_{\;0} \ge 1$, $\det l = 1$;
      2. improper, orthochronous: $l^0_{\;0} \ge 1$, $\det l = -1$;
      3. proper, antiorthochronous: $l^0_{\;0} \le -1$, $\det l = 1$;
      4. improper, antiorthochronous: $l^0_{\;0} \le -1$, $\det l = -1$.
      In fact, each of these components is individually connected. Display simple examples of transformations in each class. Explain why the set (or subgroup) of proper, orthochronous Lorentz transformations is the same as the set of transformations which can be built up from infinitesimal transformations.
   4. Any 4-vector $x$ may be classified as lying within one of the categories:
      1. timelike, future-directed: $x^2 \lt 0$, $x^0 \gt 0$;
      2. timelike, past-directed: $x^2 \lt 0$, $x^0 \lt 0$;
      3. null, future-directed: $x^2 = 0$, $x^0 \gt 0$;
      4. null, past-directed: $x^2 = 0$, $x^0 \lt 0$;
      5. spacelike: $x^2 \gt 0$;
      6. zero: $x = 0$.
      (``Null'' is a synonym for ``light-like''.) Display a concrete example of a 4-vector in each category. Prove that orthochronous Lorentz transformations transform a 4-vector in any one of these categories into a vector in the same category. In other words, prove that these categories are invariant under orthochronous transformations. Show that this fails if one divides the spacelike category into spacelike, future-directed and spacelike, past-directed. In other words, show that there is no invariant distinction between past and future directed spacelike vectors. Show that antiorthochronous transformations map categories (i) $\leftrightarrow$ (ii) and (iii) $\leftrightarrow$ (iv).
4. (Extra credit) Under the action of an arbitrary Lorentz transformation $l^\mu_{\;\nu}$, a scalar field $\phi(x)$ is defined to transform as $\phi(x) \to \bar\phi(x)$ where $\bar\phi(\bar x) \equiv \phi(x)$ for $\bar x^\mu \equiv l^\mu_{\;\nu} \, x^\nu$. Similarly, a vector field $v^\alpha(x)$ is defined to transform as $v^\alpha(x) \to \bar v^\alpha(x)$ where $\bar v^\alpha(\bar x) \equiv l^\alpha_{\;\beta} \, v^\beta(x)$.

   Justify the assertion that the current density $j^\mu(x) \equiv (c \rho(x), \vec \jmath(x))$ does, in fact, transform as a vector field under Lorentz boosts. Use an operational definition for charge density (in terms of the amount of charge in an infinitesimal volume) and for components of the spatial current density (in terms of the amount of charge flowing through an infinitesimal surface per unit time). Explain how, from the perspective of a moving observer, the effects of Lorentz contraction and time dilation precisely conspire to produce the claimed transformation.