Exercises

1. Consider two Cartesian reference frames, $S$ and $S'$ , in the standard configuration. Suppose that $S'$ moves with constant velocity $v<c$ with respect to $S$ along their common $x$ -axis. Demonstrate that the Lorentz transformation between coordinates in the two frames can be written

   $$x' = x\,\cosh\varphi - c\,t\,\sinh\varphi,$$

   $$y' = y,$$

   $$z' = z,$$

   $$c\,t' = c\,t\,\cosh\varphi - x\,\sinh\varphi,$$

   
   
   where $\tanh\varphi = v/c$ . Show that the previous transformation is equivalent to a rotation through an angle ${\rm i}\,\varphi$ , in the $x$ - ${\rm i}\,c\,t$ plane, in ($x$ , $y$ , $z$ , ${\rm i}\,c\,t$ ) space.

2. Show that, in the standard configuration, two successive Lorentz transformations with velocities $v_1$ and $v_2$ are equivalent to a single Lorentz transformation with velocity
   $$v = \frac{v_1+v_2}{1+v_1\,v_2/c^{\,2}}.$$

3. Let ${\bf r}$ and ${\bf r}'$ be the displacement vectors of some particle in the Cartesian reference frames $S$ and $S'$ , respectively. Suppose that frame $S'$ moves with velocity ${\bf v}$ with respect to frame $S$ . Demonstrate that a general Lorentz transformation takes the form

   $${\bf r}' = {\bf r} + \left[\frac{(\gamma-1)\,{\bf r}\cdot{\bf v}}{v^{\,2}} - \gamma\,t\right]{\bf v},$$

   $$t' = \gamma\left(t - \frac{{\bf r}\cdot{\bf v}}{c^{\,2}}\right),$$ (1982)

   
   
   where $\gamma = (1-v^{\,2}/c^{\,2})^{-1/2}$ . If ${\bf u} =d{\bf r}/dt$ and ${\bf u}'=d{\bf r}'/dt'$ are the particle's velocities in the two reference frames, respectively, demonstrate that a general velocity transformation is written
   $${\bf u}' = \frac{{\bf u}+ \left[(\gamma-1)\,{\bf u}\cdot{\bf v}/c^{\,2} - \gamma\right]{\bf v}}{\gamma\,(1-{\bf u}\cdot{\bf v}/c^{\,2})}.$$

4. Let $v$ be the Earth's approximately constant orbital speed. Demonstrate that the direction of starlight incident at right-angles to the Earth's instantaneous direction of motion appears slightly shifted in the Earth's instantaneous rest frame by an angle $\theta =\sin^{-1}(v/c)$ . This effect is known as the abberation of starlight. Estimate the magnitude of $\theta$ (in arc seconds).

5. Let ${\bf E}$ and ${\bf B}$ be the electric and magnetic field, respectively, in some Cartesian reference frame $S$ . Likewise, let ${\bf E}'$ and ${\bf B}'$ be the electric and magnetic field, respectively, in some other Cartesian frame $S'$ , which moves with velocity ${\bf v}$ with respect to $S$ . Demonstrate that the general transformation of fields takes the form

   $${\bf E}' = \gamma\,{\bf E} + \frac{1-\gamma}{v^{\,2}}\,({\bf v}\cdot{\bf E})\,{\bf v} + \gamma\,({\bf v}\times {\bf B}),$$

   $${\bf B}' =\gamma\,{\bf B} + \frac{1-\gamma}{v^{\,2}}\,({\bf v}\cdot{\bf B})\,{\bf v} - \frac{\gamma}{c^{\,2}}\,({\bf v}\times {\bf E}),$$

   
   
   where $\gamma = (1-v^{\,2}/c^{\,2})^{-1/2}$ .

6. A particle of rest mass $m$ and charge $e$ moves relativistically in a uniform magnetic field of strength $B$ . Show that the particle's trajectory is a helix aligned along the direction of the field, and that the particle drifts parallel to the field at a uniform velocity, and gyrates in the plane perpendicular to the field with constant angular velocity
   $${\mit\Omega} = \frac{e\,B}{\gamma\,m}.$$
   Here, $\gamma = (1-v^{\,2}/c^{\,2})^{-1/2}$ , and $v$ is the particle's (constant) speed.

7. Let $P={\bf E}\cdot{\bf B}$ and $Q =c^{\,2}\,B^{\,2}-E^{\,2}$ . Prove the following statements, assuming that $E$ and $B$ are not both zero.
   1. At any given event, ${\bf E}$ is perpendicular to ${\bf B}$ either in all frames of reference, or in none. Moreover, each of the three relations $E>c\,B$ , $E=c\,B$ , and $E<c\,B$ holds in all frames or in none.

   2. If $P=Q=0$ then the field is said to be null. For a null field, ${\bf E}$ is perpendicular to ${\bf B}$ , and $E=c\,B$ , in all frames.
   3. If $P=0$ and $Q\neq 0$ then there are infinitely many frames (with a common relative direction of motion) in which $E=0$ or $B=0$ , according as $Q>0$ or $Q<0$ , and none other. Precisely one of these frames moves in the direction ${\bf E}\times {\bf B}$ , its velocity being $E/B$ or $c^{\,2}\,B/E$ , respectively.

   4. If $P\neq 0$ then there are infinitely many frames (with a common direction of motion) in which ${\bf E}$ is parallel to ${\bf B}$ , and none other. Precisely one of these moves in the direction ${\bf E}\times {\bf B}$ , its velocity being given by the smaller root of the quadratic equation $\beta^{\,2}-R\,\beta+1=0$ , where $\beta = v/c$ , and $R = (E^{\,2}+c^{\,2}\,B^{\,2})/\vert{\bf E}\times c{\bf B}\vert$ . In order for $\beta$ to be real we require $R>2$ . Demonstrate that this is always the case.

8. In the rest frame of a conducting medium, the current density satisfies Ohm's law ${\bf j}' = \sigma \,{\bf E}'$ , where $\sigma$ is the conductivity, and primes denote quantities in the rest frame.
   1. Taking into account the possibility of convection currents, as well as conduction currents, show that the covariant generalization of Ohm's law is
      $$J^{\,\mu} - \frac{1}{c^{\,2}}(U_\nu \,J^{\,\nu})\, U^{\,\mu}= \frac{\sigma}{c} F^{\,\mu\nu} U_\nu,$$
      where $U^{\,\mu}$ is the 4-velocity of the medium, $J^{\,\mu}$ the 4-current, and $F^{\,\mu\nu}$ the electromagnetic field tensor.
   2. Show that if the medium has a velocity ${\bf v}=c\,$$\beta$ with respect to some inertial frame then the 3-vector current in that frame is
      $${\bf j} = \gamma\,\sigma\, [{\bf E} +$$   $\beta$ $$\times c\,{\bf B} -($$$\beta$ $$\cdot{\bf E})\,$$$\beta$ $$] +\rho \,{\bf v}$$
      where $\rho$ is the charge density observed in the inertial frame.

9. Consider the relativistically covariant form of Maxwell's equations in the presence of magnetic monopoles. Demonstrate that it is possible to define a proper-4-current
   $$J^{\,\mu} = ({\bf j},\,\rho\,c),$$
   and a pseudo-4-current
   $$J_m = ({\bf j}_m,\,\rho_m\,c),$$
   where ${\bf j}$ and $\rho$ are the flux and density of electric charges, respectively, whereas ${\bf j}_m$ and $\rho_m$ are the flux and density of magnetic monopoles, respectively. Show that the conservation laws for electric charges and magnetic monopoles take the form

   $$\partial_\mu J^{\,\mu} =0,$$

   $$\partial_\mu J_m^{\,\mu} =0,$$

   
   
   respectively. Finally, if $F^{\,\mu\nu}$ is the electromagnetic field tensor, and $G^{\,\mu\nu}$ its dual, show that Maxwell's equations are equivalent to

   $$\partial_\mu F^{\,\mu\nu} = \frac{J^{\,\nu}}{\epsilon_0\,c},$$

   $$\partial_\mu G^{\,\mu\nu} = \frac{J_m^{\,\nu}}{\epsilon_0\,c}.$$

   
   

10. Prove that the electromagnetic energy tensor satisfies the following two identities:
    $${T^{\,\mu}}_\mu = 0,$$
    and
    $${T^{\,\mu}}_\sigma \,{T^{\,\sigma}}_\nu = \frac{I^{\,2}}{4} \,\delta^{\,\mu}_\nu,$$
    where
    $$I^{\,2} = \left(\frac{B^{\,2}}{\mu_0}- \epsilon_0 \,E^{\,2}\right)^2 + \frac{4\,\epsilon_0}{\mu_0} \,({\bf E}\cdot{\bf B})^2.$$

11. A charge $e$ moves in simple harmonic motion along the $z$ axis, such that its retarded position is $z(t')=a\, \cos(\omega_0 \,t')$ .
    1. Show that the instantaneous power radiated per unit solid angle is
       $$\frac{dP(t')}{d{\mit\Omega}} = \frac{e^{\,2}\,c \,\beta^{\,4}}{16... ...ta \,\cos^2(\omega_0 \,t')}{[1+\beta\,\cos\theta \,\sin(\omega_0 \,t')]^{\,5}}$$
       where $\beta = a\,\omega_0/c$ , and $\theta$ is a standard spherical polar coordinate.
    2. By time averaging, show that the average power radiated per unit solid angle is
       $$\frac{dP}{d{\mit\Omega}} = \frac{e^{\,2} \,c\, \beta^{\,4}}{128\p... ...,2}\,\cos^2\theta}{(1-\beta^{\,2}\,\cos^2\theta)^{\,7/2}}\right] \sin^2\theta.$$

    3. Sketch the angular distribution of the radiation for non-relativistic and ultra-relativistic motion.

12. The trajectory of a relativistic particle of charge $e$ and rest mass $m$ in a uniform magnetic field ${\bf B}$ is a helix aligned with the field. Let the pitch angle of the helix be $\alpha$ (so, $\alpha=0$ corresponds to circular motion). By arguments similar to those used for synchrotron radiation, show that an observer far from the charge would detect radiation with a fundamental frequency
    $$\omega_0 = \frac{{\mit\Omega}}{\cos^2\alpha},$$
    where ${\mit\Omega} = e\,B/(\gamma \,m)$ , and that the spectrum would extend up to frequencies of order
    $$\omega_c = \gamma^{\,3} \,{\mit\Omega}\, \cos\alpha.$$