Reading:

Taylor, Archibald and Wheeler chapter 6.
Rolnick section 4.1 (Note that Rolnick uses natural units, setting $c=1$.)

Problems:

1. In class we used Lorentz boost matrices to relate measurement of an event's spacetime coordinates in different inertial frames.
   1. Show that $\gamma$ which appears in the Lorentz boost is always greater or equal to 1.

   2. For two spacelike separated events, there is always an inertial frame in which they appear to be simultaneous. What is the distance between the events in that frame, in terms of some Lorentz invariant object?

   3. For two timelike separated events, there is always an inertial frame in which they appear to occur at the same place. What is the time between the events in that frame, in terms of some Lorentz invariant object?

2. Take A and B to be simultaneous events in the lab frame, with $B$ a distance $\Delta s$ to the right of A (along the x-axis). By using specific Lorentz boosts,
   1. Show that different observers view B as occuring variously before, or after, A.
   2. Show that B is always perceived to occur to the right of A, at a distance of at least $\Delta s$.

3. Take A and B to occur at the same spatial point in the lab frame, with $B$ occurring a time $\Delta \tau$ after A. By using specific Lorentz boosts,
   1. Show that different observers view B as occuring variously to the left, or to the right, of A (along, say, the x-axis).
   2. Show that B is always perceived to occur after A, after a time of at least $\Delta \tau$.

4. 1. Using the usual definition of the position 4-vector $x^\mu$, write out its covector $x_\mu$ explicitly as a row vector. Be sure to show how you used the metric $\eta_{\mu\nu}$ to obtain the components.
   2. Calculate the inner product $x_\mu \, x^\mu$ explicitly, in terms of spacetime components $ct, x, y,$ and $z$.

5. For electromagnetism, we can show that charge density $\rho$ and current $\vec{j}= \rho\, \vec{v}$ transform under Lorentz transformations as a 4-vector:
   
   $$j^{\, \mu} = \left(\ \begin{array}{c}c\rho\\ \vec{j}\end{array}\ \right)\ \ .$$
   
   
   We can also show that $j^{\, \mu}$ obeys the following (Lorentz invariant) equation:
   
   $$\partial_\mu\,j^{\, \mu} = 0\ \ .$$
   
   
   1. Evaluate the sum over $\mu$ explicitly, to rewrite the equation $\partial_\mu\,j^{\, \mu} = 0$ as a sum of terms involving the individual components of $\partial_\mu$ and $j^{\, \mu}$.
   2. Explain why this equation corresponds to local charge conservation: that is, why it states that charge in a volume changes only by locally flowing in and out of the volume.

6. Also for electromagnetism (the first known Lorentz-covariant theory), we find that electric and magnetic fields transform into each other under Lorentz transformations, in such a way that the object
   
   $$F^{\mu\nu} = \left(\ \begin{array}{rrrr} 0&-E_x&-E_y&-E_z\\ E... ...-B_z&B_y\\ E_y&B_z&0&-B_x\\ E_z&-B_y&B_x&0 \end{array}\ \right)$$
   
   
   is an antisymmetric rank $(2,0)$ tensor, written in row-column order ($\mu$ indexes rows, $\nu$ indexes columns). (This object is so complicated because electric and magnetic fields are not really fundamental objects from a Lorentz transformation point of view.) We find that they obey the Lorentz-covariant equation
   
   $$\partial_\mu\, F^{\mu\nu} = j^\nu/c\ .$$
   
   
   Explicitly evaluate the sum over $\mu$ to show
   1. for $\nu = 0$, that you obtain Gauss' law.
   2. for $\nu = i$ (the spatial indices), that you obtain Ampere's law.

7. 1. Identify whether each of the following is invariant, or transforms under Lorentz transformations. If the object transforms, identify how it transforms -- as a 4-vector, 4-covector, rank (x,y) tensor; or by mixing with what other observables.
      1. $a_\mu b^\mu$
      2. $a_\mu b^\nu$
      3. $T_{\mu\nu}\ v^\nu$
      4. $T_{\mu\nu}T^{\mu\nu}$
      5. $\vec{E}$ (electric field)
      6. $\vec{j}$ (electric current)

   2. Are the following Lorentz-covariant equations? State why or why not.
      1. $\vec{F}= m\vec{a}$
      2. $\frac{d^2 x^\mu}{dt^2} = F^\mu$
      3. $\frac{d^2 x^\mu}{d\tau^2} = F^\mu$
      4. $\partial _\mu\partial^\mu \phi + m^2 \phi = 0$
      5. $p_\mu p^\mu = m^2$
      6. $F_{\mu\nu} = \partial _\mu A_\nu - \partial _\nu A_\mu$