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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 13}\\%
\vspace {3mm} (April 8, 2009) \\
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\item{(46)} For light from some galaxy the Balmer spectrum of the 
hydrogen atom is found at wavelengths $\lambda=(729.2\,{\rm nm})\,
m^2/(m^2-4)$, $m=3,\,4,\,5,\,\dots~.$ Neglect expansion of space 
and find the speed at which the galaxy moves away or towards us.  
Due April 14 before class (4~points).  \medskip

\item{(47)} Derive for the momentum in the rest frame $p^0=c\,m_0$,
where $m_0$ is the rest mass, from the relativistic transformation
laws and the known definition of momentum in the non-relativistic
limit. Due April 12 in class (4~points).  \medskip

\item{(48)} (1)~Taylor expand $\sqrt{x+\epsilon}$ to leading order 
in $\epsilon$. (2)~Substitute $x=c^4m_0^2$ and $\epsilon=c^2
\vec{p}^{\,2}$. (3)~Compare with $m v^2/2$.
Due April 12 in class (4~points).  \medskip

\item{(49)} End of the spaceship journey: Assume that the spaceship
moves by transforming mass into light, which is exhausted.
\begin{enumerate}
\item Derive an expression for $m(\tau )$, the (remaining) mass
      of the spaceship at proper time $\tau$. (Hint: Use momentum
      conservation in the instanteneous rest frame.)
\item Which fraction of the original mass is left, after the 
      spacetrip has been performed? 
\end{enumerate}
Due April~19 before class (8~points). 
 
\item{(50)} (1) Write down the definition of $\partial_{\alpha}$.
Due April 14 in class (1~point).  

(2) Let $A^{\alpha}$ be a vector field. Write down the scalar 
field from this and $\partial_{\alpha}$ (not the scalar operator).
Due April 16 in class (1~point).  

(3) Use the quantities of assignments (1) and (2) to write down a
rank two tensor $T^{\alpha\beta}$ field (not a tensor operator).
Due April 16 in class (1~point).  

(4) Write the equation
$$ \partial_{\alpha}\,F^{\alpha\beta} = \frac{4\pi}{c}\,J^{\beta} $$
explicitly out for $\beta = 0,\,1,\,2,\,3$.
Due April 16 in class (4~points).  

(5) Compare the $\beta=0$ result with $\nabla\vec{E}=4\pi\rho$ and
the $\beta=i=1,\,2,\,3$ results with $\nabla\times\vec{B}-
c^{-1}\left(\partial\vec{E}/\partial t\right) = 4\pi\vec{J}/c$. 
Due April 16 in class (4~points).  
\medskip

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