% Bernd Berg Oct 13 2017.
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\centerline{\bf Special and General Relativity (PHZ 4601/5606) Fall 2017 
Solutions} \medskip

\centerline{\bf Set 9} 

\begin{description} % After Rindler 10.2.
\item {29. Are $t=var$, $x^i=const$ lines geodesics?}
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\noindent
The metric is $d\vec{s}\,^2=dt^2-dl^2$ and $dx^i=0$ holds for $t=var$.
Hence, $ds^2=dt^2 \Rightarrow s=t$. So, $\dot{t}=1$, $\ddot{t}=0$ and
$\dot{x}^i=0$, $\ddot{x}^i=0$.

\noindent
Now, let us check the geodesic equations of $L=\dot{t}^2-g_{ij}\,
\dot{x}^i\,\dot{x}^j$:
$$ \frac{d~}{dt}\,\frac{\partial L}{\partial\,\dot{t}} - 
   \frac{\partial L}{\partial\,t} = 0~~\Rightarrow~~ 
   2\,\ddot{t} - g_{ij,4}\,\dot{x}^i\,\dot{x}^j = 0 $$
which is solved by $\ddot{t}=0$, $\dot{x}^i=0$. Next,
$$ \frac{d~}{dt}\,\frac{\partial L}{\partial\,\dot{x}^i} - 
   \frac{\partial L}{\partial\,x^i} = 0~~\Rightarrow~~ 
   -2\,\frac{d~}{dt}\,\left(g_{ij}\,\dot{x}^j\right) - 
    g_{jk,i}\,\dot{x}^j\,\dot{x}^k = 0\,.$$
The geodesic equations are satisfied.

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