How to find the trajectory of stationary action of a photon
I am trying to work through a "check your understanding" question on p.393 of the textbook From Photon to Neuron by <PERSON>. The action functional S of a photon's trajectory is proposed to be
$S [\underline X^\mu(\xi),e(\xi)] =\frac{\hbar}{2} \int d\xi (e^{-1}||\dot {\underline X}||^{2} )$
where $\mu$ = 0,1,2, or 3. $\underline X^\mu$ are spacetime coordinates; $\underline X^0$ is $ct$. The parameter $\xi$ is some index along the trajectory. The variable $e$ is described as a metric necessary so that different parameterizations of the same path give the same action. 394
The dot denotes differentiation with respect to $\xi$ and $||\dot {\underline X}||^{2} $ is the Lorentz invariant interval $ -(c \dot t)^2 + (\dot r)^2 $.
The reader is asked to show that the equation above is stationary under any trajectory that satisfies
$0 = e^{-2} ||\dot {\underline X}||^{2}$ and
$e^{-1} \dot {\underline X} =$ constant.
It's not clear to me if the second equation is implying an <PERSON> summation.
Here is what I have tried: I introduced a small variation in both the spacetime coordinates and metric along a path and examined the action of this new worldline:
$S [\underline X^\mu+ \Delta \underline X^\mu,e + \Delta e] =\frac{\hbar}{2} \int d\xi \bigl[(e +\Delta e )^{-1}||\dot {\underline X}+\Delta\dot {\underline X} ||^{2} \bigr]$
$\approx \frac{\hbar}{2} \int d\xi (1 -\frac{\Delta e}{e} )\bigl( ||\dot {\underline X}||^2+2 \dot {\underline X}\Delta\dot {\underline X}^{2} + ||\Delta\dot {\underline X}^{2}||\bigr) $
I then subtracted the action of the original worldline to get
$ \Delta S \approx \frac{\hbar}{2} \int d\xi \bigl(\frac {2\dot {\underline X}\Delta \dot X}{e}
+\frac {{\Delta \dot X}^2}{e}
$ $ +\frac {2 \dot X \Delta \dot X \Delta e}{e^2} $ $ +\frac {\Delta \dot X^2 \Delta e}{e^2} $ $ + \frac {|| \dot X||^2 \Delta e}{e^2} \bigr) $
I figured I could neglect terms that were second order in $\Delta$. That leaves the first and last term in the equation above. The last term must equal zero; this satifies one of the two conditions I set out to show. But shouldn't the first also have to equal zero, not a constant?
I am hoping someone can help me track down my error(s), explain if one must vary $e$ in addition to $X$, and clarify if the second condition stated above is an <PERSON> summation, i.e. if $ \dot {\underline X} = c \dot t - \dot r$. Thanks! 955