The acceleration of a particle moving at fixed Rindler coordinate $X=\frac{1}{g}$ Let us define the transformation from the Minkowski coordinates $x^\alpha=(t,x)$ to the Rindler coordinates $x^{\bar\alpha}(T,X)$ by
$$t=X\sinh T,\\
x=X\cosh T.$$
If an object is moving at fixed Rindler coordinate $X=1/g$ and $ds^2$ for Minkowski coordinates is given by $ds^2=-dt^2+dx^2$, I'd like to show that the 2-acceleration $a^{\bar\alpha}$ is given by
$$a^{\bar\alpha}=(a^T,a^X)=(0,g).$$
I tried to figure it out by using covariant derivatives, because I have successfully obtained the Christoffel symbols for the Rindler coordinates:
$$\Gamma^X_{TT}=X,\Gamma^T_{TX}=\Gamma^T_{XT}=\frac{1}{X}$$
After observing the complicated formula
$$a^{\bar\alpha}=\frac{d^2}{d\tau^2}x^{\bar\alpha}+\Gamma^{\bar\alpha}_{\bar\nu\bar\lambda}\frac{dx^{\bar\nu}}{d\tau}\frac{dx^{\bar\lambda}}{d\tau},$$
though, I didn't see any promising future for this approach. Am I on the right track? Thank you.
 A: In your notation $T$ is proper time and your $\tau$ is redundant because it equals $T\,.$ Some further notation of yours is a bit confusing. Did you mean $X=1/a$ (title) or $X=1/g$ (text)? In any case the coordinate change from Rindler $(T,X)$ to  Minkovski $(t,x)$  that I am familiar with is
\begin{align}
t&=X\sinh(\rho\,T)\,,\\
x&=X\cosh(\rho\,T)\,.
\end{align}
The world line of the object in Rindler coordinates is $x^{\overline{\alpha}}=(T,X)$ where $X$ is fixed. It is known [1] that $\rho=1/X$ is the proper acceleration
of the object. To check this observe that
the metric tensor is
$$
g_{\alpha\delta}=\left(\begin{matrix}-\rho^2X^2 & 0\\0&1\end{matrix}\right)\,,\quad
g^{\alpha\delta}=\left(\begin{matrix}-\frac{1}{\rho^2X^2} & 0\\0&1\end{matrix}\right)
$$
so that $g_{TT,X}=-2\rho^2X$ and all other partial derivatives being zero. Therefore,
$$
\Gamma^\delta_{\beta\gamma}=\frac{1}{2}g^{\alpha\delta}\left(g_{\alpha\beta,\gamma}+g_{\alpha\gamma,\beta}-g_{\beta\gamma,\alpha}\right)
$$
becomes
\begin{align}
\Gamma^T_{TX}&=\frac{1}{X}\,,&\quad\delta=\beta=T,\gamma=X\,,\\
\Gamma^T_{XT}&=\frac{1}{X}\,,&\quad\delta=\gamma=T,\beta=X\,,\\
\Gamma^X_{TX}&=0\,,&\quad\beta=T,\delta=\gamma=X\,,\\
\Gamma^X_{XT}&=0\,,&\quad\gamma=T,\delta=\beta=X\,,\\
\Gamma^T_{TT}&=0\,,&\quad\beta=\delta=\gamma=T\,,\\
\Gamma^X_{TT}&=\rho^2X\,,&\quad\beta=\gamma=T,\delta=X\,,\\
\Gamma^X_{XX}&=0\,,&\quad\beta=\gamma=\delta=X\,,\\
\Gamma^T_{XX}&=0\,,&\quad\beta=\gamma=X,\delta=T\,.
\end{align}
Note the factor $\rho^2$ in $\Gamma^X_{TT}$ that you missed.
Therefore,
\begin{align}
a^T&=\underbrace{\frac{d^2T}{dT^2}}_{0}+\underbrace{\Gamma^T_{TX}\frac{dT}{dT}\frac{dX}{dT}}_{\frac{1}{X}\cdot 1\cdot 0=0}+\underbrace{\Gamma^T_{XT}\frac{dX}{dT}\frac{dT}{dT}}_{\frac{1}{X}\cdot 0\cdot 1=0}=0\,,\\
a^X&=\underbrace{\frac{d^2X}{dT^2}}_{0}+\underbrace{\Gamma^X_{TT}\frac{dT}{dT}\frac{dT}{dT}}_{\rho^2X\cdot 1\cdot 1}=\rho^2X\,.
\end{align}
Since $X$ is constant equal to $1/\rho$ we get the 2-acceleration as
$$
\boldsymbol{a}=\left(\begin{matrix}0\\\rho\end{matrix}\right)
$$
as expected.
[1] Rindler Coordinates.
