# Covariant derivative in semi-log coordinates

Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Problem P17.5.

As another example of using the geodesic equation to calculate Christoffel symbols, we’ll consider the semi-log coordinates introduced earlier:

 $\displaystyle p$ $\displaystyle =$ $\displaystyle x\ \ \ \ \ (1)$ $\displaystyle q$ $\displaystyle =$ $\displaystyle e^{by} \ \ \ \ \ (2)$

The invariant interval is

 $\displaystyle ds^{2}$ $\displaystyle =$ $\displaystyle dx^{2}+dy^{2}\ \ \ \ \ (3)$ $\displaystyle$ $\displaystyle =$ $\displaystyle dp^{2}+\frac{1}{\left(bq\right)^{2}}dq^{2} \ \ \ \ \ (4)$

so the metric is

$\displaystyle g_{ij}=\left[\begin{array}{cc} 1 & 0\\ 0 & \frac{1}{\left(bq\right)^{2}} \end{array}\right] \ \ \ \ \ (5)$

We compare the geodesic equation:

$\displaystyle g_{aj}\ddot{x}^{j}+\left(\partial_{i}g_{aj}-\frac{1}{2}\partial_{a}g_{ij}\right)\dot{x}^{j}\dot{x}^{i}=0 \ \ \ \ \ (6)$

with the expression for the Christoffel symbols:

$\displaystyle \ddot{x}^{m}+\Gamma_{\; ij}^{m}\dot{x}^{j}\dot{x}^{i}=0 \ \ \ \ \ (7)$

We get

 $\displaystyle \ddot{p}$ $\displaystyle =$ $\displaystyle 0\ \ \ \ \ (8)$ $\displaystyle \frac{1}{\left(bq\right)^{2}}\ddot{q}-\frac{2}{b^{2}q^{3}}\dot{q}^{2}-\frac{1}{2}\left(-\frac{2}{b^{2}q^{3}}\dot{q}^{2}\right)$ $\displaystyle =$ $\displaystyle 0\ \ \ \ \ (9)$ $\displaystyle \ddot{q}-\frac{1}{q}\dot{q}^{2}$ $\displaystyle =$ $\displaystyle 0 \ \ \ \ \ (10)$

The Christoffel symbols are thus

 $\displaystyle \Gamma_{\; ij}^{p}$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]\ \ \ \ \ (11)$ $\displaystyle \Gamma_{\; ij}^{q}$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 0 & 0\\ 0 & -\frac{1}{q} \end{array}\right] \ \ \ \ \ (12)$

Now consider a vector field ${A^{i}=\left[0,Cx\right]}$ (where ${C}$ is a constant) in rectangular coordinates. In these coordinates, its covariant derivative is just the normal derivative, so

 $\displaystyle \nabla_{i}A^{j}$ $\displaystyle =$ $\displaystyle \partial_{i}A^{j}\ \ \ \ \ (13)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 0 & C\\ 0 & 0 \end{array}\right] \ \ \ \ \ (14)$

In the semi-log system, we have

$\displaystyle \nabla_{i}A^{j}=\partial_{i}A^{j}+\Gamma_{\; ik}^{j}A^{k} \ \ \ \ \ (15)$

To use this formula we need ${A^{i}}$ in the ${pq}$ system, which we can get by the usual transformation, using:

 $\displaystyle A^{p}$ $\displaystyle =$ $\displaystyle A^{i}\partial_{i}p=A^{x}=0\ \ \ \ \ (16)$ $\displaystyle A^{q}$ $\displaystyle =$ $\displaystyle A^{i}\partial_{i}q=A^{y}\left(be^{by}\right)=Cbpq \ \ \ \ \ (17)$

With these transformations, we get

 $\displaystyle \nabla_{i}A^{j}$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 0 & Cbq\\ 0 & Cbp-Cbp \end{array}\right]\ \ \ \ \ (18)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left[\begin{array}{cc} 0 & Cbq\\ 0 & 0 \end{array}\right] \ \ \ \ \ (19)$

The difference in gradients is because of the different scales used in the vertical direction. Consider first the vertical component of ${A}$. In rectangular coordinates this has a constant value for a given value of ${x}$ namely ${A^{y}=Cx}$. However, if we use ${q}$ to measure vertical distance, a unit change in ${y}$ results in a larger and larger change in ${q}$ the higher up the vertical axis we go, so ${A^{q}}$ must increase as ${q}$ increases, even if ${x=p}$ is held constant. This is reflected by 17, where ${A^{q}}$ is proportional to ${q}$ as well as ${p}$.

If we used ordinary derivatives to calculate the gradient of ${A}$ in the ${pq}$ system, we would get ${\partial_{q}A^{q}=Cbp}$. However, the ‘true’ value of the vertical component of ${A}$ doesn’t change as we move up or down a vertical line, and this is reflected by the ${\Gamma_{\; ik}^{q}A^{k}=-Cbp}$ correction term that is present in the covariant derivative 15, with the result that ${\nabla_{q}A^{q}=\partial_{q}A^{q}+\Gamma_{\; qk}^{q}A^{k}=Cbp-Cbp=0}$.

The value of ${\nabla_{p}A^{q}=Cbq}$ again reflects the fact that a vertical component that is constant in rectangular coordinates must get numerically larger with increasing height in the ${pq}$ system.

As a final test that all is well, we can use the standard tensor transformation to transform 19 back to rectangular coordinates, where we denote rectangular coordinates by ${r^{i}}$ and the semi-log ${pq}$ system by ${s}$:

$\displaystyle \left[\nabla_{i}A^{j}\right]_{r}=\left[\nabla_{a}A^{b}\right]_{s}\frac{\partial s^{a}}{\partial r^{i}}\frac{\partial r^{j}}{\partial s^{b}} \ \ \ \ \ (20)$

The only non-zero component of ${\left[\nabla_{a}A^{b}\right]_{s}}$ is ${\left[\nabla_{p}A^{q}\right]_{s}=Cbq}$, so the RHS has only one non-zero term, which occurs when ${a=p}$, ${i=x}$, ${b=q}$ and ${j=y}$. For this term we have

 $\displaystyle \left[\nabla_{x}A^{y}\right]_{r}$ $\displaystyle =$ $\displaystyle Cbq\frac{\partial p}{\partial x}\frac{\partial y}{\partial q}\ \ \ \ \ (21)$ $\displaystyle$ $\displaystyle =$ $\displaystyle Cbq\left(1\right)\frac{1}{bq}\ \ \ \ \ (22)$ $\displaystyle$ $\displaystyle =$ $\displaystyle C \ \ \ \ \ (23)$

The overall transformation thus gives 14 back again.