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.

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