- #1

chroot

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## Main Question or Discussion Point

Hi all, I'm trying to solve Exercise 1.4.3 in Foster & Nightingale's "A Short Course in General Relativity."

The question essentially provides the metric for Euclidean space in spherical coordinates and the matrix representing the coordinate transformation from spherical to cylindrical coordinates. The question prompts the deduction of the metric for Euclidean space in cylindrical coordinates using this information and the rule

[tex]g_{i^\prime j^\prime}

= U^k_{i^\prime}

U^l_{j^\prime}

g_{kl}

[/tex]

where primed coordinates are cylindrical, and unprimed coordinates are spherical.

The matrices involved are

1) the metric of Euclidean space in spherical coordinates

[tex][g_{ij}] =

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & r^2 & 0\\

0 & 0 & r^2 \sin^2 \theta

\end{array}

\right]

[/tex]

2) the coordinate transformation matrix between spherical and cylindrical coordinates:

[tex][U^{i^\prime}_j] =

\left[

\begin{array}{ccc}

\sin \theta & r \cos \theta & 0\\

0 & 0 & 1\\

\cos \theta & -r \sin \theta & 0

\end{array}

\right]

[/tex]

Now, it seems to me that, in matrix notation:

[tex]

[g_{i^\prime j^\prime}] =

The question essentially provides the metric for Euclidean space in spherical coordinates and the matrix representing the coordinate transformation from spherical to cylindrical coordinates. The question prompts the deduction of the metric for Euclidean space in cylindrical coordinates using this information and the rule

[tex]g_{i^\prime j^\prime}

= U^k_{i^\prime}

U^l_{j^\prime}

g_{kl}

[/tex]

where primed coordinates are cylindrical, and unprimed coordinates are spherical.

The matrices involved are

1) the metric of Euclidean space in spherical coordinates

[tex][g_{ij}] =

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & r^2 & 0\\

0 & 0 & r^2 \sin^2 \theta

\end{array}

\right]

[/tex]

2) the coordinate transformation matrix between spherical and cylindrical coordinates:

[tex][U^{i^\prime}_j] =

\left[

\begin{array}{ccc}

\sin \theta & r \cos \theta & 0\\

0 & 0 & 1\\

\cos \theta & -r \sin \theta & 0

\end{array}

\right]

[/tex]

Now, it seems to me that, in matrix notation:

[tex]

[g_{i^\prime j^\prime}] =

__[g_{ij}] [U^T]__

[/tex]

But this doesn't seem to be correct. I should end up with the metric for Euclidean space in cylindrical coordinates:

[tex]

g_{i^\prime j^\prime} =

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & r^2 \sin^2 \theta & 0\\

0 & 0 & 1

\end{array}

\right]

=

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & \rho^2 & 0\\

0 & 0 & 1

\end{array}

\right]

[/tex]

If my thinking *is* correct, somewhere I'm just getting confused. Can anyone help me figure out what's going on? Thanks in advance.

- Warren[/tex]

But this doesn't seem to be correct. I should end up with the metric for Euclidean space in cylindrical coordinates:

[tex]

g_{i^\prime j^\prime} =

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & r^2 \sin^2 \theta & 0\\

0 & 0 & 1

\end{array}

\right]

=

\left[

\begin{array}{ccc}

1 & 0 & 0\\

0 & \rho^2 & 0\\

0 & 0 & 1

\end{array}

\right]

[/tex]

If my thinking *is* correct, somewhere I'm just getting confused. Can anyone help me figure out what's going on? Thanks in advance.

- Warren