I can't seem to get my results to match D'Inverno's electromagnetic tensor for a charged point (page 239 of his book - *Introducing Einstein's Relativity*).

**Here are D'Inverno's steps:**

- The line element in spherical coordinates is (\(\eta\) and \(\lambda\) are functions of \(r\) only)

\(\mathrm{d}s^2 = \mathrm{e}^\eta \mathrm{d}t^2 - \mathrm{e}^\lambda \mathrm{d}r^2 - r^2 (\mathrm{d}\theta^2 + \sin^2\!\theta\ \mathrm{d}\phi^2)\)

- He defines this covariant electromagnetic field tensor:

\(F_{\mu\nu} = E(r) \begin{pmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}\)

- He then proceeds to find the electric field, and consequently the electromagnetic field tensor by using the source-free Maxwell equations:

\(\begin{align}
\nabla_\nu F^{\mu\nu} & = 0 \\
\partial_{[\lambda} F_{\mu\nu]} & = 0.
\end{align}\)

- Solving the differential equation that appears from the equations above, he finds the electric field:

\(E(r) = \mathrm{e}^{(\eta+\lambda)/2} \varepsilon/r^2\)

- He then notes that this field is that of a point charge at infinity (\(\eta\) and \(\lambda\) go to zero at infinity) where \(\varepsilon\) is the electric charge. I managed to reproduce all these steps.

**Now, here are my steps, using the four-potential procedure (the line element is the same):**

- I define my contravariant four-potential (there is just the first element which is the electric potential of a point charge, just as D'Inverno found):

\(A^\mu = (\varepsilon/r, 0, 0, 0)\)

- Then I lower the index of this four-potential to find the covariant one:

\(A_\mu = (\mathrm{e}^\eta \varepsilon/r, 0, 0, 0)\)

- Finally I apply this equation to build the covariant electromagnetic tensor:

\(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\)

- The result is

\(F_{\mu\nu} = \frac{\mathrm{e}^\eta \varepsilon}{r^2}\! \left(r\frac{\partial\eta}{\partial r}-1\right) \begin{pmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}\)

- Where:

\(\frac{\mathrm{e}^\eta \varepsilon}{r^2}\! \left(r\frac{\partial\eta}{\partial r}-1\right) = E(r)\)

And this is different from D'Inverno's electric field. I don't know what I am doing wrong. The calculations are not difficult for this simple case.

The question is, due to these calculations:

Do my contravariant four-potential needs to contain my metric funcions in some way? I was assuming it is just the four-potential for a electric charge in the flat space:

\(A^\mu = (\varepsilon/r, 0, 0, 0)\)

If everything is right, the wrong assumption must be here.