Components of The Electromagnetic Field Strength Tensor

[darida]

Source: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf

Section 2
Page: 2
Eq. (15)
The radial component of the magnetic field is given by
$$B_r = g_{11} ε^{01μν} F_{μν}$$
Where does this equation come from?

Section 4
Page 3
Similar to the electric charges, the Gauss's flux theorem for the magnetic field gives
$$const = P/√4π$$
where P is the total magnetic charge of black hole.

Why does the magnetic charge exist here?

 

[robphy]

1. You can think of it this way: The magnetic field is part of the space-space submatrix of the Field Tensor.
(The electric field is part of the time-space submatrix of the Field Tensor, as in Eq.12.)
Alternatively, one could say that the magnetic field is part of the time-space submatrix of the Hodge-dual of the Field Tensor.
(From poking around... check out
http://gmammado.mysite.syr.edu/notes/Electromagnetic_Field_Strength_Tensor.pdf )

2. The abstract says that magnetic monopoles are assumed.

 

[darida]

Thank you!

 

[darida]

Ok now I get confused. So, I am trying to find the radial component of the magnetic field from the Hodge-dual of the Field Tensor, but then end up like this

$$*F_{\mu\nu}=\frac{1}{2} \epsilon_{\mu\nu\lambda\rho}F^{\lambda\rho}= \begin{bmatrix} 0 B_x B_y B_z \\ -B_x 0 -E_z E_y \\ -B_y E_z 0 -E_x \\ -B_z -E_y E_x 0 \end{bmatrix}$$
$$*F_{01}=B_r=\frac{1}{2} \epsilon_{01\lambda\rho}F^{\lambda\rho}=\frac{1}{2} g_{11}\epsilon^{01\lambda\rho}F_{\lambda\rho}g_{00}$$
which is different from Eq. (15)
$$B_r=g_{11}\epsilon^{01\mu\nu}F_{\mu\nu}$$
What did I do wrong?

 

[darida]

Never mind, just found the answer

 

[RRM]

Hi Darida, How did you find the answer?