Components of The Electromagnetic Field Strength Tensor

  • #1
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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
[tex]B_r = g_{11} ε^{01μν} F_{μν}[/tex]
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
[tex]const = P/√4π[/tex]
where P is the total magnetic charge of black hole.

Why does the magnetic charge exist here?
 

Answers and Replies

  • #2
robphy
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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.
 
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  • #3
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Thank you!
 
  • #4
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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

[tex]*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}[/tex]
[tex]*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}[/tex]
which is different from Eq. (15)
[tex]B_r=g_{11}\epsilon^{01\mu\nu}F_{\mu\nu}[/tex]
What did I do wrong?
 
  • #5
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Never mind, just found the answer
 
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  • #6
RRM
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Hi Darida, How did you find the answer?
 

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