- #1

carpinus

- 11

- 1

1. according to Robert Wald, General Relativity, equation (4.2.22)

the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by

## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ##

where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for more detail on the latter the reader is referred to the appendix).

I conclude for ## v^b ## = 0 (I am not sure about the sign of ##v^0##):

## B_a = - \frac {1}{2} {ϵ_{a0}}^{cd} F_{cd}

= \frac {1}{2} {ϵ_{0a}}^{cd} F_{cd} ##

2. I assume the common identities

##B_1 =F_{23},B_2 = F_{31}, B_3 = F_{12}##

are true only for cartesian coordinates -

because if they were true for general coordinates then from:

## ∂_1 F_{23} + ∂_2 F_{31} + ∂_3 F_{12} =0 ##

(one of four homogenous Maxwell-equations, true for general coordinates)

it would follow: ## ∂_1 B_1 +∂_2 B_2 +∂_3 B_3 = 0 ## for general coordinates.

The latter, however, I think is true only in cartesian coordinates.

my question:

does equation (4.2.22) allow for a magnetic field B deviating from ##(F_{23},F_{31}, F_{12})## and how? Can B according to (4.2.22) be seen to satisfy div B = 0 (E,B according to (4.2.21)f to satisfy the homogenous equations) ?

thank you for any comment. I just see that an article by robphy covering this sort of question can at present be found at the top of the forum page. So I should study this first. Or is there a somewhat shorter answer to my question?