I understand that the Electromagnetic Tensor is given by
Fμν↦(0−Ex−Ey−EzEx0−BzByEyBz0−BxEz−ByBx0)
where μ, ν can take the values {0,1,2,3} or {t,x,y,z}.
So, for example F01=Ftx=−Ex
My question is, what would the following expression be?
Ftρ=?
where ρ=√x2+y2 is the radial coordinate in cylindrical coordinates?
And more generally, how can we construct the Electromagnetic Tensor in cylindrical coordinates? Where μ, ν now take the values {t,ρ,φ,z}.
Answer
Just use the Jacobian of the coordinate system transformation. If your Cartesian coordinates are μ and ν and your cylindrical coordinates are μ′,ν′, then there is a Jacobian fμμ′ that allows you to write
Fμ′ν′=Fμνfμμ′fνν′
where the Jacobian is given by
fμμ′=∂xμ′∂xμ
Now that's all well and good, but you might be thinking it's a bit abstract, and...it is. There's another way to do this instead, using what's called geometric algebra.
In geometric algebra, the EM tensor is called a bivector, taking on the form
F=Ftxet∧ex+Ftyet∧ey+…=12Fμνeμ∧eν
where eμ represent basis covectors. What we've used here is called a wedge product, and orthogonal basis vectors will anticommute under it.
To extract the components in a new basis, you have a couple choices: (1) you can write the basis covectors in terms of the cylindrical basis and simplify. So that would entail writing ex and ey in terms of eρ and eϕ. This is equivalent to finding the inverse Jacobian.
However, there is another choice (2), which is to simply take the inner product of the basis vectors eρ∧et,eϕ∧et and so on with F. This requires a little more knowledge of geometric algebra, but you can write eρ∧et in terms of ex∧et,ey∧et, and so on, which may be an easier computation.
I'll do the latter here to demonstrate the technique. See that eρ=excosϕ+eysinϕ. We can then find Ftρ as:
Ftρ=F⋅(eρ∧et)=F⋅(ex∧etcosϕ+ey∧etsinϕ)=Ftxcosϕ+Ftysinϕ
This is no more exotic that finding the components of a vector in a new basis by finding the projection of the vector on each new basis vector.
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