Saturday, 22 October 2016

homework and exercises - Electromagnetic Tensor in Cylindrical Coordinates


I understand that the Electromagnetic Tensor is given by


Fμν(0ExEyEzEx0BzByEyBz0BxEzByBx0)


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=? or Fzρ=?


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


Fμν=Fμνfμμfνν


where the Jacobian is given by


fμμ=xμxμ




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=Ftxetex+Ftyetey+=12Fμνeμeν


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 eρet,eϕet and so on with F. This requires a little more knowledge of geometric algebra, but you can write eρet in terms of exet,eyet, 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(eρet)=F(exetcosϕ+eyetsinϕ)=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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