Electromagnetic tensor in cylindrical coordinates

[ngkamsengpeter]

I can find the metric tensor in cylindrical coordinates to be [1,-1,-1/r^2,-1] but how about the electromagnetic field tensor and thus the energy stress tensor?
Is it just change the Ex,Ey,Ez to Eρ,Eθ,Ez?
Is F^σρF_σρ still equal to 2(B^2-E^2)

 

[Mentz114]

The field tensor will give the same contraction whatever the coordinates. This is what defines tensors. Changing the coordinate basis from x^a -> x^A
$$F^{AB}F_{AB} = \Lambda^A_a\Lambda^B_bF^{ab}\Lambda^a_A\Lambda^b_B F_{ab}=F^{ab}F_{ab},\ \ \Lambda^a_A=dx^a/dx^A$$The flat space cylindrical metric is ds²= dt² - dz² - dr² - r²dθ², I think.

 

[ngkamsengpeter]

The field tensor will give the same contraction whatever the coordinates. This is what defines tensors. Changing the coordinate basis from x^a -> x^A
$$F^{AB}F_{AB} = \Lambda^A_a\Lambda^B_bF^{ab}\Lambda^a_A\Lambda^b_B F_{ab}=F^{ab}F_{ab},\ \ \Lambda^a_A=dx^a/dx^A$$The flat space cylindrical metric is ds²= dt² - dz² - dr² - r²dθ², I think.

Yes. I guess it should be the same in cylindcal coordinates but I try to work it by using

$$n_{\mu \nu }= \left[ \begin{array}{cccc}c^2& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$$
$$n^{\mu \nu }= \left[ \begin{array}{cccc} \frac{1}{c^2} & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & \frac{-1}{r^2} & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$$

$$F_{\mu \nu }= \left[ \begin{array}{cccc} 0 & -E_{\rho} & -E_{\phi} & -E_{z} \\ E_{\rho} & 0 & B_{z} &-B_{\phi} \\E_{\phi} & -B_{z} & 0 & B_{\rho} \\E_{z} & B_{\phi} & -B_{\rho} & 0 \end{array} \right]$$

and I get
$$F^{\mu \nu }= \left[ \begin{array}{cccc} 0 & \frac{E_{\rho}}{c^2} & \frac{E_{\phi}}{c^2 r^2} & \frac{E_{z}}{c^2} \\ \frac{-E_{\rho}}{c^2} & 0 & \frac{B_{z}}{r^2} &-B_{\phi} \\\frac{-E_{\phi}}{c^2 r^2} & \frac{-B_{z}}{r^2} & 0 & \frac{B_{\rho}}{r^2} \\\frac{-E_{z}}{c^2} & B_{\phi} & \frac{-B_{\rho}}{r^2} & 0 \end{array} \right]$$

And thus
$$F^{\mu \nu }F_{\mu \nu }≠2(B^2-\frac{E^2}{c^2})$$

But since it is tensor, so I guess it should be equal to 2(B^2-E^2/c^2) also in cylindrical or any coordinates right? Can anyone tell me which part I didt it incorrectly?

Thanks

 

[Mentz114]

Use this to calculate F
$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$

 

[ngkamsengpeter]

Use this to calculate F
$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$

I am new to this, can you show me some steps and the final answer so that I can check with my one?

Thanks.

 

[Mentz114]

I am new to this, can you show me some steps and the final answer so that I can check with my one?

Thanks.

When you transform from rectangular coordinates to cylindrical you must transform the differential operators also. This will remove the factor of r² that is giving you the wrong answer.

See here for the transformed operators
http://people.rit.edu/pnveme/pigf/Coords/Diff_operators_cyl.html

http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

 

[Mentz114]

The transformation (t,x,y,z) -> (T,R,θ,Z)
$$t \rightarrow T,\ \ x \rightarrow R \cos(\theta),\ \ y \rightarrow R \sin(\theta),\ \ z \rightarrow Z$$
is done by this matrix
$$\left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & cos\left( \theta\right) & sin\left(\theta\right) & 0\\ 0 & -\frac{sin\left( \theta\right) }{R} & \frac{cos\left(\theta\right) }{R} & 0\\ 0 & 0 & 0 & 1 \end{array} \right]$$

The transformed field tensor is
$$F^{mn}=\left[ \begin{array}{cccc} 0 & sin\left( \theta\right) \,Ey+cos\left( \theta\right) \,Ex & \frac{cos\left( \theta\right) \,Ey-sin\left( \theta\right) \,Ex}{R} & Ez\\ -sin\left( \theta\right) \,Ey-cos\left( \theta\right) \,Ex & 0 & -\frac{Bz}{R} & cos\left( \theta\right) \,By+sin\left( \theta\right) \,Bx\\ -\frac{cos\left( \theta\right) \,Ey-sin\left( \theta\right) \,Ex}{R} & \frac{Bz}{R} & 0 & -\frac{sin\left( \theta\right) \,By-cos\left( \theta\right) \,Bx}{R}\\ -Ez & -cos\left( \theta\right) \,By-sin\left( \theta\right) \,Bx & \frac{sin\left( \theta\right) \,By-cos\left( \theta\right) \,Bx}{R} & 0 \end{array} \right]$$

and the contraction $g_{an}g_{bm}F^{ab}F^{mn} = -2 E^2 + 2B^2$, where
$$g_{mn}=\left[ \begin{array}{cccc} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & {R}^{2} & 0\\ 0 & 0 & 0 & 1 \end{array} \right]$$

 

[ngkamsengpeter]

The transformation (t,x,y,z) -> (T,R,θ,Z)
$$t \rightarrow T,\ \ x \rightarrow R \cos(\theta),\ \ y \rightarrow R \sin(\theta),\ \ z \rightarrow Z...$$

Let me see if I understand it correctly. the tensor transformation rule is
$\widetilde{A^\mu}=\frac{∂\widetilde{x^\mu}}{∂x^\nu}A^\nu$
So the transformation matrix you given is $$\frac{∂\widetilde{x^\mu}}{∂x^\nu}$$right?
However, I try to work it out the matrix, I have the one different from your one. for example the 1,1 component, is $$\frac{∂\widetilde{x^1}}{∂x^1} =\frac{∂R}{∂x}=\frac{1}{Cos(\theta)}$$

And I don't quite understand how the Field tensor transform. I would think of it transform with the same matrix with the following formula:
$$\widetilde{F^{\mu\nu}}=\frac{∂\widetilde{x^\mu}}{∂x^ρ}\frac{∂\widetilde{x^\nu}}{∂x^σ}=F^{ρσ}$$

However, when I do the matrix multiplication using the matrix you given, I cannot get the answer same as your one. Is it transform with another different matrix?

If I want to write the tensor in terms of E_R,E_θ,E_z instead of Ex,Ey,Ez, how should I do that?

Many thanks for your help.

 

[Mentz114]

If I want to write the tensor in terms of E_R,E_θ,E_z instead of Ex,Ey,Ez, how should I do that?

Many thanks for your help.

(I haven't helped much so far ).
What you've written is correct, but the matrix I've given transforms the inverse of the cylindrical metric to the inverse of the Minkowski metric, so it is not used correctly in my calculation. Anyhow, it wouldn't answer your question.

To find F directly in cylindrical coordinates is not as simple as I thought. Starting the 4-potential A_μ the field tensor is $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$.

If
$$\frac{\partial}{\partial {x^2}}=\frac{1}{R}\frac{\partial}{\partial \theta}$$
and $g^{\mu\nu}=diag(-1,1,1/R^2,1)$ then $g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}F_\mu\nu$ has factors of R and is not what is wanted. Maybe the potential A needs to be written differently.

I'll think about this. Maybe I've made another mistake. My calculation above is horribly wrong .

 

[Mentz114]

I think I've solved this. By replacing A_θ with A_θ/R and using the spatial curl operator defined below for the magnetic fields it works out correctly. The curl is the spatial ( and temporal ) parts of $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$.

The curl operator in cylindrical coords is equation (117) on this page
http://mathworld.wolfram.com/CylindricalCoordinates.html


So far I've done the calculation on a piece of paper and it gives the right contraction but I'll set up a script when I can to check it. As I said earlier, the differential operators must be correct for the coordinates used.

[added later]
Using equation (117) the magnetic fields are
$$\begin{align} B_R &= \frac{1}{R}\frac{\partial A_z}{\partial \theta}-\frac{1}{R}\frac{\partial A_\theta}{ \partial Z}\\ B_\theta &= \frac{\partial A_R}{\partial z}-\frac{\partial A_z}{ \partial R}\\ B_Z &= \frac{1}{R} \left[ \frac{\partial A_\theta}{\partial r}-\frac{\partial A_Z}{ \partial \theta} \right] \end{align}$$
and the electric fields are
$$\begin{align} E_\theta &= \frac{1}{R} \left[\frac{\partial A_\theta}{\partial t}-\frac{\partial A_t}{ \partial \theta} \right]\\ E_R &=\frac{\partial A_R}{\partial t}-\frac{\partial A_t}{ \partial R}\\ E_Z &=\frac{\partial A_Z}{\partial t}-\frac{\partial A_t}{ \partial Z} \end{align}$$
now we can write
$$F^{\mu\nu}= \left[ \begin{array}{cccc} 0 & \frac{\partial}{\partial\,t}\,A_R-\frac{\partial}{\partial\,R}\,A_t & \frac{\frac{\partial}{\partial\,t}\,A_\theta-\frac{\partial}{\partial\,\theta}\,A_t}{R} & \frac{\partial}{\partial\,t}\,A_z-\frac{\partial}{\partial\,z}\,A_t\\ \frac{\partial}{\partial\,R}\,A_t-\frac{\partial}{\partial\,t}\,A_R & 0 & \frac{\frac{\partial}{\partial\,R}\,A_\theta-\frac{\partial}{\partial\,\theta}\,A_R}{R} & \frac{\partial}{\partial\,R}\,A_z-\frac{\partial}{\partial\,z}\,A_R\\ \frac{\frac{\partial}{\partial\,\theta}\,A_t-\frac{\partial}{\partial\,t}\,A_\theta}{R} & \frac{\frac{\partial}{\partial\,\theta}\,A_R-\frac{\partial}{\partial\,R}\,A_\theta}{R} & 0 & \frac{\frac{\partial}{\partial\,\theta}\,A_z-\frac{\partial}{\partial\,z}\,A_\theta}{R}\\ \frac{\partial}{\partial\,z}\,A_t-\frac{\partial}{\partial\,t}\,A_z & \frac{\partial}{\partial\,z}\,A_R-\frac{\partial}{\partial\,R}\,A_z & \frac{\frac{\partial}{\partial\,z}\,A_\theta-\frac{\partial}{\partial\,\theta}\,A_z}{R} & 0 \end{array} \right]$$
and (finally ?) $g_{ma}g_{bn}F^{ab}F^{mn}= 2(-E^2+B^2)$ where $g_{mn}=diag(-1,1,R^2,1)$.

 

[ngkamsengpeter]

I think I've solved this. By replacing A_θ with A_θ/R and using the spatial curl operator defined below for the magnetic fields it works out correctly. The curl is the spatial ( and temporal ) parts of $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$.

The curl operator in cylindrical coords is equation (117) on this page
http://mathworld.wolfram.com/CylindricalCoordinates.html

First of all, why replace A_θ with A_θ/R. We must have a reason for that, cannot just replace it in order to get the answer.
Assume all your calculation correctly, then the final answer is just replaced the E_x,E_y,E_z by E_r,E_θ,E_z only right? That is your final answer should be
$$F^{\mu \nu }= \left[ \begin{array}{cccc} 0 & E_{\rho} & E_{\phi} & E_{z} \\ -E_{\rho} & 0 & -B_{z} &B_{\phi} \\-E_{\phi} & B_{z} & 0 & -B_{\rho} \\-E_{z} & -B_{\phi} & B_{\rho} & 0 \end{array} \right]$$
Isn't that same as the first attempt I made?

And I am sure that it does not contract correctly. What is your final answer of F that give the correct contraction as you mentioned?

 

[Mentz114]

First of all, why replace A_θ with A_θ/R. We must have a reason for that, cannot just replace it in order to get the answer.

We have to raise the index of A_μ with the inverse of the cylindrical metric because the curl operators I'm using are contravariant. That's why the curls are components of a contravariant rank-2 tensor.

Assume all your calculation correctly, then the final answer is just replaced the E_x,E_y,E_z by E_r,E_θ,E_z only right? That is your final answer should be
$$F^{\mu \nu }= \left[ \begin{array}{cccc} 0 & E_{\rho} & E_{\phi} & E_{z} \\ -E_{\rho} & 0 & -B_{z} &B_{\phi} \\-E_{\phi} & B_{z} & 0 & -B_{\rho} \\-E_{z} & -B_{\phi} & B_{\rho} & 0 \end{array} \right]$$
Isn't that same as the first attempt I made?

And I am sure that it does not contract correctly. What is your final answer of F that give the correct contraction as you mentioned?

My calculation is spelt out. F and g are given and the results are correct because there are no coordinates in the contraction. It just depends on how you define Ex, Ey etc.

I've had enough and I'm satisfied, it was interesting.

 

[ngkamsengpeter]

We have to raise the index of A_μ with the inverse of the cylindrical metric because the curl operators I'm using are contravariant. That's why the curls are components of a contravariant rank-2 tensor.



My calculation is spelt out. F and g are given and the results are correct because there are no coordinates in the contraction. It just depends on how you define Ex, Ey etc.

I've had enough and I'm satisfied, it was interesting.

Yes. It was interesting. I can understand the A_θ/r now but I still don't quite understand with your calculation. What is the final answer of F in terms of E_r,E_θ,E_z?
Many thanks for your help.

 

[Mentz114]

Yes. It was interesting. I can understand the A_θ/r now but I still don't quite understand with your calculation. What is the final answer of F in terms of E_r,E_θ,E_z?
Many thanks for your help.

I think the contraction is
$$-2 ( -R^{-2}{E_\theta}^2-{E_r}^2-{E_z}^2+{B_\theta}^2 +R^{-2}{B_r}^2+R^{-2}{B_z}^2)=-2( -\vec{E}^2+\vec{B}^2) )$$

$$-2 ( -{E_\theta}^2-{E_r}^2-{E_z}^2+{B_\theta}^2 +{B_r}^2+{B_z}^2)=-2( -\vec{E}^2+\vec{B}^2) )$$

depending how you define the E and B vectors.

 

[ngkamsengpeter]

I think the contraction is
$$-2 ( -R^{-2}{E_\theta}^2-{E_r}^2-{E_z}^2+{B_\theta}^2 +R^{-2}{B_r}^2+R^{-2}{B_z}^2)=-2( -\vec{E}^2+\vec{B}^2) )$$

$$-2 ( -{E_\theta}^2-{E_r}^2-{E_z}^2+{B_\theta}^2 +{B_r}^2+{B_z}^2)=-2( -\vec{E}^2+\vec{B}^2) )$$

depending how you define the E and B vectors.

Yes. I get this answer also but how can we simply define vectors? In cylindrical coordinate, vector should be defined as
$$\vec{E}=E_r \hat{r}+ E_\theta \hat{\theta}+E_z \hat{z}$$
Is that mathematically correct if we simply defined the vector just to fits into the answer?

 

[ngkamsengpeter]

Can anyone verified if what Mentz114 did is correct or not? Can we defined the E and B vector like that?
Thanks.