Field strength tensor / matrix

[Abigale]

In my note,
we have written the field strength tensor as:

F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = <br /> <br /> <br /> <br /> \begin{pmatrix}<br /> 0&amp;E_x &amp;E_y&amp;E_z \\<br /> -E_x&amp;0 &amp;B_z &amp;-B_y \\<br /> -E_y&amp;-B_z &amp;0 &amp;B_x \\<br /> -E_z&amp;B_y &amp;-B_x&amp;0 <br /> \end{pmatrix}<br /> <br /> <br /> <br />

But if I look into another book or wiki it is written as:


<br /> F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = <br /> \begin{pmatrix}<br /> 0&amp;-E_x &amp;-E_y&amp;-E_z \\<br /> E_x&amp;0 &amp;-B_z &amp;B_y \\<br /> E_y&amp;B_z &amp;0 &amp;-B_x \\<br /> E_z&amp;-B_y &amp;B_x&amp;0 <br /> \end{pmatrix}<br /> <br /> <br />


Why is it possible to write the field strength tensor in both notations?
And are both notations really equal?

THX
Abby

 

[ChrisVer]

It depends on how you define your metric and thus the ∂^{μ}

 

[Abigale]

I think it comes from that $$F^{\alpha \beta} $$ is an antysymmetric tensor. So I can use $$F^{\beta \alpha} =-F^{ \alpha \beta}$$ but I am not sure.
Need help^^

Both definitions of $$\partial ^\mu$$ are in both cases equal.

 

[ChrisVer]

Well then try for example to write the first line:
F^{0i}=∂^{0}A^{i}-∂^{i}A^{0}
Now I am not sure about minus/plus conventions I would write it:
F^{0i}=\frac{∂A^{i}}{∂t}-∇_{i}Φ= Ε^{i}
So that's what I'd use...
it has to do I guess with how you define covariant and contravariant vectors.

 

[samalkhaiat]

In my note,
we have written the field strength tensor as:

F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = <br /> <br /> <br /> <br /> \begin{pmatrix}<br /> 0&amp;E_x &amp;E_y&amp;E_z \\<br /> -E_x&amp;0 &amp;B_z &amp;-B_y \\<br /> -E_y&amp;-B_z &amp;0 &amp;B_x \\<br /> -E_z&amp;B_y &amp;-B_x&amp;0 <br /> \end{pmatrix}<br /> <br /> <br /> <br />

But if I look into another book or wiki it is written as:


<br /> F^{\mu\nu} =\partial ^\mu A^\nu -\partial ^\nu A^\mu = <br /> \begin{pmatrix}<br /> 0&amp;-E_x &amp;-E_y&amp;-E_z \\<br /> E_x&amp;0 &amp;-B_z &amp;B_y \\<br /> E_y&amp;B_z &amp;0 &amp;-B_x \\<br /> E_z&amp;-B_y &amp;B_x&amp;0 <br /> \end{pmatrix}<br /> <br /> <br />


Why is it possible to write the field strength tensor in both notations?

Because you can use either one to write the correct form of Maxwell equations:
\partial_{ \mu } F_{ 1 }^{ \mu \nu } = - \partial_{ \mu } F_{ 2 }^{ \mu \nu } = - e J^{ \nu }

And are both notations really equal?

How can they be equal? F_{ 1 }^{ \mu \nu } = - F_{ 2 }^{ \mu \nu }.

 

[ChrisVer]

well the problem is that both notations are practically equivalent- you will just have to define differently the current 4vector...
In the Lagrangian what appears is the F_{μν}F^{μν}, and whether you have them with a minus each, they will lead in the same equations of motion...