How Can Wedges Help Simplify Tensor Notation for the Electromagnetic Field?

[Albereo]

This isn't actually coursework, I'm doing some studying on my own. These are my very preliminary attempts to wrangle with tensor notation, so please be patient with me. I'm trying to get the components of the electromagnetic field tensor from

[itex]\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/itex]

But I'm having a problem with my signs. For example, when I do

[itex]F^{10}=\partial^{1}A^{0}-\partial^{0}A^{1}=-\frac{\partial A_{x}}{\partial t}+\frac{\partial \phi}{\partial x}[/itex]

I think the first term should be positive, so I get [itex]-E_{x}[/itex] as that entry. What am doing wrong?

 

[gabbagabbahey]

This isn't actually coursework, I'm doing some studying on my own. These are my very preliminary attempts to wrangle with tensor notation, so please be patient with me. I'm trying to get the components of the electromagnetic field tensor from

[itex]\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/itex]

But I'm having a problem with my signs. For example, when I do

[itex]F^{10}=\partial^{1}A^{0}-\partial^{0}A^{1}=-\frac{\partial A_{x}}{\partial t}+\frac{\partial \phi}{\partial x}[/itex]

I think the first term should be positive, so I get [itex]-E_{x}[/itex] as that entry. What am doing wrong?

[tex]\partial^{\mu}A^{\nu}=\eta_{ \mu \gamma} \partial_{ \gamma } A^{\nu}[/tex]

 

[Albereo]

[tex]\partial^{\mu}A^{\nu}=\eta_{ \mu \gamma} \partial_{ \gamma } A^{\nu}[/tex]

Ah, thanks! Index gymnastics...going to be fumbling with those for a while.

 

[Dickfore]

[tex]
\partial_{\mu} \equiv \left(\frac{1}{c} \, \frac{\partial}{\partial t}, \nabla \right)
[/tex]
[tex]
\partial^{\mu} = \eta^{\mu \nu} \, \partial_{\nu} = \left(\frac{1}{c} \, \frac{\partial}{\partial t}, -\nabla \right)
[/tex]

 

[Muphrid]

[tex]F = \nabla \wedge A = e^t \wedge e^x (\partial_t A_x - \partial_x A_t) + \ldots[/tex]

Extract up or down components the same way you would as with a vector. No index wrangling required. (Not that I think wedges are any "easier" to the uninitiated, but it bears pointing out there do exist alternatives to the nightmare that is abstract index notation.)

 

[Albereo]

I haven't seen that alternative notation before, are there any advantages (besides the avoidance of hand-cramps from writing indices) to using it?

Also, can it be used easily instead of the usual notation in QFT and GR?

 

[Muphrid]

I'm not as familiar with QFT, but with GR, yes, you can absolutely avoid index notation with the proper tools. Understand that it isn't nearly as common, though--it's still good to be familiar or conversant in index notation anyway, but for personal use, I wouldn't do this stuff any other way.

EM in special relativity is still one of the easiest contexts to talk about this, though. Take the electric and magnetic fields to be, for example, [itex]E \equiv E_i e^i[/itex] and similarly for the magnetic field. Then, the Faraday field is

[tex]F = E \wedge e^t + (e^x \wedge e^y \wedge e^z) \cdot B[/tex]

And you see that Maxwell's equations (outside of matter) just reduce to

[tex]\nabla \cdot F = - \mu_0 J, \quad \nabla \wedge F = 0[/tex]

Or, these equations can be married together. The [itex]\nabla \cdot F[/itex] equation must produce a rank-1 tensor. The [itex]\nabla \wedge F[/itex] equation must produce a rank-3 tensor. Both equations can be seen as parts or components of a single equation.

[tex]\nabla F \equiv \nabla \cdot F + \nabla \wedge F = -\mu_0 J[/tex]

Wedges are really the missing piece to being able to talk about higher-ranking tensors in a way that is more clearly an extension of traditional vector operations. The wedge is the proper extension of the cross product from 3D; it's defined in arbitrary dimensions, and not only is it antisymmetric but it's associative as well, and the combined "geometric" product formed by marrying the dot and wedge products by addition is associative, too. This is extremely powerful for dealing with objects beyond simple vectors.