Divide in space and time component

[aries0152]

For $$-\frac{1} {4} F_{\mu\nu} F^{\mu\nu}$$ We can write $$-\frac{1} {4} F_{i j} F^{ij} -\frac{1}{2}F_{0i} F^{0i}$$ Where $$F_{\mu\nu} \equiv \partial_\mu W_\nu-\partial_\nu W\mu$$
If there are 3 indices how can I separate them like this?
I want to separate $$\frac{1} {12} G_{\mu\nu\rho} G^{\mu\nu\rho}$$ into time and space component . Where $$G_{\mu\nu\rho}\equiv\partial_{\mu}\phi_{\nu\rho}+ \partial_{\nu}\phi_{\rho\mu}+\partial_{\rho}\phi_{\mu\nu}$$

How can I do it?

 

[granpa]

I don't know the answer to your question but have you looked into clifford algebra?

http://geocalc.clas.asu.edu/

http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html

 

[sweet springs]

Hi.

What is Φ　with two indexes? I have not seen it before. Thank you in advance.

 

[aries0152]

Hi.

What is Φ　with two indexes? I have not seen it before. Thank you in advance.

$$\phi_{\nu\rho}$$ is a antisymmetric tensor field.

 

[aries0152]

I don't know the answer to your question but have you looked into clifford algebra?

http://geocalc.clas.asu.edu/

http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html

Actually I want to separate this in space and time component. There is some hint in the "Classical Electrodynamics by Jackson" section 11.6

I can separate the space and time component for two indices (like: $$F_{\mu\nu}$$ ) but I am not sure how to do it when there are three indices.
can anybody help?

 

[sweet springs]

Hi. aries.

$$\phi_{\nu\rho}$$ is a antisymmetric tensor field.

I see. so G is antisymmetric tensor. Exchange of any pair of indexes changes signature. Among 4^6 = 64 components, only four components are independent, i.e. 012, 013, 023 and 123. So the formula you are looking for is

1/2 { G_012 G^012 + ( similar other three terms ) }

Regards

 

[aries0152]

sweet springs
many many Thanx :-)