- #1

Gene Naden

- 321

- 64

## Homework Statement

From Misner, Thorne and Wheeler's text

*Gravitation*(MTW), exercise 3.15:

Show that, if F is the EM field tensor, then ##\nabla \cdot *F## is a geometric, frame-independent version of the Maxwell equation ##F_{\alpha,\beta,\gamma}+F_{\beta\gamma,\alpha}+F_{\gamma\alpha,\beta}=0##

(and similarly for the current equations)

##*F## is the dual of ##F##

## Homework Equations

The Maxwell equation ##F_{\alpha,\beta,\gamma}+F_{\beta\gamma,\alpha}+F_{\gamma\alpha,\beta}=0##

The dual of a vector ##J## is defined ##*J_{\alpha\beta\gamma}=J^\mu \epsilon_{\mu\alpha\beta\gamma}##

where ##\epsilon## is the Levi-Civita symbol.

The dual of a second-rank antisymmetric tensor F is defined by ##*F_{\alpha\beta}=F^{\mu\nu}\epsilon_{\mu\nu\alpha\beta}##

The dual of a third-rank antisymmetric tensor ##B## is defined by ##*B_\alpha=B^{\lambda\mu\nu}\epsilon_{\lambda\mu\nu\alpha}##

## The Attempt at a Solution

##(\nabla \cdot *F)_\beta=(*F)_{\alpha\beta,\alpha}=(\eta^{\alpha\gamma}F^{\mu\nu}\epsilon_{\mu\nu\gamma\beta})_{,\alpha}##

##=\eta^{\alpha\gamma}\epsilon_{\mu\nu\gamma\beta}F^{\mu\nu,\alpha}##

In the second expression, the first ##\alpha## should be in the upper position. In the last expression, the final ##,\alpha## should be in the lower position. But I don't know how to get Physics Forum to accept that. On my local machine I declare \usepackage{tensor} and say F\indices{...} but that doesn't seem to work on the forum.

I know that the Maxwell equation as given above is actually 64 equations, mostly redundant. I In MTW ##F## has two indices for the components, which leads me to think ##*F## is second-rank. But when I take the divergence of the second-rank tensor I get a vector as shown in my attempt at a solution. This seems to yield four equations rather than 64.

In equation (3.3) of MTW, it has the expression ##F(u)## which makes me think F is a first-rank tensor. I am unclear what is the rank of ##F## in this problem.