Okay, some background. This was a project I started in 2011. About the time I retired I said to myself, self, wouldn't it be good for me to understand gravitational wave detection in terms of received power. In radio communication (and measurements in general) there is no free lunch. Signal detection reduces to received power and how this power compares to system and background noise power. I reasoned similar things must hold for the detection of gravitational waves since one ultimately detects electrical signals of some form. Along with this goal I wanted to look at possible high frequency detection schemes and needed a figure of merit with which to compare these (largely dumb) ideas. Basically, this figure of merit boils down to a power conversion area measure in ##m^2##. Knowing the conversion area, ##A##, for a detector the signal power received is then,
##P = A P_I##
where, ##P_I##, is the watts per square meter of gravitational wave illumination. What I soon learned, though, is femto-barns[1] are perhaps a more reasonable unit.
So the basic view I stated with is how does gravitational radiation give rise to electromagnetic radiation in a typical household environment for a given detector system. Household environment means we're working in the weak field limit where,
##g_{\mu\nu} = \eta_{mu\nu} + h_{\mu,\nu}##
where, ##\eta_{\mu\nu}##, is the usual Minkowski metric. I insisted upon keeping all constants like the speed of light and such because I want numbers in the end and I'm not the sharpest pencil in the box.
There are 4 fields in EM, ##E##, ##B## and their hillbilly siblings, ##H## and ##D##. It wasn't until I was faced with EM in an industrial setting that the truly awesome properties of ##D## and ##H## become apparent. So, Maxwells equations group into a set connecting ##E## and ##B## and a second independent set connecting ##H## and ##D##. It is a well known fact that neither set depend at all on the metric, weakly curved or otherwise. This has to do with the symmetry of the covariant derivative. Clearly though, EM fields will be effected by curvature so what gives?
The answer is one must supply a constitutive relation between (##E##, ##B##) and the other less popular fields, (##H##,##D##), in order for Maxwell's equations to form a dynamical system. For linear isotropic materials, people love to write the relations as,
##D = \epsilon E##
##B = \mu H##
When written in tensor for (##E##,##B##) are the components of an antisymmetric tensor MTW call the
Faraday, ##F_{\mu\nu}## while the (##H##,##D##) become the components of the other antisymmetric tensor, ##M_{\mu\nu}##, call the
Maxwell. As written above, the vacuum constitutive relations are not a tensor. The actual connection between fields in vacuum is,
(1) ##M_{\mu\nu} = -\sqrt{\frac{\epsilon_o}{\mu_o}}\left(\frac{1}{2\sqrt{-g}}g_{\mu\alpha}g_{\nu\beta}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\right)##
which does depend on the metric and is a tensor relationship.
So, what I did from here is expand (1) to first order in the metric strain, ##h##. I'm tired of typing so I'll post the answer I got later.
[1] A Barn being ##10^{-28}m^2##
