# The problem of determining the surface current density induced on an antenna surface placed in a nonlinear inhomogeneous and anisotropic medium taking gravitational effects into account

Statement of the problem: Let e^(w, *r,*** F) **denote the field dependent permittivitypermeability tensor. Here,

*E*up = A_{p} *n —*

is the covariant em field tensor with A_{/(} as the covariant em four potential, the dependence of on the em Held tensor F = ((F_{MI/})) shows that the medium is nonlinear and its dependence on the space-time coordinates *x* and the fact that this tensor is generally non-diagonal shows that the medium is nonlinear, inhomogeneous and anisotropic. The Maxwell field equations in such a medium are

(^(_{W},r,F)F“^)_{(IZ} = ^(_{W},_{æ})

yields the nonlinear wave equation in such a medium taking space-time curvature into account. We assume the Lorentz gauge conditions

= 0

Here,

*A>‘ = g^A _{v}, = g^g^F_{aa}*

We are here assuming that the gravitational field metric tensor is time independent and hence we are operating completely in the temporal frequency domain. Thus where-ever *do* occurs in the above system, we replace it by the multiplication operator *jw.*

= A_{M}(w,r),

■^Or *j^A _{r} —* A

_{Or},

*9nv* ^{=} P^(^{r})

where *r = (x,y, z).* The above Maxwell equations can be expressed as

>$(u>,r,F)F^y^ + (e^F^^),_{m} = 0

To simplify further, we must make use of the gauge condition:

*{Ayy^gg^Y^ =* 0

gives

*jcjg ^{Ov} y/^gA_{v} + {A_{v}g^{mv} y/^gYm* = 0

If we use the synchronous reference system, then *go™* = 0 and *goo* = 1- Then the metric can be expressed as

*dr ^{2} = dt^{2} — 'Y_{rs}(r)dx^{r} dx^{s}*

This equation implies that *g ^{Om} =* 0 and

*g°°*= 1. So we get for the gauge condition,

*juy/^Ao - (A _{s}y^{ms}yYm =* 0

where ((q^{rs})) = ((7r_{S}))^{-1}-

Reference: L.D.Landau and E.M.Lifshitz, ’’The classical theory of fields”, Butterworth and Heinemann.

Acknowledgements: I am grateful to Prof.Malay Ranjan Tripathi for suggesting me to write this article for a workshop conducted by him.