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,

Eup = 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_vg^mv y/^gYm = 0

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

dr² = dt² — '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_sy^msyYm = 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.