SetDirectory["~/KappaLib/"];
<<kappaLib-1.2.m
<<helper.m

Define medium 

    kappa = C1 ast_g + rho overline (A) otimes A + C2 Id

where g is a Lorentz metric.

In suitable coordinates

Metric=k DiagonalMatrix[{-1,1,1,1}];

(* Since hodge operator is conformally invariant,we may assume that k=1. *)
Metric=Metric/.k->1;

AA=emMatrix["a",4,Structure->"AntiSymmetric"];

(* rho = scalar density of weight 1 *)
(* C1 = non-zero constant *)
(* C2 = constant *)

kappa=C1 emHodge[Metric]+ emBiProduct[rho,AA,AA]+ C2 emIdentityKappa[];

Compute Fresnel polynomial

vars={x0,x1,x2,x3};
fresnel=FullSimplify[emKappaToFresnel[kappa,vars]];

HH=Table[
C1 Metric[[i]][[j]] -2 rho Sum[AA[[i]][[a]] Metric[[a]][[b]] AA[[b]][[j]],{a,1,4},{b,1,4}]
,
{i,1,4},
{j,1,4}
];

factor1=vars.Metric.vars;
factor2=vars.HH.vars;
fresnelExp=- C1^2factor1 factor2;

FullSimplify[(fresnel- fresnelExp)]

Symbolic expression for det(kappa)

detKappa=FullSimplify[FullSimplify[emDet[kappa]]]

sub={
EEE->2C1 rho(-(a12^2)-(a13^2)-(a14^2)+(a23^2)+(a24^2)+(a34^2)) 
};

detKappaAlt=(C1^2+C2^2)^2((C1^2)+C2^2+ EEE+C2 emTrace[emBiProduct[rho,AA,AA]]);
Simplify[(detKappa-detKappaAlt)/.sub]

Symbolic expression for det h

FullSimplify[Det[HH]]

DetMet2=-((((C1^2)+EEE-(1/2 emTrace[emBiProduct[rho,AA,AA]])^2 ))^2);

FullSimplify[(Det[HH]-DetMet2)/.sub]

Suppose Metric and Metric2 are propertional. Show that this implies that A =0 or rho = 0.

deltaMat=Metric-Const HH;

(* consider only diagonal elements *)
eqs=Table[deltaMat[[i]][[i]],{i,1,4}];

sub={};
Case=FullSimplify[eqs//.sub];
show[Case]

The second equation shows that Const != 0. Solving for C1 yields

sub=Append[sub,C1->(1/Const)+2 ((a12^2)+(a13^2)+(a14^2)) rho];

Case=FullSimplify[Case//.sub];
show[Case]

Since Const != 0, it follows that rho = 0 or A = 0.

Extra: Alternative proof that A= 0 using a Gröbner basis

eqs=Union[Flatten[deltaMat]];
gb=GroebnerBasis[eqs,Variables[eqs]];//Timing

show[simp[gb]]

The first three equations imply that rho = 0 or A = 0.