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

In this notebook we define a kappa such that:

1) kappa is algebraically decomposable.

2) beta^2 -alpha gamma = -1.

3) the Fresnel polynomial is product of a two irreducible quadratic forms of signatures + + - -.

4) The nonlinear equation for D has no solution

Define medium

kappa=emMatrixToKappa[(({{0, 0, -2, 0, -1, 0}, {-1, -1, -1, (8/3), -1, 0}, {1, 0, 0, -(14/3), 0, -1}, {0, 0, 0, 0, 4, 1}, {1, 0, -1, 0, 2, 0}, {1, 0, -1, 0, 0, -1}}))];

(* kappa is invertible *)
FullSimplify[emDet[kappa]]

(* kappa has no axion component *)
Simplify[emTrace[kappa]]

(* Since this does not simplify to {0}, kappa has a skewon component *)
Union[Flatten[kappa-emPoincare[kappa]]]

(* Since this does not simplify to {0}, kappa has a principal component *)
Union[Flatten[kappa+emPoincare[kappa]]]

Define constants alpha, beta, gamma and bivectors A and B

alpha= 2;
beta = 1;
gamma=1;
rho=1;
Abivector=(({{0, 0, 0, 0}, {0, 0, 0, (7/2)}, {0, 0, 0, 0}, {0, -(7/2), 0, 0}}));
Bbivector=(({{0, 0, -(1/7), (10/7)}, {0, 0, (1/7), -1}, {(1/7), -(1/7), 0, -1}, {-(10/7), 1, 1, 0}}));

Note these satisfy beta^2-alpha gamma = 0

beta^2-alpha gamma

Verify that kappa is algebraically decomposable

LHS =alpha emIdentityKappa[] + beta (kappa +emPoincare[kappa]) + gamma emCompose[emPoincare[kappa],kappa]; 
RHS=emBiProduct[rho,Abivector,Bbivector]+emBiProduct[rho,Bbivector,Abivector];
Union[Simplify[Flatten[(LHS-RHS)]]]

Fresnel polynomial is product of two quadratic forms of signatures (+ + - -)

coords={xi0,xi1,xi2,xi3};
fresnel=FullSimplify[emKappaToFresnel[kappa,coords]];

gPlus=(({{0, 3, 0, -3}, {3, 6, 3, 3}, {0, 3, 0, -14}, {-3, 3, -14, -16}}));
gMinus=(({{0, 3, -1, -4}, {3, 6, 5, -6}, {-1, 5, 0, -7}, {-4, -6, -7, -2}}));

frExp=-((1/4)/(3 )) (coords.gPlus.coords)(coords.gMinus.coords);

Simplify[fresnel-frExp]

Sort[Eigenvalues[gPlus]//N]
Sort[Eigenvalues[gMinus]//N]

Show that the quadratic factors are irreducible in the complex polynomials

v={v0,v1,v2,v3};
w={w0,w1,w2,w3};
delta=gPlus - (coords.v)(coords.w);
eqs=Union[Flatten[CoefficientList[delta,coords]]];
GroebnerBasis[eqs,Variables[eqs]]

delta=gMinus - (coords.v)(coords.w);
eqs=Union[Flatten[CoefficientList[delta,coords]]];
GroebnerBasis[eqs,Variables[eqs]]

Show that equation

   D (kappa + beta Id) = 1/2 trace (rho bar(D) otimes D)  A + B
   
has no (complex) solution for D

(*
 * If D=1/2 D^ij d/dx^i /\ d/dx^j is a bivector we represent 
 * the coefficients by the anti-symmetric matrix with coefficients 
 * of (D^ij)_ij. If kappa is an antisymmetric (2,2)-tensor,
 * then this routine returns coefficients of bivector D (kappa).
 *)
contract[biv_,kappa_] := Table[
 1/2 Sum[biv[[i]][[j]]emReadNormal[kappa,a,b,i,j]
      ,
      {i, 1, 4},{j, 1, 4}
    ]
 ,
 {a, 1, 4}, {b, 1, 4}
]

Dbivector=emMatrix["d",4,Structure->"AntiSymmetric"];
dLHS = contract[Dbivector,kappa+beta emIdentityKappa[]];
dRHS = 1/2 emTrace[emBiProduct[rho,Dbivector,Dbivector]]Abivector+Bbivector;
deqs=Union[Flatten[Simplify[dLHS-dRHS]]];
GroebnerBasis[deqs,Variables[deqs]]