Proof of Proposition 3.4

SetDirectory["~/writing/WIP/KappaLib/"];
<<kappaLib10.m

Suppose g is a Riemann metric of arbitrary signature and kappa is the Hodge operator. We show that 
the Tamm-Rubilar tensor density satisfies

    G^abcd xi_a xi_b xi_c xi_d = sgn(det g) sqrt(abs(det(g))) (g(xi,xi))^2

invMetric=(({{h11, h12, h13, h14}, {h12, h22, h23, h24}, {h13, h23, h33, h34}, {h14, h24, h34, h44}}));

Union[Flatten[invMetric-Transpose[invMetric]]]

(*
 *  The below is adopted from emHodge with the difference that the below
*  computes the Hodge star operator as a function of sqrt(abs(det g)) 
*  and g^-1.
 * 
 *)
emHodgeLow[detSq_, invMetric_] := Module[
	{resKappa,   i,j ,m,n, a,b},

	resKappa = emZeroKappa[];
	
	For[i=1, i<=4, i++,
		For[j=i+1, j<=4, j++,
			For[m=1, m<=4, m++,
				For[n=m+1, n<=4, n++,
					resKappa[[i]][[j]][[m]][[n]] = detSq Sum[ 
							invMetric[[i]][[a]] invMetric[[j]][[b]] Signature[{a,b,m,n}],
								{a,1,4},{b,1,4}
							];
				];
			];
		];
	];

	resKappa

]

(* Compute (2,2)-tensor induced by the above metric tensor *)
kappa = emHodgeLow[detSq, invMetric];

(* Compute and simplify Fresnel equation *)
xi={xi0,xi1,xi2,xi3};
fresnel =emKappaToFresnel[kappa,xi];
FullSimplify[fresnel]

fresnelClosedForm=detSq^3 Det[invMetric](xi.invMetric.xi)^2;

Simplify[fresnel-fresnelClosedForm]

Check that trace kappa = 0

Simplify[emTrace[kappa]]