On the theory of the skewon field: From electrodynamics to gravity
  The Maxwell equations expressed in terms of the excitation $\H=({\cal H}, {\cal D})$ and the field strength $F=(E,B)$ are metric-free and require an additional constitutive law in order to represent a complete set of field equations. In vacuum, we call this law the ``spacetime relation''. We assume it to be local and linear. Then $\H=\H(F)$ encompasses 36 permittivity/permeability functions characterizing the electromagnetic properties of the vacuum. These 36 functions can be grouped into 20+15+1 functions. Thereof, 20 functions finally yield the dilaton field and the metric of spacetime, 1 function represents the axion field, and 15 functions the (traceless) skewon field $\notS_i{}^j$ (S slash), with $i,j=0,1,2,3$. The hypothesis of the existence of $\notS_i{}^j$ was proposed by three of us in 2002. In this paper we discuss some of the properties of the skewon field, like its electromagnetic energy density, its possible coupling to Einstein-Cartan gravity, and its corresponding gravitational energy. 
F.W. Hehl, Yu.N. Obukhov, G.F. Rubilar, and M. Blagojevic