Relativistic geometrical optics (Q1181825)

Let \(M\) be a smooth manifold. A metric \[ g_{ij}(x,V(x))=\gamma_{ij}(x)+(1-1/n^2(x,V(x)))V_iV_j, \tag{1} \] where \(\gamma_{ij}\) is a Riemannian metric, \(V^ i(x)\) a vector field on \(M\), \(V_i=\gamma_{ij}V^j\) and \(n(x,V(x))\) is the index of refraction of a medium, was extensively studied by \textit{J. L. Synge} [Relativity. The general theory Amsterdam: North-Holland (1960; Zbl 0090.18504) (2nd ed. 1966)] and applied to the propagation of electromagnetic waves in a medium. Regarding \(V^i\) as a section \(S_V\) of the tangent bundle the authors remark that the metric (1) is the restriction to \(S_V(M)\) of the \(d\)-tensor field \[ g_{ij}(x,y)=\gamma_{ij}(x)+(1-1/n^2(x,y))y_ iy_ j, \quad y_i=\gamma_{ij}(x)y^j. \tag{2} \] Thus they relate the Synge metric (1) to generalized Lagrange spaces introduced and studied by \textit{R. Miron} [An. Stiint. Univ. Al. I. Cuza Iaşi, N. Ser., Sect. Ia 32, No. 2, 37--62 (1986; Zbl 0619.53021)]. The generalized Lagrange space \((M,g_{ij}(x,y))\) given by (2) is studied in detail. The interrelation to Synge's metric is fully exploited and some applications are pointed out. So an interesting and substantial geometrical model for relativistic optics is obtained.