 Solution

In the rest frame $u^{\mu }=(1,0,0,0)$ and $h_{\mu \nu }=\mathrm{diag}(0,1,1,1)$. The two material contractions in the printed action become $-{\mathbf{P}}^2$ and $-2{\mathbf{M}}^2$. Consequently its material Lagrangian density isThe Euler-Lagrange field equations are algebraic, since no derivatives of the material variables occur. They giveThese also follow from the $i0$ and $ij$ components of the corrected equation in part (b). If instead the intended starting point was the PDF's displayed equation in part (b), its predictions would beThe two magnetic answers express the documented sign error, rather than different choices of frame. Here the inverse coefficients in the action require nonzero $\kappa ,\chi$; zero response is obtained as a limiting constitutive law.