Einstein-Cartan-Dirac equations in the Newman-Penrose formalism

We formulate the Einstein-Cartan-Dirac equations in the Newman-Penrose (NP) formalism, thereby presenting a more accurate and explicit analysis of previous such studies. The equations show in a transparent way how the Einstein-Dirac equations are modified by the inclusion of torsion. In particular, the Hehl-Datta equation is presented in NP notation. We then describe a few solutions of the Hehl-Datta equation on Minkowski space-time, and in particular report a solitonic solution which removes the unphysical behavior of the corresponding Dirac solution. The present work serves as a prelude to similar studies for nondegenerate Poincaré gauge gravity.

Figure 1

Case I and case VI. The left column shows plots for case I with $w=-2$. The right column shows plots for case 6 with $w=+2$. Both the cases have unphysical solutions.

Figure 2

Solutions to the linear (torsionless) Dirac equations. Only the plane-wave solutions (Cases I, VI) are physical.

Figure 3

Case III and case IV. Case III on the left, with $w=-0.5$. Case IV on the right, with $w=0$. Both the cases have unphysical solutions.

Figure 4

Case V. In all plots: green: $w=0.75$; orange: $w=0.5$; blue: $w=0.25$. For probability density plot, we have two subcases. Case V(b) has global maxima at origin (2 candidates of this case are shown in orange and green). Case V(a) has local minima at origin and two maximas at the two symmetrically opposite sides of origin at nonzero p (blue graph represents this case). Both cases V(a) and V(b) are asymptotically vanishing.

Figure 5

Plots for ${\mu }_{-}$ and ${\mu }_{+}$ as a function of $L_{\mathrm{CS}}$ for various values of $l_0$.