Spherical symmetric solutions for gravity+dirac Concerning only $g_{\mu\nu}$ and fermion $\psi$,  I can write Lagrangian as
\begin{align}
\sqrt{-g} (R + \bar{\psi} \gamma^\mu  D_\mu \psi + m \bar{\psi}\psi)
\end{align}
Now i am curious about static-spherical solution of this Lagrangian. 
(I.e, (?) extended version of Schwarzschild solution) 
are there any known solution for this kinds of case? 
[any comment will be helpful!] 
 A: In 1975 I published a paper (Ann. of Physics 91, 40-57) in which I showed that the most general time independent, spherically symmetric, Hermetian, time reversal invariant Dirac equation has three possible interaction terms (scalar, vector (time-like component) and tensor) all dependent on the radial coordinate only.  Since gravity is a spin-2 theory (from a QFT perspective) the tensor term would satisfy your requirement.
The resulting radial Dirac Hamiltonian (with tensor interaction only) has the form: $$H_r=\not{h}c\frac{d}{dr}\begin{pmatrix}0 &-1\\1&0\\\end{pmatrix}+[iU_t(r)+\omega(J+\frac{1}{2})/r]\begin{pmatrix}0 & 1\\1 & 0\\\end{pmatrix}+mc^2\begin{pmatrix}1 & 0\\0 &-1\\\end{pmatrix}$$ where $J$ and $\omega$ are the angular momentum and parity quantum numbers ($P=(-1)^{J+\frac{\omega}{2}}$, $\omega=+ or -1$).  The radial Dirac wave function upon which this Hamiltonian would operate is: $$\psi_r=\begin{pmatrix}F(r)\\G(r)\\\end{pmatrix}$$ The fumnctions $F(r)$ and $G(r)$ are normally referred to as the large and small components (for positive energy solutions).
