Hawking and Penrose's theorem on the singularities of gravitational collapse is applied to Riemann-Cartan space-time ($U_4$) generated by Dirac spinor fields. The energy condition of the theorem is modified into
$R_{ij}t^it^j+2K_{ikl}K_j^{ki}t^it^j \geq 0$
for $U_4$ where $R_{ij}$ is Ricci tensor in $U_4$ and $K_{ijk}$ is contortion tensor. The contribution of spin-spin interaction in the modified energy condition is calculated;
$(\ell^4/16k)[2(t^ia_i)^2+a^2a_i$
where $a^i=\overline{\Psi} \gamma_5 \gamma^i \Psi$ is spin axial vector. It is shown that this contribution can take both positive and negative value.
Particularly, if the parallel component of spin with timelike vector $t^i$ and the orthogonal component satisfy $\mid{a_{\bot}}\mid > \sqrt{3} \mid{a_{\|}\mid$ and $(\bar{\Psi}\Psi) \gtrsim m/\ell^2$ than the modified conditions is violated and space-time singularities are prevented.
Dirac spinor field가 주어진 Riemann-Cartan space-time $(U_4)$에 대하여 Singularity 정리를 적용하였다.
정리의 energy 조건은 임의의 unit timelike vector $t^i$ 에 대하여 $U_4$의 Ricci tensor $R_{ij}$와 contortion tensor $K_{ijk}$ 는
$R_{ij}t^it^j + 2K_{ikl}K_j^{kl} t^it^j ≥ 0$
를 만족해야 되는 것으로 고쳐진다. 이중 spin-spin interaction에 의한 기여는
$\frac{\ell^4}{16k}[2(t^i a_i)^2 + a^i a_i]$
$(a^i = \bar{\Psi} \gamma_5 \gamma^i \Psi, k = \frac{8\pi}{c^4} \gamma = 2\times 10^{-48} dyn^{-1}, \ell^2 = \hslashck)$로써 spin axial vector $a^i$의 $t^i$에 평행한 성분 $a_{\parallel}$와 수직한 성분 $a \bot$ 사이에 $\mid {a_{\bot}} \mid ＞ \sqrt{3} \mid {a_{\|}} \mid$ 일 때 (-) 가 된다. 동시에 $(\bar{\Psi} \Psi) \gtrsim m/\ell^2(\hslashm/c$는 Dirac 입자의 질량)일 때 수정된 energy 조건에 어긋나 singularity를 막을 수 있게 된다.
