Non-existence of time-periodic solutions of the Dirac equation in a Reissner- Nordström black hole background (Q2737926)

The authors of this interesting paper show analytically that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordström black hole background. In particular, there are no static solutions of the Dirac equation NEWLINE\[NEWLINE \biggl (iG^j(x)\partial /\partial x^j+(i/2)(\nabla _jG^j)(x)+eG^j(x)A_j(x)\biggr)\Psi (x)=m\Psi (x) NEWLINE\]NEWLINE in such a background metric; \(m\) is the rest mass of the particle, \(A=A_jdx^j\) is the electromagnetic potential, \(e\) is the electromagnetic coupling constant; the Dirac matrices \(G^j(x)\) are real linear combinations of the usual \(\gamma -\)matrices. The physical interpretation is that Dirac particles can either disappear into the black hole or escape to infinity, but they cannot stay on a periodic orbit around the black hole.