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As a simple application of the Newman-Penrose spin coefficient formalism, useful for beginners, we find the vacuum spherical symmetry (Schwarzschild) solution. The calculations also show that all spherically symmetric metrics are Petrov typeD.

An exact solution to the Einstein-Dirac equations is presented for a static, plane-symmetric spacetime generated by neutrinos. We find the neutrino field to be nonzero and correspond to a neutrino current along the symmetry axis of the space. The neutrinos yield a zero energy-momentum tensor and therefore the gravitational field is exactly the same as for the vacuum case. A comparison with other solutions is presented along with a discussion of the possible physical significance of this "ghost neutrino" field.

An exact solution to the Einstein−Dirac equations is presented for a static, cylindrically−symmetric spacetime. The neutrino field is nonzero and corresponds to a neutrino current in the radial direction. The neutrinos yield a zero energy−momentum tensor and therefore the gravitational field is the same as for the vacuum case. The neutrinos in these static cylindrically−symmetric spacetimes can exist only if the spacetimes are locally equivalent to static plane−symmetric spacetimes. This type of ’’ghost neutrino’’ solution is already known to exist in plane−symmetric spacetimes.