On the metric tensor of empty space in general relativity

The problem of deducing the metric of empty space in special relativity from the equations of general relativity has been solved by means of the works by Serini, Einstein and Pauli, Lichnerowitz. In the present paper the same problem is considered when in the equations of general relativity the cosmological term is included. The solution is obtained by showing how, by means of a suitable choice of co-ordinates, it is possible to transform the de Sitter universe (solution of the equations of general relativity with cosmological term in the empty space) in the Castelnuovo universe. This latter can be interpreted as the geodesical representation of the de Sitter universe on a flat manifold (extension of Beltrami’s theorem); furthermore, in analogy with Minkowski’s universe, it has some geometrical properties which make the interpretation of the physics of special relativity possible. Finally, by the method followed in this paper, the derivation of a new proof of the theorem of Serini, Einstein and Pauli, Lichnerowitz is made possible as a zero limiting case (i.e. when the cosmological constant tends to zero).