We state a variational principle which allows the variational characterization of the class of torsion_less affme connections on a Riemannian manifold, as well as of any subclass of it determined by a suitable set of constraints on the metricity of the connection.
1. Introduction
This paper is devoted to the study of gravitational variational principles in the presence of constraints. This is suggested by the remark that many variational problems do not yield all the necessary field equations. A well-known example is the Palatini principle [2], which requires some subsidiary conditions in order to determine the equations for both the metric and the connection, these quantities being regarded as independent variables.** In other words, recalling that a connection on a Riemannian manifold is determined by its torsion and metricity fields T and Q, we may say that the variational principle in the arguments qs, T, Q (q5 is the fundamental form) based on the Hilbert-Palatini Lagrangian alone does not provide a complete set of equations for as, T and Q. This argument may be generalized to the study of variational principles, based on linear Lagrangians, in the presence of constraints. Among the various possible choices of the constraints, the most meaningful fall within two categories: algebraic constraints on the torsion tensor, and algebraic constraints on the metricity tensor. The first possibility, which admits the Palatini method as a special case, was examined by E. Massa [1].
The second possibility will form the subject of the present paper. We shall state a variational principle which always yields the condition T = 0 and the usual empty-space Einstein equations.
No conditions on the metricity arise. In this sense, the variational principle characterizes the class of the torsionless connections over the manifold. Furthermore, the principle may be completed by whatever constraint on the metricity, thus allowing the variational characterization of arbitrary, a-priori chosen, subsets of the class of torsionless connections. * Work performed under the auspices of the Italian Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR). ** A discussion of some modifications of Palatini principle which do not present such a drawback may be found in [7][8][9], [1].