Abstract
Quadratic invariants of the Riemann-Christoffel curvature tensor and its contractions in a four-dimensional Riemann space are used as the Lagrangians in three variational principles. The field equations are derived by treating the metric tensor and the arbitrary symmetric affine connection as independent variables (following the method of Palatini), and specializing to the Christoffel connection after the variation. It is shown that the field equations derived from two of these variational principles in this way have as a class of solutions all solutions of Einstein’s equations with cosmological term, whilst all three sets of field equations are satisfied by the Schwarzschild metric and have vanishing divergence. This suggests alternative forms of the field equations for gravitation, quadratic in the Riemann-Christoffel tensor and with zero trace, which give the same results for the three «crucial tests» of general relativity as Einstein’s equationsR ik =0.
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