On the naturality of the Einstein equation

We characterize the Einstein tensor as the only, up to a constant factor, 2covariant tensor naturally constructed from a semiriemannian metric which is divergence-free and invariant under constant rescalings of the metric. Since these two conditions are also satisfied by the energy-impulse tensor of a relativistic space-time, we discuss how this theorem and its variations can be used to derive the field equation of General Relativity. In General Relativity, it is assumed that the mass-energy content of the Universe is determined by its geometry. Therefore, it is supposed a field equation of the following type: G2(g) = T2 (1) where T2 is the energy-impulse tensor of the matter, g is the Lorentz metric of space-time and G2(g) is a suitable tensor constructed from g. Since the energy-impulse tensor T2 is symmetric and divergence-free, we are led to choose for the left-hand side of (1) a tensor G2(g) satisfying these two properties. The choice of G2(g) is then suggested by a beautiful classical result, first published by Vermeil ([13]) and developed by Cartan ([3]) and Weyl ([14]), which characterizes the Einstein tensor G2(g) := R2(g) − 1 2 r(g)g of a semiriemannian metric g as the only, up to a constant factor and the addition of a cosmological term Λg, 2-covariant symmetric and divergence-free tensor whose coefficients are functions of the coefficients of the metric, its first and second derivatives and are linear in these second derivatives. Since the very first days of the theory, this theorem has been the cornerstone to justify the field equation of General Relativity. University of Extremadura Email addresses: [email protected], [email protected] The first author has been supported by a Spanish FPU grant, AP2006-02414