The Universality of Einstein Equations
M. Ferraris, M. Francaviglia, I. Volovich
TL;DR
The paper addresses whether Einstein equations persist as a universal backbone across a wide class of nonlinear gravity theories formulated with a metric and independent connection (Palatini formalism). By examining actions $S(\Gamma,g)=\int_M L(R)\sqrt{g}\,d^nx$ and the universal trace condition $L'(R)R-\frac{n}{2}L(R)=0$, it shows that for $n>2$ the field equations reduce to Einstein dynamics with a cosmological constant $\Lambda= c_i/n$ for real roots $R=c_i$, while in $n=2$ dimensions the system yields a Weyl connection and a constant-curvature constraint; exceptional Lagrangians such as $L(R)=|R|^{n/2}$ and cases with $L'(c_i)=0$ give alternative universal relations. The work provides explicit examples (e.g., $L(R)=aR^2+bR+c$ and $L(R)=R+aR^k$) illustrating bifurcations in the space of Lagrangians and the emergence or modification of Einstein equations, highlighting the role of conformal invariance and degenerate branches. Overall, the results establish a robust link between a broad class of nonlinear Lagrangians and Einstein gravity, with implications for higher-dimensional theories, conformal structures, and potential quantization frameworks.
Abstract
It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.