The Universality of Einstein Equations

Details The Universality of Einstein Equations The Universality of Einstein Equations

1993-03-02

arxiv.org/abs/gr-qc/9303007v2

Class.Quant.Grav.11:1505-1517,1994

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

15 pages, LaTeX, (Extended Version), TO-JLL-P1/93

Scientific paper

10.1088/0264-9381/11/6/015

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.

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