EINSTEIN EQUATIONS DO NOT ADMIT A GENERIC POTENTIAL

J.F. Pommaret

Key words: Differential geometry, Einstein equations,
        differential operator, duality.

1991 Math. Subject Classification: 35N10, 53C80, 58A20,
        58G05, 83C05, 83C20.

Abstract: In 1970, J. Wheeler, at Princeton, offered
a 1000\$ award to know wether Einstein equations in vacuum
should admit or not a generic potential, that is to say
if it could be possible to express a solution metric by
means of a few arbitrary functions and their derivatives
up to a certain order, in such a way that this metric
should become a solution of Einstein equations and no
others.

Such a problem is a particular case of the following
one: how to recognize when a given linear partial
differential operator is expressing all the compatibility
conditions of a preceding one in a differential
sequence.

The purpose of our lecture is to give an {\it effective
answer} to this question by means of the new technique of
{\it differential duality} and to apply it to Einstein
equations, thus solving the above challenge negatively.