From Differential Sequences to Black Holes

Abstract

When E. Beltrami introduced in 1892 the six stress functions Φ i j = Φ j i wearing his name and allowing to parametrize the Cauchy stress equations of elasticity theory in space, he surely did not know he was using the Einstein operator introduced by A. Einstein in 1915 for general relativity in space-time, both ignoring that it was self-adjoint in the framework of differential double duality and confusing therefore stress functions with the variation Ω i j = Ω j i of the metric ω . When I proved in 1995 that the Einstein equations in vacuum could not be parametrized like the Maxwell equations, solving thus negatively a 1000 dollars challenge of J. Wheeler in 1970, I did not imagine that such a purely mathematical result could also prove that the equations of the gravitational waves were not coherent with differential homological algebra. The purpose of this paper is to prove that this result is also showing that black holes cannot exist, not for a problem of detection but because their existence should contradict the link existing between the only two canonical differential sequences existing in the literature, namely the Janet sequence and the Spencer sequence, a result showing that the important object is not the metric but its group of invariance. Indeed, the last sequence is isomorphic to the tensor product of the Poincaré sequence by a Lie algebra of extremely small dimension when dealing with the differential resolutions of Killing vector fields while using successively the Minkowski (M), the Schwarzschild (S) and Kerr (K) metrics with respective parameters ( 0 , 0 ) , ( m , 0 ) and ( m , a ) . The comparison with other explicit motivating examples also provided needs no comment.