Gavriel Segre
Einstein's lifts and topologies: topological investigations on the Principle of Equivalence
(63K, Latex 2 \epsilon, revtex4)

ABSTRACT.  The gedanken-experiment of Einstein's lift is analyzed in order 
of determining whether the free-falling observer inside the 
lift can detect the eventual topological non-triviality of 
space-time, as it would seem considering a non-globally-hamiltonian action of the symmetry group of the observer's action (that, unfortunately, can be obtained only submitting the lift also to a suitable electromagnetic field) and considering that the observer can locally detect the topological alteration of the constants-of-motion's algebra. 
It follows that a problem exists in formalizing the Principle of 
Equivalence, owing to its indetermination as to the topology of the reference's flat space-time defining the special relativistic laws to which, up to first order terms in the normal coordinates of the lift's Lorentz moving inertial frame, all the non-gravitational Laws of Physics 
have to collapse. 
It is then shown how the problem may be avoided getting rid of the Principle of Equivalence following the Hawking-Ellis' axiomatization of 
General Relativity purely based on the assumption of the Einstein-Hilbert's action. 
Connes' axiomatization of General Relativity having as 
only dynamical variable the spectrum of Dirac's operator is 
then used to discuss the initial topological question concerning 
Einstein's lift in the language of Spectral Geometry, 
explicitly showing its inter-relation with the celebrated 
Marc Kac's issue whether one can hear the shape of a drum, and 
showing how Index Theory is the natural framework in which some 
partial answer may be obtained. 
The whole issue is then analyzed in Connes' Quantum Gravity, 
suggesting how Noncommutative Geometry allows, through Noncommutative Index Theory, to get some insight along the footsteps followed in the 
commutative case. 
Some attempt of relating the issue to Anandan's remark on the 
difference among the holonomies of General Relativity and the holonomies of Yang-Mills' theories is finally reported.