02-327 Gavriel Segre

Einstein's lifts and topologies: topological investigations on the Principle of Equivalence (63K, Latex 2 \epsilon, revtex4) Jul 26, 02

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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.

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