 02327 Gavriel Segre
 Einstein's lifts and topologies: topological investigations on the Principle of Equivalence
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Jul 26, 02

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Abstract. The gedankenexperiment of Einstein's lift is analyzed in order
of determining whether the freefalling observer inside the
lift can detect the eventual topological nontriviality of
spacetime, as it would seem considering a nongloballyhamiltonian 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 constantsofmotion'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 spacetime 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 nongravitational 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 HawkingEllis' axiomatization of
General Relativity purely based on the assumption of the EinsteinHilbert'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 interrelation 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 YangMills' theories is finally reported.
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