Abstract
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out here that the two effects are mutually exclusive when compared with experimental bounds. Moreover, the former stems from modified Poisson brackets obtained from a wrong classical limit of the deformed canonical commutators.
Highlights
It is well known [1] that the Equivalence Principle (EP; namely the equality between gravitational and inertial mass) dictates that the equation of motion of test particles in a gravitational field be of the form d2xλ dτ 2
In successive studies [3, 4], Einstein and collaborators obtained a result of considerable importance: the equation of motion of point particles, that is the geodesic equation (1), can be derived from the gravitational field equations (2). 2 In other words, the field equations determine uniquely the equation of motion for bodies in a gravitational field which are not subjected to other forces, and the ensuing trajectories are geodesics of the corresponding metric
This finding is in full agreement with the postulate of geodesic motion, which appears as a consequence of the field equations, and not as an independent axiom of the theory
Summary
It is well known [1] that the (weak) Equivalence Principle (EP; namely the equality between gravitational and inertial mass) dictates that the equation of motion of test particles in a gravitational field be of the form d2xλ dτ 2. In successive studies [3, 4], Einstein and collaborators obtained a result of considerable importance: the equation of motion of point particles, that is the geodesic equation (1), can be derived from the gravitational field equations (2). 2 In other words, the field equations determine uniquely the equation of motion for bodies in a gravitational field which are not subjected to other forces, and the ensuing trajectories are geodesics of the corresponding metric This finding is in full agreement with the postulate of geodesic motion, which appears as a consequence of the field equations, and not as an independent axiom of the theory. It is important here to remark that the starting point is the conservation of the energy-momentum tensor, to wit In this way, it appears as a consistency condition for the field equations. One cannot modify or renounce to either of them
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