A proposal for Heisenberg uncertainty principle and STUR for curved backgrounds: an application to white dwarf, neutrons stars and black holes

After a critical overview of the generalized uncertainty principle (GUP) applied to compact objects, we propose a texture of Heisenberg uncertainty principle in curved spacetimes (CHUP). CHUP allows to write down physically motivated STUR (spacetime uncertainty relations) in a generic background for a non commutative spacetime in terms of tetrad variables. In order to study possible quantum effects for compact astrophysical objects as white dwarf, neutron stars and black holes, an expression for quantum fluctuations is outlined. As a result, contrary to GUP-based claims, we found no evidence for quantum effects concerning equilibrium equation and critical mass Mc for white dwarf and neutron stars. Conversely, our expression for CHUP confirms that general relativistic effects strongly reduce the Oppenheimer–Volkoff Newtonian limit for very compact astrophysical objects as neutron stars. In particular, we found that for a degenerate relativistic Fermi gas, the maximum mass decreases for increasing compactness of the star with a minimum critical mass Mc ≃ 0.59M⊙ at the Buchdahl limit. Finally, we study possible non commutative effects near the event horizon of a black hole.