Abstract
We obtain bounds for the minimum and maximum mass/radius ratio of a stable, charged, spherically symmetric compact object in a $D$-dimensional space-time in the framework of general relativity, and in the presence of dark energy. The total energy, including the gravitational component, and the stability of objects with minimum mass/radius ratio is also investigated. The minimum energy condition leads to a representation of the mass and radius of the charged objects with minimum mass/radius ratio in terms of the charge and vacuum energy only. As applied to the electron in the four-dimensional case, this procedure allows one to re-obtain the classical electron radius from purely general relativistic considerations. By combining the lower mass bound, in four space-time dimensions, with minimum length uncertainty relations (MLUR) motivated by quantum gravity, we obtain an alternative bound for the maximum charge/mass ratio of a stable, gravitating, charged quantum mechanical object, expressed in terms of fundamental constants. Evaluating this limit numerically, we obtain again the correct order of magnitude value for the charge/mass ratio of the electron, as required by the stability conditions. This suggests that, if the electron were either less massive (with the same charge) or if its charge were any higher (for fixed mass), a combination of electrostatic and dark energy repulsion would destabilize the Compton radius. In other words, the electron would blow itself apart. Our results suggest the existence of a deep connection between gravity, the presence of the cosmological constant, and the stability of fundamental particles.
Highlights
The existence of a minimum length is an important prediction of phenomenological quantum gravity
One way to introduce a minimum length is via a Generalized Uncertainty Principle (GUP) that extends the usual Heisenberg Uncertainty Principle (HUP) to include nonlinear terms, which may be interpreted as quantum gravity effects
Lower bounds on the ratios of physical quantities do occur naturally in classical general relativity, as a form of stability condition for compact objects. Two such bounds are of particular interest for both astrophysics/cosmology and for the study of subatomic particles: the minimum mass/radius ratio for a compact object in the presence of dark energy and for a charged compact object
Summary
The existence of a minimum length is an important prediction of phenomenological quantum gravity. Lower bounds on the ratios of physical quantities do occur naturally in classical general relativity, as a form of stability condition for compact objects Two such bounds are of particular interest for both astrophysics/cosmology and for the study of subatomic particles: the minimum mass/radius ratio for a compact object in the presence of dark energy and for a charged compact object. Classical (3 + 1)-dimensional general relativity, with no dark energy component ( = 0), imposes an upper bound on the mass/radius ratio of any compact object, the Buchdahl limit [7], which requires a sphere of matter with arbitrary equation of state to satisfy the stability constraint If this condition is violated, the object will inevitably collapse under its own gravity to form a black hole.
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