Abstract
The classical Raychaudhuri equation predicts the formation of conjugate points for a congruence of geodesics, in a finite proper time. This in conjunction with the Hawking-Penrose singularity theorems predicts the incompleteness of geodesics and thereby the singular nature of practically all spacetimes. We compute the generic corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole, arising from modifications to the algebra inspired by the generalized uncertainty principle (GUP) theories. Then we study four specific models of GUP, compute their effective dynamics as well as their expansion and its rate of change using the Raychaudhuri equation. We show that the modification from GUP in two of these models, where such modifications are dependent of the configuration variables, lead to finite Kretchmann scalar, expansion and its rate, hence implying the resolution of the singularity. However, the other two models for which the modifications depend on the momenta still retain their singularities even in the effective regime.
Highlights
These parameters set minimal scales of the model which determine the onset of quantum gravitational effects
We compute the generic corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole, arising from modifications to the algebra inspired by the generalized uncertainty principle (GUP) theories
We investigate the modified dynamics of the interior of the Schwarzschild black hole using Ashtekar-Barbero variables but using modified algebra inspired by GUP
Summary
Using the form of the lapse function (2.19), we can derive the equations of motion for b, c as [16, 32, 82] These show that, classically, b is proportional to the rate of change of the square root of the physical area of S2, and c is proportional to the rate of change of the physical length of I. To obtain the classical dynamics of the interior, we choose a different gauge The advantage of this lapse function is that the equations of motion of c, pc decouple from those of b, pb as we will see in a moment and it makes it possible to solve them. All diverge, signaling the presence of a physical singularity for pc → 0 as expected
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