The Radius of the Schwarzschild Black Hole in Noncommutative Space with GUP

Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so called Generalized Uncertainty Principle (GUP) which predict the existence of a minimum observable length or a maximum observable momentum. On the other hand noncommutativity also corresponds to a minimum measurable length. So we expect that there is a relation between GUP and noncommutativite space in the range of the Planck scale. Here, we are going to derive a relation between the noncommutative coordinate and a commutative coordinate space with GUP. Then we use this new relation to compute the event horizon of a quantum Schwarzschild black hole in the Plank scale.