Black holes and phase-space noncommutativity

We use the solutions of the noncommutative Wheeler-DeWitt equation arising from a Kantowski-Sachs cosmological model to compute thermodynamic properties of the Schwarzschild black hole. We show that the noncommutativity in the momentum sector introduces a quadratic term in the potential function of the black hole minisuperspace model. This potential has a local minimum and thus the partition function can be computed by resorting to a saddle point evaluation in the neighborhood of the minimum. The thermodynamics of the black hole is derived and the corrections to the usual Hawking temperature and entropy exhibit a dependence on the momentum noncommutative parameter, $\eta$. Moreover, we study the $t=r=0$ singularity in the noncommutative regime and show that in this case the wave function of the system vanishes in the neighborhood of $t=r=0$.

Figure 1

Potential function for some typical values of $\eta$ and $c$: (a) the potential function for the noncommutative case, $\theta \ne 0$ and $\eta =0$; and (b) the potential function with $\eta \ne 0$.