Black Hole Entropy from complex Ashtekar variables

Abstract

In loop quantum gravity, the number $N_{\hat{I}}“(A,{\hat{I}}^3)$ of microstates of a black hole for a given discrete geometry $\hat{I}“$ depends on the so-called Barbero-Immirzi parameter ${\hat{I}}^3$. Using a suitable analytic continuation of ${\hat{I}}^3$ to complex values, we show that the number $N_{\hat{I}}“(A,\pm i)$ of microstates behaves as $\exp (A/(4{\ell }_{\text{Pl}}^2))$ for large area $A$ in the large spin semiclassical limit. Such a correspondence with the semiclassical Bekenstein-Hawking entropy law points towards an unanticipated and remarkable feature of the original complex Ashtekar variables for quantum gravity.

5 pages. New point of view on the analytic continuation, which is now made rigorous by analytically-continuing the SU(2) spins in addition to the Barbero-Immirzi parameter