Black Hole Entropy from complex Ashtekar variables

Abstract

In loop quantum gravity, the number $N_{\Gamma }(A,\gamma )$ of microstates of a black hole for a given discrete geometry $\Gamma$ depends on the so-called Barbero-Immirzi parameter $\gamma$. Using a suitable analytic continuation of $\gamma$ to complex values, we show that the number $N_{\Gamma }(A,\pm i)$ 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

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