The two canonical conjugate pairs at the horizon of a D1D5 black hole

Abstract

The Euclidian opening angle at the $r-t_E$ surface, ${\Theta }_{r-t_E}$ at the horizon of a black hole is canonically conjugate to the black hole entropy. We prove that for a $D1D5$ black hole there exists in addition to this pair, another canonical pair: the opening angle at the $r-y$ surface, ${\Theta }_{r-y}$ and a Wald like term $S_{Wr-y}$. This leads to an uncertainty at ${\Theta }_{r-y}$ which suggests that the surface $r-y$ is actually a superposition of surfaces with different conical singularities. This corresponds to the same type of singularities obtained by string theory excitations of a $D1D5$ black hole.