Quantum Corrections to the Mass of a Black Hole coupled to $N$scalars

Einstein gravity coupled minimally or conformally to $N$scalar fields has well-known static and spherically symmetric classical black hole solutions of Schwarzschild and extremal Reissner-Nordstr\"{o}m type,respectively. These classical solutions depend on a single integration constant corresponding to their Schwarzschild radius $NR$. Assuming that this system can be considered in isolation and/or other mass scales may be neglected, the mass $m$ of such a configuration is of the form $m(N\sim \infty )=\frac{c^2NR}{2G}+\chi \frac{\hbar }{cR}+O\left( {{G\hbar ^2} \mathord{\left/ {\vphantom {{G\hbar ^2} {(c^4NR^3)}}} \right. \kern-\nulldelimiterspace} {(c^4NR^3)}} \right)$, where$\ell _P =\sqrt {G\hbar \mathord{\left/ {\vphantom {\hbar {c^3}}} \right. \kern-\nulldelimiterspace} {c^3}} $is the Planck length and $R$corresponds to the Schwarzschild radius for a single scalar.Only the first two terms of the expansion are relevant in the formal asymptotic limit of an infinite number of only gravitationally interacting scalars forming a black hole whose mass essentially is proportional to the number of degrees of freedom,. The correction to the classical mass that is inversely proportional to$R$ may be interpreted as due to the change in vacuum energy caused by forming a black hole of radius $NR$, i.e. as a Casimir effect. The dimensionless constant $\chi $escribing this correction is estimated semi-classically using periodic orbit theory. The value (and sign) of $\chi $ in this approximation is determined by the unstable classical periodic orbits on the photon sphere of the black hole.