Towards a supersymmetric generalization of the Schwarzschild-(anti)de Sitter space-times

Abstract

The Wheeler-DeWitt (WDW) equation for the Lambda-Kantowski-Sachs model can also be understood as the WDW-equation corresponding to the Schwarzschild-(anti)deSitter space-times, due to the well-known diffeomorphism between these two metrics. This equation ignores the coordinate patch one chooses and only by imposing coordinate conditions will it be possible to distinguish between black hole or cosmological models. At that point, the foliation parameter $t$ or $r$ will appear in the solution of interest. In this work we supersymmetrize this WDW-equation obtaining an extra term in the potential with two possible signs. We have already, in a similar manner, proposed and analyzed a supersymmetric generalization of a Schwarzschild black hole. The WKB method is then applied, given rise to four classical equations, two for the Schwarzschild-deSitter space-time and another two for the Schwarzschild-anti-deSitter space-time. One can then study the asymptotic cases in which one of two potential terms arising in each Hamiltonian, for each of these space-times, dominates the behavior. One of these limiting (bosonic) cases gives the usual Schwarzschild-(anti)deSitter space-times. We will study here the other four asymptotic regions; they provide six solutions. For the Schwarzschild-deSitter space-time we get two solutions which have singularities at $r=0$ and ${r}_{0}^{s}$, and depending on an integration constant $C$ and the sign at the potential due to these SUSY region, they can also present another two singularities in ${r}_{h}^{s}$ and ${r}_{c}^{s}$. For the Schwarzschild-anti-deSitter space-time the solutions have a singularity at $r=0$ and depending on the integration constant and the sign of the SUSY potential another singularity can appear at ${r}_{+}$. We find associated masses and, based on the holographic principle, we find also the entropies for the bosonic region, which coincide with the ones obtained by the usual methods. We apply this same procedure to get entropies associated to the supersymmetric asymptotic solutions. Even though we were unable to obtain the complete solution to the model, it is shown that horizons can never be reached because when one would approach the standard horizons (the bosonic region), the relevant term in the potential is the one due to supersymmetry and as mentioned in this asymptotic limit one does not have horizons but instead singularities.