The effects of massive graviton on the equilibrium between the black hole and radiation gas in an isolated box

It is well known that the black hole can has temperature and radiate the particles with black body spectrum, i.e. Hawking radiation. Therefore, if the black hole is surrounded by an isolated box, there is a thermal equilibrium between the black hole and radiation gas. A simple case considering the thermal equilibrium between the Schwarzschild black hole and radiation gas in an isolated box has been well investigated previously in detail, i.e. taking the conservation of energy and principle of maximal entropy for the isolated system into account. In this paper, following the above spirit, the effects of massive graviton on the thermal equilibrium will be investigated. For the gravity with massive graviton, we will use the de Rham-Gabadadze-Tolley (dRGT) massive gravity which has been proven to be ghost free. Because the graviton mass depends on two parameters in the dRGT massive gravity, here we just investigate two simple cases related to the two parameters, respectively. Our results show that in the first case the massive graviton can suppress or increase the condensation of black hole in the radiation gas although the $T-E$ diagram is similar like the Schwarzschild black hole case. For the second case, a new $T-E$ diagram has been obtained. Moreover, an interesting and important prediction is that the condensation of black hole just increases from the zero radius of horizon in this case, which is very different from the Schwarzschild black hole case.Comment: 9 pages, 4 figure

Quantum phase transitional patterns in the SD-pair shell model

Patterns of shape-phase transition in the proton-neutron coupled systems are studied within the $SD$-pair shell model. The results show that some transitional patterns in the $SD$-pair shell model are similar to the $U(5)-SU(3)$, $U(5)-SO(6)$ transitions with signatures of the critical point symmetry of the interacting boson model.Comment: 13 pages, 7 figure