On Thermodynamic Properties Of Bosons And Fermions In Spherically Symmetric Spacetimes

Since Bekenstein and other pioneers proposed the concept that black holes also have thermal entropy,the research on the thermal properties of black holes has attracted extensive attention of scientists.Hawking gives the specific expression of black hole temperature and black hole entropy(Bekenstein-Hawking entropy)by the method of semi-classical quantum field theory,so that black hole can be studied as a real thermodynamic system.In recent years,with the great development of black hole thermodynamics research,it has been found that black holes have thermal behaviors such as van der Waals-like phase transition,fluctuation and dissipation similar to ordinary thermal systems.These results not only demonstrate that black holes can behave like ordinary thermodynamic system which has rich thermodynamic behaviors,but also can well demonstrate the unique properties of black holes as strong gravitational objects.Black hole thermodynamics is a theory that combines gravity,quantum mechanics and thermodynamics,so the relevant research results are expected to be an effective window to understand quantum gravity.Although the study of black hole thermodynamics has yielded fruitful results,the microscopic origins of black hole entropy and other related thermodynamic properties are still poorly understood.As for the study to reveal microscopic origin of black hole entropy,the string theory,Fuzzball theory,Cardy formula and etc make some useful attempt,which discusses the area dependence of black hole entropy.However,these studies of black hole entropy are far from enough for us.Since we know few about the information inside black holes,it is a good strategy to study the thermodynamic properties of the matter around black holes.Through assuming that the black hole entropy is generated by the thermal radiation field around the black hole,there are some work including brick-wall model,thin-film model and calculating black hole entropy by generalized uncertainty relation.But there are still some shortcomings in these research works still.Taking into account the generalized covariance principle,Padmanabhan developed a set of statistical mechanics research methods for curved spacetimes,which are used to study the thermodynamic properties of matter near the horizon in curved spacetimes.Based on this,there is a research work which gives the thermal properties of the two-dimensional Bose and Fermi systems on the stretched horizon in the high-dimensional black hole background.But unfortunately this work is limited to the special case where the system is two-dimensional.In view of the fact that two-dimensional systems cannot well reflect the properties of high-dimensional black holes,it is necessary to study higher-dimensional Bose and Fermi systems.Therefore,the author studies the thermal properties of arbitrary-dimensional Bose and Fermi systems in spherically symmetric spacetimes at the near-horizon limit of the black hole.This work solves the difficulty that previous studies on Bose and Fermi systems were limited to two-dimension.Our work is divided into the following parts: 1)Starting from the partition function of the system,we study the Bose-Einstein condensation(BEC)of the Bose-gas shell near the Schwarzschild black hole horizon.We find that the critical temperature of the BEC is affected by the radius of the black hole’s event horizon under the influence of strong gravity near horizon.2)As a special case,we give the entropy of the photon gas near the black hole horizon,and compare it with the Bekenstein-Hawking entropy of the black hole itself.We find that the entropy of the photon gas and the black hole entropy have similar characteristics.3)We calculated the Fermi energy of the Fermi gas near the horizon of Schwarzschild black hole.And we find that under the influence of strong gravity,the Fermi energy is not only proportional to the radius of the black hole’s horizon,but also proportional to the Hawking temperature of the black hole.4)We calculated the entropy of Fermi gas at the near horizon limit of Schwarzschild black hole,and find that its entropy is area-dependent.5)Finally,we generalized the results obtained in Schwarzschild spacetime to spherically symmetric spacetimes of arbitrary dimensions.All the results above still hold for arbitrary dimensional spacetimes.In this paper,the related research on the Bose and Fermi systems at the near horizon limit of black holes reveals the intrinsic unity of statistical mechanical properties between black holes and their surrounding matter,which further shows that black holes are not only a geometry given by general relativity but also a potential research object for the statistical mechanics of curved spacetimes.In addition,the BEC of the shell-like Bose gas and the shell-like degenerate electron gas we consider will provide a theoretical reference for the study of black hole thermodynamics.