Topological Structure of Entropy of 4-Dimensional Axisymmetric Black Holes

Using the relationship between the entropy and the Euler characteristic, an entropy density is introduced to describe the inner topological structure of the entropy of 4-dimensional axisymmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of spacetime, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, which are topological invariants. At last, Kerr–Newman black hole as an example of axisymmetric black holes is given. What’s more, the entropy and the latent heat of the topological phase transition of the black hole mentioned above are calculated and the latent heat just lies in the range of the energy of gamma ray bursts. This work is supported in part by the NSFs of China under Grant No. 10575068 and of Shanghai Municipal Committee of Science and Technology under Grant No. 04ZR14059 and Shanghai Leading Academic Discipline Project under Project Number: T0104.