Abstract
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 (3+1)-dimensional spherically symmetric black holes. It is pointed out that the density of entropy is determined by the singularities of the timelike Killing vector field of space–time, and these singularities carry the topological numbers, Hopf indices and Brouwer degrees, naturally, which are topological invariants. Taking account of the physical meaning in statistics, the entropy of black holes is given by the Hopf indices merely, which will lead to the increasing principle of entropy of black holes.
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