The Space-time Structure Of The Maximum Extension Space-time,Photon Sphere,and Timelike Circular Orbit Of Ca(?)ate-Bergliaffa Black Hole

The spacetime maximum continuation of black hole,photon sphere and particle time-like circle orbits are important ways to understand the spacetime of black hole.The spacetime maximum continuation of black hole is of great significance to the overall physical properties of black hole and the study of singularities.The study of photon spheres and time-like circular orbits gives astrophysics extremely rich phenomenology,so its research has important theoretical significance.This paper mainly studies the maximum extension of Ca(?)ate-Bergliaffa black hole spacetime,the photon sphere and time-like circular orbit structure under the 3+1-dimensional extended scalar-tensor-Gauss-Bonnet theory.The main results are as follows:(1)The maximum extension spacetime of the Ca(?)ate-Bergliaffa black hole is isomorphic to that of the RN black hole,which is an infinite number of asymptotically flat space-time regions connected by black hole tunnels.When r>r+or r<r-,r is the space-like coordinate.When r-<r<r+,r is the time-like coordinate;it is impossible to cover the entire space-time manifold with only one coordinate system.In the Kruska coordinate system(U+,V+),there is a coordinate singularity r=r-;in the Kruska coordinate system(U-,V-),there are coordinate singularity r=r+and intrinsic singularity r=0,and the intrinsic singularity is time-like.(2)For the Ca(?)ate-Bergliaffa black hole,there is always an unstable photon sphere;for the compact massive objects in Ca(?)ate-Bergliaffa spacetime,the existence of the photon sphere is precisely depending on the value of q/m.When m/q>-(?)/2,the photon sphere does not exist,otherwise,the photon sphere exists.The stable/unstable dividing point of the photon sphere is strictly located at the extreme black hole horizon,that is,when rps/m>2,the photon sphere is unstable,otherwise,the photon sphere is stable.(3)For Ca(?)ate-Bergliaffa spacetime,when rc/q<-(?)or rc/m>2,there are extreme stable timelike circular orbits(ESCO),due to the multi-valued nature of the ESCO curve,there may be an innermost stable circular orbit(ISCO)or outermost stable circular orbit(OSCO).When(?)<rc/q<-(?),or 2<rc/m<8/3(-5+2(?)),ESCO is represented as OSCO;otherwise,it is represented as ISCO.Since the Ca(?)ate-Bergliaffa black hole satisfies and only satisfies the conditions for the existence of ISCO,there is always only ISCO,not OSCO.Moreover,the ESCO curve is also closely related to the photon sphere:the ESCO curve ends at the extreme value of the photon sphere curve,and the unstable circular orbit is always limited between the ESCO curve and the unstable photon sphere curve.