Research On Time-like And Null Geodesic Structures In Hayward Black Hole Space-time

The research on the geodesic structure of black holes can make people indirectly understand the gravitational effects at the core of black holes and comprehend the geometric structure around black holes space-time.Through analyzing the behavior of effective potential curves,combining the phase plane method and solving the equation of orbital motion numerically,we investigated the time-like and null geodesic structures in regular Hayward black hole space-time and modified Hayward black hole space-time.The results show that:(1)In the regular Hayward black hole space-time,all possible orbits of test particles are investigated by using phase plane method.Our results show that the orbits of massive particles are divided into four types:stable and unstable circular orbits,stable hyperbolic orbits and elliptical orbits in regular Hayward black hole space-time.The number of orbital types is influenced by the change of angular momentum b.The orbital properties vary with the change of l(is related with the central energy density 3/8?l2).If l=1/3 and b<3.45321,the test particles which moving toward the black hole will definitely be plunging into the black hole.In addition,it is obtained that the innermost stable circular orbit happens at rmin=5.93055 for b=3.45321.(2)In the regular Hayward black hole space-time,there are three kinds of horizons:no horizon?single horizon and double horizon with different l.The results show that there may exist four orbital types in the time-like geodesics structure:planetary orbits,circular orbits,escape orbits and absorbing orbits.The circular orbits have two kinds of orbital types:stable circular orbits and unstable circular orbits.In addition,when l equals to 0.6M,and b is 3.9512M as a extreme value of angular momentum,escape orbits exist only under b>3.9512M.When b<3.9512M,there are not escape orbits.When b=3.9512M,there exists a special orbit type which is consist of unstable circular orbits and bound orbits.The change of angular momentum has an obvious influence on the radius of stable and unstable circular orbits and the corresponding energy levels,the energy range of bound orbits and absorbing orbits,the curve peak of effective potential and the velocity and direction of bound orbits.The precession direction is associated with values of b.With b=3.70M the bound orbits shift clockwise but counter-clockwise with b=5.00M in the regular Hayward black hole space-time.In addition,the energy level does not change the number of orbital types,but has an obvious effect on the same orbital type,for example,the precession velocity of bound orbits and the bending degree of escape orbits.(3)Comparing the time-like geodesic structure in the regular Hayward black hole,null geodesics is simpler than that of time-like geodesics.There only exist three kinds of orbits(unstable circle orbits,escape orbits and absorbing orbits).The bound orbits and stable circular orbits are not included.In the null geodesic structure,comparing to the angular momentum b,the energy level has more influence on orbital motions.The absorbing orbits can be invited into the plunging orbits of high energy level and the near orbits of low level by the peak of effective potential.The change of energy level mainly affects the bending degree of photon escape orbits and absorbing orbits.The type of orbital motion of test particles in regular Hayward black hole space-time depends on angular momentum b,energy initial position and outside disturbance.(4)Comparing the regular Hayward black hole,when the massive test particles moves in modified Hayward black hole space-time,the property of orbital motion is not only associated with angular momentum b,energy initial position and outside perturbation but also with the modifications ?(? is associated with the time delay)and ?(? is related to 1-loop quantum corrections).The extreme angular momentum b=4.016M,the modifications make the extreme angular momentum increase.There are no escape orbits when b<4.016M,?=0.50 and ?=1.00.The limit value of angular momentum equals to 3.493M which is boundary between the stable and unstable orbits.For fixed ?=0.50 and?=1.007 when b<3.493M,there only exist unstable orbits;when b=3.493M,there exists the innermost stable circular orbit(ISCO),rISCO=5.997.Introducing the modifications makes the radius of the innermost stable circular orbit(ISCO)and the corresponding angular momentum become large.Comparing to the massive particles with the same angular momentum,the unstable circular radii of the regular and modified Hayward black hole space-times equal to 3.27 and 3.40,the corresponding energy levels equal to 1.33 and 1.30,respectively.The modifications makes the radius become larger,the corresponding energy level become low.The stable circular radii of the regular and modified Hayward black hole space-times equal to 21.52 and 21.44,the corresponding energy levels equal to 0.9560 and 0.9559,respectively.The modified terms make the radius of stable circular orbit and the corresponding energy level both become small.(5)For the null geodesic structure of modified Hayward black hole space-time,the parameter b has an obvious influence on the photon effective potential curves than the parameters ? and ?.All possible orbital types in modified Hayward black hole space-time:escape orbits,circle orbits and absorbing orbits.The circular orbits only have the unstable circular orbits.The radius of unstable circular orbit equals to 2.8790 for fixed l=0.60M,b=4.00M,?=0.50,?=1.00 and M=1.00.In the null geodesic structure,when the parameters l=0.60M,b=4.00M,?=0.50 and ?=1.00,comparing with the time-like geodesic,the unstable circular radius of photon is smaller than that of the massive particles and the corresponding energy level is smaller.