Circular Orbits And Gravitational Lensing In Curved Spacetime

In this thesis,we are going to study the particle motion in curved spacetimes,specifically,the circular orbits and gravitational lensing.In the work of circular orbits,the existence and stability of timelike and null cir-cular orbits(COs)in the equatorial plane of a general static and axisymmetric(SAS)spacetime are investigated.Using the fixed-point approach,we first obtained a neces-sary and sufficient condition for the non-existence of timelike COs.Some other sufficient conditions on the stability of timelike COs are also solved.We then found the neces-sary and sufficient condition on the existence of null COs.It is then proven that there will always exist timelike COs in an asymptotically flat SAS spacetime with a positive ADM mass and moreover,for large enough radius these timelike COs are stable.It is generally shown that the existence of timelike COs in SAS spacetime does not imply the existence of null COs,and vice-versa,regardless whether the spacetime is asymp-totically flat or the ADM mass is positive or not.These results are then used to show the existence of timelike COs and their stability in an SAS Einstein-Yang-Mills-Dilaton spacetimes whose metric is not completely known.We also used the theorems to deduce the existence of timelike and null COs in some known SAS spacetimes.In the work of gravitational lensing,we studied the deflection angleΔφand gravi-tational lensing of both lightlike and timelike neutral rays in Reissner-Nordstr(?)m(RN)spacetimes.The exact deflection angle is found as an elliptical function of the impact parameter b and velocity v of the ray,and the charge Q of the spacetime.In obtaining this angle,we found the critical impact parameter b_cand radius of particle sphere r_cthat are also dependent on v and Q.In general,both the increase of velocity and charge reduces the b_cas well as r_c.To study the effect of v and Q on the deflection angleΔφ,its weak and strong deflection limits,relativistic and non-relativistic limits,and small charge and extremal RN limits are analyzed carefully.It is found that both the increase of velocity and charge reduces the deflection angle.For weak deflection,the velocity and charge corrections appear respectively in the O(1/b)and O(1/b~2)orders.For strong deflections,these two corrections appear in the same order.The apparent angles and magnifications of weak and strong regular lensing,and retro-lensing are studied for both lightlike and timelike rays.In general,in all cases the increase of velocity or charge will decrease the apparent angle of any order.We show that velocity correction is much larger than that of charge in the weak lensing case,while their effects in the strong regular lensing and retro-lensing are comparable.It is further shown that the apparent angle and magnification in strong regular lensing and retro-lensing can be effectively unified.These observables at different orders in these two kinds of lensing are stag-gered:the apparent angles can be ordered in a staggered way and the magnifications form two staggered geometric series.Finally,we argue that the correction of v and Q on the apparent angle can be correlated to mass or mass hierarchy of timelike particles with certain energy.In addition,the effects of v and Q on shadow size of black holes are discussed.