Dynamics Of Test Particle In Curved Space-time

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress-energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto 3-D space.It is extremely important for astrophysics to understand the space-time geometry around a black hole, one of the best way is to study the time-like and null geodesies of the black hole. The accretion model of the black hole is often approximately to assume that the accreting matter moves along the geodesies. The radiation near the black hole falls into the black hole or propagates to infinite observers along the null geodesic. By analyzing the effective potential of the test particles or photons, we find all possible kinds of geodesic orbits which correspond to the energy levels of the test particle, at the same time by numerically solving the geodesic equation, we simulate all kinds of motion orbits for the particle. These figures visually display the nature of the black hole space-time.In Chapter II, we investigate the spherically symmetric exterior solution in the brane-world which can be used to explain the galaxy rotation curves without assuming the existence of dark matter. By analysing the particle effective potential, We mainly take account of how the cosmological constant parameter a and the stellar pressure parameter β affect the time-like geodesic structure of the black hole. We find that the radial particle falls to the singularity from a finite distance or plunges into the singularity, depending on its initial conditions. But the non-radial time-like geodesic structure is more complex than the radial case. We find that the particle moves on the bound orbit or stable (unstable) circle orbit or plunges into the singularity, or reflects to infinity, depending on its energy and initial conditions. By comparing the particle effective potential curves for different values of the stellar pressure parameter β and the cosmological constant parameter a. we find that the stellar pressure parameter β does not affect the time-like geodesic structure of the black hole, but the cosmological constant a has an impact on the time-like geodesic structure.In Chapter Ⅲ, by analyzing the behavior of the effective potential for the particles and photons, we investigate the time-like and null geodesic structures of the Bardeen space-time, which describes a regular space-time, i.e. a singularity-free black hole space-time. At the same time, all kinds of allowed orbits, according to the energy levels of the effective potentials, are numerically simulated in detail. We find many-world bound orbits, two-world escape orbits and escape orbits in this space-time. We also find the procession directions of the bound orbits are opposite each other and their precession velocities are different, i.e. the inner bound orbits shift along counter-clockwise trajectories with high velocity while the exterior bound orbits shift along clockwise trajectories with low velocity.In Chapter IV we study the geodesic structure of the Janis-Newman-Winicour space-time which contains a strong curvature naked singularity. This metric is an extension of the Schwarzschild geometry when a massless scalar field is included. we show the parameter μ can be divided into three regimes where structure of the geodesics is qualitatively different. By solving the geodesic equation and analyz-ing the behavior of effective potential, we investigate all geodesic types of the test particle and the photon in the JNW space-time.