Phase-plane Analysis Of Particles Motion In Static Curved Spacetimes

In this paper, we investigate the motion of a test particle in curved spacetimes using the effective potentials and the phase-plane method. We extend the treatments from that of the Schwarzschild spacetime to spcetimes with mass quadrupoles σ and spherically symmetric dilaton spacetimes with an arbitrary coupling parameter a. We derive the equations of motion and the existence conditions for stable circular orbits. For spacetimes with different σ and α we discuss the effects of the energy E and angular momentum b of a test particle on the orbital motion and its stability. We obtain numerically the innermost circular orbits and their corresponding angular momentum b. The main results are:(1)There are stable and unstable orbits in Schwarzschild spacetimes. The types of the orbits and their stability depend on E and b of the test particle. We use the phase plane method to obtain numerical results which are consistent with known results of Schwarzschild spacetimes in the literature: Orbits of particles with b < 3.464M (where M is the ADM mass of the Schwarzschild spacetime.) cannot be stable;they can only spiral into the center of gravity. The innermost stable circular orbit is at r = 6M in Schiwarzschild coordinate r. For 3.464M < b < 4M, all stable orbits are elliptical. For b > 4M, there are stable elliptical and hyperbolical orbits.(2)In a spacetime with mass quadrupoles, both stable orbits and unstable orbits exist. The type and stability of the orbits are related not only to energy E and angular momentum b, but also to the mass quadrupole moment σ. When b < 3.760M, no stable orbit exist, and the innermost stable circular orbit occurs at r = 7.439M. When b = 8M and σ ≥ 3410, there is also no stable orbits. The particle will fall into the black hole eventually. Comparing with the case of the Schwarzschild spacetime, the existence of the mass quadrupole increases both the critical value of the angular momentum b and the radius of the innermost stable circular orbit.(3)For dilaton spacetime, the orbit type and its stability depend on the energy E , angular momentum b and coupling constant α. When α = 2 and Q = 0.8M,the innermost stable circular orbit occurs at pmin = 6.07M. When b < 3.159M, there is no stable orbit, and the particles will fall eventually into the black hole. For the case of a = 2, we study how the angular momentum b affects the critical value of the orbital type and how the coupling constant a affect the motion orbits of particles. We found that the radius of both stable and unstable orbits decrease when a increases.