Motion of Test Particle in Generalized Schwarzschild Geometry

By the Hamilton-Jacobi formalism, the features of orbits of a test particle moving in generalized Schwarzschild geometries with the parameter 0 < λ ⩽ 1 are studied, where the intensity of λ corresponds to the contribution of massless scalar field. In special case λ = 1, it is reduced to the Schwarzschild metric. It is found that λ = 1/2 is a critical point, when 1/2 ⩽ λ < 1 the qualitative features are similar to Schwarzschild geometry whereas the case of 0 < λ < 1/2 is different from the case of λ = 1.