Lorentz-invariant method of gravitational perturbation

A Lorentz-invariant method for obtaining equations of motion out of Einstein's gravitational field equations is developed. The method is based on the assumption that the metric tensor as well as every field function can be expanded into a power series in the gravitational constant k. The world-line of each particle is described parametrically as x ^μ (τ) where τ measures the Minkowskian distance along the path of the particle which is considered as a curve in the flat background metric. Each world-line is further characterized in a geometrical fashion by its three scalar curvatures and it is assumed that the first curvature is of the order of magnitude of the gravitational constant k. Equations of motion are derived explicitly in linear approximation, which can be considered as a Lorentz-invariant extension of Newton's law, and implicitly in the second approximation. The method is then applied to the simple problem of motion of a test particle moving in a field of a large particle, where the forces are determined up to second order in k. A detailed discussion is given in comparing the results with those obtained by other methods in connection with the problem of perihelion advance. We show that gravitational radiation effects found in other methods are not relevant in the linear approximation as was thought before. It seems that the equations of motion previously obtained are not consistent as to «order»; the use of the Frenet curvatures in our method does give a systematic order for the equations obtained. No attempt was made to extend the method to describe hydrodynamical motion.