Gravitational mass and Newton's universal gravitational law under relativistic conditions

We discuss the predictions of Newton's universal gravitational law when using the gravitational, m_g, rather than the rest masses, m_o, of the attracting particles. According to the equivalence principle, the gravitational mass equals the inertial mass, m_i, and the latter which can be directly computed from special relativity, is an increasing function of the Lorentz factor, γ, and thus of the particle velocity. We consider gravitationally bound rotating composite states, and we show that the ratio of the gravitational force for gravitationally bound rotational states to the force corresponding to low (γ ≈ 1) particle velocities is of the order of (m_Pl/m_o)² where mpi is the Planck mass (ħc/G)^1/2. We also obtain a similar result, within a factor of two, by employing the derivative of the effective potential of the Schwarzschild geodesics of GR. Finally, we show that for certain macroscopic systems, such as the perihelion precession of planets, the predictions of this relativistic Newtonian gravitational law differ again by only a factor of two from the predictions of GR.