Nonlinear newtonian theory and the effect of time dilatation

Abstract

The fact that the energy density ρ_g of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function$f\left( \varphi \right) = e^{\varphi /c^2 }$, which is determined by the nonlinear differential equation

$$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$

Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt _φ and the other in a position with potential φ=0-i.e., dt ₀, as

$${\text{d}}t_\varphi = f{\text{d}}t_0 .$$

When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.