Free fall equation with a signal delay (Q2773567)

To derive the simplest free fall equation of a body of unit mass onto a body of mass \(M\) of the form \(\ddot{x}(t)=-M/x^2(t)\), one has to adopt that the velocity of the gravity interaction is infinite. Assuming that the velocity of the gravity interaction is a finite number \(c\) and that the fraction \(x/c\) is small, the author obtains another free fall equation \(\ddot{x}(t)=-Mx^{-2}(t)\bigl(t-2c^{-1}x(t)\bigr)^{-1}\). He finds the following conditions for existence of smooth solutions to the new equation with delay: \(\dot{x}/c>1\) and \(R_g/x>1\).