The 1+3-Newton Equations for gravitation on a Lorentzian manifold

1+3 formulation of Newton's equations

We present in this paper a 4-dimensional formulation of the newton equations for gravitation on a 4-dimensional lorentzian manifold, inspired from the 1+3 and 3+1 formalisms of general relativity.We show that these equations can be recovered from general relativity with a newtonian limit performed in the rest frames of the relativistic fluid.We also define a dictionary between the 1+3 formulation of the equations and general relativity.This dictionary is defined in the restframes of the dust fluid, i.e.A non-accelerating observer.We show that the parabolic free-fall solution in 1+3-newton exactly translates into the schwarzschild spacetime, and this without any approximations.