We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones).
Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.
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Simonetta Frittelli (Fellow of CONICOR)
Present address: Physics Department, University of Pittsburgh, 100 Allen Hall, 15260, Pittsburgh, PA, USA
FaMAF, Medina Allende y Haya de la Torre, Civdad Universitaria, 5000, Córdoba, Argentina
Simonetta Frittelli (Fellow of CONICOR) & Oscar Reula (Member of CONICET)
Communicated by S-T. Yau
About this article Cite this articleFrittelli, S., Reula, O. On the Newtonian limit of general relativity. Commun.Math. Phys. 166, 221–235 (1994). https://doi.org/10.1007/BF02112314
Received: 25 March 1992
Revised: 14 April 1994
Issue Date: December 1994
DOI: https://doi.org/10.1007/BF02112314
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