Existence of families of spacetimes with a Newtonian limit

  title={Existence of families of spacetimes with a Newtonian limit},
  author={Todd A. Oliynyk and Bernd Schmidt},
  journal={General Relativity and Gravitation},
Jürgen Ehlers developed frame theory to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter λ, which can be thought of as 1/c2, where c is the speed of light. By construction, frame theory is equivalent to general relativity for λ > 0, and reduces to Newtonian gravity for λ = 0. Moreover, by setting $${\epsilon=\sqrt{\lambda}}$$ , frame theory provides a framework to study the Newtonian limit $${\epsilon \searrow 0 \,{\rm (i.e… 
Editorial note to: On the Newtonian limit of Einstein’s theory of gravitation (by Jürgen Ehlers)
We give an overview of literature related to Jurgen Ehlers’ pioneering 1981 paper on Frame theory—a theoretical framework for the unification of general relativity and the equations of classical
The Newtonian limit of geometrostatics
The Newtonian limit of General Relativity for static isolated systems with compactly supported matter is discussed, new quasi-local notions of mass and center of mass are introduced, and several uniqueness claims in geometrostatics as well as other geometric and physical properties of these systems are proved.
Motion of small bodies in general relativity: Foundations and implementations of the self-force
Extreme mass-ratio inspirals, in which solar-mass compact bodies spiral into supermassive black holes, are an important potential source for gravitational wave detectors. Because of the extreme
On slowly rotating axisymmetric solutions of the Einstein-Euler equations
  • T. Makino
  • Mathematics, Physics
    Journal of Mathematical Physics
  • 2018
In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which give mathematical models of slowly uniformly rotating gaseous stars. We try to extend this result to
Singular perturbation techniques in the gravitational self-force problem
Much of the progress in the gravitational self-force problem has involved the use of singular perturbation techniques. Yet the formalism underlying these techniques is not widely known. I remedy this


Cosmological Post-Newtonian Expansions to Arbitrary Order
We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter $${\epsilon=v_T/c}$$$${(0< \epsilon <
On the existence of rotating stars in general relativity
AbstractThe Newtonian equations of motion, and Newton's law of gravitation can be obtained by a limit $$\lambda = \frac{1}{{c^2 }} \to 0$$ of Einstein's equations. For a sufficiently small constant
Post-Newtonian Expansions for Perfect Fluids
We prove the existence of a large class of dynamical solutions to the Einstein-Euler equations that have a first post-Newtonian expansion. The results here are based on the elliptic-hyperbolic
The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system
It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are
On Newton-Cartan Cosmology
After a brief summary of the Newton-Cartan theory in a form which emphasizes its close analogy to general relativity, we illustrate the theory with selective applications in cosmology. The
Examples of Newtonian limits of relativistic spacetimes
A frame theory encompassing general relativity and Newton - Cartan theory is reviewed. With its help, a definition is given for a one-parameter family of general relativistic spacetimes to have a
Covariant Newtonian limit of Lorentz space-times
The formulation of this limit given by Dautcourt [1] is slightly improved using the notions of Galilei manifold and Newtonian connection. It is then shown under what conditions the conservation
On a Local Existence Theorem for the Evolution Equation of Gaseous Stars
The Newtonian Limit for Perfect Fluids
We prove that there exists a class of non-stationary solutions to the Einstein-Euler equations which have a Newtonian limit. The proof of this result is based on a symmetric hyperbolic formulation of
On the definition of post-newtonian approximations
  • A. Rendall
  • Physics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1992
A definition of post-newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms. This establishes a link between the existing precise formulation of the