Cosmological Post-Newtonian Expansions to Arbitrary Order

Abstract

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 ǫ = v T /c (0 < ǫ < ǫ 0 ), where c is the speed of light, and v T is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab M ∼ = [0, T ) × T 3 , and converge as ǫ ց 0 to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter ǫ to any specified order with expansion coefficients that satisfy ǫ-independent (nonlocal) symmetric hyperbolic equations.

Newtonian Limits of Isolated Cosmological Systems on Long Time Scales

“…Given a vector-valued map f (u), where u is a vector, we use Df and D u f interchangeably to denote the derivative with respect to the vector u, and use the standard notation Df (u) · δu := d dt t=0 f (u + tδu) 2 To convert the 1-parameter solutions to the Einstein-Euler equations from [47,48] to solutions of (1.1)-(1.2), the metric, four-velocity, time coordinate and spatial coordinates must each be rescaled by an appropriate powers of ǫ, after which the rescaled time coordinate must be transformed according to the formula (1.9).…”

“…For any fixed basis {e I } N I=1 of V , we follow [47] and define a subspace of H s (T n , V ) bȳ Specializing to n = 3, we define, for fixed ǫ 0 > 0 and r > 0, the spaces…”

On slowly rotating axisymmetric solutions of the Einstein-Euler equations

“…This formulation of the Einstein's equations contains the perspective of a post-Newtonian approximation already. The basic exposition of this Ehlers' frame theory can be found in [11], while an introductory review can be found in [18], and recent developments can be found in [19], [20], [21]. U. Heilig [3] claims that the metric is constructed globally, but the result seems not to be a solution of the matter-vacuum matching problem.…”

The Newtonian Limit on Cosmological Scales

“…To do so, we adapt the approach employed in [38]. We begin by introducing the conformal fluid four-velocitȳ 32) and stating a few useful identities. First, we observe that 6…”

“…For isolated dynamical systems, rigorous results concerning the relationship between solutions have been established in [32,33,39]. In the articles [34,35], we adapted the approach taken in [32,33] to the cosmological setting, and we were able to construct 1-parameter families of ǫ-dependent solutions to the Einstein-Euler equations that limit as ǫ ց 0 to solutions of the cosmological Poisson-Euler equations of Newtonian gravity. However, as pointed out in [18], the class of solutions that we constructed were not valid on cosmological scales, and therefore did not address the relationship between Newtonian gravity and General Relativity on cosmological scales.…”