On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion (original) (raw)

THE CAUCHY PROBLEM IN GENERAL RELATIVITY

The Einstein equations may safely be regarded as one of the highest triumphs of 20 th century physics. Through them , a deep and non-trivial connection is established between the curvature of spacetime and the matter and energy content of the universe: Rµν − 1 2 Rgµν = 8πGTµν These tensorial equations form the cornerstone of the theory of general relativity, much like Newton's F = mα does for Newtonian theory. One of the most fruitful strategies that were adopted to understand those equations was studying them through an initial value problem. This viewpoint culminated, in 1969, in the proof of the existence of a maximal globally hyperbolic development (MGHD) for suitable initial data. It is the purpose of this essay to discuss the meaning of the above sentence and provide the tools and theorems necessary to present its proof. The penultimate chapter of the essay focuses on a new proof that does away with an often frowned-upon characteristic of solutions to mathematical problems, namely Zorn's lemma.

Existence and Uniqueness of Solutions of the Semiclassical Einstein Equation in Cosmological Models

Annales Henri Poincaré, 2021

We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time, and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper, we show...

Geodesic completeness and homogeneity condition for cosmic inflation

There are two disjointed problems in cosmology within General Relativity (GR), which can be addressed simultaneously by studying the nature of geodesics around t → 0, where t is the physical time. One is related to the past geodesic completeness of the inflationary trajectory due to the presence of a cosmological singularity, and the other one is related to the homogeneity condition required to inflate a local space-time patch of the universe. We will show that both the problems have a common origin, arising from how the causal structure of null and timelike geodesics are structured within GR. In particular, we will show how a non-local extension of GR can address both problems, while satisfying the null energy condition for the matter sources.

The Infinite Distance Horizon and the Hyperbolic Inflation in the Hyperbolic Universe

ABSTRACT The Universe is globally hyperbolic as we did prove mathematically [S. A. Mabkhout, Phys. Essays 25, 112 (2012)]. The solution predicts the equation of state of cosmology ( P = -𝝆 ). Penrose said if the curvature is not equal to zero, then inflation is out. The hyperbolic universe inflates exponentially produces an accelerated expansion of the universe without cosmological constant or scalar field, as we have shown [S. A. Mabkhout, Phys. Essays 26,422 (2013)]. “In testing the validity of any scientific paradigm, the key criterion is whether measurements agree with what is expected given the paradigm”1. The hyperbolic time evolution equation of the universe (the hyperbolic scale factor) fits the observed data and successfully predicts the Planck length (10-33 cm) at micro-cosmos scale as well as it predicts the current observed large structure (1028 cm) at macro-cosmos scale, thus connecting both General Relativity and Quantum Theory. The distance horizon in the flat universe is 14 Gpc. We found the distance horizon in the hyperbolic universe to be infinitely, thus solves the horizon problem. Although the perspective for nearby objects in hyperbolic space is very nearly identical to Euclidean space (i.e., the universe locally is approximately flat consistent with local observations), the apparent angular size of distant objects falls off much more rapidly, in fact exponentially.The universe is globally hyperbolic.Locally the spacetime is approximately flat, by means of the hyperbolic inflation. Locally any deviations from flatness will be hyperbolically suppressed by the hyperbolic expansion of the scale factor, and the flatness problem is solved.